RATE CONSERVATION LAWS FOR ... - CiteSeerX

RATE CONSERVATION LAWS FOR MULTIDIMENSIONAL PROCESSES OF BOUNDED VARIATION WITH APPLICATIONS TO PRIORITY QUEUEING SYSTEMS FABRICE GUILLEMIN AND RAVI MAZUMDAR Abstract. In this paper we derive a multidimensional version of the Rate Conservation Law (RCL) for c` adl` ag processes of bounded variation. These results are then used to analyze queueing models which have a natural multidimensional characterization, such as priority queues. In particular the RCL is used to establish certain conservation laws between the idle probabilities for such queues. We use the relations to provide a detailed analysis of preemptive resume priority queues with M/G inputs. Special attention is paid to the validity of the so-called reduced service rate approximation.

1. Introduction Recently there has been a great amount of interest in computing the tails of buffer distributions in queueing systems with multiple-buffers and appropriate scheduling of the server in order to characterize loss. In the context of Internet applications there are two service disciplines of primary interest namely the Head of Line (HOL) priority service used in the so-called Differentiated Services (Diffserv) architectures and the Weighted Fair Queueing (WFQ) discipline (also called the Generalized Processor Sharing (GPS) service schedule). In the context of both disciplines a number of approximations based on the so-called Reduced Service Rate (RSR) have been proposed whereby the service rate associated with a particular buffer (of a priority class in HOL) is taken to be the service rate from which the average load in the case of HOL or the guaranteed rate for WFQ of the traffic in the other (higher priority in HOL) buffers is subtracted. It is thus natural to verify the validity of such an approximation. While there have been a number of papers devoted to the study of multi-dimensional queueing systems there has been no systematic approach for the derivation of the distribution of the buffer contents. For the priority case most of the focus has been on the study of the waiting time distributions for the low-priority classes, see the paper of Abate and Whitt [1]. Detailed derivations of the Laplace transforms of the waiting time distributions can be found in Takagi [21] and the classic reference of Jaiswal [11]. These rely on the fact that lower-priority buffers only receive service during the periods of idleness of all the higher priority buffers and thus depend on the busy-period analysis of the higher priority buffers. The case of systems with WFQ has received much less attention. The basic motivation of this paper is to formalize the derivation of joint distribution of the buffer content of multi-dimensional queueing systems in order to study the tail behavior of the buffer content distributions. The approach we take is via an extension of the rate conservation law (RCL) for c` adl` ag processes of bounded variation to the multidimensional case. Rather than resort to ad-hoc arguments, such an approach exploits directly the sample-path evolution. This approach yields the analog of the Fokker-Planck equation from which we can derive the Laplace transform of the individual buffer content distribution. The RCL, originally introduced by Miyazawa [17] in the context of queueing theory, provides a relationship between the continuous and discontinuous parts of a c` adl` ag stochastic process with paths of bounded variation. For a stationary c` adl` ag process, RCL specifically establishes a connection between the mean value of the right derivative of the process at an arbitrary instant and the mean value of the jumps of the process. For example, consider a stationary stochastic 1

2

F. GUILLEMIN AND R. R. MAZUMDAR

process {Yt } satisfying the following evolution equation: Z t X (Ys − Ys− ), Yt = Y 0 + Ys0 ds + 0

0≤s≤t

where Yt0 denotes the right derivative of the function t → Y t at point t. Let {Nt } be the point process counting the jumps of the process {Yt }. RCL reads (1)

E[Y00 ] + λE0N [Y0 − Y0− ] = 0,

where λ is the mean intensity of the point process {N t } and E0N denotes the expectation with respect to the Palm probability associated with the point process {N t } [4]. It has been shown by Mazumdar et al [16] that the the balance equations for Markov chains are one form of the rate conservation law. Different generalizations of RCL have been investigated in the literature, for instance by establishing RCL for non-stationary stochastic processes with or without a diffusion term [14, 15, 18] (see [19] for a survey of RCL for one-dimensional stochastic processes). When considering the workload of an M/G/1 queue, it is worth noting that as shown in [15], RCL encompasses the so-called Tak´ acs integro-differential formula (see for instance [5]): for x > 0, Z x ∂ ∂ B(x − y)dy Ft (y), Ft (x) = Ft (x) − λFt (x) + λ ∂t ∂x 0

where Ft (x) = P{Wt ≤ x | W0 = w0 } with Wt denoting the workload in the queue at time t, λ is the mean intensity of the input Poisson process, and B(x) is the probability distribution function of the service time. It can also be shown that RCL is the integrated form of the well-known Fokker-Planck equation for diffusion processes. In this paper, we establish RCL for multidimensional processes of bounded variation (without diffusion terms). We then show that an integro-differential formula of the Fokker-Planck type can be established for the density probability function of the process. To get rid of differentiability issues, this function is considered as a distribution. The resulting equation is then applied to a queueing system with preemptive resume priority. This paper is organized as follows: In Section 2, we establish RCL as well as Fokker-Planck equation for multidimensional stochastic processes of bounded variation. Multidimensional RCL is then applied to the M/G/1 queue with preemptive resume priority in Section 3. Special attention is paid in the analysis of this queueing system to validity of the so-called Reduced Service Rate (RSR) approximation, which is often used in the literature (see [2] for instance) to approximate the behavior of the low priority queue. The RSR approximation specifically states that everything happens as if the service rate of the low priority queue were reduced up to the offered traffic of the high priority stream. We conclude the paper with an application of the RCL to derive the buffer content distributions for GPS queues. 2. RCL for multidimensional processes of bounded variation

We assume that all processes considered throughout this paper are defined on some reference filtered probability space (Ω, F, P, {Ft }), where the filtration {Ft } satisfies the so-called “usual” conditions [10]. Consider a (optional and σ-finite) random measure µ(ω, dt; dx) taking values in R d+ and a function r(s, x) = (r1 (s, x), . . . , rd (s, x)) defined on R+ × Rd and taking values in Rd . Let ν(dt; dx) denote the compensator of the random measure µ(dt; dx), which is unique up to P-null sets (see [10] for details). There exist a predictable measure dA t (ω) defined on R+ and a transition kernel K(ω, t; dx) such that (2)

ν(dt; dx) = dAt K(t; dx),

In the following, we assume that dAt has the form (3)

dAt = λt dt

MULTIDIMENSIONAL RCL

3

where {λt } is a predictable process, referred to as the stochastic intensity of the point process counting the jumps of the random measure µ(dt, dx) [4, 10]. Moreover, we assume that K(t; dx) is a Markovian transition kernel, which has the form: K(ω, t; dx) = K(Xt− (ω); dx).

(4)

where K is a transition kernel. Finally, let Φ(t) denote the joint distribution of the couple (λ t , Xt ) at time t and let ϕt (dx) be defined by Z ∞ λΦt (dλ, dx). (5) ϕt (dx) = 0

We consider the unique 1 solution ({Xt }, {Zt }) to the reflection problem: Z tZ Z t µ(ds, dx) + Zt , r(s, Xs )ds + (P) X t = X0 + Rd

0

0

where Xt = (Xt1 , . . . , Xtd ) takes values in Rd+ and Zt = (Zt1 , . . . , Ztd ) is such that Zti increases on the set {t : Xti = 0} only. Specifically, we have Z t i (6) Zt = − 1I{Xsi =0} ri (s, Xs )ds. 0

Theorem 1. Let ({Xt }, {Zt }), where Xt = (Xt1 , . . . , Xtd ) and Zt = (Zt1 , . . . , Ztd ), be the ddimensional processes, which are the solutions of problem (P). Let f : Rd → R be a function with left hand derivatives. The rate conservation law for the process {X t } reads:   Z Z d X ∂ (7) E[f (Xt )] = Dj− f (Xt )rj (t, Xt )1I{X j >0}  , [f (u + v) − f (u)]K(u, dv)ϕt (du) + E  t d ∂t Rd R + + j=1 where Dj− denotes the left derivative of f with respect to the jth component.

Proof. Let us write the simplified version of Itˆ o-Stieltjes formula when there are no diffusion terms:   d Z t d X X X f (Xs ) − f (Xs− ) − Dj− f (Xs− )∆Xsj  . f (Xt ) = f (X0 ) + Dj− f (Xs− )dXsj + 0

j=1

j=1

s≤t

Since

d Z X j=1

t 0

Dj− f (Xs− )dXsj −

d XX

Dj− f (Xs− )∆Xsj =

d Z X j=1

s≤t j=1

t 0

Dj− f (Xs )rj (s, Xs )ds +

d Z X j=1

t 0

Dj− f (Xs )dZsj ,

we have f (Xt ) = f (X0 ) +

d Z X j=1

t 0

Dj− f (Xs )rj (s, Xs )ds

+

d Z X j=1

t 0

Dj− f (Xs )dZsj +

X s≤t

[f (Xs ) − f (Xs− )] .

1This assertion is used in the following without proof. The existence and uniqueness of the solution can be straightforwardly established by using the proof of the Skorokhod reflection problem (see for instance the analysis in [12])

4


The last term in the above equality can be rewritten as Z tZ X [f (Xs ) − f (Xs− )] = [f (Xs− + v) − f (Xs− )] µ(ds, dv) 0

s≤t

= +

Z tZ

Rd

0

Rd

[f (Xs− + v) − f (Xs− )] (µ(ds, dv) − ν(ds, dv))

0

Rd

[f (Xs− + v) − f (Xs− )] ν(ds, dv).

Z tZ

By definition of the compensator ν(dt, dx)), the process {U t } defined by Z t (8) Ut = [f (Xs− + v) − f (Xs− )] (µ(ds, dv) − ν(ds, dv))} 0

is a martingale, with null expectation. Furthermore, Z t Z t Dj− f (Xs )1I{Xsj =0} rj (s, Xs )ds. Dj− f (Xs− )dZsj = − 0

0

It follows that

f (Xt ) = f (X0 ) + Ut +

Z

t 0

[f (Xs− + v) − f (Xs− )] ν(ds, dv) +

d Z X j=1

t 0

Dj− f (Xs )r(s, Xs )1I{Xsj >0} ds.

Taking expectations, we obtain Z t (9) E[f (Xt )] = E[f (X0 )] + E [f (Xs− + v) − f (Xs− )] ν(ds, dv) 0   d Z t X Dj− f (Xs )rj (s, Xs )1I{Xsj >0} ds . + E j=1

0

Since ν(dt, dx) = λt dtK(Xt , dx), we have Z tZ Z E[f (Xt )] = E[f (X0 )] + [f (u + v) − f (u)]K(u, dv)ϕs (du)ds 0

Rd +

Rd +



+ E

d Z X j=1

and the result follows by taking derivatives with respect to t.

t 0



Dj− f (Xs )rj (s, Xs )1I{Xsj >0} ds ,

An easy consequence of the above theorem is the following multi-dimensional stationary RCL. We assume in this case that the function r does not depend on time t. Moreover, we use the concept of Palm probability measure associated with a multidimensional random measure (see [6]). Corollary 1 (Multidimensional RCL). Let ({Xt }, {Zt }), where Xt = (Xt1 , . . . , Xtd ) and Zt = (Zt1 , . . . , Ztd ), be the d-dimensional processes, which are the solutions of problem (P). Assume that the process {Xt } is stationary. Then, it satisfies the stationary RCL: h i (10) ΛE0µ [X0 − X0− ] + E r1 (X0 )1I{X01 >0} , . . . , rd (X0 )1I{X0d >0} = 0, where E0µ is the expectation with respect to the Palm probability associated with the jumps of the process {Xt } and Λ is the mean intensity of the point process counting jumps.


5

Proof. When the process {Xt } is stationary, we apply equation (9) to f (x) = xj for j = 1, . . . , d, and we get Z t Z Z t µ(ds, dx) = 0, (r1 (Xs )1I{Xs1 >0} , . . . , r1 (Xs )1I{Xs1 >0} )ds + E E since

E

Rd

0

0

"Z Z t 0

d Rd + ×R+

#

(v1 , . . . , vd )K(u, dv)ϕs (du)ds = E

Now, by using the relation ΛtE0µ [X0

− X 0− ] = E

Z t Z 0

Rd

"Z Z t 0

Rd +

#

µ(ds, dx) .

µ(ds, dx) ,

and by taking derivatives with respect to t, the result follows. This completes the proof.

In the following, we assume that the random variable X t has a probability density function, defined in the sense of distributions (i.e., which can include masses at some points). Let pt (x) = pt (x1 , . . . , xd ) denote the probability density function of X t ; pt is the derivative in the sense of distributions of the probability distribution function F t (x) = Ft (x1 , . . . , xd ) = P{X1 ≤ x1 , . . . , Xd ≤ xd }. In other words, pt (x) =

∂d Ft (x). ∂x1 . . . ∂xd

Theorem 2 (Fokker Planck equation). The density probability distribution function p t satisfies the following forward Fokker Planck equation: Z d X ∂ ∂ rj (t, x)1I{xj >0} pt (x) + Q pt (x) = − k(u, x − u)ϕt (du) − ϕ0t (x) (11) ∂t ∂x j j [0,xj ] j=1

where k(u, .) is the density of the transition kernel K(u, .) and ϕ 0t is the derivative of ϕt in the sense of distributions. Q Proof. Specializing relation (7) to the case f (y1 , . . . , yd ) = j (xj − yj )+ for (x1 , . . . , xd ) ∈ Rd+ , we obtain     d d X Y ∂ Y E rj (t, Xt )1I{X j >0} 1I{X j ≤xj } (xj − Xtj )+  = − E (xk − Xtk )+  t t ∂t j=1 j=1 k6=j   Z d d Y Y  (xj − yj − zj )+ − (xj − yj )+  K(y; dz1 , . . . , dzd )ϕt (dy), + d Rd + ×R+

Dj− f (y)

j=1

j=1

Q

since = −1I{yj ≤xj } k6=j (xk − yk )+ . By differentiating twice (in the sense of distributions) the above equation with respect to the variables x j , j = 1, . . . , d, we easily get equation (11). This completes the proof. Assume now that the process {Xt } admits a stationary regime. In that case, the release rate r as well as the functions p and ϕ do not depend on time t. A simple consequence of the previous result is the following partial differential equation. Theorem 3. The stationary density probability distribution p of the process {X t } satisfies Z d X ∂ k(u, x − u)ϕt (du) − ϕ0 (x) = 0, r(x)1I{xj >0} p(x) + Q (12) − ∂x j j [0,xj ] j=1

where k and ϕ0 are the density functions of the transition kernel K and the marginal probability distribution function ϕt , respectively.

6


Formula (12), which is the multidimensional equivalent of Tak´ acs formula, can be applied to a wide variety of applications arising in queueing and risk theory, and more generally, the class of so-called piecewise deterministic Markov processes [8]. In systems composed of multiple queues which can conveniently be described by means of the vector whose components are the workloads in the different queues, Formula (12) then directly yields the partial differential equation satisfied by the joint probability density functions of the workloads. Such a partial differential equation can then be solved by using standard techniques. The ideal candidates for such a procedure are queueing systems with preemptive resume priority and systems with coupled servers such as the generalized processor sharing systems (also referred to as weighted fair queueing systems in the literature). These two systems will be studied in the next sections in the case of two customer classes, one specific queue being associated with one customer class. The multidimensional Tak´ acs equation (12) can also be applied to more complex systems. For instance when the service rate of a queue is proportional to the workload of the queue. For instance, when the service rate of queue #i is given by xφ P i i k x k φk

P where xi is the workload in queue #i and the coefficient φi are some weight such that i φi = 1. This system is very close to the so-called discriminatory processor sharing system but the service rate of a queue is proportional to the workload and not to the number of customers in the queue. Such a service discipline can be used for instance to offer a weighted fairness in terms of delay between different customer classes. In such a case, equation (12) directly gives the partial differential equation satisfied by the joint probability density distribution of the workload but the analysis of this equation is more complicated than in the preemptive resume and generalized processor sharing systems.

3. Application of multidimensional RCL to preemptive resume priority queues As a first application of the multi-dimensional RCL we consider a priority queue with head of line priority. As mentioned in the introduction, much of the work in the literature has focused on determining the waiting time characteristics in the lower priority queue, see for instance [21]. In [1], the authors study the tail asymptotics for the lower-priority queue in the M/G/1 setting and show that the asymptotics can be non-exponential. Since the waiting time distribution is different from the buffer content distribution in the lower-priority queue, the asymptotics cannot be directly determined from the results in [1]. We are interested in the buffer asymptotics since they play a key role in the loss asymptotics and we wish to study the range of validity of the RSR approximation. In the case when the service times are exponentially distributed, we compute the exact tail asymptotics of the low priority buffer in terms of the parameters.

3.1. Application of RCL and consequences. Consider a single server, infinite capacity queueing system with two customer classes, where customers of class #0 have preemptive head of line (HoL) priority over customers of class #1. A specific queue is associated with each customer class and queue #1 is served only when there are no customers in queue #0. If a class #0 customer arrives while a class #1 customer is in service, the server immediately interrupts the service of the class #1 customer in order to serve the class #0 customer. Once queue #0 has emptied, the server resumes the service of the class #1 customer, which was in service. Let Xt0 and Xt1 denote the workloads in queues #0 and #1 at time t, respectively. Suppose that the customers of class #j arrive according to a Poisson process with intensity λ j and require service times with general distribution Kj and mean 1/µj . Under the assumption that the system is stable, which amounts to assuming that the total offered load ρ = λ 0 /µ0 + λ1 /µ1 < 1, the joint


7

distribution of the stationary workloads X00 and X01 satisfies the RCL equation: (13)

∂ ∂ 1I{x0 >0} p(x0 , x1 ) + 1I{x0 =0} 1I{x1 >0} p(x0 , x1 ) − (λ0 + λ1 )p(x0 , x1 ) ∂x0 ∂x1 Z x0 Z x1 Z x0 Z x1 k1 (x1 − u1 )p(u0 , u1 )du0 du1 = 0, k0 (x0 − u0 )p(u0 , u1 )du0 du1 + λ1 + λ0 0

0

0

0

Let L(p) denote the Laplace transform of the density probability distribution p defined for