l and its ramifications

The queuing system M/G"/I and its ramifications I. Introduction

M.L. C H A U D H R Y

Department of Mathematics. Royal Military Collegeof Canada

Several authors have discussed one aspect or another of the queuing system M/Grill in which customers arrive singly at random, form a queue before a single server, and are served in order o f arrival in batches of fLxed m a x i m u m size B, e.g., Bailey [ 1 ], using the imbedded Markov chain technique, considers the pgf P+(z), of the limiting distribution of the number in the system at departure epochs; Downton [4] discusses the waiting time distribution and later [5] considers the numerical evaluation of certain operational parameters of the system. For the less general system MIEn~ 1 wherein ER indicates a modified Erlangian distribution,

and J.G.C. TEMPLETON

Department of bldustrial Engh~eering, Universityof Toronto Received January 1979 Revised August 1979 This paper deals with the bulk service queuing system

M/GB/I and its ramifications. In the system M/GB/I, customers arrive singly by a Poisson process and are served by a single server in order of arrival in batches of fixed maximum size 6'. The distribution of service time is general and an idle server begins service immediately ff one or more customers await service. The main aims of the paper are: (i) to generalize Jaiswal's queuing system M/EB/I ; (ii) to unify Bailey's and Jaiswal's generalized results and obtain relationships between the pgf's (probability generating functions) oo

P,,(z)--

oo

Cr{(.ut)r-l/(r - 1)!}/~ e x p ( - / a t ) , r=l

Jaiswal [8,9] using a modified Erlangian technique obtains the transient and steady-state distribution of the number in the queue and waiting time distm-,ution in the steady-state case. In the steady-state case of the less general system M/E~/I, he connects, though incorrectly (see equation (29) of [8]), random point probabilities with the corresponding departure epoch probabilities. The correct relation for the more general case M/Grill is presented in this paper. In queuing problems, one o f the chief interests, lies in getting Pn (Pn, l in our case - defined later) - the probability that there are n customers in the system at a random epoch, or its probability generating function (pgf) when it is not easy to get Pn in explicit form. However, to simplify analysis, Kendall, in his two important papers, proposed the use of imbedded Markov chains which led to derivation o f P n+ - the probability that there are n customers in the system at a departure epoch (or its pgf) or other such probabilities by appropriately defining tile regeneration point,;, e.g. see the discussion of the system MX/G/1 with bulk arrivals in Gross and Harris [7]. Bloemena [2] studies, among other things, the distribution of Pn+ (expressed in our notation) for the queuing system M/Grill by using the m e t h o d of collective marks. Although his results are more general

oo

e n , l 7."

n=O

and

P+ (z) = n=O

+

where Pn,l and Pn are the limiting probabilities of n customers in the queue (at random epoch) with server being busy and n in the queue immediately after a departure epoch respectively; (iii) to obtain a simple derivation of the transform of the distribution of waiting time first obtained by Downton and later by others, and to suggest ways of getting computational results in the case when expected values are considered at random epochs rather than at departure epochs as discussed by Bailey and Downton. The supplementary variable technique is used in discussing the above results. It is also demonstrated that Little's formula is true for M/GB/I as it should, since it is independent of the distributions involved. Some other operational parameters of this are also discussed. Examples of extensive numerical results for P~ and L + axe presented in tabular form. The authors wish to thank Major Fraser Holman for pointing out some of the errors in a first draft, and to a referee for various suggestions for improvements. The research for this paper was supported (in part) by the Defenee Research Board of Canada, Grant Number 3610-603, and by the National Research Council of Canada, under Grant Number A5639. © North-HoUand Publishing Company European Journal of Operational Research 6 (1981) 5 6 - 6 0

56

M.L. Chuudhry, J. G,C Templeton / The queuing system M/G B/I

than those obtained by Jaiswa! [9], he derives only the imbedded Markov chain probabilities P,+,.,which serve as an approximation to random point results, when the latter are difficult to obtain. It can be inferred from his results that although the assumptions in his model and those of Bailey's model are slightly different, the imbedded Markov chain results discussed by Bailey and him turn out to be the same, an observation made by Jaiswal [8,9] in discussing a somewhat less general model M/EBR/I. However, one cannot find probabilities such as Pn for the queuing system M/GB/1 from Bloemena's results. Our purpose here is not only to derive the pgf of the distribution of the number in the queae M/Grill with the server being busy, but also to deduce Bloemena's and hence Bailey's results. In fact, we find the ratio Pq(z)/P+(z) from which one can derive moments of queue size at a random epoch in terms of moments of queue size at a departure epoch; in particular see the formula relating the two means of queue size. The approach proposed in this paper not only unifies all the rest, Its due to Bailey, Jaiswal's generalized results and Bloemena, but also it helps us in correcting an error in Jaiswal's paper [8]. in addition, our method gives a simple derivation of the waiting distribution, first obtained by Downton and later by other authors. Once a relation between the pgf's of Pn or Pn, l (defined later) and Pn+ is obtained, it will be seen that computational aspect of various operational parameters, which has neither been discussed by Bloemena nor by Jaiswal, can easily be handled by using the numerical results due to Downton and some additional results due to Taylor and Russell [13]. Since the assumptions made by Bloemena and Jaiswal (and in this paper) are the same, viz., an idle server starts service instantaneously if one or more customers await service, we confirm Bloemena's findings that the pgf of P,~ is the same as the one obtained by Bailey. In view of what has been said above, it may be remarked that the supplementary variable technique that is being emoloyed in this paper is more versatile than the techniques used by other authors in that it not only unifies their results, but produces them in an elegant way. The last section derives the expected busy period using an alternating renewal process. It is then expressed in terms of,%. This approach is quite general in that it can be used in several other single server queues with bulk arrivals or bulk services. It may be appropriate here, though not directly related to the results being discussed in this paper,

57

to mention another bulk service queuing system, it is

an M/Gk/I queuing system in which batch size, k, is fixed. Foster and Perera [6], using renewal theoretic arguments, discuss, the relation between the pgCs of Pn and P+. The method proposed in this paper shows that many of the results concerning single server queuing systems with bulk arrival and/or bulk service can be unified. In fact, the method has successfully been employed not only in deriving appropriate measures for the queuing system M/Gk/1, but also in finding results for more complicated bulk service queuing systems, such as the one discussed by Neuts [12]. These topics are the subject matter of another paper.

2. Limiting distribution of queue size Consider an M/GB/I queue in which customers F l arrive singly at epochs 0 = o~, Ol, ..., Or, ..., and groups of customers depart at epochs ol, o2 .... , ? F On, ... • The interarrival times (or+l - or), r >1O, are exponentially distributed with mean 1/~,. The service times of groups of customers are independently distributed rv's with common distribution function B(v) and mean service time 1//a = fo°°V dB(v). Then the density b(v) and conditional service rate r/(v) are given by dB(v) b(v)=--do-

d

and

71(v)=-~-o-ln[l-O(v)].

Also B(o)= 1 - e x p [ - f o 'v r/(x) dx ] and B(v)= r/(v)[l - B ( o ) ] . We consider the distribution of queue size, defined here to denote the number of customers waiting for service (not including those in service). Note that Foster and Perera [6] use the term 'queue size' to denote the number of customers in the system, including those in service, and also that they " consider a system with service in batches of fixed size. Let Nq(t) be the queue size at time t, and X(t) the elapsed service time of tile group under sc~'ice at time t. The supplementary variable X(t) is introduced in order to obtain a bivariate Markov prgcess {Nq(t), X(t)}. Define P.,I(x)= lim

Pa, l(x,t)

t - + Do

and Poo = lira Poo(t)

n>O

M.L. Chaudhry, Z G . C Tet.zpleton / The queuing system M/GB/I

58

and the normalizing condition

where

e,,,~(x, t) dx =eiNq(t) =n, x < X(t) < x + dx 1,

oo

n>~O, Poo(t) = P[Nq(t) = 0 and the server is idle] .

too + ,:..i

We then have

From equations (2), (5) and (6) we obtain a differential equation for Pq(z;x)whose solution is

oo

Pn = lira P[Nq(t)=nl =f pn,,(x)dx, n :>0 t'-~°°

(9)

P,,,,(x ) dx = l

n=O 0

(1)

0

Thus P,~. l (x, t) is a probability and density for the state in which there are n customers waiting for service (not including those in service) at epoch t, and the elapsed service time is x. Poo(t) is the probability that there are no customers waiting at epoch t, and the server is idle (and will remain idle until a customer is present). Define the pgf's Pq(z:x) and P(x) by

Pq(z;x)=Pq(z;O)[1-B(x)] e x p [ - k ( l - z ) x ] .

(10)

From (7) and (8) we obtain B

oo

( : - zk) f e~.~f~) n¢~) d~ k=O

Vq(z;O)=

0

zB-bC A- ~z)

(11)

where b'(a) = fo °* e -ax dB(x). Hence

oo

Pq(z;x) ~ Pn,~(x)z"

(2)

?1=0

oo

pq(z) = f pq(z;x) ar o

(3)

e(z)= f Pq(z;x)dx +Poo 0

We proceed, following Cox [3] and Kosten [ 111], ~o obtain Pq(z; x) and P(z). The balance equations are:

Xeoo= f eo,~(x)n(x)dx

(4)

0

d

~. Po.~(x)= -Ca + )7(x))?o,, (x)

(5)

=Pq(z;O)[1 - b(X -- ~z,)]/[X(l - z ) ] .

(12)

An explicit solution for Pq(z), and hence for P(z), can be found by considering the roots of the denominator of equation (11)inside the unit circle. We omit details of this solution, which are unnecessary for our present purpose. The method of solution is similar to that given in detail by Downton [41 , for the M/E~/I queue considered at departure epochs.

3. Number of customers waiting at a departure epoch

d

~xt',,.l(x)

Let =-CA+~(x))Pn, I(X)+~Pn_I,I(X),

for t / ~ 1.

p;~- lim t'[Nq(ok + O)= n l.

(6) These equations are to be solved under the following boundary conditions at x = 0: t:o

&~(o)- f t',,÷a,l(X)n(x)ax,

n >o

(7)

That is, P+n is the limiting probability that there are n customers in the queue just before a departure and hence n customers in the system just after the departure. Define also the pgfP+(z) = Z~=oP+zn. We have

0

oo

? ~ - o f &~(x)n(x)dx,

B

Po, l(0) = ~

;

k=l 0

Pk, l(x)rl(x)dx + LPoo

(8)

n>~O,

0

where D is a normalizing constant. Hence~ using (11)

M.L Chaudhry, J.G.C Templeton / I'he queuing ~ystem M/GBfl

of the service times. For the M/G/I queue, we recover the known r~:sult L* = Lq + 0 = L, since Poe = 1 - p and L, the expected number of customers in the system, is equal to p + (p2 + 32o2)/[2 ( 1 .... p)].

and (12) oo

oo

P+(z)=D f

Z"Pn,I(X)~(X)(~I~

~

" O

n-O

=D :

59

5. Distribution of waiting time

Pq(z;x) rt(x) dx

o

= DPq (z; 0) b(?, -- hz )

(13)

B

b--(X-~) ~ e~,(:- ?) k=O

-.

,r

z B - ~(x - Xz)

,,14)

Equation (14) agrees with Bailey's result for intermittently available server. From (10) and (13), using the normalizing conditions P{ 1) = 1 = P+(1 ), we find D = [ta(l - Poo)] - l , and hence

Pq(z)_ P+(z)

l-~O,-Xz)

(1 - z ) b ( X - kz)

•

1-Boo

(15)

Bp

Equation (15) may be compared to a formula stated without proof as equation (29) of Jaiswal [8], attributed to an anonymous referee. From Jaiswal's Equation ( 2 9 ) w e get, in our notation,Pq(z)= (1 - Poo)P+(z), which reduces to (15) if and only if bCa - kz) = [ 1 + Bp(1 - z)] -1, that is, if and only if b-(cx) = M ( a + ta), which implies that the service time distribution is exponential. It follows that equation (29) of Jaiswal is correct only in the case of exponential service and not in the system M]E~/I as stated in Jaiswal's paper.

Downton [4,5] obtained the conditional waiting time distribution for those customers who wait in the M/GB/I queue. Downton derivation makes use (if the probability that a customer is the rth arrival in its service batch, and of the conditional distributio~i of waiting time for customers which ale rth ar,:ivals. A simpler derivation of the condi(iunal waiti:~g time distributioo can, however, be given, using Equation (I 5 ) a n d the well-known result p(z) = ~ q f a - Xz)

(16)

where ~q(o~) is the Laplace-Stieltjes Transform of the waiting-time distribution W(t) defined by ~q(Ot) = fo °° e - s t d W(t). We note in passing that, by differentiating equation (16) and setting z = l, we obtain at once Little's formula Lq = XWq. From ( 1 6 ) w e get &q(~) = Poo + tall - Pool 11 - E(~)]

(l 7)

XP+[ 1 - ( ~ / X ) l / [ o £ ( ~ ) ] .

The conditional distribution of waiting time for those who have to wait can be found from ( i 7). The result is

ffOq(O~l Vq > 0 ) = = tall - b-(~)l e+I1 -- (~/;~)l/[a6(~)l

4. Mean queue size

Cross-multiplying in (15), differentiating twice and setting z = 1, we obtain L q = P 'q( I ) = [ 1 - Pool " [L + + (~.[2) ta(o 2 - (1/ta)2)], where o ~- is the variance

where Vq is the waiting time (in queue) of a customer. This agrees with results given by Downton [4] and also by Jaiswal [9]. Setting z = 0 in (15), we have Pq(O) = [( 1 - b(X)}/bCh)l [(I -- Poo)/(BP)] Po. Similarly from (4), ( l 1 ) and ( 12 ) we obtain Pa(O) =

Table 1 p - 0.01

e~ k 1 2 3

B

L+

1

2

3

4

5

1

2

3

4

5

0.9000 0.9000 0.9000

0.829 0,824 0.822

0.767 0.755 0.751

0.713 0.694 0.687

0.666 0.640 0.630

0.111 0J08 0.107

0.206 0.203 0.203

0.304 0.302 0.301

0.403 0.4131 0.400

0.502 0.500 0.500

M.L. Chaudhry,Z G.C Templeton/ The queuingsystemM/GBfl

60 Table 2 0=0.9 P~ k

B

1 2 3

L+

I

2

3

4

5

1

0.100 0.100 0.100

0.068 0.061 0.057

0.052 0.041 0.035

0.042 0.029 0.023

0.035 0.022 0.016

9.000 6.975 6.300

[ ( 1 - b('h))}/b'('A)] Poo. Equating the two expressions for Pq(O) gives

too ao+e

(18)

Since P~ can be obtained from an imbedded Markov chain solution, Poo and hence ~q(Ot), and mean queue size Lq at random epoch, can be found from a solution for queue size at departure epochs. Some numerical work for evaluating P~, and hence Poo, in the case when G = Ek, has been undertaken by Russell and Taylor [13].

2

3

4

5

13.659 18.319 22.979 27.639 9.646 12.327 15.012 17.698 8.314 10.344 12.379 14.419

the k[Erlang distribution is a good approximation to systems with service time distributions having coefficient of variation less than one, this is the one whose numerical results are presented here. Computational work for P~ and L + was undertaken by Russell and Taylor [13], for various combinations of k and B. This is because P~ easily gives E(X) and Poo whereas Poo and L ÷ together lead to Lq. Tables 1 and 2 give various values in units of mean service time for p = 0 . 1 a n d p = 0.9,B = l(1)5, k = 1(1)3.

References 6. Expected busy period An interesting result which seems to fall outside the preceding results, but is really related to the probability Poo, or P~, is the expected busy-period of the server. Jaiswal [ 10] discusses the busy-period distribution of the queuing system MIEn~1 and gives expressions for the mean and variance for the simpler case M/Ee[I. One can, in principal, find these moments for the queuing system MIEn~1. However, the derivation of the expected busy-period for the more general c~w M/Grill using an alternating renewal process is sim!~i:..r. It is this approach that we adopt here. Sftnce idleTeriods and busy periods generate an alternating renewal process,

E(X)/E(Y) "- (1 - Poo)/Poo, where E(X) and E(Y) are the expected busy and idle periods respectively and Poo is given by (18). Now, since in M/GB/I, by using the Markov property of the exponential, E(Y) = 1[X,

E(X) - (1 - Poo) _ l/[/.tP~].

Xeoo 7. Numerical calculations By specifying a service time distribution we can discuss some numerical results of this system. Since

[ 1] N.T.J. Bailey, On queuing processes with bulk service, J. Roy. Statist. Soc. Ser. B. 16 (1954) 80-87. [2] A.R. Bloemena, A queuing process with a certain type of bulk service, Bull. Inst. Internat. Statist. 37 (1960) 219-226. [3] D.R. Cox, The analysis of non-Markovian stochastic processes by the inclusion of supplementary variables, Proc. Cambridge Phil. Soc. 51 (1955) 433-441. [4] F. Downton, Waiting time in bulk service queues, J. Roy. Statist. Soc. Ser. B. 17 (1955) 256-261. [5] F. Downton, On limiting distributions arising in bulk service queues, J. Roy. Statist. Soc. Ser. B. 18 (1956) 265 -274. [6] F.G. Foster and A.G.A.D. Perera, Queues with batch departures II, Ann. Math. Statist. 35 (1964) 1147-1156. [ 7] D. Gross and C.R. Harris, Fundamentals of Queuing Theory (Wiley, New York, 1974). [8] N.K. Jaiswal, Time-dependent solution of the bulkservice queuing problem, Operations Res. 8 (1960) 773-781. [9l N.K. Jaiswal, Bulk-servicequeuing problem, Operations Res. 8 (1960) 139-143. [10] N.K. Jaiswal, Distribution of busy periods for the bulk service queuing problem, Defence Sci. J. 12 (1962) 309-316. [11] L. Kosten, Stochastic theory of service systems (Pergamon Press, Oxford, 1973). [121 M.F. Neuts, A general class of bulk queues with Poisson input, Ann. Math. Stat. 38 (1967) 759-770. [131 W.A. Russell and R.G. Taylor, Numerical analysis and relationships for the queuing models M/ESk/1,E~M/I and M/E~/I,a fourth year Engineering and Management thesis, Royal Military College, Kinston, Ont., Canada