QUEUE AND ITS APPLICATION TO APPROXIMATION. M. Miyazawa. Science University of Tokyo ABSTRACT

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1 COMMUN. STATIST.-STOCHASTIC MODELS, 3(1), (1987) A GENERALIZED POLLACZEK-KHINCHINE FORMULA FOR THE GI/GI/l/K QUEUE AND ITS APPLICATION TO APPROXIMATION M. Miyazawa Science University of Tokyo ABSTRACT This paper deals with >the steady-state waiting time distribution in the GZ/GZ/l/k (GZ/GZ/l with k waiting places) queue. MarshallCBl's generalized Pollaczek-Khinchine formula is generalized to the GZ/GZ/l/k model. From the resulting, formulas, we obtain approximation formulas for the waiting time distribution and for the probabilities of the system being empty and of customer loss. The quality of those approximations is 'shown by numerical examples. 1. INTRODUCTION We consider the steady-state distribution of the waiting time in the GZ/GZ/l/k queue (the GZ/GZ/l queue with k waiting places). When k is finite, customers who find k+l customers in the system are lost. The queuing discipline is assumed to be FCFS (First come First served). For GZGZl MarshallCBl derived the Laplace-Stieltjes transform of the waiting time distribution (the generalized Pollaczek-Khinchine formula, see Sec.8.4 of KleinrocktBI). This formula are useful for approximations and stochastic order relations, see e.g. Stoyantl41. In this paper, we further generalize this result to the GI/GI/l/k queue, and show that it is useful for approximations.

2 5 4 MIYAZAWA In general, the analysis of GI/GI/l/k is more difficult than for the GI/GI/l model. The method of MarshallC81 is difficult to apply to GZ/GI/l/k. Therefore, in this paper, we use the method of MiyazawaC121, where approximations for the queue length distributions are discussed. We first derive a version of the steady-state equations called the basic equations, and we transform them to get a formula for the Laplace- Stieltjes transform of the waiting time distribution. This formula involves two unknown distributions and agrees with Marshall's formula when k is infinite. The probabilistic meaning of those distributions is clear and we can therefore derive approximation formulas by assuming suitable distributions for them. In Section 2, we discuss the derivation of the basic equations by using simple results from the theory of point processes. From the basic equations, we derive our main result (Theorem 2.2). In Section 3, we derive explicit approximation formulas for the probabilities of the system being empty and of the arriving customer being lost. The quality of the approximations is illustrated by numerical examples. 2. THE BASIC EQUATIONS FOR A GI/GI/l/K QUEUE In this section, we derive the steady-state equations for the GI/GI/l/k queue, which are refinements of relations between time and customer stationary characteristics given in Ktlnig and SchmidtC71 and MiyazawaC91. We call those equations "the basic equations" to emphasize the differ- ence with the usual steady-state equations. From them, we obtain a useful expression for the Laplace-Stieltjes transform of the waiting time distributions. We assume that the arrival process of customers starts from -- and that the customers are numbered in order of their arrivals so that the first customer after time is numbered 1. Let T, be the interarrival time between the n-th and (n+l)-th customers and Sn be the service time of the n-th customer. From the model assumptions, the sequences {T,) and (Sn) are i.i.d. and independent of each other. We denote their univariate distribution functions by F and G respectively. The Laplace- Stieltjes transform of a distribution is indicated by adding a tilde. ET and ES are the means of T, and Sn, A=ET-' and P=hES. The following characteristics are defined for all times t.

3 GI/GI/l/k QUEUE 55 u(t) : the residual arrival time to the next arrival of a customer, r(t) : the residual service time of a customer being served, l(t) : the number of customers in the system, v(t) : the virtual waiting time. Let tn be the arrival epoch of the n-th customer. Then, the waiting time of the n-th customer Wn equals to v(tn-1. Our essential assumption is that {(u(t),r(t),l(t))} is a stationary process with respect to a suitably chosen probability measure P. When k is infinite, it is well-known that this holds when P<1 C91. When k is finite, this is more difficult to ensure it in general. Under mild conditions such as F or G having densities, this holds by the theory of Markov processes. We refer to BorovkovC13 for other conditions. We also use some fundamental notions of the theory of point processes. For a detailed treatment, we refer to Franken et al.c31 and MiyazawaC91. Firstly we introduce the point processes of the arrival and departure epochs of customers, denoted by No and N1 respectively. Ni(A) denotes the number of time points in the Bore1 set A on R=(--,+-). The lost customers are also counted as the departures. Then, the processes No, N1 and ((u(t),r(t),l(t))} are jointly stationary with respect to P. From the theory of point processes and our inclusion of the lost customers, we have where E denotes expectation with respect to P. Next we define, for i=,1, where 9 is the a-field induced by the process, Ts denotes a s time shift of an event and IA is the indicator function of the set A. Pi is called a Palm distribution with respect to Ni. Note that Pi is the conditional distribution of P given that there is a point of Ni at time. Now let CT(t)) be a process Jointly stationary with No and N1 with respect to P, and satisfying the following conditions. (a) For all t, X(t) has a right-hand derivative, which is denoted by XYt). (b) All discontinuities of X(t) are at points of No or N1 The processes u(t), r(t), l(t) and v(t) all satisfy these conditions. The next basic lemma is a version of Corollary 3.1 of MiyazawaC11.

4 5 6 MIYAZAWA Lemma 2.1 If EIX'()I and EOIX(O-1-X(O+)I are finite, then we have EW()) = h 3 E~ [XO-1-x(o+)], (3) i=o where Ei denotes expectation with respect to Pi for i=,1. We now derive the basic equations in terms of l(t), u(t) and r(t) by using Lemma 2.1. Let where 9 and S are real numbers. The processes Xj clearly satisfy the conditions (a) and (b). Thus we may apply Lemma 2.1 with X(t) = Xj(t). At first glance, the definition of Xj(t) may seem strange, but we shall see that it is appropriate to obtain the Laplace-Stieltjes transforms of several distributions. We use the following notation. where I=I(O), If=1(*), u=u(o), r=r(o) and r-=r(-). Then, for j=1,2,---,k, For j=k+l, (5) holds if we change EIXj(O+) to This last equation follows from the fact that for l=k+l an arriving customer leaves the system immediately. We further note that

5 ~1/GI/l/k QUEUE and finally for j=, we have Exyo) = E+j(E)~ j, E# j(-1 = P j Ed(j(O+) = EIXj(O-) =, (6) EIXJ(O+) = d;(f )pf By substituting (4) in Lemma 2.1, Theorem 2.1 follows from (51, (6) and the fact that pj = p; (see Franken et al.131). Theorem 2.1 In the GI/GI/l/k queue, We call (7)-(9) the basic equations. The special case E=O was obtained for M/GI/s/k in Miyazawall21. To obtain a compact form of the basic equations, we set and Corollary 2.1 In the GI/GI/l/k queue,

6 5 8 MIYAZAWA Let $(O) be the Laplace-Stieltjes transform of the waiting time W, of a customer who is not lost. Clearly, ite) = -~~p~+(+~(e,e(e))-p~)/g(e) I. (11) l-pk+l Theorem 2.2 In the GI/GI/l/k queue, where pk+l =, when k is infinite. Proof. By setting x=&8) and F=-8 in (lo), we obtain the expression for +(8,k8)). Substituting in (ll), we obtain (12). Note that, for finite k, (12) agrees with the Pollaczek-Khinchine's formula of MarshallC81. Equation (12) is exact, but contains the unknown quantities po, pk+l, $O(-B,O) and +k+l, We shall show that the first two characteristics can be obtained from the other two for finite k. Let t be a non-trivial solution of the equation where V=O if and only if is a multiple root of the equation. We assume the existence of such 7), which is satisfied if there exist some negative numbers BO and for which keo) and kel) are finite. We easily see that t= if and only if P=l. Note that k)=1 and that the numerator of (12) must be zero for = t since ke) has no singular point for all 8 such that &e) exists. Hence, we have, for PZ1, and for P=l, Thus po and are determined from 9(e,) and 9k1. We note that V is the limiting exponential rate of the tail distribution of the waiting time

7 GI/GI/l/k QUEUE 5 9 in GI/GI/l (see Section XII.5 in Fellert21). Hence, the expression of (12) is very natural, and it is interesting that d() is expressed by the two boundary conditions at I= and I=k+l. We conclude this section by discussing the relation between F and i. Theorem 2.3 In GI/GI/l/k, let kg) be the Laplace-Stieltjes transform of the virtual waiting time v(o), then Proof. Substituting E=O and x=&9) in (lo), we have From this, using (11) and the equality (17) follows. This theorem can also be obtained from the invariance relation between and d of G/G/s with variable service rate (cf. Konig and Schmidtl71 and Miyazawatlll). From (17), we can get an expression similar to (12) for k8). Further, by using Corollary 2.1, we can derive the expression for, 9,, i.e., the joint distribution between the residual arrival time and the virtual waiting time. This is obtained by setting x=&n in (1) and using the relations (11) and (12). Finally, some remarks on Theorem 2.1 are given. That result can also be obtained by using the backward Kolmogorov equations since {(u(t),r(t),l(t))) is a Markov process. In the literature, this method is called the supplementary variable method. However, it requires considerably complicated calculations and unnecessary regular conditions, such as F and G have densities. Also the notion of the conditional distributions as defined by (2) is needed to derive Theorem 2.1 from the Kolmogorov equations. A further merit of our approach is that its results are easily extended to non-markov processes as Markov properties are not used in Lemma 2.1 (See Miyazawatlll). For example, from (13) and (161, we have the well-known formula,

8 6 MIYAZAWA and it can be established for G/G/l/k, i.e., a queue with a stationary input. This formula can also be obtained intuitively by applying the work conservation law, but our derivation is more formal. 3. APPROXIMATION FORMULAS Let us now derive approximations for kg). By Theorem 2.2 and (13)-(16). it suffices to give approximations for do(e,o) and dk+l(8). Firstly, we derive approximation formulas leaving them unknown. Their proper forms are considered later on. Henceforth, we indicate approximate quantities by (app), such as ke)(app). For notational convenience, we set The Laplace-Stieltjes transforms of the stationary residual distribu- tions of T and S are The following theorem is a direct consequence of (12)-(16). Theorem 3.1 In the GZ/GI/l/k queue, if there exists a non-trivial? such that hf(-?)=l, then approximations for 9 and yields: and for P=l, From this theorem, we derive approximations for the mean waiting time.

9 Corollary 3.1 In the GI/GI/l/k queue, we have for Pfl under the assumptions that E(S2), E(T% al and bl are finite, and for P=l under the assumptions that E(S3), E(T3), a2 and b2 are finite, where p(app) and pk+l(app) are given by (19)-(21). Proof From (18) of Theorem 3.1. we have (22) follows by differentiating twice with respect to and letting tend to. Similarly, (23) is obtained by differentiating three times. We now consider A(E) and B(). Natural approximations for their distributions are stationary residual versions. So we assume for B, B() = ae(e), b1 = E(S2)/(2ES), b2 = E(S3)/(3ES). But for ACE), that is, for $(5,), 'we note the relation (71, which implies that its distribution is a stationary residual version of the idle time distribution (its Laplace-Stieltjes transform So it is better to determine A from $:. We consider the following two cases. (Case 2) $;(E,o)(app) = ug(~)-f(u)) (u-exi-fm ' Case 1 is simple, but it is known to be quite good for GI/Gl/l (See Case 2 is obtained by approximating the idle time distribution by C43). the conditional distribution of T-S given that T>S, in which S is assumed to be exponential distributed with a mean ES. The last assumption is for the sake of analytical convenience only. We easily calculate A(f), al and a2. In Case 1, we have

10 MIYAZAWA and, in Case 2, We note that all our expressions for A and B are exact for M/M/l/k, and then so are the approximation formulas (18)-(21). In Tables 1, 2 and 3, we compared those approximations with the exact values, where Ei denotes an Erlang distribution with phase i and H2 denotes a hyperexponential distribution of order 2 with density where ql = ztl+ -1 (S2 = Var(T)/E(Tl)), q2 = 1-ql, B1 = 2q1/ET and B2 = 2q2/ET (See page 6 of Seelen et al.cl51). Tables 1.1 and 1.2 are concerned with moderate cases. On the other hand, Tables 2 and 3 deal with rather extreme cases. In all cases, we set h = 1. In the tables, EL is the mean number of customers in the system, and (Casel) and (Case2) refer to our approximations by Theorem 3.1 for Cases 1 and 2 respectively. The exact values of the tables are quoted from OhsoneC131 for Tables 1.1 and 1.2 and from Seelen et a1.[151 for Tables 2 and 3. In the former, Pi+l and EW are not given but we can calculated them using (13) tor (16)) and Little's formula. Similarly, we calculate EW, p and EL from the tables of Seelen et al.cl51 to obtain our tables. 7) is easily calculated by Newton's method. From the tables, we can see that Case 1 gives good approximations except for low traffics (P<.5). On the other hand, Case 2 is worse than Case 1 except when P is small and T is hyper-exponential. Though we give no numerical details here, we also tested the case A = Fe. That case is considerably worse than Cases 1 and 2. Thus we recommend Case 1, but we should use it cautiously when P is less than.5. For the light traffic case, the behavior of GI/GZ/l/k is much like to that of GI/GI/l, and so the problem becomes to approximate the GI/GI/l queue. No satisfactory

11 GI/GIIl/k QUEUE 63 Table 1.1 The cases of exponential inter-arrival or service times Po E3/M/1/4 Pk+l EL (exact) (Casel) (Case (exact) (Casel) (Case (exact) (Casel) (Case (exact) (Casel) (Case2) Table 1.2 The case of E3/E3/1/ (exact) (Casel) (Case (exact) (Casel) (Case2) (exact) (Casel) (Case (exact) (Casel) (Case21 Table 2 The cases of Elo/E2/l/k ( v~~(?')/e~t=o.~ ) k=1 Po Pk+l EL (exact) (Case (Case (exact) (Casel) (Case (exact) (Casel) (Case (exact) (Casel) (Case (exact) (Casel) (Case (exact) (Casel) (Case21

12 MIYAZAWA Table 3 The cases of H2/E2/l/k.2525 (exact) (Casel).2791 (Case (exact).433 (Casel).791 (Case2) (exact) (Casel) (Case2) (exact) (Casel) (Case2) (exact) (Casel) (Case (exact) (Casel) (Case21 approximations have yet been obtained for the waiting time distributions when P is small. This is left to future investigation. Finally, we note that KimuraC51 recently tried to approximate the queue length distribution of GI/GI/l/k by refining diffusion approxima- tions. There is no numerical example given for the case of a finite k. The weak point of his method is to infer the loss probability pk+l by heuristic consideration. Our results may be helpful to this approach also. ACKNOWLEDGEMENTS The author owes much to Professor M. F. Neuts who read the manuscript and gave many helpful comments. The author also thanks Professor D. Konig for his advice and useful discussions and to Professor T. Kimura and referees for their helpful comments. REFERENCES C11 A.A. Borovkov, Stochastic Processes in Queueing Theory, Springer- Verlag, New York Heidelberg Berlin (1976). C21 W. Feller, An Introduction to Probability Theory and Its Applica- tions, Vol. 11, John ~iley 8 Sons, New York (197).

13 ~1/GI/l/k QUEUE 65 P. Franken, D. Konig, U. Arndt and V. Schmidt, Queues and Point Processes, Akademie-Verlag and Wiley & Sons Ltd. New York (1982). S. Halfin, Delays in queues, properties and approximations, The proceedings of llth ITC Congress, Vol. 1, (1985). T. Kimura, Refining diffusion approximations for GI/G/l queues: A Tight discretization method, The proceedings of llth ITC Congress, Vol. 1, 3.1A-2-1 (1985). L. Kleinrock, QUEUEING SYSTEMS Volume 1: THEORY, John Wiley & Sons, New York (1974). D. Konig and V. Schmidt, Imbedded an non-imbedded stationary characteristics of queueing systems with varying service rate and point processes, J. Appl. Prob. 17, K.T. Marshall, Some relationships between the distributions of waiting time, idle time, and interoutput time in GI/GI/l queue, SIAM Journal Appied Math., 16, (1968). M. Miyazawa, A formal approach to queueing processes in the steady state and their applications, J. Appl. Prob. 16, (1979). M. Miyazawa, The derivation of invariance relations in complex queueing systems with stationary inputs, Adu. Appl. Prob. 15, (1983). M. Miyazawa, The intensity conservation law for queues with randomly changed service rate, J. Appl. Prob. 22, No. 2, 48-41'8 (1985). M. Miyazawa, Approximations of the queue length distribution of an M/GI/s queue by the basic equations, J. Appl. Prob. 23, No. 2, (1986). T. Ohsone, The GI/E,/l queue with finite waiting room, J. Opns. Res. Soc. Japan 24, (1981). D. Stoyan, Comparison methods for queues and other stochastic models, edited with revisions by D.J. Daley, John Wiley 4 Sons, New York (1983). L.P. Seelen, H.C. Tijms and M.H. Van Hoorn, TABLES FOR MULTI-SERVER QUEUES, North-Ho 1 land, Amsterdam (1985). Recommended by Dieter Konig, Editor Received: 8/1/1985 Revised: 8/25/1986