THE M/G/1 PROCESSOR-SHARING QUEUE ... - Semantic Scholar

THE M/G/1 PROCESSOR-SHARING QUEUE WITH LONG and SHORT JOBS Ward Whitt AT&T Labs1 September 3, 1998

AT&T Labs, Room A117, 180 Park Avenue, Building 103, Florham Park, NJ 07932-0971; email: [email protected] 1

Abstract In this paper we study the classical M/G/1 processor-sharing queue under the assumption that there tend to be some jobs much longer than others, as occurs with a heavy-tailed service-requirement distribution. We suggest analyzing the evolution of the long and short jobs in dierent time scales. To study the short jobs in a short time scale, we act as if a speci ed number of long jobs are permanently in the system. We use the steady-state results with k permanent jobs as an approximation for the shorter-term steady-state behavior that should prevail while a given number of especially long jobs are in the system. We also describe the evolution of the longer jobs in a longer time scale, assuming that the long jobs arrive in a Poisson process with small arrival rate and have large nite mean service requirements. If the arrival rate and the reciprocal of the mean service requirement of the long jobs approach zero, with the trac intensity held xed, then in the time scale of the mean long-job service requirement it is possible to act as if the number of short jobs in the system is always equal to its conditional steady-state mean given the number of long jobs present. Hence, in the long time scale, it is possible to determine the transient and steady-state behavior of the long jobs, without considering the uctuations of the short jobs. Thus we can approximately describe the eect of imposing an upper limit on the number of long jobs allowed in the system. The long jobs exceeding this upper limit might be blocked or delayed.

Key words: processor-sharing queues, heavy-tailed service distributions, exceptionally long service times, separation of time scales

1. Introduction Measurements of communication and computer systems have shown that the distributions of service requirements of jobs often have very heavy tails; i.e., there tend to be a few exceptionally large jobs; e.g., see Crovella and Bestavros [8], Harchol-Balter and Downey [12], Heyman and Lakshman [13], Jelenkovic, Lazar and Semret [16], Leland, Taqqu, Willinger and Wilson [20], and Paxson and Floyd [23]. Thus there has been considerable interest in the behavior of queues with heavy-tailed service-time distributions; e.g., see Abate, Choudhury and Whitt [1], Asmussen and Mller [4], Boxma and Cohen [6], Choudhury and Whitt [7], Jelenkovic and Lazar [15], and Zwart and Boxma [31]. However, the heavy-tailed service-time distributions pose problems in model interpretation. The standard steady-state descriptions tend to be of questionable value because steady-state is approached very slowly and itself tends to be highly variable, depending critically on the exceptionally long jobs. Here we suggest that in some situations it may be possible to obtain a more useful description in a shorter time scale by treating the jobs with exceptionally long service requirements separately from the remaining jobs. In this paper we study the classical M/G/1/PS queueing model from this perspective. In particular, we condition on a speci ed number of jobs with exceptionally long service times being in the system and examine the \temporary" or \quasi" steady-state behavior of the other jobs. Just as in nearly decomposable Markov chains, e.g. see Courtois [9], there tends to be a short-term steady state between successive changes in the number of exceptionally large jobs. By examining this short-term steady-state behavior, we can assess the impact of the long jobs upon the short jobs, and if necessary determine an appropriate upper limit on the number of long jobs that can be in service. This upper limit might be enforced by blocking or delaying long jobs exceeding the upper limit. Moreover, we show that, as the xed number of long jobs in the system increases, the total number of jobs becomes more predictable (the ratio of the standard deviation to the mean decreases), so that it becomes possible to accurately predict the prevailing processing rate received by each job during that regime. The M/G/1/PS model has a Poisson arrival process with rate , independent and identically distributed (iid) service requirements with cumulative distribution function (cdf) G having nite mean m1, one server, unlimited waiting room and the (egalitarian) processor-sharing (PS) service discipline. With the PS discipline, each job in the system receives service at rate 1=n when there are n jobs in the system. The PS discipline is a frequently used approximation for (and limit of) the round robin discipline used in computers, in which each job receives a small quantum of service, after which it goes to the end of the queue if more service is required. The PS discipline may also reasonably approximate head-of-the-line

(per- ow) fair queueing disciplines used in communication network routers when the number of packets in the router from any one ow is not large, e.g., see Demers, Keshav and Shenker [10] and Parekh and Gallager [22]. We investigate the impact of exceptionally long jobs by assuming that there are k permanent jobs in the system (with in nite service requirements). Thus, the service-requirement cdf G is the cdf of the service requirement of the remaining jobs. We think of obtaining this cdf G by simply truncating the original service-requirement cdf, so that the service requirements become bounded and G has moments of all orders. We assume that   m1 < 1, so that the conditional steady-state is well de ned. We also study the evolution of the longer jobs in a longer time scale. For this, we assume that the long jobs arrive according to a Poisson process with arrival rate L  and have service requirements with mean mL1=, where we allow  ! 0. We study the evolution of the long jobs in the long time scale ?1 . We show that, asymptotically as  ! 0, we can act as if the number of short jobs is always equal to its conditional steady-state mean value given the number of long jobs present. This provides a separating of time scales, enabling us to analyze the transient behavior of the long jobs. As an application, we can approximately describe the impact of imposing an upper limit on the number of long jobs admitted to the system. We then assume that the long jobs can be identi ed upon arrival or relatively soon thereafter and are blocked or delayed if their limit is exceeded. It is natural to regard a job as a long job after it has been in the system a certain length of time. For heavy-tailed distributions, the expected conditional remaining time given the elapsed time tends to increase with the elapsed time. Here is how the rest of this paper is organized. In Section 2 we characterize the steady-state distribution of the number of jobs in the system and their residual service requirements when there are k permanent jobs. We observe that the classical product-form insensitivity result extends from k = 0 to positive k. In Section 3 we extend the elementary argument of Asare and Foster [3] to characterize the conditional expected response time. In Section 4 we extend Ott's [21] derivation of the joint steady-state distribution of the conditional response time and the number of jobs left behind upon departure to positive k and show that the distribution is the (k + 1)-fold convolution of the distribution when k = 0. In Section 5 we make stochastic comparisons showing how variability in the service-requirement cdf G aects the conditional mean response times. In Section 6 we study the evolution of the extra long jobs in the longer time scale. We also consider the eect of imposing an upper limit on the number of long jobs that can be in the system.

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2. The Conditional Steady-State Distribution In this section we describe the conditional steady-state given a certain number of long jobs. If the long jobs are suitably long, then a new temporary steady-state will be achieved during their presence. To describe this temporary steady-state, we stipulate that there are k permanent jobs with in nite service requirements. We let Q(t) be the number of other jobs in the system at time t and let Sj (t) be the amount of completed (or remaining) service of jobs j , 1  j  Q(t). It is clear that the vector process f(Q(t); S1(t); : : :; SQ(t)(t)): t  0g is a Markov process. Let Ge be the stationary-excess cdf associated with the service-requirement cdf G, i.e., Zt ? 1 Ge(t) = m1 Gc (u)du; t  0 ; (2.1) 0

where Gc (t)  1 ? G(t) is the complementary cdf (ccdf). Let ) denote convergence in distribution.

Theorem 2.1. Consider an M/G/1/PS system with  < 1 modi ed by initially having k permanent jobs with in nite service requirements. Then

[Q(t); S1(t); : : :; SQ(t)(t)] ) [Q; S1; : : :; SQ] as t ! 1 ;

(2.2)

where, conditional on fQ = ng, S1 ; : : :; SQ are distributed as n iid random variables with stationary-excess cdf Ge in (2.1) and Q has the negative binomial distribution   k+1 n + k P (Q = n) = (1 ? ) n ; n  0 ; (2.3)

k

with

EQ = (k + 1) 1 ?  and V ar Q = (k + 1) (1 ? )2 :

(2.4)

Proof. First suppose that the service requirements are exponential. Then Q(t) evolves as a birth-anddeath process with birth rates  = , n  0, and death rates  = n=(n + k), n  1. As usual, the n

n

steady-state distribution of this birth-and-death process is 1 X P (Q = n) = b(n)= b(j ) ;

(2.5)

j=0

where

n?1 b(n) = 0 : :: : : : ; 1

n

e.g., see Chapter 4 of Heyman and Sobel [14], but here   n + k b(n) = k! n

k

3

(2.6) (2.7)

P b(n) = k!(1 ? )?(k+1) , as can be seen by considering the kth derivative with respect to  of and 1 n=0 P n. The distribution (2.3) is the familiar negative binomial distribution with mean and (1 ? )?1 = 1 n=0 variance in (2.4); e.g., see Chapter 5 of Johnson and Kotz [17]. Next, consider a general service-requirement cdf G. Then the vector [Q(t); S1(t); : : :; SQ(t)(t)] is a Markov process. In this case the convergence, the insensitivity of the distribution of Q to the distribution of G beyond its mean, and the distribution of (S1; : : :; SQ) follow by the same argument used to treat the standard PS model with k = 0. In particular, we can apply insensitivity results for generalized semi-Markov processes with speeds in Schassberger [24]. Alternatively, we can apply insensitivity results for symmetric queues in Section 3.3 of Kelly [18]. See Chapter 6 of Franken et al. [11] for further discussion. It is signi cant that the negative binomial distribution is the (k + 1)-fold convolution of the geometric distribution arising when k = 0. This convolution property is a persistent theme. It is well known and easy to verify that there is a law of large numbers (LLN) and a central limit theorem (CLT) for the negative binomial distribution in (2.3) as k increases. This shows that the prevailing processing rate available to each job becomes predictable as k increases; it tends to be (1 ? )=(k + 1). We rst obtain a CLT and LLN for Q. Let N (m;  2) denote a random variable that is normally distributed with mean m and variance  2.

Theorem 2.2. In the setting of Theorem 2.1, Q=(k + 1) ) =(1 ? ) as k ! 1 and

(k + 1)?1=2[Q ? (k + 1)=(1 ? )] ) N (0; =(1 ? )2) as k ! 1 :

Proof. The classical CLT and LLN can be applied after observing that the negative binomial distribution in (2.3) is the (k + 1)-fold convolution of the geometric distribution with mean =(1 ? ) and variance =(1 ? ) . 2

The prevailing processing rate is 1=Q. We can apply Theorem 2.2 to obtain a CLT and LLN for 1=Q.

Theorem 2.3. In the setting of Theorem 2.1, (k + 1)=Q ) (1 ? )= as k ! 1 and

(k + 1)3=2[(1=Q) ? (1 ? )=(k + 1)] ) N (0; (1 ? )2=3) as k ! 1 : 4

Proof. Note that

  [ Q ? ( k + 1) = (1 ?  )] ? (1 ?  )( k + 1) p (k + 1) [(1=Q) ? (1 ? )=(k + 1)] = Q k+1 ) N (0; =(1 ? )2)((1 ? )2=2) =d N (0; (1 ? )2=3) 3=2

by the two parts of Theorem 2.2, applying Theorems 4.4 and 5.1 of Billingsley [5]. Alternatively, apply Theorem A on p. 118 of Ser ing [25]. By Theorem 2.3, the mean and standard deviation of 1=Q approach (1 ? )=(k +1) and (1 ? )=[(k + p 1)]3=2, so that the ratio SD(1=Q)=E (1=Q) is asymptotically 1= (k + 1).

3. The Conditional Expected Response Time As a consequence of Theorem 2.1, the average number of jobs in the system when there is a new arrival (including the new arrival) is k + 1 + (k + 1)=(1 ? ) = (k + 1)=(1 ? ). Assuming that this expected value remains present throughout a new arrival's service, the new arrival with service requirement  should expect to have a response time (k + 1)=(1 ? ). We now formalize this rough argument. For this purpose, let mk;n ( ) be the conditional expected response time of a job with service requirement  given that there are k jobs with in nite service requirements and n other jobs in the system upon arrival. Let mk ( ) be the corresponding mean without conditioning on n. A nice elementary derivation by Asare and Foster [3] in the case k = 0 extends to positive k.

Theorem 3.1. In the M/G/1/PS system modi ed by having k permanent jobs with in nite service re-

quirements,

 Z t mk;n ( ) = (k1+?1)  + n ? (k1+?1)  W c (u)du ;

where W c (t) is the M/G/1 steady-state workload ccdf, and mk ( ) = (k1+?1)  :

0

(3.1) (3.2)

Proof. The argument of Asare and Foster [3] extends directly to yield (3.1). In particular, their (4)

should become rn (x) = (k + 1) + n (x). From Theorem 2.1, we know that the expected number of other jobs seen by an arrival is (k + 1)=(1 ? ), so that the second term of (3.1) drops out when we uncondition on n to compute (3.2), just as when k = 0.

4. The Conditional Response-Time Distribution The conditional response-time distribution given the service requirement in the case k = 0 was derived by Kitaev and Yashkov [19] and Yashkov [28]; e.g., see Yashkov [29], [30]. The joint distribution of the 5

conditional response time T  T ( ) and the number of customers left behind by a departure in the case k = 0 was derived by Ott [21]. We now show that there are simple extensions to the case of k  1. Let N  N (T +) be the number of customers left behind at departure by the tagged customer (himself excluded), assuming that there are k permanent customers (with in nite service requirements, who are not themselves counted).

Theorem 4.1. Consider the M/G/1/PS system with  < 1 modi ed by having k extra permanent customers with in nite service requirements. Then

E [e?sT zN jk;  ] = E [e?sT z N j0;  ]k+1 ;

(4.1)

where E [e?sT z N j0;  ] is given in Theorem 2.2 of Ott [21].

Proof. Modify the proof of Theorem 2.2 of Ott [21], getting ? p) E [e? z jk; x] = h(s;zz; x) (1 ? (1 pH (s; z; x)) sT

k+1

N

k+1

k+1

k+1

= E [e?sT z N j0; x]k+1

(4.2)

instead of his (2.26) in his notation.

Corollary. In the setting above, the conditional sojourn-time distribution given k and  is the (k +1)-fold convolution of the conditional sojourn time distribution with k = 0 and

2

V ar[T jk;  ] = (k + 1)V ar[T j0;  ] = (k + 1) (1 ? )2

Z

0



( ? u)W c (u)du :

(4.3)

Paralleling Theorems 2.2 and 2.3, we can establish CLTs and LLNs for the variables T  T ( ) and N as k ! 1; we do not state them.

5. Stochastic Comparisons for Conditional Means We now make stochastic comparisons showing how the variability of the service-requirement cdf G aects conditional means. First we have the conditional mean mk;n ( ) in (3.1). Theorem 2.1 of Ott [21] also displays the conditional mean E [T ( )j; k; n; x1; : : :; xn?1 ] where there are n ? 1 other jobs with remaining service requirements x1 ; : : :; xn?1 in addition to the one new job with requirement  . In particular, ? X E [T ( )j; k; n; x1; : : :; xn?1 ] = (k + 1)B ( ) + [B (1)( ) ? B (1)( ? xj )] n 1

(1)

i=1

6

(5.1)

where

B

(1)

(x) = (1 ? )?1

Z x_0 0

W (u)du :

(5.2)

Equations (5.1) and (5.2) are actually valid for all  > 0, including   1, if we rst appropriately cancel out (1 ? ) terms; i.e., we should replace (1 ? )?1 W (t) in (5.2) with 1 X (5.3) R(t) = 1 + nGne (t) ; n=1

where Gne  (t) is the cdf of the n-fold convolution of Ge in (2.1) with itself. To make the comparisons, we use notions of stochastic order; e.g., see Chapter 1 of Stoyan [26]. We say that one random variable X1 with cdf G1 is stochastically less than or equal to another random variable X2 with cdf G2, and write X1 st X2 or G1 st G2, if Ef (X1)  Ef (X2) for all nondecreasing real-valued functions f for which the expectations are de ned or, equivalently, if Gc1 (t)  Gc2(t) for all t. We say that X1 is less than or equal to X2 in the convex stochastic order and write X1 c X2 or G1 c G2 if Ef (X1)  Ef (X2) for all convex real-valued functions for which the expectations are de ned. (Since f (x) = x and f (x) = ?x are both convex, X1 c X2 requires that EX1 = EX2.)

Theorem 5.1. Consider two M/G/1/PS systems with common arrival rate and service-requirement cdf's ordered by G  G . 1

c

2

(a) For all  , k, n, x1; : : :; xn?1 ,

E [T1( )j; k; n; x1; : : :; xn?1]  E [T2( )j; k; n; x1; : : :; xn?1] : (b) If n  ()(k + 1)=(1 ? ), then

(5.4)

m(1) ( )  () m(2) ( ) : k;n k;n

(5.5)

Proof. By Theorem 5.2.3(a) on p. 82 of Stoyan [26], W  W since G  G . Hence W (t)  W (t) 1

st

2

1

c

2

1

2

for all t. For part (a), apply (5.1) and (5.2). For part (b), observe that (5.5) follows directly from (3.1) given the ordering on W1(t)  W2(t). Theorem 5.1 shows that for any given information (and n  (k + 1)=(1 ? ) in (b)), more variable service requirements lead to shorter conditional mean response times. The increased variability means more short service requirements as well as more long ones. Evidently the shorter service requirements help more than the longer ones hurt in these cases. However, there is a counteracting eect. If G1 c G2, then G1e st G2e ; e.g., see Whitt [27]. Hence, when G becomes more variable in the convex order, the residual service requirements tend to grow. The fact 7

that the steady-state mean response time is (k + 1)=(1 ? ), independent of the service-time distribution beyond its mean shows that the two eects | greater remaining service times and shorter conditional mean for given remaining service times | exactly cancel out.

6. Evolution of the Long Jobs We now describe the evolution of the longer jobs in a longer time scale, assuming that the long jobs arrive in an independent Poisson process and have iid service requirements. First we observe that we can apply classical results to describe the steady-state behavior of the two-class model; e.g., see Section 3.3 of Kelly [18]. Let S and L be the arrival rates of the short and long jobs. Let mS1 , mL1 , L  S mS1 and S  L mL1 be the associated mean service requirements and trac intensities. Let   S + L and   S + L be the aggregate arrival rate and trac intensity. We assume that  < 1, so that the system is stable. The two-class model is again an M/G/1/PS model. In this framework, the steady-state number of customers in service at any time has the geometric distribution P (Q = n) = (1 ? )n, n  0. In addition, the classical theory concludes that, conditional on the total number being n, the steady-state probability that there are k long jobs in service is binomial b(n; p; k) where p  L=. We now describe the time-dependent behavior of the long jobs in a longer time scale. Somewhat surprisingly, we can treat the longer jobs much as we treated the shorter jobs in Section 2. In a longer time scale, we can act as if the number of short jobs in the system is constant between changes in the number of long jobs. We thus achieve a separation of time scales. To formally introduce the long time scale, we assume that the exceptional long jobs arrive according to a Poisson process with rate L and have iid service requirements having mean mL1 =. Note that the contribution to the overall trac intensity by the long jobs is L = L mL1, independent of . Let L (t) be the number of long jobs in the system at time t as a function of . We focus on the time-scaled process L(t=) as  ! 0. By Theorem 2.1, the expected total number of regular (shorter) jobs in the system is (k +1)S =(1 ? S ) when there are k long jobs in the system. We will show that as  ! 0 we can think of this average number there continuously. In this long time scale, we can apply a law of large numbers as  ! 0 to conclude that the number of shorter jobs is asymptotically equal to its mean. We rst describe a single transition in the long-jobs process.

Theorem 6.1. Consider the system above starting with k long jobs with remaining long-job service requirements distributed as T1 =; : : :; Tk =. Let X be the time until the next long-job arrival or departure.

8

Then

X ) minfT0; (k + S )T=(1 ? S )g as  ! 0 ;

where T = minfT1; : : :; Tk g and T0 is an exponential random variable with mean ?L 1 independent of (T1; : : :; Tk ). The probability of the next long-job event being an arrival is asymptotically

Z1 0

P (T > (1 ? S )t=(k + S ))Le?Lt dt :

Proof. At rst ignoring possible arrivals, the time until the next long-job departure is D =

Z

T = 0

[k + Q(t)]dt

where Q(t) is the number of short jobs in the system at time t. By Theorem 2.1 and the law of large numbers, as  ! 0,     ( k + 1)  k +  S S D ) k + 1 ?  T = 1? T : S

S

For the law of large numbers, we exploit regenerative structure; i.e., the emptiness epochs for the short jobs serve as regeneration epochs. Since the time until the next long-job arrival is independent and exponentially distributed with mean 1=L, it is easily incorporated to yield the results. We obtain additional birth-and-death process structure if we assume that the long jobs have exponential service requirements.

Theorem 6.2. If in addition the long-job service-requirement cdf is exponential, then L (t=) ) L(t) as  ! 0, where fL(t) : t  0g is a birth-and-death process with birth rates    and death rates   k(1 ?  )=m (k +  ). 

k

k

S

L1

L

S

Proof. The extra exponential assumption makes T ; : : :; T iid exponential random variables with mean 1

k

mL1. Thus T has an exponential distribution with mean mL1 =k. The resulting process describing the number of long jobs thus evolves as a birth-and-death process with the indicated rates in the limit as  ! 0. For example, we can apply Theorem 6.2 to calculate rst-passage-time distributions from one level to another for the long jobs in the long time scale. We can exploit numerical transform inversion with Section 5 of Abate and Whitt [2] to obtain explicit numerical results. We can also describe the steady-state behavior, for which we have insensitivity to the long-job service-requirement cdf. The insensitivity follows from Theorems 6.1 and 6.2 by the same argument used in Theorem 2.1. 9

Theorem 6.3. Let Y be a long-job service-requirement and assume that Y ) Y as  ! 0, where Y has cdf G with mean m . Then L (t=) ) L(t), where fL(t) : t  0g has a unique stationary distribution, 

L1





distributed as L, with L being negative binomial; i.e.,

P (L = n) = (1 ? ^) with where

1+S

  n + S ^n; n  0 ; n

(6.1)

EL = (1 + S ) 1 ?^ ^ = (1 +1 ?S)L and V ar L = (1 + S ) (1 ?^^)2 ;

(6.2)

^ = 1 ?L < 1 :

(6.3)

S

Note that as S ! 0 the negative binomial distribution in (6.1) approaches the geometric distribution which prevails when there are no short jobs. Also note that the asymptotic steady-state behavior described in Theorem 6.3 does not agree with the two-class description at the beginning of this section. By (6.2), EL = (1+ S )L=(1 ? ) instead of L =(1 ? ). Given that EL = L (1 ? ), the average of the steady-state means in Theorem 2.1 is S =(1 ? ), as it should be. Evidently the order of the two limits t ! 1 and  ! 0 makes a dierence. We can apply the asymptotic framework above to consider the eect of imposing an upper limit on the number of long jobs that can be in the system.

Theorem 6.4. Suppose that long jobs can be identi ed upon arrival and are rejected whenever the number of long jobs in the system exceeds n. As  ! 0, L (t=) ) L(t), where fL(t) : t  0g is a birth-and-death 

process with stationary distribution

P (L = k) = b(k)=

n X

b(j ); 0  k  n ;

j=0

where b(n) is de ned in (2.6) with k = L and k = k(1 ? S )=mL1(k + S ) as in Theorem 6.2.

With this upper limit on the number of long jobs, the short-term conditional steady-state distribution of the number of short jobs in the system given the number of long jobs is given by Theorem 2.1, just as before. An approximate long-term steady-state distribution for the short jobs is obtained by averaging the distribution from Theorem 2.1 for each k weighted by the steady-state probability of the long jobs from Theorem 6.4. The upper limit for the number of long jobs can be determined by considering the worst-case average processing rate and conditional expected response time for the short jobs and the blocking probability for the long jobs. 10

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