The Fluid Limit of a Heavily Loaded Processor Sharing ... - CiteSeerX

The Fluid Limit of a Heavily Loaded Processor Sharing Queue H. Christian Gromoll EURANDOM Den Dolech 2 5612 AZ Eindhoven The Netherlands [email protected]

Amber L. Puha Department of Mathematics California State University, San Marcos San Marcos, CA 92096{0001 [email protected]

Ruth J. Williamsy Department of Mathematics University of California, San Diego La Jolla, CA 92093-0112 [email protected] November 27, 2001

c H. C. Gromoll, A. L. Puha and R. J. Williams 2001. All print and Copyright electronic rights reserved. Personal scienti c non-commercial use only for individuals with permission from the authors ([email protected]). Short title: Fluid Limit of a Processor Sharing Queue AMS 2000 Subject Classi cations: Primary 60K25, Secondary 68M20, 90B22. Key words and phrases: Processor sharing queue, heavy trac, uid model, renewal equation, measure valued process.

Research supported in part by a University of California Oce of the President Postdoctoral Fellowship and an NSF Mathematical Sciences Postdoctoral Fellowship. y Research supported in part by NSF grants DMS 9703891 & 0071408, and a gift from the David and Holly Mendel fund. 

1

Abstract

Consider a single server queue with renewal arrivals and i.i.d. service times in which the server operates under a processor sharing service discipline. To describe the evolution of this system, we use a measure valued process that keeps track of the residual service times of all jobs in the system at any given time. From this measure valued process, one can recover the traditional performance processes, including queue length and workload. We propose and study a critical uid model (or formal law of large numbers approximation) for a heavily loaded processor sharing queue. The uid model state descriptor is a measure valued function whose dynamics are governed by a nonlinear integral equation. Under mild assumptions, we prove existence and uniqueness of uid model solutions. Furthermore, we justify the critical uid model as a rst order approximation of a heavily loaded processor sharing queue by showing that, when appropriately rescaled, the measure valued processes corresponding to a sequence of heavily loaded processor sharing queues converge in distribution to a limit that is almost surely a uid model solution.

2

Contents

1 Introduction 2 The Processor Sharing Queue

2.1 Primitive Processes and Initial Condition . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Performance Processes and Descriptive Equations . . . . . . . . . . . . . . . . . . . . 2.3 Measure Valued State Descriptor . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 The Critical Fluid Model 3.1 3.2 3.3 3.4

De nition of Fluid Model Solutions . . . . . . . . . . . . . Existence and Uniqueness Result . . . . . . . . . . . . . . A Sequence of Heavily Loaded Processor Sharing Queues Fluid Limit Result . . . . . . . . . . . . . . . . . . . . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

4 7 7 8 9

11

11 13 13 15

4 Existence and Uniqueness of Fluid Model Solutions

15

5 Convergence to Fluid Model Solutions

32

4.1 Proof of Uniqueness of Fluid Model Solutions . . . . . . . . . . . . . . . . . . . . . . 17 4.2 Proof of Existence of Fluid Model Solutions . . . . . . . . . . . . . . . . . . . . . . . 24 4.3 Dependence on the Initial Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 5.1 5.2 5.3 5.4

Preliminaries . . . . . . . . . . . . . . . . . . . Proof of Tightness . . . . . . . . . . . . . . . . Proof of Limit Point Properties . . . . . . . . . Proof of Convergence to Fluid Model Solutions

A Appendix

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

33 38 44 51

52

3

1 Introduction Consider a single server with an in nite capacity buer to which jobs arrive according to a delayed renewal process. The ith such arrival requires an amount of processing time that is the ith member of a sequence of independent and identically distributed strictly positive random variables. The server, rather than providing service to just one job at a time, operates under a processor sharing discipline, i.e., it works simultaneously on all jobs currently in the system, providing an equal fraction of its attention to each. Thus, at any given time that the system is nonempty, each job in the system is being processed at a rate that is the reciprocal of the number of jobs in the system. When the server has ful lled a given job's service requirement, the job exits the system. This system is known as a processor sharing queue. The processor sharing service discipline can be viewed as an idealization of a round-robin or timesharing protocol used in computer and communication systems. Although there is a considerable literature on processor sharing queues (see Yashkov [25] for a survey up through 1987), much of that work assumes either Poisson arrivals or exponential service times. Indeed, there are few papers on the GI/GI/1 processor sharing queue. The most general results known to us are those of Jean-Marie and Robert [20] for transient (overloaded) queues, Baccelli and Towsley [1] who establish a positive correlation type of property for the delays incurred by customers, Grishechkin [18] who obtains asymptotics of the steady state distribution as the trac intensity approaches one, and Chen, Kella and Weiss [9] who consider a uid approximation for the queue length process. In particular, there is no heavy trac diusion approximation for the queue length process of a GI/GI/1 processor sharing queue. (Note that since the workload process in a GI/GI/1 queue is the same for all nonidling service disciplines, the heavy trac approximation for the workload process under a processor sharing service discipline is the same as the well known approximation under a FIFO ( rst-in- rst-out) service discipline [19]. However, this simple relationship does not hold for the queue length process.) In this paper, we study the uid (or law of large numbers) approximation for a measure valued process that keeps track of all of the residual service times of the jobs in a heavily loaded processor sharing queue. Our main motivation for studying this so-called critical uid model is our conjecture (to be studied in subsequent work) that an understanding of the asymptotic behavior of solutions of this model is likely to play a key role in establishing a heavy trac diusion approximation for the queue length process in a processor sharing queue. This conjecture represents an extrapolation of the known situation for open multiclass networks with a HL (head-of-the-line) service discipline, which we brie y summarize here. Recently, Bramson [7] showed that if the critical uid model for an open multiclass HL-network has a certain asymptotic property, then state space collapse holds. Roughly speaking, state space collapse implies that in the heavy trac limit, the queue length process can be recovered from the workload process by an appropriate lifting map. In a companion work to [7], Williams [24] showed that state space collapse, plus an algebraic condition on the queueing model data, is sucient to imply a heavy trac diusion approximation for an open multiclass network with a HL service discipline. To illustrate this modular approach, Bramson [7] and Williams [24] applied their results to obtain new heavy trac limit theorems for FIFO networks of Kelly type and for networks with a HLPPS (head-of-the-line proportional processor sharing) service discipline. Processor sharing, as considered in this paper, is not a HL service discipline. However, we seek to mimic the modular approach of [7, 24] for a processor sharing queue. The study of a suitable critical uid model is the rst step in this process. The work presented in [7] indicates that it is important to use a suciently detailed state 4

descriptor in formulating a uid model. For example, in FIFO networks, one needs to keep track of the order in which customers arrive to the queue. In a processor sharing queue, each job in the system at time t has an associated residual service time. This is given by the total amount of service originally requested by the job minus the total amount of service it has received up to time t. It is necessary to keep track of all of the residual service times in order to adequately describe the state of the system at any given time. In fact, it is natural to use a measure valued process f(t) : t  0g such that the measure (t) at time t is a nite, nonnegative Borel measure on R+ = [0; 1) that puts a unit of mass at the residual service time of each job in the system at time t. From such a measure valued process, one can recover information about the performance of the system. For example, the number of jobs in the system, or queue length, at time t is obtained by integrating (t) against the function that is identically one. This measure valued process will be the object of central interest in this paper, and we refer to it as the state descriptor. This terminology is slightly disingenuous since the measure is not necessarily a Markovian state descriptor. In particular, it does not include the residual interarrival time|the time remaining until the next job arrives to the system. Since the residual interarrival time will play no role in our analysis, we choose not to include it in the state descriptor. Section 2.3 contains a precise description of the state descriptor (
). The process (
) was previously used by Grishechkin [18], along with other measure valued descriptors, in his heavy trac analysis of the steady state distribution of a processor sharing queue. More recently, Doytchinov, Lehoczky, and Shreve [13] used a measure valued descriptor in the context of a queueing system with deadlines. Our critical uid model has two parameters,  2 (0; 1) and a Borel probability measure  on R+ that has a nite rst moment and that does not charge the origin. These parameters are limits of parameters in the queueing system, where  corresponds to the rate at which jobs arrive to the system and the probability measure  corresponds to the distribution of the i.i.d. service times for those jobs. The quali er critical refers to the fact that we are interested in the limiting regime where the service and arrival rates are equal, i.e.,

?1

Z

x (dx) : R The model is de ned by considering a formal limit (
) of the measure valued state descriptor under law of large numbers scaling. This limit takes values in MF , the set of nite, nonnegative Borel measures on R+, which is endowed with the topology of weak convergence. For  2 MF , the real valued projection of  associated with a bounded, real valued, Borel measurable function g de ned R on R+ is denoted by hg;  i = R g (x) (dx). The uid model equations describe the dynamics of the real valued projections of (
) over the class of functions C = fg 2 C1b (R+) : g(0) = 0; g0(0) = 0g: Here, C1b (R+) denotes the space of once continuously dierentiable real valued functions de ned on R+ that, together with their rst derivatives, are bounded on R+. The requirement that g and g0 vanish at the origin is imposed to avoid singular behavior of hg; (
)i associated with mass in the

uid model abruptly disappearing as it reaches the origin. The latter corresponds to jobs in the queueing system abruptly departing when residual service times reach zero. A uid model solution associated with the critical data (;  ) is a continuous function  : [0; 1) ?! MF such that (t) does not charge the origin for all t  0, and which for each g 2 C satis es Zt 0 hg; (t)i = hg; (0)i ? hhg1;;((ss))ii ds + thg;  i; 0 =

+

+

5

for all t prior to the time that (
) rst reaches the zero measure. Once (
) reaches the zero measure, it is identically equal to the zero measure thereafter. We will in fact show that under mild assumptions, a uid model solution with (0) 6= 0 does not reach the zero measure in nite time. However, in formulating the notion of a uid model solution, we have not assumed this a priori. (For a more detailed account, see Section 3.1.) A dierent uid approximation for a processor sharing queue, which focussed on approximating the queue length process, was proposed in [9]. The idea there was to uid scale what is called the busy time equation (see equation (4) in [9]), and to pass to a limit to obtain equation (31) in [9]. However, passage to the limit in the proof of Theorem 16 in [9] seems to implicitly assume that all limit points are deterministic, or in other words, that all possible limit points are the same. Our approach in fact provides a proof that this assumption is valid under the mild conditions identi ed here (cf. Theorem 3.2). This paper contains two main results, Theorems 3.1 and 3.2. Theorem 3.1 states that under mild assumptions on the initial condition, a uid model solution exists and is unique. Theorem 3.2 says that, again under mild conditions, the measure valued state descriptors for a sequence of heavily loaded processor sharing queues under law of large numbers scaling converge in distribution to a measure valued process, which we refer to as the uid limit. Moreover, sample paths of the

uid limit are almost surely uid model solutions. Hence it is appropriate to regard the critical

uid model as a rst order approximation of a heavily loaded processor sharing queue. The paper is organized as follows. Section 2 is devoted to a precise description of the dynamics of a processor sharing queue. Two important quantities are introduced there, namely, the cumulative service process and the measure valued state descriptor. In addition, an equation that the state descriptor satis es is presented in Section 2.3. This is used to motivate the de nition of the critical

uid model, which is contained in Section 3. That section also contains the precise statements of the two main theorems, Theorems 3.1 and 3.2. Section 4 contains the proof of Theorem 3.1 and Section 5 contains the proof of Theorem 3.2. The proof of uniqueness for Theorem 3.1 and the proof of tightness, which is used to prove Theorem 3.2, bene ted from some ideas in Chen, Kella and Weiss [9] (cf. Lemmas 4.4 and 5.4 below). The proof of tightness also bene ted from some ideas of Grishechkin [18] (cf. the proof of Lemma 5.3 below). The following notation will be used throughout the paper. Let R denote the set of real numbers. For a; b 2 R, we write a _ b for the maximum of a and b, a ^ b for the minimum of a and b, a+ and a? for the positive and negative parts of a respectively, bac for the largest integer less than or equal to a, and dae for the smallest integer greater than or equal to a. The nonnegative real numbers [0; 1) will be denoted by R+. For a function g : R+ ?! R, let kg k1 = supx2R jg (x)j and kg kK = supx2[0;K ] jg (x)j for each K  0. We de ne the positive and negative parts of such a function g by g +(x) = g (x) _ 0 and g ? (x) = (?g )(x) _ 0 for all x 2 R+. Recall that MF is the set of nite, nonnegative Borel measures on R+. Consider  2 MF and a Borel to  . We de ne R R measurable function g : R+ ?! R which is integrable with respect hg;  i = R g(x) (dx): Our equations will involve expressions of the form [a;1) g(x ? a) (dx), for a > 0. To ease notation throughout, we write this as hg(
? a);  i, making the convention that such a g is always extended to be identically zero on (?1; 0). As previously noted, MF is endowed with w the topology of weak convergence of measures, i.e., for n ;  2 MF , n = 1; 2; : : : , we have n ?!  if and only if hg; ni ?! hg;  i as n ! 1, for all g : R+ ?! R that are bounded and continuous. With this topology, MF is a Polish space (cf. [22]). We denote the zero measure in MF by 0 and the measure in MF that puts one unit of mass at the point x 2 R+ by x . For a set B  R+, we denote the indicator of the set B by 1B . We also de ne the following +

+

6

real valued functions on R+: (x) = x for x 2 R+, and '(x) = 1=x for x 2 (0; 1) with '(0) = 0. For a topological space A, denote by Cb (A) the set of continuous, bounded, real valued functions de ned on A. In addition, for an interval I  R, C1b (I ) is the set of once continuously dierentiable, real valued functions de ned on I that together with their rst derivatives are bounded on I . For g 2 C1b (I ) we write g 0(x) = dxd g(x), x 2 I . We will use \=)" to denote convergence in distribution of random elements of a metric space. Following Billingsley [2], we will use P and E respectively to denote the probability measure and expectation operator associated with whatever space the relevant random element is de ned on. All stochastic processes used in this paper will be assumed to have paths that are right continuous with nite left limits (r.c.l.l.). For a Polish space S , we denote by D([0; 1); S ) the space of r.c.l.l. functions from [0; 1) into S , and we endow this space with the usual Skorohod J1 -topology (cf. [15]).

2 The Processor Sharing Queue Here we describe the processor sharing queueing system more precisely. The primitive stochastic processes and initial condition for our model are introduced in Section 2.1. The system dynamics and performance processes are described in Section 2.2. Here an important quantity for the processor sharing queue is introduced, namely the cumulative service process. In Section 2.3, we introduce the measure valued state descriptor and a dynamic equation associated with its evolution in time. The state descriptor and the associated dynamic equation play a fundamental role in motivating the de nition of the critical uid model and in justifying it as a rst order approximation of the heavily loaded processor sharing queue.

2.1 Primitive Processes and Initial Condition

The exogenous arrival process E (
) is a rate  delayed renewal process. The arrival rate  is assumed to be strictly positive and nite. Jump times of this process correspond to times at which jobs enter the system. This renewal process is de ned from a sequence of interarrival times fuig1 i=1 , where u1 denotes the time at which the rst job to arrive after time zero enters the system and ui, i  2, denotes the time between the arrival of the (i ? 1)st and the ith jobs to enter the system after time zero. Frequently,P we will simply refer to the ith job to enter the system after time zero as th the i arrival. Thus, Ui = ij =1 uj is the time at which the ith arrival enters the system, which is interpreted as zero if i = 0, and E (t) = supfi  0 : Ui  tg is the number of exogenous arrivals by time t. We assume that the sequence fuig1 i=2 is an i.i.d. sequence of nonnegative random variables with E [u2 ] = 1= < 1. The random variable u1 is associated with an initial delay preceding the rst arrival and is assumed to be strictly positive with nite mean and to be independent of fuig1 i=2 , but otherwise can have an arbitrary distribution. We refer to u1 as the initial residual interarrival time. A typical situation in which this relaxed assumption on u1 comes into play is when one would like to apply the results of this paper to a processor sharing queue that has been operating for some time in the past. Although the interarrival times of such a queue may be governed by an i.i.d. sequence, the time at which one begins to observe the system (t = 0) is arbitrary, and so in general does not coincide with the arrival of a job to the system. As a result, the residual interarrival time u1 will in general have a dierent distribution from the subsequent interarrival times fuig1 i=2 and, in particular, may depend on the initial state of the system. The service process, fV (i); i = 1; 2; : : : g, is such that V (i) records the total amount of service 7

required from the server by the rst i arrivals. More precisely, fvi g1 i=1 denotes an i.i.d. sequence of strictly positive random variables with common distribution given by a Borel probability measure  on R+. We interpret vi as the amount of processing time that the ith arrival requires from the server. The vi 'sPare known as the service times and  is known as the service time distribution. Then, V (i) = ij =1 vj , which is taken to be zero if i = 0. It is assumed that v1 > 0 a.s. and E [v1] < 1. In terms of  , these assumptions are expressed by saying that  does not charge the origin ( (f0g) = 0) and h;  i < 1. Recall that (x) = x for all x 2 R+. The two processes E (
) and V (
) are called the primitive processes, since they provide the primitive stochastic inputs for the model. Note that E (
) and V (
) are not assumed to be independent of one another. An independence assumption is not necessary since our interest is in asymptotic behavior under uid scaling where only laws of large numbers come into play. The initial condition speci es Z (0), the number of jobs present in the system at time zero, and the service requirement for each of these jobs. Here Z (0) is assumed to be a nonnegative, integer valued random variable with nite mean. The service times for these jobs are taken to be the rst Z (0) elements of a sequence fv~j g1 j =1 of strictly positive random variables. The job present in the system at time zero requiring v~j units of service time will be referred hPZto(0)as ithe j th initial job. It is assumed that the initial workload has a nite mean, i.e., that E j =1 v~j < 1 . The random variables Z (0) and fv~j g1 j =1 are not assumed to be independent of one another, nor are they assumed to be independent of the primitive processes.

2.2 Performance Processes and Descriptive Equations

As a processor sharing queue evolves in time, certain r.c.l.l. stochastic processes are used to track important measures of performance for the system such as queue length, workload, and idle time. Let Z (t) denote the queue length at time t, which is the total number of jobs in the system at time t. Also, let W (t) denote the (immediate) workload at time t, which is the total amount of time that the server must work in order to satisfy the remaining service requirement of each job present in the system at time t, ignoring future arrivals. Finally, let Y (t) denote the cumulative amount of time that the server has been idle up to time t. The processes W (
), Y (
), and Z (
) are called performance processes. These processes satisfy a set of descriptive equations, which we now present. We begin with the familiar equations for the workload W (
) and idle time Y (
) processes, which are valid for any nonidling service discipline, including processor sharing. For t  0, we have

W (0) =

ZX (0) j =1

v~j ;

(2.1)

W (t) = W (0) + V (E (t)) ? t + Y (t); Y (t) = supf(W (0) + V (E (s)) ? s)? : 0  s < tg:

(2.2) (2.3)

Recall that E (t) is the total number of arrivals up to time t. Thus, W (0) + V (E (t)) is the total service time required by the initial jobs plus that required by the jobs arriving in (0; t]. Since the server completes t ? Y (t) units of work in [0; t], (2.2) represents the remaining work in the system at time t. A set of equations that describes the queue length process Z (
) under a processor sharing service 8

discipline is the following. For t  0,

Z (t) = Z (0) + E (t) ? D(t); D(t) = S (t) =

ZX (0)

Zj=1t 0

1fv~j S (t)g +

EX (t) i=1

(2.4)

1fvi S (t)?S (Ui )g;

'(Z (s))ds:

(2.5) (2.6)

Recall that '(x) = 1=x if x > 0, and '(0) = 0. The process D(
) is the departure process, where D(t) represents the total number of jobs that have departed from the system by time t. The process S (
) is known as the cumulative service process, and S (t) represents the cumulative amount of service time allocated per job up to time t. The cumulative service process S (
) will play a particularly important role in our analysis. We will nd it convenient to have notation for the increments of this process. For t; h  0, de ne the cumulative service per job in [t; t + h] by

St;t+h = S (t + h) ? S (t) =

Z t+h t

'(Z (s))ds:

(2.7)

Then the ith arrival receives an amount of service equal to vi ^ SUi ;t by time t, for t  Ui . De ne the residual service times at time t  0 of the ith arrival, i 2 f1; : : :; E (t)g, and of the j th initial job, j 2 f1; : : :; Z (0)g, by Ri (t) = (vi ? SUi;t )+ and R~ j (t) = (~vj ? S (t))+ ; (2.8) respectively. The quantity Ri(t) (resp. R~ j (t)) represents the remaining amount of service time required by the ith arrival (resp. j th initial job) at time t. When a residual service time reaches zero, the associated job departs the system. Notice that Ri (t) does not generally correspond to the amount of time that the ith arrival will stay in the system beyond time t, since this job will receive service at less than full rate whenever there are other jobs in the system. The quantity Ri (t) should rather be thought of as the amount of work for the system, measured in units of remaining required service time, embodied in the ith arrival at time t. A similar interpretation holds for R~ j (t) as well. In fact, it can be shown that the workload at time t  0 can be rewritten as

W (t) =

ZX (0) j =1

R~ j (t) +

2.3 Measure Valued State Descriptor

EX (t) i=1

Ri(t):

(2.9)

A major diculty in analyzing the performance of a processor sharing queue lies in the fact that it could experience large drops in its queue length process during an interval [t; t + h], which cannot be predicted if the only information known about the state of the system at time t is the queue length Z (t) and the workload W (t). For instance, consider a processor sharing queue which, at a particular time t, has a large number of \low time" jobs in its buer, all having very small residual service times, and a small number of \high time" jobs, all having very large residual service times. The workload W (t) in this example could be made arbitrarily large, by increasing the residual 9

service times of the high time jobs. The queue length Z (t) is almost entirely embodied in the low time jobs, all of which will depart in the near future (say by time t + h) due to the simultaneous processing of jobs. In this situation, the imminent drop in the queue length during [t; t + h] would not be evident from Z (t) and W (t), since both could be arbitrarily large. Thus, while these two processes are very useful measures of the overall performance of the system, they do not encode enough information about the state of the system to facilitate a proper analysis of its dynamics. We will require a richer description of the state of the system. As the above example illustrates, this description should include information about the residual service times of all jobs in the system at any given time. The use of such a state descriptor will be essential to our analysis. An eective way to keep track of the residual service times, as well as the aforementioned performance processes and the initial condition, is to use a certain measure valued process which we call the state descriptor. For each t  0, let (t) be the random, nite, Borel measure on R+ = [0; 1) given by

(t) =

ZX (0)

1(0;1)(R~ j (t)) R~j (t) +

j =1

EX (t) i=1

1(0;1)(Ri(t)) Ri(t) :

(2.10)

Recall that x is the measure that puts a single unit of mass at x for x 2 R+. Thus, the random measure (t) has a unit of mass at the residual service time of each job that is in the system at time t, or in other words, for each 0 < a < b < 1, the measure that (t) assigns to the interval (a; b) is the number of residual service times that lie in (a; b) at time t. The indicator functions in the above de nition serve to eliminate jobs with zero residual service times from the description of the system state, since such jobs have departed the system. The queue length and workload at time t can be obtained from (t) by integrating against an appropriate function. In particular, for t  0, Z (t) = h1; (t)i and W (t) = h; (t)i : (2.11) Recall that (x) = x for all x 2 R+. Furthermore, given the primitive processes and the initial condition, one can recover D(
) and S (
) from Z (
), and Y (
) from W (
). Notice that the information given by the initial condition is described by the random initial measure (0). In particular, our assumptions on the initial condition imply that E [Z (0)] = E [h1; (0)i] < 1 and E [W (0)] = E [h; (0)i] < 1: (2.12) The assumptions on the initial condition together with the processor sharing dynamics imply that the random measure (t) takes values in the space MF of nite, nonnegative Borel measures on R+. It is straightforward valued stochastic process with sample ? to see that
(
) is a measure paths in the Polish space D [0; 1); M? F of functions from [0 ; 1) into MF that are right continuous
with nite left limits. The space D [0; 1); MF is endowed with the Skorohod J1 -topology (cf. [15]). An equivalent formulation of (2.10) uses the real valued processes hg; (
)i, for a suitable class of functions g : R+ ?! R. In fact, (2.10) holds for all t  0 if and only if the following holds for each bounded, Borel measurable function g : R+ ?! R,

hg; (t)i =

ZX (0) ? j =1

1(0;1)

EX (t)
?1 ~ g (R (t)) + j

i=1

(0;1)g


(R (t)); i

t  0:

(2.13)

This set of equations, or equivalently equation (2.10), will be the starting point for our analysis of processor sharing queues. 10

3 The Critical Fluid Model The purpose of this section is twofold, to introduce the critical uid model and to state our results. We begin with a precise description of a uid model solution for critical data (;  ) in Section 3.1. This is followed in Section 3.2 by the statement of Theorem 3.1, which gives existence and uniqueness of such uid model solutions under mild conditions. Next our attention turns to justifying the critical uid model as a rst order approximation of a processor sharing queue operating in heavy trac. In order to do that, we describe, in Section 3.3, a sequence of heavily loaded processor sharing queues under uid scaling. This prepares us to state our uid limit result, Theorem 3.2, in Section 3.4.

3.1 De nition of Fluid Model Solutions

The critical uid model is formulated by considering a formal limit (
) of the measure valued state descriptors under law of large numbers scaling. The dynamics of (
) are prescribed through a set of equations satis ed by the real valued projections hg; (t)i, t  0, for a suitable class of functions g. To avoid singular behavior associated with the abrupt departure of mass at the origin, this class is chosen so that the functions together with their rst derivatives, vanish at the origin. Speci cally, we work with the class

C = fg 2 C1b (R+) : g(0) = 0; g0(0) = 0g:

(3.1)

Motivated by the fact that the measure valued state descriptors do not charge the origin, we require the same of any uid model solution (see (ii) below). Using this condition, we are able to prove existence and uniqueness of uid model solutions under mild assumptions, despite the restriction to functions that vanish at the origin. The critical uid model depends on two parameters. The rst is  2 (0; 1), which corresponds to the renewal arrival rate to the queue. The second is a Borel probability measure  on R+ satisfying  (f0g) = 0, which corresponds to the distribution of the service times of arrivals to the queue. It is assumed that

 h;  i = 1;

(3.2)

which simply means that the arrival and service rates coincide, or equivalently that the model is critical. The pair (;  ) is referred to as the data of the critical uid model, or simply the critical data. A uid model solution for the critical data (;  ) is a function  : [0; 1) ?! MF such that (i) (
) is continuous, (ii) h1f0g; (t)i = 0 for all t  0, (iii) for all g 2 C , (
) satis es

hg; (t)i = hg; (0)i ?

Z t hg0; (s)i

for all t < t = inf fs  0 : (s) = 0g, and (iv) for all t  t , (t) = 0. 11

0

h1; (s)i ds + thg;  i;

(3.3)

The equations in (3.3) (one for each g 2 C ) are called the uid model equations. Condition (i) is natural in light of Theorem 3.2 below, which implies that, under mild assumptions on the limiting initial condition, uid limit points are a.s. continuous. Note that continuity of (
) is equivalent to continuity of hg; (
)i for all g 2 Cb (R+). The fact that the measures are precluded from charging the origin in condition (ii) stems from the fact that, at the level of the queueing system, zero residual service times correspond to jobs that have departed the system. The

uid model equations in condition (iii) will be derived from (2.13) by passing to the uid limit. As we will see in Section 5.3, this requires some work. However, it is possible to provide an informal explanation of (3.3). We interpret the second and third terms on the right side of (3.3) as accounting for changes to the measure valued function (
) due to the uid dynamics. On uid scale, mass is being added to the system at a constant rate of  and is being distributed according to the service time probability measure  . Thus, the third term describes changes resulting purely from arrivals. The second term describes changes resulting from the service dynamics. Recall that the server works at rate one at any given time that the system is nonempty. Since the server provides an equal fraction of its attention to each unit of mass in the system, the service rate per unit of mass in the system equals the reciprocal of the total mass in the system. Thus, the server shifts the mass towards the origin at rate 1= h1; (s)i at time s. Consequently, hg (
? h= h1; (s)i); (s)i approximates the portion of hg; (s + h)i resulting from the service dynamics over the time interval [s; s + h]. In particular, (hg (
? h= h1; (s)i); (s)i ? hg (
); (s)i) =h approximates the average rate of change due to the service dynamics. Since, as h tends to zero, this dierence quotient approaches the integrand of the second term on the right side of (3.3), this explains the form of that term. Condition (iv) simply re ects the fact that the data is critical, i.e., the arrival and service rates are balanced, and therefore mass shouldn't build up when starting from a zero initial measure. We will nd it convenient to refer to a uid model solution for the critical data (;  ) as simply a uid model solution with the understanding that the data under consideration always satis es (3.2). Given a uid model solution (
), the uid analogue of the queue length is de ned by Z(t) = h1; (t)i; for all t  0: (3.4) For obvious reasons, Z (t) is referred to as the total mass at time t. Due to the assumed continuity of uid model solutions, Z(
) is continuous. Also, the uid analogue of the cumulative service per job is de ned by

S(t) =

Zt 0

'(Z(s))ds;

for all t  0:

Since Z (
) is strictly positive and continuous for t 2 [0; t), S(
) 2 C1 ([0; t)) with S0 (t) = dtd S(t) = Z1(t) ; for all t 2 [0; t): Finally, the uid analogue of the workload process is given by W (t) = h; (t)i; for all t  0:

(3.5)

(3.6) (3.7)

Since  is not bounded, W (t) cannot be assumed to be continuous (or even nite) for an arbitrary

uid model solution. However, we will show that, under a mild assumption on the initial condition, W (
) is in fact constant, although this constant equals in nity if h; (0)i = 1 (cf. Theorem 3.1). 12

3.2 Existence and Uniqueness Result

Theorem 3.1 Let  2 MF be such that (fxg) = 0 for all x 2 R+. A uid model solution (
) for the critical data (;  ) with (0) =  exists, is unique and satis es W (t) = h;  i for all t  0. In particular, if  = 6 0, then the associated uid model solution never reaches the zero measure (t = 1) and thus satis es (3.3) for all time. Notice that Theorem 3.1 asserts that the uid analogue of the workload process is constant and equals its initial value. This holds even if the initial measure  has an in nite rst moment, in which case W (t) = 1 for all t  0. The \no atoms" condition on  is readily explained by the assumed continuity of uid model solutions. Indeed, if  were to have an atom at x 2 R+, then since all of the mass initially at x departs the system simultaneously, the total mass would have a downward jump (i.e., a discontinuity) at that departure time. Section 4.1 contains the proof that there is at most one uid model solution (
) such that (0) =  , that the workload is constant, and that t = 1 if  6= 0. Section 4.2 contains the proof of existence. Remark. Theorem 3.1 and its proof can be extended in a straightforward manner to situations in which the uid model data is either strictly subcritical (h;  i < 1), or strictly supercritical (h;  i > 1) and the initial measure  6= 0. In particular, for strictly subcritical data (h;  i < 1) and  2 MF that has no atoms, there is a unique  : [0; 1) ?! MF satisfying (0) =  and conditions (i){(iv) of the de nition of a uid model solution. Moreover, t = h;  i= (1 ? h;  i) and W (t) = W (0) + ( h;  i ? 1)t for t 2 [0; t); W (t) = 0 for t 2 [t; 1): In the strictly supercritical case ( h;  i > 1), condition (iv) is inappropriate since mass should build up from the zero initial measure. Indeed, in this case, for  2 MF that has no atoms and satis es  6= 0, there is a unique solution  : [0; 1) ?! MF satisfying conditions (i){(iii) of the de nition of a uid model solution and t = 1 there. The case h;  i > 1 and  = 0 is the only one for which the result and proof of Theorem 3.1 do not immediately generalize. In a separate paper, Puha, Stolyar, and Williams [23] formulate and analyze a strictly supercritical, measure valued uid model. In particular, a natural uid workload equation is added to equations (i){(iii) of the de nition of a uid model solution, to govern how mass builds up from zero. With this additional condition, uniqueness is established for solutions of the supercritical uid model starting from the zero measure.

3.3 A Sequence of Heavily Loaded Processor Sharing Queues

In this section, we specify the assumptions under which the uid limit result will be proved. Consider a sequence of processor sharing queueing models indexed by r, where r increases to 1 through a sequence in (0; 1). Each model in the sequence may be de ned on a separate probability space, however we use P and E respectively for the probability and expectation operator on each of these spaces. The rth model is de ned as in Section 2, except that all accompanying parameters and processes have a superscript r appended to them. In particular, the performance processes associated with the rth system are W r (
), Y r (
), and Z r (
), and the state descriptor is r (
). Recall that (cf. (2.12)) for each r > 0 the initial condition r (0) satis es

E[h1; r(0)i] < 1 and E[h; r (0)i] < 1: 13

(3.8)

The uid limit result concerns the behavior of processor sharing queues on law of large numbers scale, or uid scale. Accordingly, we de ne the uid scaled processes Zr (t) = 1 Z r (rt) (3.9)

r

W r (t) = r1 W r (rt)

(3.10)

r (t) = r1 r (rt)

E r(t) = r1 E r(rt)

(3.11)

r St;tr +h = Srt;r (t+h) =

Z r(t+h) rt

'(h1; r(s)i)ds =

Z t+h t

(3.12)

'(h1; r(s)i)ds;

(3.13)

for all t 2 [0; 1), h  0. Let (;  ) be critical data as in Section 3.1. In order to obtain convergence in distribution of the uid scaled state descriptors r (
) to a process that is a.s. a uid model solution for the critical data (;  ), we impose the following asymptotic assumptions on the sequence of processor sharing queues. For the primitive processes, assume that as r ! 1,

r ?! ; w  r ?! ; r h;  i ?! h;  i; E[ur1]=r ?! 0; E[ur2; ur2 > r] ?! 0:

(3.14) (3.15) (3.16) (3.17) (3.18)

Recall that by (3.2), h;  i = 1. Thus, assumptions (3.14) and (3.16) guarantee that as r ! 1, the systems become heavily loaded. Indeed if we de ne the trac intensity parameter r = r h;  r i, then the assumptions imply that r ?! 1 as r ! 1. Assumption (3.17) implies that the initial residual interarrival time vanishes on uid scale. We assume (3.18) in order to provide uniform control over the tail of the distribution of ur2 , which is used to obtain a weak law of large numbers for a triangular array (cf. Lemma A.2). In applications, one often has a stronger condition than (3.18). For example, (3.18) holds if lim supr!1 E[(ur2 )1+ ] < 1 for some  > 0. For the uid scaled initial measures, we assume that for some random measure  taking values in MF , we have

?r (0); h; r (0)i
=) ?; h; i
;

as r ! 1;

(3.19)

and that  satis es

E[h1; i] < 1; E [h; i] < 1;

1fxg;  = 0; for all x 2 R+; a.s.

(3.20) (3.21) (3.22)

In (3.19), we are assuming that the uid scaled initial measures and their rst moments converge jointly in distribution to some limit. Since, for any g 2 Cb (R+), g : MF ?! R de ned by g ( ) = hg;  i is continuous, by the continuous mapping theorem (cf. [2], Thm. 5.1), hg; r(0)i =) hg; i 14

as r ! 1. The second component of (3.19) implies that the uid scaled initial workload converges in distribution, i.e., W r (0) =) h; i as r ! 1. Assumptions (3.20) and (3.21) require the total mass and rst moment of  to have nite expected values. Assumption (3.22) states that a.s.  has no atoms, which is used to prove tightness of fr (
)gr>0 and to show that uid limit points are continuous paths a.s. The \no atoms" assumption will be used in the equivalent form

1 ;  <  lim P sup [x;x+] #0 4 x2R +

!

= 1; for all  > 0:

(3.23)

The equivalence of (3.22) and (3.23) is proved in the Appendix (cf. Lemma A.1).

3.4 Fluid Limit Result

Theorem 3.2 Consider a sequence of processor sharing queueing models as de ned in Section 3.3, satisfying assumptions (3.14){(3.22). Then the sequence of uid scaled state descriptors fr (
)g converges in distribution as r ! 1 to a measure valued process ? (
) taking values in MF such that ? (0) is equal in distribution to  and almost surely ? (
) is a uid model solution for the critical data (;  ).

We refer to the limiting process ? (
) as the uid limit of the sequence fr (
)g of uid scaled state descriptors. Notice that, by Theorem 3.1, once an initial measure  2 MF is speci ed, a uid model solution for the critical data (;  ) is a deterministic path taking values in MF . Thus we see by Theorem 3.2 that although the initial measure ? (0) of the limiting process may be random (cf. (3.19)), for each ! in some set of probability one, ? (t)(! ) is determined for all t > 0 by the initial measure ? (0)(! ). Theorem 3.2 is proved in Section 5.

4 Existence and Uniqueness of Fluid Model Solutions This section contains the proof of Theorem 3.1 and a related result, Lemma 4.9. Before proceeding with the proofs, it will be convenient to introduce some notation. Recall that the critical data (;  ) satis es  2 (0; 1),  (f0g) = 0, and  h;  i = 1. Let F denote the cumulative distribution function associated with the probability measure  . Since  does not charge the origin, F (0) = 0. The cumulative distribution function F has associated with it an excess lifetime cumulative distribution R function Fe , which is given by Fe (x) =  0x (1 ? F (y )) dy , x 2 R+. In particular, Fe has probability density function fe (x) =  (1 ? F (x)), x 2 R+. Here the fact that h;  i = 1 was used to simplify the form of the normalizing constant. In Theorem 3.1, the initial measure  2 MF is assumed to have no atoms. Denote the set of all such measures by McF = f 2 MF : (fxg) = 0 for all x 2 R+g: Here c stands for continuous, which re ects the relationship between the \no atoms" condition and the continuity of uid model solutions. If the initial measure  = 0, then clearly (
)  0 is the unique uid model solution such that (0) =  . Thus, the main interest is in proving existence and uniqueness for nonzero initial measures. For this, let c Mc;p F = f 2 MF :  6= 0g:

15

Here p stands for positive. Given  2 Mc;p F , let

H (x) =

Z x

0



x 2 R+:

1(y;1) ;  dy;

(4.1)

As we will see in Sections 4.1 and 4.2, H plays an important role in the proof of Theorem 3.1 for  2 Mc;p F . Since  has no atoms, the integrand in (4.1) is continuous. Thus, H (
) is continuously dierentiable with

 H0 (x) = 1(x;1);  ; x 2 R+: (4.2) The proof of Theorem 3.1 for  2 Mc;p F takes advantage of the behavior of solutions to certain convolution equations. In order to prepare the reader, the facts that will be needed are brie y reviewed here. For a more detailed account, the reader is referred to [17], Section XI.1. Given a locally bounded Borel measurable function g : R+ ?! R+ and a nondecreasing, right continuous function U : R+ ?! R+, let (g  U ) (u) =

Z

[0;u]

g(u ? s)dU (s);

u  0:

Note that by convention the contribution to the above integral is g (u)U (0) at s = 0 whenever U (0) 6= 0. For  2 Mc;p F , consider the convolution equations M (u) = H (u) + (M  Fe ) (u); u  0; (4.3) 0 N (u) = H (u) + (N  Fe ) (u); u  0: (4.4) Since H (
) and H0 (
) are locally bounded on R+, each equation has a unique bounded, Borel measurable solution on [0; b] for each b 2 R+. In fact, there is an explicit representation for these solutions in terms of the renewal function 1 ?
X Fei (u); Ue (u) =  (i?1) i=0

u  0;

(4.5)

 Fe (
), i  1. Indeed, M (u) = (H  Ue) (u); u  0;

(4.6)

where Fe0 (
)  1 and Fei (
) = Fe

is the unique locally bounded solution of (4.3), and
? (4.7) N (u) = H0  Ue (u); u  0; is the unique locally bounded solution of (4.4). Observe that Ue 2 C(R+) since Fe 2 C(R+). Also note that (H  Ue )(0) = 0. Since  6= 0, it is easily veri ed that H  Ue is strictly increasing and limu!1 (H  Ue )(u) = 1. Moreover, since Ue 2 C(R+), H 2 C1(R+), and H (0) = 0, it follows that H  Ue 2 C1 (R+) with (H  Ue )0(u) = (H0  Ue )(u), u  0. We are now ready to proceed with the proofs. This section is organized in the following manner. Section 4.1 contains a proof that for  2 Mc;p F there is at most one uid model solution (
) such that (0) =  , and that the uid analogue of the workload is constant for such a solution. Section 4.2 contains the proof that for  2 Mc;p F , there exists a uid model solution (
) such that (0) =  . Finally, in Section 4.3, it is proved that the mapping from the initial measure to the uid model solution that it determines is measurable. In fact, it will be shown that this map restricted to Mc;p F is continuous. For a uid model solution (
), the reader is reminded that the functions Z (
), S(
) and W (
) are de ned by (3.4), (3.5), and (3.7) respectively. 16

4.1 Proof of Uniqueness of Fluid Model Solutions

Fix  2 Mc;p F . Here, we will prove that there is at most one uid model solution (
) with (0) =  and that for such a solution the workload is constant. In order to do that, we will derive, for t < t , an expression for h1(0;w); (t)i, w 2 (0; 1], in terms of S(
) and  |see (4.14) below. This is used to reduce the problem of proving uniqueness of uid model solutions to proving that t = 1 and that S(
) is uniquely determined by  . Then, we will utilize (4.14) below with w = 1 and the assumption that (t)(f0g) = 0 for all t  0 to obtain an equation for the total mass Z (t) = h1; (t)i of (t) for t < t . A key aspect of the uniqueness proof will be to perform a time change in this equation, thereby reducing it to the convolution equation in (4.4) and relating the time change of Z(
) to the solution of (4.4). Indeed, we will see that this relationship between the time changed Z(
) and the unique solution of (4.4) forces t = 1 and results in uniqueness of S(
), given . Despite the intuitive nature of equation (4.14), it is not completely straightforward to derive it from (3.3). This is due to the fact that in (3.3) one can only consider continuously dierentiable functions that together with their rst derivatives vanish at the origin. In particular, the function g = 1(0;w) cannot be directly substituted into (3.3) and consequently must be approximated. If one were to simply substitute into (3.3) a sequence of functions in C that approximate 1(0;w), one would be faced with the problem of controlling errors involving the rst derivative, which can be large near the origin. In order to circumvent this dicultly, we will derive a version of (3.3) for functions that depend on both time and space|see (4.8) below. This yields a larger set of equations satis ed by any uid model solution. In order to prove (4.14), we take g 2 C and compose with a timedependent spatial shift to create a time-dependent function for which the integrals in (4.8) involving the rst partial derivatives cancel one another|see (4.15) and (4.16). The resulting equation (4.18) does not involve g 0. Therefore, g 2 C can increase to 1(0;w) in (4.18) irrespective of the behavior of g0 near the origin, and this yields (4.14). We begin by deriving the version of (3.3) that holds for a class of functions of both time (denoted by [0; 1)) and space (denoted by R+). The class will be a subset of C1b ([0; 1) R+), the set of once continuously dierentiable functions de ned on [0; 1)  R+ that together with their rst partial derivatives, denoted by fs (s; x) = @s@ f (s; x) and fx (s; x) = @x@ f (s; x), are bounded. Notice that, if (
) is a uid model solution such that (0) = , then since  6= 0 and (
) is continuous, t > 0.

Lemma 4.1 If (
) is a uid model solution such that (0) = , then for all f 2 C1b ([0; 1)  R+) such that f (
; 0)  0 and fx (
; 0)  0, (
) satis es Z t hfx(s;
); (s)i Zt Zt hf (t;
); (t)i = hf (0;
); i + hfs(s;
); (s)i ds ? h1; (s)i ds +  0 hf (s;
);  i ds; 0 0

(4.8)

for all t < t = inf fs : (s) = 0g.

The next proposition will be used in the proof of Lemma 4.1.

Proposition 4.2 Let  : [0; 1) ?! MF be continuous. For each f 2 Cb([0; 1)  R+), t ?! hf (t;
); (t)i is a continuous function of t 2 [0; 1). 17

Proof. Let f 2 Cb([0; 1)  R+). For xed t 2 [0; 1) and h 2 (?1; 1) such that t + h  0, hf (t + h;
); (t + h)i ? hf (t;
); (t)i  hf (t + h;
); (t + h)i ? hf (t;
); (t + h)i + hf (t;
); (t + h)i ? hf (t;
); (t)i : (4.9) The last term on the right side of the above converges to zero as h tends to zero since (
) is continuous. Furthermore, by this same continuity and the fact that (t) 2 MF , given  > 0, there exists an M < 1, and h1 > 0 such that

1 ; (t + h) < ; h 2 [? min(t; h ); h ]: 1 1 (M;1) Since f (
;
) is uniformly continuous on [(t ? h1 )+ ; t + h1]  [0; M ], there exists h2 2 (0; h1) such that jf (t + h; x) ? f (t; x)j   for all h 2 [? min(t; h2 ); h2] and x 2 [0; M ]. Therefore, for h 2 [? min(t; h2 ); h2],

hf (t + h;
); (t + h)i?hf (t;
); (t + h)i   1[0;M ]; (t + h) +2kf k1    h1; (t + h)i+2kf k1 : Letting h and then  tend to zero in (4.9) and using the above estimates, we see that the left side of (4.9) tends to zero as h tends to zero.  1 Proof of Lemma 4.1. Let f 2 Cb ([0; 1)  R+) be such that f (
; 0)  0 and fx(
; 0)  0. Fix 0  t < t and consider h 2 (?1; 1) such that 0  t + h < t . Then hf (t + h;
); (t + h)i ? hf (t;
); (t)i = hf (t + h;
); (t + h)i ? hf (t;
); (t + h)i + hf (t;
); (t + h)i ? hf (t;
); (t)i : (4.10) Rewrite the rst two terms on the right side of (4.10) as

hf (t + h;
); (t + h)i ? hf (t;
); (t + h)i = =

Z t+h

Z 1

=h

t

Z0 1 0

fs (u;
)du; (t + h)



fs (t + hv;
)hdv; (t + h)



hfs(t + hv;
); (t + h)i dv:

For each v 2 [0; 1], de ne a function f v : [0; 1)  R+ ?! R by f v (u; x) = fs (t + (u ? t)v; x). Then f v 2 Cb ([0; 1)  R+), and so, by Proposition 4.2, u ?! hf v (u;
); (u)i is a continuous function of u 2 [0; 1). Moreover, f v (t + h;
) = fs (t + hv;
). Therefore, for each v 2 [0; 1], lim hf (t + hv;
); (t + h)i = hlim hf v (t + h;
); (t + h)i = hf v (t;
); (t)i = hfs (t;
); (t)i : h!0 s !0 This together with bounded convergence implies that Z1 h f ( t + h;
) ;  ( t + h ) i ? h f ( t;
) ;  ( t + h ) i lim = hfs (t;
); (t)i dv = hfs (t;
); (t)i : h!0 h 0

(4.11)

Now consider the last two terms on the right side of (4.10). Since t 2 [0; t) and h 2 (?1; 1) is such that t + h 2 [0; t), and since f (t;
) 2 C , we can use (3.3) with g (
) = f (t;
) to obtain Z t+h hfx(t;
); (s)i (4.12) hf (t;
); (t + h)i ? hf (t;
); (t)i = ? h1; (s)i ds + h hf (t;
);  i : t 18

Since (
) is assumed to be continuous and h1; (s)i > 0 for s 2 [0; t), s ?! hfxh(1t;; 
)(;s)(is)i ; is a continuous function of s 2 [0; t). It follows that

Z t+h hfx(t;
); (s)i hfx(t;
); (t)i 1 lim (4.13) h!0 h t h1; (s)i ds = h1; (t)i : Combining (4.10), (4.11), (4.12), and (4.13), we obtain d hf (t;
); (t)i = hf (t;
); (t)i ? hfx(t;
); (t)i +  hf (t;
);  i ; t < t : s dt h1; (t)i Since f , fs , and fx are continuous and bounded, and since 1=Z (
) is continuous on [0; t), by Proposition 4.2, each term on the right side of the above equation is continuous for t 2 [0; t). Thus, (4.8) follows.  Next the uid model equations for time-dependent functions, i.e., (4.8), are used to derive (4.14) below. Recall that this serves two purposes. Firstly, it reduces proving uniqueness of (
) to proving that t = 1 and that S(
) is uniquely determined by  . Secondly, in the proof of Lemma 4.4, (4.14) with w = 1 will be used to show that a time change of Z(
) is related to the unique solution of (4.4). Lemma 4.3 Suppose that (
) is a uid model solution such that (0) = . Then for all w 2 (0; 1],

1

(0;w)



; (t) = 1

 (0;w)(
? S(t));  + 

for all t < t = inf fs : (s) = 0g.

Z t

0

 1(0;w)(
? (S(t) ? S(s)));  ds;

(4.14)

Proof. For t = 0, (4.14) holds trivially. Fix t such that 0 < t < t . Consider g 2 C1b (R) with g(x) = 0 for all x  0. Note that g 0(x) = 0 for all x  0. Let ?
f (s; x) = g x ? S(t) + S(s) ; s 2 [0; t]; x 2 R+: (4.15) Since S(
) 2 C1([0; t)) with S0(s) = 1=Z (s) for s 2 [0; t), it follows that f 2 C1b ([0; t]  R+) and 0 ?x ? S(t) + S(s)
?
s 2 [0; t]; x 2 R : g f (s; x) = and f (s; x) = g 0 x ? S(t) + S(s) ; s

x + Z(s) Also, for s 2 [0; t], f (s; 0) = 0 and fx (s; 0) = 0 because g (x) = 0 and g 0(x) = 0 for all x  0. Fix 0 <  < t. We wish to construct a function f  that satis es the conditions in Lemma 4.1 and that closely approximates f on [0; t]  R+. For this, choose a cut-o function h 2 C1b ([0; 1))

such that

(

h (s) = 1; s 2 [0; t ? ]; 0; s 2 [t ? =2; 1): Extend f to be identically equal to zero on (t; 1)  R+ and de ne f  (s; x) = f (s; x)h(s); s 2 [0; 1); x 2 R+: 19

Thus, f  2 C1b ([0; 1)  R+) with f  (
; 0)  0, fx (
; 0)  0, and f  = f on [0; t ? ]  R+. Now substitute f  into (4.8), and use the fact that f  (s;
)  f (s;
) for 0  s  t ?  to obtain

Z s ?
 (4.16) hf (s;
); (s)i = hf (0;
); i + g0
? S(t) + S(v) ; (v) Z(v)?1dv 0 Z s ? Z s ?

 ? g0
? S(t) + S(v) ; (v) Z(v)?1dv +  g
? S(t) + S(v) ;  dv; 0

0

for all s 2 [0; t ? ], for each  2 (0; t). The rst and second integral terms in the above expression cancel one another. Since  2 (0; t) was arbitrary, it follows that for all s 2 [0; t),

Z s

   hf (s;
); (s)i = g(
? S(t));  +  g(
? S(t) + S(v));  dv: 0

(4.17)

We wish to let s " t in (4.17). It is clear what eect that has on the right side. To see that the left side tends to hg; (t)i, note that by a minor modi cation of the proof of Proposition 4.2, hf (s;
); (s)i converges to hf (t;
); (t)i as s increases to t. Moreover, by the de nition of f , hf (t;
); (t)i = hg; (t)i. So letting s " t in (4.17) gives, for all g 2 C1b (R) such that g(x) = 0 for all x  0,

Z t

   hg; (t)i = g(
? S(t));  +  g(
? S(t) + S(v));  dv; 0

(4.18)

for all t 2 (0; t). To obtain (4.14) from (4.18), consider a sequence of nonnegative functions fgng  C1b (R) that increases to 1(0;w) pointwise on R and apply monotone convergence.  As a consequence of Lemma 4.3 and the assumption that (t) does not charge the origin for all t  0, in order to prove uniqueness, it suces to show that t = 1 and that S(
) is uniquely determined by  . Observe that S(
) is continuous and strictly increasing prior to the time t at which (
) reaches the zero measure. Thus, the continuous inverse of S(
) on [0; t) is given by T(u) = S?1 (u) = inf ft  0 : S(t) > ug; u < u = lim S(t): (4.19) t"t

In (4.19), the superscript ?1 denotes the functional inverse. Recall that, since (
) is continuous, S(
) 2 C1([0; t)) with S0(
) = 1=Z(
). This means that T(
) 2 C1([0; u)) with

T0(u) = S0(T1(u)) ;

u 2 [0; u):

(4.20)

Our strategy is to use (4.14) with w = 1 to show that M (
) = T(
) is the unique locally bounded solution of (4.3). Notice that assumption (3.2) has not yet been used, but it is needed for the statement of the following lemma. Lemma 4.4 Suppose that (
) is a uid model solution for the critical data (;  ) with (0) = . Then u = 1 and t = 1. Moreover, for T(
) de ned by (4.19), M (
) = T(
) is the unique locally bounded solution of (4.3). In particular, T(
) = (H  Ue ) (
) with H (
) de ned by (4.1) and Ue (
) de ned by (4.5). The statement of Lemma 4.4 is much like the statement of Theorem 17 in [9]. In fact, reading Theorem 17 in [9] suggested to us that proving something like Lemma 4.4 for our uid model might be useful. In the remark following the proof of Lemma 4.4, we explain the connection between our work and that in [9] more precisely. Before proceeding with the proof of Lemma 4.4, we will need to verify the following proposition, which is a consequence of Lemma 4.3. 20

Proposition 4.5 Suppose that (
) is a uid model solution with (0) = . Then u < 1 implies t = 1. Proof. Suppose that u < 1. Substituting w = 1 into (4.14) and using the fact that (t)(f0g) = 0 for all t  0 gives, for t < t ,

 Z(t) = 1(0;1)(
? S(t));  +   

Z t

Z t

0

 1(0;1)(
? S(t) + S(s));  ds

 1(0;1)(
? S(t) + S(s));  ds

Z0 t

0

 1(0;1)(
? u + S(s));  ds:

 ~t < t such that for all s 2 [t~; t ), Fix 0 <  < 1 and choose  > 0 such that 1(;1) ;   . Choose       ~ u ? S (s)  . Then, for s 2 [t; t ), 1(0;1)(
? u + S(s));   . In particular, for t 2 [t~; t), Z(t)  (t ? ~t): Therefore, u < 1 implies that Z (t) is bounded away from zero as t increases to t . If in addition t < 1, then Z(
) is discontinuous at t because, by de nition of t , Z (t) = 0 for all t > t . But Z(
) is continuous since (
) is continuous, and therefore it must be the case that t = 1 whenever u < 1.  Proof of Lemma 4.4. Observe that, since (t)(f0g) = 0 for all t  0, (4.14) with w = 1 becomes

Z(t) = H0 (S(t)) + 

Z t

 1 ? F (S(t) ? S(s)) ds;

0

t < t ;

(4.21)

where H0 (
) is de ned by (4.2). In (4.21), let u = S(t) and perform the change of variables v = S(s) to obtain Zu 0   Z (T (u)) = H(u) +  [1 ? F (u ? v)] dT(v); u < u : 0

Since S0(s) = 1=Z(s) for s < t and T(u) < t for u < u. This together with (4.20) gives

T0 (u) = H0 (u) + 

Zu 0

u < u , the left side is in fact 1=S0(T(u)) for

[1 ? F (u ? v )] dT(v );

u < u :

Using a change of variables and the de nition of fe (
), we have

T0 (u) = H0 (u) +

Zu 0


? T0(u ? v)fe (v)dv = H0 (u) + T0  Fe (u);

u < u :

(4.22)

Thus, N (
) = T0 (
) is a locally bounded solution of (4.4) on [0; u). It follows from the uniqueness of locally bounded solutions of (4.4) on [0; u) that T0 (u) is given by the right side of (4.7) for u 2 [0; u), i.e.,
? (4.23) T0(u) = H0  Ue (u); u < u : 21

Since T 2 C1 ([0; u)) and T(0) = 0, and since H  Ue 2 C1(R+), (H  Ue ) (0) = 0, and (H  Ue)0 (
) = (H0  Ue)(
), integrating (4.23) yields T(u) = (H  Ue) (u); u < u : (4.24)   The nal task is to show that t = 1 and u = 1. By (4.24) and the properties of (H  Ue )(
), limu"u T(u) < 1 if and only if u < 1. By de nition (4.19), we have t = limu"u T(u). Thus, t < 1 if and only if u < 1. But this can only be consistent with the statement of Proposition 4.5 if both u = 1 and t = 1.  Remark. A result like Lemma 4.4 holds for the uid approximation studied in [9] as well (cf. Theorem 17 in [9]). This follows almost as an immediate consequence of one of their uid approximation equations. In particular, they consider a uid approximation of the so-called busy time equation given by

Zt?


  t = S(t) ^ ;  +  h S(t) ? S(s) ^ ;  ids; 0

t < t ;

(4.25)

(cf. (31) in [9]). The right side of (4.25) can be interpreted as the total amount of work performed in the uid limit by the server by time t, for t < t . It is easily veri ed that H (x) = hx ^ ;  i and Fe (x) =  hx ^ ;  i, x 2 R+. This, together with a time change, shows that (4.25) is equivalent to (4.3) with M (
) = T(
). Since our state descriptor tracks more information than just S(
), it is perhaps not surprising that we are able to derive (4.25) from (3.3). However, as the proofs of Lemmas 4.1, 4.3, and 4.4 demonstrate, some work is required for this. While reading [9], we realized that deriving a version of (4.25) for our uid model could provide an important component of our uniqueness proof. Lemma 4.4 will now be used in conjunction with Lemma 4.3 to prove both uniqueness of uid model solutions and the constant workload property. Proof of Uniqueness for Theorem 3.1,  2 M . Suppose that (
) is a uid model solution for the critical data (;  ) with (0) =  . By Lemma 4.3 and the fact that (t)(f0g) = 0 for all t  0, h1[0;w); (t)i is uniquely determined by S(
),  , and (;  ) for each w 2 (0; 1] and for each t 2 [0; t). Since intervals of the form [0; w), w 2 (0; 1], generate the Borel  -algebra on R+, this uniquely determines (t) for each t 2 [0; t). By Lemma 4.4, t = 1, so (
) is uniquely determined by S(
),  , and (;  ). By (4.19) and Lemma 4.4, S(
) is the inverse of (H  Ue )(
). Since (H  Ue )(
), is determined by  and (;  ), the function S(
), and therefore the uid model solution (
), is uniquely determined by  and (;  ).  Proof that W (
) is constant for Theorem 3.1,  2 M . Suppose that (
) is a uid model solution for the critical data (;  ) with (0) =  . Fix t  0. By de nition, Z Z Z  dy (t)(dx): W (t) = h; (t)i = x (t)(dx) = R [0;x] R By Fubini's theorem, the order of integration can be interchanged and this gives Z

Z Z  1[y;1) ; (t) dy: (t)(dx)dy = W (t) = R R [y;1)  This together with (4.14) and the fact that t = 1 (cf. Lemma 4.4) gives c;p F

c;p F

+

+

+

+

W (t) =

Z 

1 R

+

 [y;1) (
? S(t));  + 

Z t

0

22



 1[y;1) (
? S(t) + S(s));  ds dy:

(4.26)

We begin by simplifying the rst term on the right side of (4.26). Since  has no atoms, from (4.2) it follows that Z Z Z

 0 H0 (y)dy: H (y + S(t))dy =  1[y;1) (
? S(t));  dy = [S (t);1) R R R Notice that [0;S(t)) H0 (y )dy < 1. Therefore, we may add and subtract this term on the right side R of the above equality. Also notice that [0;1) H0 (y )dy = h;  i. Thus, we have +

+

Z

1

  (0) ? [y;1) (
? S(t));  dy = W

Z

H0 (y)dy = W (0) ? H(S(t)):

(4.27) R Next we simplify the second term on the right side of (4.26). By interchanging the order of integration, we obtain +

Z

R

+



[0;S(t))

Z t

0

=

Z

 1[y;1) (
? S(t) + S(s));  ds dy

R

+

Z tZ Z t ? ?



   y + S(t) ? S(s); 1 dy ds:   y + S (t) ? S (s); 1 ds dy = 0

0

R

+

?

??

For xed 0  s  t,  y + S(t) ? S(s); 1 =  y + S(t) ? S(s); 1 for all y 2 R+ such that y + S(t) ? S(s) is not an atom for  . Since  has at most countably many atoms, there are at most countably many exceptional y 2 R+ (where the exceptional set may depend on s and t). This, together with the de nition of F , a change of variables, and the de nition of Fe gives Z

R

+



Z t

=

 1[y;1) (
? S(t) + S(s));  ds dy

Z0 t Z 1

S(t)?S(s)

0

 (1 ? F (z)) dz ds =

Z t?

Zt

0

0


1 ? Fe (S(t) ? S(s)) ds = t ?

Fe(S(t) ? S(s))ds:

Recall that T(
) de ned in (4.19) is the functional inverse of S(
) and that t = u = 1 (cf. Lemma 4.4). Using the change of variables v = S(s) gives

Z

R

+



Z t

0

=t?

 1[y;1) (
? S(t) + S(s));  dsdy Z S(t) 0


?
? Fe(S(t) ? v)dT(v) = t ? Fe  T (S(t)) = t ? T  Fe (S(t)):

(4.28)

Combining (4.26), (4.27) and (4.28), we have  
? W (t) = W (0) + t ? H (S(t)) + T  Fe (S(t)) : Observe that the third term on the right side is simply the right side of equation (4.3) for M (
) = T(
) evaluated at u = S(t). By Lemma 4.4, M (
) = T(
) is the unique locally bounded solution of

(4.3). Thus,

W (t) = W (0) + t ? T(S(t)) = W (0) + t ? t = W (0); which holds even if W (0) = 1. 23



4.2 Proof of Existence of Fluid Model Solutions

Fix  2 Mc;p F . We wish to prove that a uid model solution (
) with (0) =  exists. The uniqueness proof suggests that (4.14) might be used to de ne a suitable candidate uid model solution. However, for that one needs to nd a suitable candidate for S(
). The uniqueness proof is also helpful here, since it suggests that S(
) should be the inverse of the unique locally bounded solution of (4.3): see (4.19) and Lemma 4.4. Our assumptions on  imply that such a solution exists, so let T(
) be the unique locally bounded solution of (4.3), i.e., T(
) = (H  Ue )(
) where H (
) is de ned by (4.1). Then, T(
) 2 C1(R+) and satis es T(0) = 0, T(
) is strictly increasing, and limu!1 T(u) = 1. It follows that S(t) = T?1 (t) for t  0 exists, is strictly increasing, S(
) 2 C1(R+), and (4.29) S0(t) = T 0(S1(t)) t  0: For each t  0, let (t) be the unique element of MF such that h1f0g; (t)i = 0 and

1

(0;w)



; (t) = 1

 (0;w)(
? S(t));  + 

Z t

0

 1(0;w)(
? (S(t) ? S(s)));  ds;

(4.30)

for all w 2 (0; 1]. Note that (0) =  and h1(0;1); (t)i  h1;  i + t for all t  0. Proposition 4.6 Let  : [0; 1) ?! MF be de ned via (4.30) and let t = inf ft  0 : (t) = 0g. Then t = 1, (t) has no atoms for each t  0, and for each w 2 (0; 1], t ?! 1(0;w); (t) is a function in C([0; 1)). In particular, (
) is continuous. Proof. In order to prove that t = 1, it suces to show that Z(t) = h1; (t)i is strictly positive for all t  0. Note that h1f0g; (t)i = 0 and so (4.30) with w = 1 becomes

Z(t) = H0 (S(t)) + 

Z t? 0




1 ? F (S(t) ? S(s)) ds;

t  0;

(4.31)

where H0 (
) is given by (4.2). Since  6= 0, Z (0) > 0. Equation (4.31) can be used to show that Z(t) > 0 for t > 0. To see this note that given  2 (0; 1), since F (0) = 0 and F is right continuous, there exists M > 0 such that 1 ? F (x) >  for all x < M . Since S(
) is continuous, for each xed t > 0, there exists t > 0 such that s  0 and jt ? sj < t imply that jS(t) ? S(s)j < M . These two facts together with (4.31) yield

Z(t)  

Zt

(t?t)+

 ds > 0;

t > 0:

So t = 1. Next we show that (t) has no atoms for each t  0. Since (0) =  , this holds for t = 0. By (4.30), we have for each t  0 and 0 < x < w  1,

1





[x;w) ; (t) = 1[x;w) (
? S(t));  + 

Z t

0



1[x;w)(
? (S(t) ? S(s)));  ds:

(4.32)

Fix x 2 (0; 1), let w # x, and use bounded convergence to obtain

1 ; (t) = 1 (
? S(t));  +  Z t 1 (
? (S(t) ? S(s)));   ds; fxg fxg fxg 0

24

(4.33)

for t  0, x 2 (0; 1). The rst term on the right side of (4.33) is clearly zero since  has no atoms. Recall that S(
) is strictly increasing and  has at most countably many atoms. Thus, for each xed (t; x) 2 [0; 1)  (0; 1), h1fxg(
? (S(t) ? S(s)));  i = 0 for (Lebesgue) almost every s 2 [0; t].

 Therefore the second term  on the right side of (4.33) is zero. Thus, 1fxg; (t) = 0 for all x > 0. By de nition 1f0g; (t) = 0, so (t) has no atoms for each t  0. In order to show that for each w 2 (0; 1], h1(0;w); (t)i is a continuous function of t 2 [0; 1), it suces to show that each term on the right side of (4.30) is continuous. This is straightforward for the rst term since S(
) is continuous and  has no atoms. For each w 2 (0; 1] and t  0, and for each s 2 [0; t] such that  does not have an atom either at S(t) ? S(s) or at w + S(t) ? S(s),



 lim 1(0;w)(
? (S(u) ? S(s)));  = 1(0;w)(
? (S(t) ? S(s)));  : u!t

Since S(
) is strictly increasing and  has at most countably many atoms, the above holds except perhaps for countably many values of s 2 [0; t], which comprise a set of Lebesgue measure zero (possibly depending on w and t). So, using bounded convergence and the fact that the integrand is bounded, one can show that, for each xed w 2 (0; 1], the second term on the right side of (4.30) is a continuous function of t 2 [0; 1). Finally, we must show that (
) is continuous. In order to do this, we must show that for each g 2 Cb (R+), hg; (t)i is a continuous function of t 2 [0; 1). Since Z (
) is strictly positive and continuous, it suces to show that hg; (t)i ; (4.34) Z(t) is a continuous function of t 2 [0; 1). In order to do this, for each x 2 R+, let

1 ; (t) G(t; x) = [0;x) ; Z(t)

t  0:

for each

Since,  t  0, (t) does not have an atom at the origin, it follows that for each x 2 R+, 1 ; (t) = 1 ; (t) for each t  0. Thus, from the results proved above, for each x 2 R , [0;x)

(0;x)

+

G(t; x) is a continuous function of t 2 [0; 1). Since (t)=Z(t) is a probability measure for each t  0, it follows that (4.34) is a continuous function of t 2 [0; 1) (cf. [2], Thm. 2.2 and use the fact that f[x; y ) : 0  x < y < 1g [ ? forms a  -system).  Thus far, it has been shown that the measure valued function (
) de ned via (4.30) satis es conditions (i) and (ii) in the de nition of a uid model solution, and that t = 1. Thus condition (iv) holds trivially. So the remaining task is to prove that (
) satis es (3.3) for all t  0. This is addressed in Lemmas 4.7 and 4.8 below. Lemma 4.7 implies that the de nitions of S(
) and (
) give rise to the appropriate relationship between S0(
) and Z (
). The proof of Lemma 4.8 uses this relationship to prove that (
) satis es a dierential form of (3.3).

Lemma 4.7 Let  : [0; 1) ?! MF be de ned via (4.30). Then, for all t  0, S0 (t) = Z1(t) :

25

(4.35)

Proof. In order to verify (4.35), perform the change of variables v = S(s) in (4.31) to obtain, for t  0, Z(t) = H0 (S(t)) + 

Z S(t) ? 0


1 ? F (S(t) ? v ) dT(v ) = H0 (S(t)) + (T0  Fe )(S(t)) = T0(S(t)):

Here the nal equality follows from the fact that N (
) = T0(
) is the unique locally bounded solution of (4.4). By (4.29), the above yields Z (t) = 1=S0(t) for all t  0.  We are now ready to prove a dierential form of (3.3), which will complete the veri cation that (
) de ned via (4.30) is a uid model solution.

Lemma 4.8 Let  : [0; 1) ?! MF be de ned via (4.30). For g 2 C , t ?! hg; (t)i is a function in C1([0; 1)) and d hg; (t)i = ? hg 0; (t)i +  hg;  i ; dt h1; (t)i

t  0:

(4.36)

Proof. Observe that for each g 2 C , hg; (t)i and the terms in the expression on the right side of (4.36) are continuous functions of t 2 [0; 1). To see this note that (
) is continuous, g and g 0 are bounded continuous functions, and Z (
) = h1; (
)i is continuous and strictly positive (since t = 1). Therefore, in order to prove Lemma 4.8, it suces to prove that for each g 2 C , t ?! hg; (t)i is a dierentiable function of t 2 [0; 1) and that (4.36) holds. Note that by (4.30),

(4.33), and a monotone class theorem,

Z t

   hg; (t)i = g(
? S (t));  +  g(
? (S(t) ? S(s)));  ds; 0

t  0;

(4.37)

for all g 2 Cb(R) such that g (x) = 0 for x  0. Fix g 2 C and extend g to be identically equal to zero on the negative half line. Then g 2 C1b (R) and g (x) = 0 and g 0(x) = 0 for x  0. In particular, (4.37) holds for both g and g 0. We will proceed by rst showing that for this xed g , each term on the right side of (4.37) is dierentiable, and then computing the derivative of each term. By (4.35), @ g(x ? S(t)) = ?g 0(x ? S(t)) ; t 2 [0; 1); x 2 R: @t Z(t) For each b < 1, the term on the right side of the above equation is continuous and bounded for (t; x) 2 [0; b]  R, where we have used the fact that 1=Z(
) is continuous and bounded on [0; b]. This together with the fact that  2 MF allows an interchange in the order of integration and dierentiation to obtain d g(
? S(t));   = ?  g 0(
? S(t)) ;   ; t  0; (4.38) dt Z(t) where the right member is a continuous function of t 2 [0; 1). Similarly, for each ( xed) s 2 [0; 1), @ g(x ? S(t) + S(s)) = ?g 0(x ? S(t) + S(s)) ; t 2 [0; 1); x 2 R; @t Z(t) 26

which for each b < 1 is again continuous and bounded (uniformly in s) for (t; x) 2 [0; b] R. Thus, for each ( xed) s 2 [0; 1), @ g(
? S(t) + S(s));   = ?  g 0(
? S(t) + S(s)) ;   ; t  0: (4.39) @t Z (t) For each b < 1, the right side is again continuous and  bounded (uniformly in s) as a function of t 2 [0; b]. Using the facts that g(
? S(t) + S(s));  and the right side of (4.39) are continuous functions of (s; t) 2 [0; 1)  [0; 1), it follows that d Z t g(
? (S(t) ? S(s)));   ds = ? Z t  g0(
? S(t) + S(s)) ;   ds + hg;  i ; t  0: (4.40) dt 0 Z(t) 0 Thus, each term on the right side of (4.37) is dierentiable and so t ?! hg; (t)i is a dierentiable function of t 2 [0; 1). Moreover, using (4.37), (4.38), and (4.40), and then using (4.37) again with g0 in place of g gives, for t  0,     d hg; (t)i = ? g0(
? S(t)) ;  ?  Z t g 0(
? (S(t) ? S(s))) ;  ds +  hg;  i  dt Z(t) 0  g0(
)Z (t)  = ?  ; (t) +  hg;  i ; Z (t)



which veri es that (4.36) holds.

Proof of Existence for Theorem 3.1,  2 M . The function (
) de ned via (4.30) clearly c;p F

satis es (0) =  and condition (ii) in the de nition of a uid model solution. By Proposition 4.6, condition (i) holds and t = 1, so that (iv) holds trivially. Finally, condition (iii) follows from Lemma 4.8 by integrating (4.36). 

4.3 Dependence on the Initial Condition

One consequence of our construction of a uid model solution in Section 4.2 is that the mapping from the initial measure to the associated uid model solution is measurable. We will need this rather technical fact in order to prove Theorem 3.2. Since the measurability argument depends on the construction, it is presented here. However, readers that are willing to accept this fact may forego an immediate reading of the proof in order to proceed to the convergence argument in Section 5. Consider the Borel  -algebra on MF associated with the topology of weak convergence. Note c;p is endowed with the M that the sets McF and Mc;p F F are measurable subsets of MF . The set relative topology of weak convergence inherited from MF . Since Mc;p is measurable, a relatively F c;p ?! D([0; 1); M ) be given by  is measurable in M . Let  : M open set in Mc;p F p ( ) = (
) F p F F c;p for  2 MF , where (
) is de ned via (4.30). Then for all t  0 and 0  x < w  1,

1

[x;w)



; (t) = 1

 [x;w) (
? S(t));  + 

Z t

0

 1[x;w)(
? (S(t) ? S(s)));  ds:

For x 6= 0, this is simply (4.32). To see that the above holds for x = 0, use (4.30) and the following facts: (t) does not charge the origin for all t  0,  has no atoms, S(
) is strictly increasing, and 27

 has at most countably many atoms. Since f[x; w) : 0  x < w < 1g [ ? forms a -system, Dynkin's  - theorem implies that, for each t  0

Zt

h1B ; (t)i = h1B (
? S(t)); i +  h1B (
? (S(t) ? S(s)));  ids; As a shorthand notation, we write

0

(t) =   S(t) +

Zt 0

for all Borel sets B  R+:

  S(t)?S(s) ds;

(4.41)

where for each Borel set B  R+, h1B ;   S(t)i = h1B (
? S(t));  i and

 Zt

 Z t

 1B ;   S(t)?S(s)ds = 1B (
? (S(t) ? S(s)));  ds: 0

0

Finally, let  : McF ?! D([0; 1); MF ) be given by ( ) = (
), where (
) = p ( ) if  6= 0 and (
)  0 if  = 0.

Lemma 4.9 The mapping p is continuous, and  is measurable. Before proceeding with the proof of Lemma 4.9, some comments are in order. Notice that the inverse image under the mapping  of the function that is identically equal to the zero measure is simply the zero measure, which is a measurable subset of McF . Therefore, in order to show that  is measurable, it suces to show that the inverse images under the mapping  of measurable subsets of D([0; 1); MF ) that do not contain the function that is identically equal to the zero measure are measurable subsets of Mc;p F . This holds if p is continuous. Thus measurability of  is an immediate consequence of the fact that p is continuous, which is veri ed in the proof of Lemma 4.9 below. This naturally raises the question as to whether or not the mapping  is continuous, i.e., is the mapping  continuous at the zero measure. This turns out to be true, however the proof requires techniques not used elsewhere in this paper. In particular, the proof exploits a certain order preservation property of the uid model dynamics. The interested reader can nd both the statement of the order preservation property and a proof that  is continuous at the zero measure in [23]. Proof of Lemma 4.9. As explained in the paragraph following the statement of Lemma 4.9, measurability of  follows immediately from continuity of p . Thus, it suces to show that p is continuous. In order to do that, consider the spaces C% = fU : [0; 1) ?! [0; 1) : U is continuous, nondecreasing, and U (0) = 0g; C" = fU 2 C% : U is strictly increasingg; and C";1 = fU 2 C" : ulim U (u) = 1g; !1 each of which is endowed with the topology of uniform convergence on compact sets. Note that, since the functions in these spaces are all nondecreasing, uniform convergence on compact sets is equivalent to pointwise convergence for each of these spaces. The proof that p is continuous proceeds by proving continuity of various intermediate maps to and from these spaces.   ?1  Let  : Mc;p F ?! C";1 be de ned by ( ) = S (
), where S (t) = T (t) = inf fu  0 : T (u) > tg,  t  0, is the functional inverse of T (
) = (H  Ue )(
) with H (
) de ned by (4.1) and Ue (
) de ned 28

by (4.5). The rst step of the proof is to show that  is continuous. In order to do this, we de ne two maps 1 and 2 given by 1 : Mc;p F ?! C";1 such that 1( ) = T(
); 2 : C";1 ?! C";1 such that 2(T(
)) = S(
): Then  = 2  1. Therefore, it suces to show that 1 and 2 are continuous. We begin by showing that 1 is continuous. In order to do this, let 1;1 : Mc;p F ?! C% be given by 1;1( ) = H (
), with H (
) de ned by (4.1). It is easily veri ed that H (x) = h ^ x;  i, x 2 R+. Note that for each xed x 2 R+, (
) ^ x 2 Cb (R+). Therefore, if n 2 Mc;p F , n  1, w , and  ?!  , it follows that H converges pointwise to H . Thus,  is continuous  2 Mc;p n n  1;1 F since pointwise convergence of functions in C% to a limit in C% implies uniform convergence on compact sets. Also, let 1;2 : C% ?! C";1 be given by 1;2(H (
)) = (H  Ue ) (
). It is perhaps not surprising that 1;2 is also continuous. To see this, consider Hn 2 C% , n  1, and H 2 C% such that Hn ?! H uniformly on compact sets as n tends to in nity. For each K  0, k (Hn  Ue) ? (H  Ue) kK  kHn ? H kK Ue(K ); which tends to zero as n tends to in nity. Thus, 1;2 is continuous. Since 1 = 1;2  1;1, it follows that 1 is continuous. Next we show that 2 is continuous. To see this, consider Tn 2 C";1 , n  1, and T 2 C";1 such that Tn ?! T uniformly on compact sets as n tends to in nity. Let Sn (
) = 2 (Tn (
)), n  1, and S(
) = 2 (T(
)). It suces to show that Sn(t) ?! S(t) as n tends to in nity for each t  0. Fix t  0. For each n  1, choose un such that Tn(un) = t and choose u such that T(u) = t. In other words, Sn (t) = un , n  1, and S(t) = u, and consequently it suces to show that un ! u. Using the fact that T(
) is strictly increasing, it can be veri ed that the sequence fun gn1 is bounded. This together with the fact that Tn ?! T uniformly on compact sets as n tends to in nity and Tn(un) = T(u), n  1, implies that limn!1 T(un) = T(u). Again using the fact that T(
) is strictly increasing, it then follows that un ! u as n tends to in nity. We now turn our attention to proving that p is continuous. Suppose that fn gn1  Mc;p F , w c;p  2 MF , and n ?! . We will show that p(n ) ?! p () in the Skorohod J1 -topology. For this, note that the Prohorov metric de ned on the set of Borel probability measures on R+ naturally extends to a metric  on MF under which MF is a Polish space. In particular, for 1; 2 2 MF , (1; 2) is given by (1; 2) = inf f > 0 : h1B ; 1i  h1B  ; 2i + ; h1B ; 2i  h1B  ; 1i +  for all closed sets B  R+g; w where B  = fx 2 R+ : inf y2B jx ? y j < g. Moreover, if fn gn1  MF and  2 MF , then n ?!  if and only if limn!1 (n ;  ) = 0 (cf. [15], Ch. 3, Thms. 1.7 & 3.1, which readily generalize from probability measures to MF .). In order to show that p (n ) ?! p ( ) in the Skorohod J1 -topology, it suces to show that there exists fn gn1  C";1 such that for each K  0, lim sup jn (t) ? tj = 0; (4.42) n!1 0tK

lim sup 

n!1 0tK

Z  (t) n

0

lim sup (n  Sn (n (t));   S(t) ) = 0;

(4.43)

  S(t)?S(s)ds = 0;

(4.44)

n!1 0tK

  Sn (n(t))?Sn(s) ds; 29

Zt 0

!

where Sn (
) = (n ), n  1, and S(
) = ( ). Here we have used the fact that (11 +12; 21 +22)  (11; 21) + (12; 22), ij 2 MF , i; j = 1; 2, (4.41) above, and Prop. 5.3 and Rmk. 5.4 in Ch. 3 of [15]. For n  1, let n (t) = Tn (S(t)) for all t  0, where Tn (
) = 1(n ). Since Tn 2 C";1 , n  1, and S 2 C";1 , it follows that n 2 C";1 for each n  1. Since 1 is continuous, for T = 1 ( ), Tn ?! T uniformly on compact sets as n ! 1. This together with T(S(t)) = t, t  0, and the fact that S 2 C";1 implies that (4.42) holds for all K  0. Note that Sn (n(t)) = S(t), t  0, which can be used to simplify the expressions in (4.43) and (4.44). We now verify (4.43). Let ~ > 0 and choose N~ such that n  N~ implies that (n ;  )  ~. For a closed set B  R+, B + S(t) is also closed, and so, for n  N~ , (4.45) h1B+S(t); ni  h1(B+S(t)) ; i + ~ and h1B+S(t); i  h1(B+S(t)) ; ni + ~: ~

~

Note that

?B + S(t)
~  ?B~ + S(t)
[ ((S(t) ? ~)+; S(t)):

(4.46)

Combining (4.45), (4.46), and the de nitions of n  S(t) and   S(t) yields that for n  N~

E D E D D 1B ; n  S(t)  1B ;   S(t) + 1((S(t)?~) E D E D D  ~

+

;S(t)); 

E

+ ~;

(4.47)

E

1B ;   S(t)  1B  ; n  S(t) + 1((S(t)?~) ;S(t)); n + ~: ~

+

(4.48)

E

D

Since (n;  )  ~ for n  N~ , and since  (f0g) = 0, it follows that 1[(S(t)?~) ;S(t)]; n  D E 1((S(t)?2~) ;S(t)+~) ;  + ~ for n  N~ . This together with (4.48) implies that for n  N~ +

+

E D

E D

D

E

1B ;   S(t)  1B  ; n  S(t) + 1((S(t)?2~) ;S(t)+~) ;  + 2~: ~

D

(4.49)

+

E

Since  has no atoms, lim~!0 supt0 1((S(t)?2~) ;S(t)+~) ;  = 0 (cf. proof of Lemma A.1). Therefore, D E given  > 0, there exists ~ > 0 such that 1((S(t)?2~) ;S(t)+~) ;  + 2~   for all t  0. Since ~   and ((S(t) ? ~)+ ; S(t))  ((S(t) ? 2~)+ ; S(t)+~), from (4.47) and (4.49) it follows that, for all t  0, n  N~ , and closed B  R+, +

+

E

E D

D

E D

D

E

1B ;   S(t)  1B  ; n  S(t) + : 1B ; n  S(t)  1B  ;   S(t) +  and Since Sn (n (t)) = S(t), t  0, and since  > 0 was arbitrary, (4.43) holds. We now verify (4.44). Fix K;  > 0. Let ~ 2 (0; ). Since Sn (n (t)) = S(t), t  0, for a Borel set B  R+, t  0, and n  1,

 Z t

 1B (
? (S(t) ? S(s)));  ds; 1B ;   S(t)?S(s) ds = * Z  (t) 0 + Z0  (t)

1 (
? (S(t) ? S (s)));   ds: 1 ;    ds =  Zt

B

n

0

n

Sn (n(t))?Sn (s)

0

B

n

By (4.42) and the continuity of , there exists N such that n  N implies that sup0tK jn (t)?tj < =2 and kSn ? SkK+=2 < ~. Fix such an N . Then, since ~ < , it follows that for n  N , 30

t 2 [0; K ], s 2 [0; t], and x 2 R+ such that x ? S(t) + S(s) 2 B, either x ? S(t) + S(s) 2 [0; ~) or x ? S(t) + Sn (s) 2 B  . Similarly, for n  N , t 2 [0; K ], s 2 [0; n(t)], and x 2 R+ such that x ? S(t) + Sn (s) 2 B, either x ? S(t) + S(s) 2 (?~; 0) or x ? S(t) + S(s) 2 B  . This gives, for n  N and t 2 [0; K ],

 Zt

 Z t

 1B ;   S(t)?S(s) ds  1B (
? (S(t) ? Sn (s)));  ds 0 0 Z t



* Z  (t) n

1B ;

0



+

+ Z

  Sn (n(t))?Sn (s)ds 

0 n (t)

0

1[0;~) (
? (S(t) ? S(s)));  ds;

1

(?~;0)[B  (
? (S(t) ? S(s))); 

 ds:

Since each of the integrands on the right side is bounded above by 1, since S(t) = Sn (n(t)), and since, for n  N and t 2 [0; K ], jn (t) ? tj < =2, it follows that for n  N and t 2 [0; K ],

 Zt 1B ;

* Z  (t) n

1B ;

0

 Z  (t)

  S(t)?S(s) ds 

0

+

Z t

 1B  (
? (Sn (n (t)) ? Sn (s)));  ds + =2

 1[0;~) (
? (S(t) ? S(s)));  ds;

(4.50)

+ Z t0 

  S ( (t))?S (s)ds  1B (
? (S(t) ? S(s)));  ds + =2 0 Z t

 n

0

n

n



n

+

0

1(?~;0)(
? (S(t) ? S(s)));  ds:

(4.51)

Thus, it suces to show that there exists ~ 2 (0; ) such that, for all t 2 [0; K ],

Z t

0

 1(?~;~)(
? (S(t) ? S(s)));  ds  =2:

This requires some care since  can have atoms. Let A PR+ denote the set containing all of the atoms of  , which is at most countable, and let d = a2A  (fag)a be the Borel measure comprised of only the

atoms of  . Then c =  ? d has no atoms. Therefore, there exists ~1 2 (0; ) such that supy2R 1(y?~ ;y+~ ) ; c < =6K (cf. the proof of Lemma A.1). Thus, for t 2 [0; K ] and ~ 2 (0; ~1], we have 1

+

Z t

0

=



1(?~;~) (
? (S(t) ? S(s)));  ds

Z tD 0

E

1(S(t)?S(s)?~;S(t)?S(s)+~); c ds +

 t+  6K

Since

1

X

a2A

 (fag)

Z tD 0

Z tD 0

E

1(S(t)?S(s)?~;S(t)?S(s)+~); d ds

E

1(S(t)?S(s)?~;S(t)?S(s)+~); a ds:

P  (fag)  1, there exists a nite set A  A such that P  a2AnA  (fag)  =6K . a2A 

31

For

t 2 [0; K ] and ~ 2 (0; ~1], we have

Z t



1(?~;~) (
? (S(t) ? S(s)));  ds

Z tD E X X   6K t +  (fag) 1(S(t)?S(s)?~;S(t)?S(s)+~) ; a ds  (fag)t + 0 a2A a2AnA Z E tD X ;  ds:   +  +  (fag) 1 0

(S(t)?S(s)?~;S(t)?S(s)+~) a 6 a2A 0 Note that for t 2 [0; K ], s 2 [0; t], and a 2 A such that a 2 (S(t) ? S(s) ? ~; S(t) ? S(s) + ~), h ?
 ?
i s 2 T S(t) ? a ? ~ + ; T S(t) ? a + ~ + : If A 6= ?, let ~ 2 (0; ~1] be such that      ?S(t) ? a + ~
+ ? T ?S(t) ? a ? ~
+   ; sup max T a2A 6jA j

6





t2[0;K ]

where jA j denotes the number of atoms in A . Otherwise, let ~ = ~1 . Then, for t 2 [0; K ],

Z t

 1(?~;~) (
? (S(t) ? S(s)));  ds   +  +  =  : 6 6 6 2 0 This together with (4.50) and (4.51) gives, for n  N and t 2 [0; K ],  Zt 1B ;

* Z  (t) 1B ;

n

0

 *

  S(t)?S(s)ds  1B  ;

Z  (t) n

+

  Sn (n(t))?Sn(s) ds + ;

+ Zt

   S ( (t))?S (s) ds  1B (
? (S(t) ? S(s)));  ds + : 0

n

n

n

0



0

Since K;  > 0 were arbitrary, (4.52) and (4.53) imply (4.44).

(4.52) (4.53)



5 Convergence to Fluid Model Solutions The objective of this section is to prove the following theorem, and then apply it to prove Theorem 3.2. Theorem 5.1 Consider a sequence of processor sharing queues as de ned in Section 3.3, satisfying assumptions (3.14){(3.22). Then the sequence of uid scaled measure valued processes fr (
)g is tight. Moreover, any limit point ? (
) is a.s. a uid model solution for the critical data (;  ). Recall that the sequence fr (
)g is tight if and only if the associated sequence of probability laws on D([0; 1); MF ) is tight. The term \limit point" in the above statement refers to any limit in distribution along some subsequence of fr (
)g. We use this terminology because our objective is to show that all such limit points have the same law. This uniqueness in law will hinge on the almost sure characterization of limit points as uid model solutions. Throughout this section we assume that the conditions in Section 3.3 hold. In Section 5.1 we lay the groundwork for our main proofs by compiling some basic results, and making some rather technical choices of various constants. Tightness of fr (
)g is proved in Section 5.2, and the fact that limit points are a.s. uid model solutions is proved in Section 5.3. Finally, the proof of Theorem 3.2 appears in Section 5.4. 32

5.1 Preliminaries

To expedite the proofs later on, we rst establish some basic consequences of the assumptions (3.14){(3.22) of Section 3.3, and make some judicious choices of various constants. For subsequent reference, the results of the following discussion are summarized at the end of this section, in Lemma 5.2. For the remainder of this section, let T > 0 and 0 < ;  < 1 be xed, and de ne ~ = =8. A dynamic equation. We begin by specifying a dynamic equation satis ed by the uid scaled state descriptor r (
). Starting with (2.13) and substituting in the de nition of the residual service times, one obtains after some simpli cation that for each r, a.s. for each bounded, Borel measurable function g : R+ ?! R, and all t; h  0,

? hg; r(t + h)i = 1

(0;1)


g (
? S r

t;t+h

 ); r (t) +

E rX (t+h) i=E r (t)+1

?1

(0;1)g


(vr ? S r

Uir ;t+h ):

i

(5.1)

Recall that we always assume g is extended to be identically zero on (?1; 0) so that functions of the form g (
? a) are well de ned on R+ for any a > 0. Notice that this equation provides a decomposition of the quantity hg; r (t + h)i into a component arising from jobs already in the system at time t, and a component resulting from the arrival of new jobs during the interval r (t; t + h]. Intuitively, since St;t +h gives the cumulative service received by each job in the system during the interval (t; t + h], the measure r (t + h) should be obtained from r (t) by shifting the r , and removing any mass that ends up in (?1; 0]. This explains latter measure to the left by St;t +h the shift and truncation of g in the rst term on the right side above. Similarly, a job arriving at time Uir 2 (t; t + h] will receive a cumulative amount of service equal to SUr ir ;t+h ^ vir up to time t + h, and so vir must be shifted to the left by this amount before contributing to the integral hg; r(t + h)i. This explains the second term on the right side. Equation (5.1) takes the following form for the uid scaled processes. For each r, a.s. for each bounded, Borel measurable function g : R+ ?! R, and all t; h  0,

? hg; r(t + h)i = 1

(0;1)


g (
? Sr

rEX (t+h)  ?1 1 r );  (t) + r

t;t+h

r i=rEr (t)+1

(0;1)g


(vr ? Sr i

Uir =r;t+h ):

(5.2)

We refer to (5.2) as the dynamic equation for r (
), and it is the equation we will use to analyze the behavior of r (
). Frequently D we will setr g E 1 in this equation, in which case the rst term on the right side will look like 1(St;tr h ;1) ;  (t) . Although we are ultimately only interested in functions g 2 C1b (R+) when we use (5.2) to show convergence in distribution of the uid scaled state descriptors, we will need to work with discontinuous functions at various stages along the way, which is why we include this possibility in equation (5.2). Functional weak laws of large numbers; the constant l. We now establish some basic functional weak law of large numbers estimates which arise from our asymptotic assumptions (3.14){(3.18) on the arrival and service processes. First note that by Lemma A.2 (cf. Appendix) and (3.14){(3.18), we have the following functional weak law: +

E (
) 1 rX g(v r) =) (
) hg;  i ; r

r

i=1

i

33

as r ! 1;

(5.3)

for each nonnegative Borel measurable function g : R+ ?! R+ that is  -a.e. continuous and satis es hg;  i < 1, hg;  ri < 1 for each r > 0, and

hg;  ri ?! hg;  i ;

as r ! 1:

(5.4)

Here (t) = t for all t  0. Moreover, since the limit in (5.3) is deterministic, the convergence in distribution there is joint with (3.19). By (3.15), any nonnegative g 2 Cb (R+) satis es (5.4). In particular, for such a g and any 0 < l  T we have

0 lim P @ r!1

r (t+l) rEX

sup r1

t2[0;T ?l] i=rEr (t)+1

With l = T and g  1, this implies

1 g(vir)  2l hg;  iA = 1:

?E r(T )  2T
= 1: lim P r!1

(5.5)

(5.6)

For the rest of this section, we x l such that 0 < l < =16 and T is an integer multiple of l. Then by (5.5), for any g 2 Cb (R+),

1 r (t+l) rEX  1 g(vir)  8 hg;  iA = 1: sup r t2[0;T ?l]

0 lim P @ r!1

i=rEr (t)+1

(5.7)

We will use (5.3) repeatedly for several choices of g below, as well as (5.7) for several choices of g 2 Cb (R+). To begin with, let g  1. Then our choice of l implies by (5.7) that lim P

r!1

!

sup (E r (t + l) ? E r (t))   = 1; 4

t2[0;T ?l]

(5.8)

where for convenience we have relaxed the bound from =8 to =4 and used the fact that  is a probability measure. Global bounds for the initial condition; the constants M0 ; MT . By (3.19){(3.21), we can choose an M0 > 0 such that

?




(5.9)

   ;  <  1 ? ~ :

(5.10)

lim inf P h1; r (0)i _ h; r (0)i < M0  1 ? ~; r!1 and



P 1[M ;1) 0

4

2

De ne MT = M0 + 2T . Fine structure bounds for the initial condition; the constant . Given  > 0, let N = dM0 =e  and de ne the nite set of overlapping closed intervals fIn gNn=0 by In = [n; (n + 2)], for n = 0; 1; : : : ; N ? 1, and IN = [N; 1). It is evident that for every x 2 R+, there is an n 2 f0; 1; : : : ; Ng such that [x; x + ]  In : 34

(5.11)

By (5.10) and (3.23) (using 2 instead of  there), we can choose 0 <  < 2Ml T so that





N P max fh1 ; ig < 4  1 ? ~: n=0 In

Now de ne the set

n o; N fh 1 A =  2 MF : max ;  ig < n=0 In 4

w and suppose that fk g1 k=1  MF with k ?!  2 A, as k ! 1. Then since each of the sets In is closed, a trivial generalization of the Portmanteau theorem (cf. [2], Thm. 2.1) to nite measures yields that for n = 0; 1; : : : ; N,

lim sup h1In ; k i  h1In ;  i <  : 4 k!1

Thus k 2 A for suciently large k, which implies that A  MF is an open set. This together with (3.19) and a second application of the Portmanteau theorem yields

?r (0) 2 A
 P? 2 A
 1 ? ~; lim inf P r!1

which implies by (5.11) that

!

  lim inf P sup 1[x;x+]; r (0) < 4  1 ? ~: r!1 x2R +

(5.12)

Convergence of the uid scaled workload processes; the constant . It is well known that, for any queue operating under a nonidling service discipline, the uid scaled workload processes W r (
) converge in distribution, under the assumptions (3.14){(3.22), to a process that a.s. equals its initial value for all time. So, since processor sharing is a nonidling service discipline, the sequence of processes fh; r (
)ig converges in distribution to a process that a.s. equals its initial value for all time. For completeness, we sketch the veri cation of this fact below. Consider the workload equation (2.2) on uid scale. If we de ne the uid scaled load processes rEX(t) Lr (t) = h; r (0)i + 1 vir ? t; r r

i=1

then the uid scaled workload equation can be rewritten for all t  0 as ?
h; r (t)i = Lr (t) + sup Lr (s) ? : 0st

(5.13)

(5.14)

Note that by (3.16), (5.3) holds with g = . This together with the fact that  h;  i = 1 implies by (5.13), (5.3) and (3.19) that L r (
) converges in distribution, as r ! 1, to the process that a.s. equals its initial value for all time, where that initial value has the same distribution as h; i. The mapping  : D([0; 1); R) ?! D([0; 1); R+) de ned by (x(
))(t) = x(t)+supf(x(s))? : 0  s  tg is continuous, so the continuous mapping theorem applied to (5.14) implies the result.

35

We make use of the above fact in the following way. Having xed l; M0; MT and , we now choose a 0 < < minf=4; l=4; T g. Then the fact that as r ! 1, h; r (
)i converges in distribution to a process that is a.s. equal to its initial value implies that

! lim P sup h; r (t)i ? h; r (0)i < =4 = 1: r!1

(5.15)

t2[0;T ]

Tail estimate for the state descriptor; the that the distribution

1constant  K?!. For

1any K;  >as0 such r  does not charge K , (3.15) implies that ;  r ! 1 , which implies by [0;K ] [0;K ]



 r (3.16) that 1(K;1) ;  ?! 1(K;1) ;  as r ! 1. So using (5.3), we have for any such K that E (
) 1 rX vr 1 r

r

1

i fvir >K g =) (
)

i=1

(K;1) ; 

;

as r ! 1.

(5.16)

Since, by (3.22),  does not charge any K a.s., we have by (3.19) that for any K > 0,

1 ; r(0) =) 1 ;  ; as r ! 1. [0;K ] [0;K ] Then, using the joint convergence in (3.19) as well as (3.21), we have for any K > 0,

1 ; r(0) =) 1 ;  ; as r ! 1. (K;1) (K;1)



 Since 1(K;1) ;  ?! 0 and E[ 1(K;1) ;  ] ?! 0 as K ! 1, we can choose a K > 0 suciently large so that

0 lim inf P @ sup r!1

1

t2[0;T ] r

 r (t) rEX

i=1

+ 1

vir 1fvir >Kg

(K;1)

1  ; r (0) < =5A  1 ? ~:

(5.17)

?




Fine structure estimate for the state descriptor; the constant ?. Now we de ne ? = K 4 ? 5 ?1 , and consider (5.7) for each member of a nite set of functions fgn gNn=0 , which we de ne in the following way. Let N = dT ?=e, and for each n = 0; 1; : : : ; N , choose gn 2 Cb(R+) such that gn is nonnegative, and for all x 2 R+, 1[n;(n+1)) (x)  gn (x)  1[(n? );(n+ )) (x). Note that for n = 0 and x 2 R+, 1[(n? );(n+ )) (x) = 1[0; ) (x). By (5.7), we have 3 2

1 2

1 2

3 2

0 N 8< \ @ lim P r!1 : n=0

which implies that

0 N 8< \ @ lim P r!1 : n=0

sup 1

3 2

91  hg ;  i=A = 1; r sup 1 g ( v )  n i 8 n ; t2[0;T ?l] r i=rE (t)+1 r (t+l) rEX r

D

r (t+l) rEX

1[n;(n+1)) (vir )  1[(n? r 8 t2[0;T ?l] i=rE r (t)+1

1 2

);(n+ 32

D

91 E= A )) ;  ; = 1; E

(5.18)

(5.19)

where for n = 0, we interpret the right side of the inequality as 8 1[0; ) ;  . Finally, having chosen the constants l; M0; MT ; ; ; K; ? in such a way that (5.6){(5.19) hold, we de ne for each r > 0, the intersection B r of all of the events appearing in (5.6), (5.8){(5.10), (5.12), (5.15), (5.17) and (5.19). Since ~ = =8, the above discussion implies that lim inf r!1 P(B r )  1? . We summarize the results of this section in the following lemma. 3 2

36

Lemma 5.2 Consider a sequence of processor sharing queues as de ned in Section 3.3, satisfying assumptions (3.14){(3.22). Let T > 0 and 0 < ;  < 1 be given. Then there exist strictly positive constants l; M0; MT ; , ; K; ?; r0 and events fB r gr>0, such that T=l is a positive integer, P(Br )  1 ? ;

for all r > r0 ;

and for each r > 0, on B r the following hold:

sup E r (t + l) ? E r (t)  4

(5.20)

t2[0;T ?l]

E r(T )  2T h1; r(0)i _ h; r (0)i < M0

1   [M ;1) ;  < 4 MT = M0 + 2T  < 2Ml T

  r sup 1[x;x+] ;  (0) < 4 x2R

< minf=4; l=4; T g r r sup h;  (t)i ? h;  (0)i < =4 0

+

t2[0;T ]

rEXt)

 1 sup r vir1fvir >Kg + 1(K;1) ; r(0) < =5 r (

t2[0;T ]

i=1

(5.21) (5.22) (5.23) (5.24) (5.25) (5.26) (5.27) (5.28) (5.29)

?1  ? = K 4 ? 5

(5.30)

for N = dT ?=e and all n 2 f0; 1; : : : ; N g, r (t+l) rEX E D 1 sup r 1[n;(n+1)) (vir )  8 1[(n? );(n+ )) ;  : t2[0;T ?l] 1 2

i=rEr (t)+1

3 2

(5.31)

As a nal note, we remark that by (5.2), sup h1; r (t)i  h1; r (0)i + E r (T );

t2[0;T ]

(5.32)

so by (5.22), (5.21), (5.27), (5.28), and (5.24), we have on B r for r > r0 ,

?




sup h1; r (t)i _ h; r (t)i  MT :

t2[0;T ]

37

(5.33)

5.2 Proof of Tightness

In this section we prove the rst half of Theorem 5.1, i.e., that the sequence of measure valued processes fr (
)g is tight. Recall that to show tightness, it suces by Jakubowski's criterion (cf. [12], Thm. 3.6.4) to show the following two properties: TC.1 For each T > 0 and 0 <  < 1, there is a compact subset CT; of MF such that ?r (t) 2 C for all t 2 [0; T ]
 1 ? : lim inf P T; r!1

TC.2 For each g 2 C1b (R+), the sequence of real valued processes fhg; r(
)ig is tight. Note that if we de ne g : MF ?! R by g ( ) = hg;  i for g 2 C1b (R+), then f g : g 2 C1b (R+)g de nes a family of continuous real valued functions on MF which is closed under addition and separates points of MF . Hence condition TC.2 above satis es condition (ii) of [12], Thm. 3.6.4. The

proof of TC.2 will follow from [15], Ch. 3, Thm. 7.2, upon verifying in Theorem 5.6 that the usual compact containment and controlled oscillation conditions are satis ed by fhg; r (
)ig. Although the compact containment condition just mentioned is an immediate consequence of TC.1, we show it separately as part of Theorem 5.6, since it is elementary and it is needed before we establish TC.1. Controlling the oscillations of hg; r (
)i for Theorem 5.6 is the major diculty in the above agenda. Proving that we have this control ultimately revolves around the fact that changes in this process over a short time interval depend on the arrival process (which is easily controlled), the magnitude of the uid scaled queue length, h1; r (
)i, and the amount of mass concentrated near zero over that time interval (cf. (5.2) and (3.13)). We estimate h1; r (
)i from the uid scaled workload, h; r (
)i, and then leverage the fact that h; r (
)i converges in distribution to a process that a.s. is equal for all time to its initial value, which has the same distribution as h; i. To do this, we consider two events for r (
): the event where the initial uid scaled workload is smaller than some threshold, and its complement. In Lemmas 5.3 and 5.4, we prove an upper bound for the uid scaled queue length on the rst event, and a lower bound for the uid scaled queue length on the second event. Then in Lemma 5.5, we give an upper bound for the amount of mass that r (t) can have concentrated near zero. Note that this is a result about the \ ne" structure of the measure r (t), as opposed to its total mass (queue length) or rst moment (workload). It is an essential ingredient for completing the proof of the controlled oscillation condition for Theorem 5.6, as well as for proving the limit point properties in Section 5.3. Finally, the proof of tightness for Theorem 5.1 appears at the end of this section. We begin by proving the three aforementioned lemmas. The rst lemma provides an upper bound for the uid scaled queue length on the event that the initial uid scaled workload is below the threshold =2. Lemma 5.3 Let T > 0 and 0 < ;  < 1. Let l; M0; MT ; ; ; K; ?; r0 be the constants, and fBr gr>0 be the events, given by Lemma 5.2. De ne D r = fh; r (0)i  =2g. Then on B r \ D r , for r > r0, sup h1; r (t)i  : t2[0;T ]

Proof. Subdivide [0; T ] into time intervals of length l (recall that l was chosen so that T is an integer multiple of l). Fix r > r0. We will prove by induction on n, that on B r \ D r , sup h1; r (t)i  ; (5.34) t2[nl;(n+1)l]

38

for n = 0; 1; : : : ; Tl ? 1. We rst verify the case n = 0. To start with we have on B r \ D r ,





h1; r(0)i = 1[0;]; r (0) + 1(;1); r (0)  4 + 1 h; r (0)i  4 + =2  ;



(5.35)

2 where the rst inequality is by (5.26) and Markov's inequality, the second uses the de nition of D r and the last is by (5.27). Now take t 2 [0; l]. Then on B r \ D r ,

h1; r(t)i  h1; r(0)i + E r(l)  2 + 4 < ;

(5.36)

h; r (nl)i  h; r (0)i + h; r (nl)i ? h; r (0)i < 2 + 4 < ;

(5.37)

by (5.32), (5.35) and (5.20). So (5.34) holds for n = 0. We now proceed by induction, having veri ed the rst step, and assume that (5.34) holds on Br \ D r for some 0  n < Tl ? 1. To show the statement holds with (n + 1) in place of n, we argue analogously to (5.36) by rst showing that h1; r ((n + 1)l)i  =2. To this end, we use an argument inspired in part by an idea of Grishechkin (cf. [18], p. 542). The idea is to consider two cases separately: the case where the queue length becomes zero during the interval [nl; (n + 1)l], and the case where it does not. We have on B r \ D r , by (5.28) and the de nition of D r . Now, if h1; r (s)i is never zero for s 2 [nl; nl + 4 ] we can write r Snl;nl +4 =

Z nl+4 nl

'(h1; r(s)i)ds =

Z nl+4 nl

h1; r (s)i?1 ds  4 =;

since we have assumed that (5.34) holds for this n and 4 < l by (5.27). This in turn implies, by (5.37), that

D

E D

1(Snl;nl ; r (nl)  1(  ;1) ; r (nl) r

;1)  4 h; r (nl)i

E

4

+4

 4 = 4 :

(5.38)

If on the other hand h1; r (s)i = 0 for some s 2 [nl; nl + 4 ], then all mass present in the system at time nl is gone by time s. More precisely, we have in this case, by (5.2), that

D

E

r 1(Snl;s r ;1) ;  (nl ) = 0;

r  Sr which, since Snl;s nl;nl+4 , also implies that

D

E

1(Snl;nl ; r(nl) = 0: r

;1) +4

39

(5.39)

Combining the above with (5.2), we see that in either case, on B r \ D r ,

D

h1; r((n + 1)l)i  1(Snl; r D n  1(Snl;nl r  2 ;

E

; r (nl) + E r((n + 1)l) ? E r(nl) l ;1) E r (nl) +  ; 

;1) 4

( +1)

+4

(5.40)

by (5.27) and (5.20) for the second inequality, and by (5.39) and (5.38) for the third. Now we can complete the proof as in (5.36). Using (5.2), we have on B r \ D r for any t 2 [(n + 1)l; (n + 2)l]:

h1; r (t)i  h1; r ((n + 1)l)i + E r((n + 2)l) ? E r((n + 1)l)  2 + 4 < ;

where the second inequality is by (5.40) and (5.20).  The next lemma provides a lower bound for the uid scaled queue length process on the event where the initial uid scaled workload is above the threshold =2. A consequence of this is an r upper bound for the rate at which St;t +h can increase as a function of h.

Lemma 5.4 Let T > 0 and 0 < ;  < 1. Let l; M0; MT ; ; ; K; ?; r0 be the constants, and fBr gr>0 be the events, given by Lemma 5.2. Let D r be the complement of D r , i.e., D r = fh; r (0)i > =2g. Then on B r \ D r , for r > r0, inf h1; r (t)i  ?1 ;

(5.41)

r sup St;t +h  h?; t2[0;T ?h]

(5.42)

t2[0;T ]

and for any 0 < h < T .

Proof. For this we have adapted the proof of Lemma 4.4 in [9]. We have on Br \ D r ,

 inf h; r (t)i 4 t2[0;T ]

? 1 ; r (t) + 1 ; r(t)
[0;K ] (K;1) t2[0;T ] 1 0  (t) rEX



 vir 1fv >Kg + 1(K;1) ; r (0) A  inf @K 1[0;K]; r (t) + 1 = inf

r

t2[0;T ]

 t2inf K h 1; r (t)i + 5 ; [0;T ]

r

i=1

r i

where the three inequalities are by (5.28) and the de nition of D r , (5.2) and (5.29) respectively. Notice that to obtain the last two terms in the second inequality above from (5.2), we have simply ignored any processing that has occurred for jobs with initial service time requirements greater than K . Now we have

? inf h1; r (t)i  4 5 = 1 : K ? t2[0;T ] 40

To prove (5.42), we have by (5.41) on B r \ D r , for 0 < h < T , r r sup St;t +h  h sup '(h1;  (t)i) t2[0;T ?h]

t2[0;T ]

h  inf t2[0;T ] h1; r (t)i  h?:



The next lemma gives, on B r \ D r , an upper bound for the amount of mass that r (t) can have concentrated near zero for t 2 [0; T ]. Note that by Lemma 5.3, on B r \ D r , the total mass of r (t)

is bounded above by  for t 2 [0; T ]. Lemma 5.5 Let T > 0 and 0 < ;  < 1. Let l; M0; MT ; ; ; K; ?; r0 be the constants, and fBr gr>0 be the events, given by Lemma 5.2. Recall that D r = fh; r (0)i > =2g. Then on B r \ D r , for r > r0 ,

 sup 1[0;] ; r (t)  =2: t2[0;T ]

Proof. In this proof we only consider realizations in Br \ D r . Recall that by Lemma 5.4, h1; r(t)i > 0 for all t 2 [0; T ]. First, consider two jobs, i < j for which vir; vjr 2 [n; (n + 1)) for some integer n  0. If Ujr =r  t and Ujr =r ? Uir =r  l, then at time t we have (vjr ? SUr jr =r;t) ? (vir ? SUr ir =r;t ) = SUr ir =r;Ujr =r + vjr ? vir Ujr =r ? Uir =r  sup ? t2[0;T ] h1; r (t)i  Ml ?  T > 2 ?  = ; where the last two inequalities are by (5.33) and (5.25) respectively. This implies that at most one of 1(0;](vir ? SUr ir =r;t) and 1(0;] (vjr ? SUr jr =r;t)

is nonzero. So for each n  0 and t 2 [0; T ], all jobs arriving by time t and satisfying 1[n;(n+1)) (vir )1(0;](vir ? SUr ir =r;t) = 1 must have arrived during the ( uid scaled) time interval (s; s + l], for some s 2 [0; t ? l]. This gives us the following estimate at time t, for each n = 0; 1; : : : ; dT ?=e:  r (t) rEX

i=1

1[n;(n+1)) (vir )1(0;](vir ? SUr ir =r;t) 

sup

r (s+l) rEX

s2[0;t?l] i=rE r (s)+1

1[n;(n+1)) (vir )1(0;](vir ? SUr ir =r;t);

which implies that for each n = 0; 1; : : : ; dT ?=e, we have

rEX(t) rEX (t+l) 1[n;(n+1)) (vir )1(0;](vir ? SUr ir =r;t)  sup 1r 1[n;(n+1)) (vir ): sup 1r t2[0;T ?l] i=rE r (t)+1 t2[0;T ] i=1 r

r

41

(5.43)

Next, for t 2 [0; T ] and Uir =r  t, we have by Lemma 5.4 that if vir  T ? + , then (vir ? SUr ir =r;t) > T ? +  ? T ? = . Thus for n  dT ?=e + 1, we have (5.44) 1[n;(n+1)) (vir )1(0;](vir ? SUr r =r;t) = 0: i

Using (5.2), we have on B r \ D r , sup

t2[0;T ]

1 ; r (t)  [0;]

sup

1

r r  (0;] (
? S0;t );  (0) +

rEX(t) sup r1 1(0;](vir ? SUr ir =r;t) r

t2[0;T ] i=1  r (t) rEX 1 X

 1 r 1[n;(n+1)) (vir )1(0;](vir ? SUr ir =r;t)  sup 1[x;x+];  (0) + sup r x2R+ n=0 t2[0;T ] i=1 r (t+l)  rEX dTX ?=e 1  1[n;(n+1)) (vir ) sup r 4+ n=0 t2[0;T ?l] i=rEr (t)+1 dTX ?=e D E  1 1  4 + 3 ;  8 [(n? 2 );(n+ 2 )) n* =0 + 1 X   1[(n? 21 );(n+ 32 )) ;  4+8 n=0 t2[0;T ]

= 4 + 4 ;

where the third line is by (5.26), (5.43) and (5.44), and the fourth line is by (5.31).  We are now ready to apply the previous three lemmas in order to verify conditions that are sucient to imply the tightness of fhg; r (
)ig, for g 2 C1b (R+). The proof of the controlled oscillation condition splits into two cases. We use Lemma 5.3 to prove it on B r \ D r and use Lemmas 5.4 and 5.5 to prove it on B r \ D r .

Theorem 5.6 Let g 2 C1b (R+), T > 0 and 0 < ;  < 1. Then there exist M;  > 0, and r0 > 0,

such that r > r0 implies

P

sup

! r P sup hg;  (t)i  M  1 ? ; t2[0;T ] ! r r sup hg;  (t + h)i ? hg;  (t)i   1 ? :

t2[0;T ?] h2[0;]

(5.45) (5.46)

Proof. De ne  = =(2(kgk1 _ 1)). Then by Lemma 5.2, there exist constants l, M0, MT , , , K , ?, r0 , and events fB r gr>0 such that r > r0 implies P(B r )  1 ?  and on B r , (5.20){(5.33) hold. De ne

M = (kgk1 _ 1)MT ;    = min T=2; l; 4?M (kg 0k _ 1) ; =?; 1 : T 1 42

(5.47)

We only consider r > r0 below. To prove (5.45), observe that on B r , by (5.33),





sup hg; r (t)i  kg k1 sup h1; r (t)i

t2[0;T ]

t2[0;T ]  kgk1 MT

 M:

To prove (5.46), consider rst D r as before. On B r \ D r , we have by Lemma 5.3, sup









sup hg; r(t + h)i ? hg; r (t)i  2 sup hg; r (t)i

t2[0;T ]  2kgk1 sup h1; r (t)i t2[0;T ]  2kgk1   :

t2[0;T ?] h2[0;]

We must show that the above estimate also holds on B r \ D r . First, observe that on B r \ D r , a rst order Taylor expansion of g gives the following estimate for all 0 < h < T , t 2 [0; T ? h] and y 2 (St;tr +h ; 1):

g(y ? Sr ) ? g(y) = ?Sr g0(w )  h?kg0k ; 1 y t;t+h t;t+h

(5.48)

r ; y ], where the inequality follows by Lemma 5.4. Now subtracting hg; r (t)i for some wy 2 [y ? St;t +h r ) = 1 r from both sides of equation (5.2) and using the fact that (1(0;1)g )(
? St;t (St;t h ;1)(
)g (
? +h St;tr +h ) for t 2 [0; T ? h] yields that on B r \ D r , +

hg; r(t + h)i ? hg; r(t)i = D1 (S

r t;t+h

+ 1r

E D

?
r r r g; r(t) ;1) (
) g (
? St;t+h ) ? g (
) ;  (t) ? 1[0;St;t h] X

rE r (t+h)



(1(0;1)g )(vir ? SUr ir =r;t+h )

E

+

D i=rE (t)+1? r


E  1(S ;1)(
) g(
? St;t+h) ? g(
) ; r(t) + kgk1 1[0;h?]; r(t) ?
+ kg k1 E r (t + h) ? E r (t)

  h?kg0k1 h1; r(t)i + kgk1 1[0;h?]; r (t)
? + kg k E r (t + h) ? E r (t) ; r

r t;t+h

1

where the second inequality is by Lemma 5.4 and the third is by (5.48).

43

Now taking the supremum over h 2 [0;  ] and t 2 [0; T ?  ], we see that on B r \ D r , sup





sup hg; r (t + h)i ? hg; r (t)i  sup

t2[0;T ?] h2[0;]



t2[0;T ?]



?kg 0k1 h1; r (t)i + kgk1 1[0;?]; r (t)





?
+ kg k1 E r (t +  ) ? E r (t)

h1; r (t)i + kgk 1 ; r (t) 1 [0;] t2[0;T ?] 4MT

 sup


? + kg k1 E r (t + l) ? E r (t)

!

 4M MT + kgk1 2 + kgk1 4 T  + + < ;

4 4 8 where the second inequality is by (5.47) and the third is by (5.33), Lemma 5.5, and (5.20). So the desired estimate holds on both B r \ D r and B r \ D r . Since P(B r )  1 ?  , this proves (5.46).  Proof of Tightness for Theorem 5.1. Recall that it suces to show conditions TC.1 and TC.2. Let T > 0 and 0 < ;  < 1. To show TC.1, de ne C~T; = f 2 MF : h1;  i _ h;  i  MT g:





1

Since h;  i  MT implies 1[K;1) ;   MT =K , we have sup

 2C~T;

[K;1); 

 ?! 0;

as K ! 1;

which implies that C~T;  MF is relatively compact (cf. [21], Thm. A 7.5). Now by (5.33), ?
lim inf P r (t) 2 C~T; for all t 2 [0; T ]  1 ? : r!1

De ne CT; to be the closure of C~T; and TC.1 is proved. Finally, TC.2 follows directly from Theorem 5.6 by applying a standard tightness criterion for real valued processes (cf. [15], Ch. 3, Cor. 7.4). 

5.3 Proof of Limit Point Properties

In this section we complete the proof of Theorem 5.1 by showing that any limit point of the sequence fr (
)g is a.s. a uid model solution for the critical data (;  ). In particular, we show that a.s., the sample paths of any such limit point ? (
) have the properties (i){(iv) of Section 3.1. Property (i) will follow from the a.s. continuity of the sample paths of any limit point of fhg; r (
)ig, for g 2 C1b (R+). Property (ii) will be a direct consequence of Lemmas 5.3 and 5.5. We will show en route that limit points ? (
) satisfy t = 1 on f? (0) 6= 0g and t = 0 on f? (0) = 0g (cf. (5.50) and (5.51) below). This is to be expected in light of Lemma 4.4. Consequently, it will suce for the proof of property (iii) to show that a.s. on the event f? (0) 6= 0g, (3.3) holds for all g 2 C and all t  0. For this, we will use the dynamic equation satis ed by r (
) (cf. (5.2)), together 44

with a Riemann integral approximation, to obtain a prelimit version of (3.3), and then we pass to the limit. Similarly for the proof of property (iv), it will suce to show that a.s. on the event f?(0) = 0g, ?(t) = 0 for all t  0. ? (
) be a limit point of fr (
)g and suppose fr0 (
)g  fr (
)g is a subsequence such that Let  r0 (
) =) ?(
) as r0 ! 1. To ease notation for the remainder of the proof, we relabel r0 as r, remembering that we have passed to a subsequence which converges in distribution to ? (
). Proof of property (i). To see that a.s., ?(
) has continuous sample paths, choose a countable set V  C1b (R+) that separates elements of MF (cf. [15], Ch. 3, Prop. 4.2 .). It follows from Theorem 5.6 (in particular (5.46)) that for each g 2 V  C1b (R+), the real valued process hg; ?(
)i a.s. has continuous sample paths. Since V is a countable separating class for MF , this implies that a.s., for every t  0, ? (t?) = ? (t).  Proof of properties (ii) and (iv). As mentioned above, it suces to show that for each T > 0 we have a.s.,

1 ; ?(t) = 0; for all t 2 [0; T ); (5.49) f0g ? ?  (t) 6= 0; for all t 2 [0; T ) on f (0) 6= 0g; (5.50) ? ?  (t) = 0; for all t 2 [0; T ) on f (0) = 0g: (5.51) To this end, let Qr and Q be the probability laws induced on D([0; 1); MF ) by r (
) and ? (
), respectively. For T > 0 xed, de ne the sets

A0 =

(

)

  (
) 2 D([0; 1); MF ) : sup 1f0g;  (t) > 0 ; t2[0;T ) ( )

A00 =  (
) 2 D([0; 1); MF ) : sup h1;  (t)i > 0 ; A000 =



t2[0;T )



 (
) 2 D([0; 1); MF ) : t2inf h1;  (t)i = 0 ; [0;T )

and let A = A0 [ (A00 \ A000 ). Clearly, any sample path of ? (
) which is not an element of A satis es (5.49){(5.51). So it suces to show that in a Q-null set. For each n = 1; 2; : : : , P A is contained choose a pair 0 < n ; n < 1 such that 1 < 1 and (n ; n ) ?! (0; 0) as n ! 1. Then by n=1 n Lemma 5.2, for each n there exist strictly positive constants ln ; M0;n; MT;n; n ; n; Kn; ?n ; r0;n and events fBnr : r > 0g such that r > r0;n implies P(Bnr )  1 ? n , and (5.20){(5.31) hold on Bnr with the above constants in place of the analogous ones appearing there. For each n, choose 0 < cn < n such that cn ?! 0 as n ! 1. Also de ne for each n

A0n = A00n = A000n =

(

( 

 (
) 2 D([0; 1); MF ) : sup

t2[0;T )

)  [0;c ) ;  (t) > n ; )

1

n

 (
) 2 D([0; 1); MF ) : sup h1;  (t)i > n ; t2[0;T )



 (
) 2 D([0; 1); MF ) : t2inf h1;  (t)i < 1=?n ; [0;T )

and let An = A0n [ (A00n \ A000n ). Recall from the de nition of the Skorohod topology, that if C is a closed subset of MF , then sets of the form f (
) 2 D([0; 1); MF ) :  (t) 2 C for all t 2 [0; T )g are 45

closed ), and so their complements are open. Since for each n, the sets f 2 MF :

1 in;  D([0; 1g,)f; M2F M n F : h1;  i  n g, and f 2 MF : h1;  i  1=?n g are all closed subsets of [0;cn ) MF , we see that A0n ; A00n; A000n and An are open sets in D([0; 1); MF ). Notice that the de nition of A implies that

A  fAn i.o.g =

1 [ 1 \

k=1 n=k

An ;

so it suces to show that Q(An i.o.) = 0. Let n be xed for the moment. Since we have chosen cn so that









sup 1[0;cn ) ; r (t)  sup 1[0;n] ; r (t)  sup h1; r (t)i

t2[0;T ]

t2[0;T ]

t2[0;T ]

for every r, combining Lemmas 5.3 and 5.5 yields the fact that on the entire event Bnr , for r > r0;n,









sup 1[0;cn) ; r (t)  sup 1[0;cn ) ; r (t)  n :

t2[0;T )

t2[0;T ]

Similarly, combining Lemmas 5.3 and 5.4 yields the fact that on Bnr for r > r0;n, either sup h1; r (t)i  sup h1; r (t)i  n ;

t2[0;T )

t2[0;T ]

or inf h1; r (t)i  inf h1; r (t)i  1=?n :

t2[0;T )

t2[0;T ]

These three facts together imply that for r > r0;n, Qr (An )  1 ? P(Bnr ). So since P(Bnr )  1 ? n w for r > r0;n, we have lim supr!1 Qr (An )  n . Now since Qr ?! Q as r ! 1, and since An is an open set, the Portmanteau theorem (cf. [2], Thm. 2.1) yields

Q(An )  lim inf Qr (An )  lim sup Qr (An )  n ; r!1 r!1

which implies by choice of the n and the Borel-Cantelli lemma that Q(An i.o.) = 0.   ? Proof of property (iii). Recall that since we have established that a.s., t = 1 on f (0) 6= 0g, and t = 0 on f? (0) = 0g, it suces to show that a.s. on f? (0) 6= 0g, (3.3) holds for all g 2 C and all t  0. We begin by restricting our attention to a slightly smaller class of functions than C . We will establish (3.3) for functions in this smaller class, and then make a simple generalization at the end of the proof. De ne C~ = fg 2 C : g0 has compact support in R+g: Recall that we always assume g is extended to be identically zero on (?1; 0) so that functions of the form g (
? a) are well de ned on R+ for any a > 0. In particular, for g 2 C~ this extension yields a function in C1b (R) that together with its rst derivative is uniformly continuous on R. We will use this fact below to simplify certain technical details involving Taylor's formula applied near zero. 46

Fix g 2 C~, and observe that since g is continuous and bounded, the assumptions of Lemma A.2 are clearly satis ed. Thus we have as r ! 1, X gr (
) =) X g (
);

P Er (t) g(vr) and X (t) = t hg;  i for all t  0. The function g  1 also satis es where Xgr (t) = r1 ir=1 g i the assumptions of Lemma A.2, and so we have E r (
) =) (
) as r ! 1. Also recall that we have passed to a subsequence so that r (
) =) ? (
) as r ! 1. Now since the limits X g (
) and (
) are deterministic, we see that the convergence in the three results just mentioned can be taken to be joint convergence in distribution (cf. [2], Thm. 4.4). Furthermore, by invoking the Skorohod representation theorem, we may assume that there is a single underlying probability space ( ; F ; P) on which all of the random elements in question are de ned, and such that the joint convergence occurs a.s. That is, we have a.s. as r ! 1, (r (
); X gr(
); E r(
)) ?! (? (
); X g(
); (
)); (5.52) where the convergence is in the product topology of D([0; 1); MF )  D([0; 1); R)  D([0; 1); R), where each of the terms in the product is endowed with the relevant Skorohod J1 -topology. Notice that since each of the limits in (5.52) is a.s. continuous, the convergence is in fact uniform on compact time intervals. Consider any ! 2 f? (0) 6= 0g such that (5.52) and properties (i) and (ii) of Section 3.1 hold, and such that t = 1; i.e., we consider any outcome for which the limiting initial uid scaled queue length h1; ?(0)i is strictly positive, and such that ! is not in any \exceptional" set. We rst show that for such an ! , (3.3) holds for all t  0 and the chosen g 2 C~. In what follows, all random elements are understood to be evaluated at this particular ! , and most references to it are suppressed. Our goal will be to derive a prelimit version of (3.3), which the path r (
) 2 D([0; 1); MF ) satis es for suciently large r. We will then pass to the limit in this relation to obtain (3.3) for ? (
). We start by noting that since g 2 C~, g 0 is uniformly continuous on R+, so there is a continuous nondecreasing function g : [0; 1) ?! [0; 1) with g (0) = 0, such that for any h 2 R,



sup g 0(x + h) ? g 0(x)  g (jhj): (5.53) x2R Fix T > 0. By (5.52), the fact that ? (
) is continuous, and the fact that for ! , ? (t) 6= 0 for all t 2 [0; T ], we may assume that r is large enough so that inf t2[0;T ] h1; r (t)i > 0. Then for such an r we always have



M r = h1; r (
)i T < 1;

mr = h1; r (
)i?1 T < 1:

(5.54) (5.55)

Now let t 2 [0; T ], and for any n = 1; 2; : : : and j = 0; 1; : : : ; n ? 1, de ne tj = jtn and tj = tj +1 .

47

Then we have

nX ?1

  r r g; r(tj ) ? hg; r (tj )i hg;  (t)i ? hg;  (0)i = j =0 nX ?1 

E  D g; r(tj ) ? g(
? Strj ;tj ); r(tj ) = j =0 nX ?1 D E  g(
? Strj ;tj ); r (tj ) ? hg; r(tj )i + j =0 r (tj ) nX ?1 1 rEX   = r g vir ? SUr ir=r;tj j =0 i=rE r (tj )+1 nX ?1 D E g(
? Strj ;tj ) ? g(
); r(tj ) ; + j =0

(5.56)

where the rst term in the last equality is by (5.2) and the fact that (1(0;1)g )  g , since g (0) = 0 for g 2 C~. We handle the two terms in (5.56) separately. To begin with, since g 2 C~ has been extended to be an element of C1b (R), we have the following rst order Taylor expansion for each j = 0; 1; : : : ; n ? 1 and each x 2 R+: (5.57) g(x ? Strj ;tj ) ? g(x) = g 0(wjx)hj ; where hj = ?Strj ;tj and wjx 2 R is in the interval [x ? Strj ;tj ; x]. Note that

r t r ?1 tmr max jh j = max S  n h1;  (
)i T = n : j 0,

1 ;  <  lim P sup [x;x+] #0 x2R +

!

= 1:

Proof. The implication (ii) ) (i) is straightforward. We prove (i) ) (ii), by contradiction. Suppose that (ii) does not hold. Then there exists an  > 0 and a sequence n such that n # 0 as n ! 1, and

!

 lim inf P sup 1[x;x+ ] ;    > 0: n!1 x2R n

+

De ne

An =

T

(A.1)

(

)

  2 MF : sup 1[x;x+ ];    ; x2R +

n

and de ne A = n An . Since An  An+1 , (A.1) implies that P( 2 A) > 0. We claim that any  2 A has an atom, which yields a contradiction to (i). To see this, let  2 A, and de ne

n

 o Bn = x 2 R+ : 1[x;x+n ] ;   2 : The sets Bn are non-empty since  2 A. Suppose fxk g1 k=1  Bn and xk ?! x 2 R+ as k ! 1. 1 Since there exists a sequence of closed T intervals fIk gk=1 such that for each k = 1; 2; : : : , Ik+1  Ik , Ik  [xk ; xk + n ] [ [x; x + n], and k Ik = [x; x + n], the fact that h1Ik ;  i  1[xk;xk +n ] ;   =2 implies that x 2 Bn . So the sets Bn are closed. Moreover,  2 MF implies that the sets Bn are T bounded, and therefore compact. Finally, Bn  Bn+1  implies that B = n Bn =6 ?. So continuity from above of  implies that for any x 2 B , 1fxg;   =2, i.e., x is an atom of  . Lemma A.2 Consider a sequence of real numbers r 2 (0; 1) such that r ! 1. For each r, r 1 let furi g1 i=1 be a sequence of independent nonnegative random variables, such that fui gi=2 is i.i.d. Assume for each r that E[ur1 ] < 1 and E[ur2]?1 = r 2 (0; 1). For each r, let fvir g1 i=1 be an i.i.d. sequence of strictly positive random variables with common distribution  r . Let  2 (0; 1), let  be a probability distribution on R+, and let g : R+ ?! R be a Borel measurable function that is  {a.e.

52

continuous and satis es hjg j;  i < 1 and hjg j;  ri < 1 for every r. Assume that the following asymptotic assumptions hold as r ! 1:

r ?! ; w r ?! 

g+;  r  ?! ; g+;   ;

g?;  r  ?! g?;   ; E[ur1]=r ?! 0; E[ur2; ur2 > r] ?! 0:

P

(A.2) (A.3) (A.4) (A.5) (A.6) (A.7)

Let U0r = 0, Uir = ij =1 urj , i = 1; 2; : : : , and for t  0, let E r (t) = supfi  0 : Uir  tg, P Er (t) g(vr), and X (t) = t hg;  i. Then as r ! 1, E r(t) = r1 E r(rt), X gr(t) = 1r ir=1 g i X gr (
) =) X g (
):

Proof. It suces to show that for each T > 0,

X r(
) ? X (
)

=) 0; as r ! 1, (A.8) g T g where k
kT denotes the supremum norm over [0; T ]. Note that both g + and g ? themselves satisfy the conditions of Lemma A.2, and that X gr (
) and X g (
) are linear in g . So it is sucient to show

that (A.8) holds with g + in place of g . We require a condition on the tails of the distributions of the members of the sequence fg + (vir )g1 i=1 which will imply a weak law of large numbers. Since g + is  -a.e. continuous, the distribution gr of g + (v1r ) converges weakly as r ! 1 to the distribution g , where g (A) =  (fx 2 R+ : g + (x) 2 Ag), for any Borel set A  R+. By (A.4), we also have as r ! 1, +

+

D

; gr

+

+

E + r +   = g ;  ?! g ;  = ; g < 1: +

(A.9)

Since g can have at most countably many atoms, we can always choose an arbitrarily large K > 0 so that 1[0;K ] is g -a.e. continuous and therefore +

+

E

D





1[0;K] ; gr ?! 1[0;K] ; g ; +

+

as r ! 1.

Combined with (A.9), this implies that

E[g+(v1r); g+(v1r) > r] ?! 0;

as r ! 1.

(A.10)

Together, (A.4) and (A.10) imply by the weak law of large numbers for triangular arrays that, for t  0 xed, brtc

 1X g +(v r) =) t g +;  ;

r i=1

i

as r ! 1.

(A.11)

Veri cation of this fact uses standard truncation arguments hinging on (A.10) (cf. [14], pp. 41{43, for example). Since the right and left members of (A.11) are nondecreasing as functions of t, and 53

the right side de nes a deterministic process that is uniformly continuous on each compact time interval, the convergence in (A.11) is actually uniform on compact time intervals, i.e., for each T > 0,

brtc X

 sup r1 g + (vir ) ? t g +;  =) 0; t2[0;T ] i=1

as r ! 1.

(A.12)

Our assumptions (A.2), (A.6) and (A.7) on furi g1 i=1 imply by the weak law of large numbers for renewal processes that, for each T > 0, E r (
) ? (
) T =) 0 as r ! 1, where (t) = t. Since (
) is deterministic, we can combine this with (A.12) using the random time change theorem (cf. [2], Section 17) to obtain for each T > 0 as r ! 1,

rE (
)

1 X g+(vr ) ? (
) g+;  

=) 0: i

r i=1

r

T

(A.13)



Acknowledgement. The authors wish to thank Pat Fitzsimmons for helpful discussions during the course of this work.

References [1] F. Baccelli and D. Towsley. The customer response times in the processor sharing queue are associated. Queueing Systems, 7 (1990), 269{282. [2] P. Billingsley. Convergence of Probability Measures. John Wiley & Sons, Inc., New York, 1968. [3] M. Bramson. Instability of FIFO queueing networks. Annals of Applied Probability, 4 (1994), 414{431. [4] M. Bramson. Instability of FIFO queueing networks with quick service times. Annals of Applied Probability, 4 (1994), 693{718. [5] M. Bramson. Convergence to equilibria for uid models of FIFO queueing networks. Queueing Systems: Theory and Applications, 22 (1996), 5{45. [6] M. Bramson. Convergence to equilibria for uid models of head-of-the-line proportional processor sharing queueing networks. Queueing Systems: Theory and Applications, 23 (1997), 1{26. [7] M. Bramson. State space collapse with applications to heavy trac limits for multiclass queueing networks. Queueing Systems: Theory and Applications, 30 (1998), 89{148. [8] M. Bramson. Stability of two families of queueing networks and a discussion of uid limits. Queueing Systems: Theory and Applications, 28 (1998), 7{31. [9] H. Chen, O. Kella and G. Weiss. Fluid approximations for a processor-sharing queue. Queueing Systems: Theory and Applications, 27 (1997), 99{125. 54

[10] J. G. Dai. On positive Harris recurrence of multiclass queueing networks: a uni ed approach via uid limit models. Annals of Applied Probability, 5 (1995), 49{77. [11] J. G. Dai. Stability of open multiclass queueing networks via uid models. In Stochastic Networks, IMA Volumes in Mathematics and Its Applications, Volume 71, F. Kelly and R. J. Williams (eds.), Springer-Verlag, New York, 1995, pp. 71{90.  e de Probabilites de Saint[12] D. A. Dawson. Measure Valued Markov Processes. In E cole d'Et Flour, Volume XXI, Lecture Notes in Math, No. 1541, P. L. Hennequin (ed.), Springer-Verlag, Berlin, 1993. [13] B. Doytchinov, J. Lehoczky and S. Shreve. Real-time queues in heavy trac with earliestdeadline- rst-queue discipline. To appear in Annals of Applied Probability. [14] R. T. Durrett. Probability: Theory and Examples, 2nd Edition. Duxbury Press, Belmont, 1996 [15] S. N. Ethier and T. G. Kurtz. Markov Processes: Characterization and Convergence. John Wiley & Sons, Inc., New York, 1986. [16] L. C. Evans and R. F. Gariepy. Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton, 1992. [17] W. Feller. An Introduction to Probability Theory and Its Applications, Volume 2, 2nd Edition. John Wiley & Sons, Inc., New York, 1971. [18] S. Grishechkin. GI/G/1 processor sharing queue in heavy trac. Advances in Applied Probability, 26 (1994), 539{555. [19] D. L. Iglehart and W. Whitt. Multiple channel queues in heavy trac I. Advances in Applied Probability, 2 (1970), 150{177. [20] A. Jean-Marie and P. Robert. On the transient behavior of the processor sharing queue. Queueing Systems, 17 (1994), 129{136. [21] O. Kallenberg. Random Measures. Academic Press, New York, 1983. [22] Yu. V. Prohorov. Convergence of random processes and limit theorems in probability theory. Theory of Probability and its Applications, 1 (1956), 157{214. [23] A. L. Puha, A. L. Stoylar and R. J. Williams. The uid limit of an overloaded processor sharing queue. Preprint. [24] R. J. Williams. Diusion approximations for open multiclass queueing networks: sucient conditions involving state space collapse. Queueing Systems: Theory and Applications, 30 (1998), 27{88. [25] S. F. Yashkov. Processor-sharing queues: some progress in analysis. Queueing Systems, 2 (1987), 1{17.

55