O~fEG4. The Inl Jl o( Mgrru $~:t Vol 6. No 5 pp 443 44x tlloS-I~t~ "x ;,~q-a~.az'~o2t~lq~ C P~:rgamon Pr~, lid I'4"S Prlnlcd nn Grc~t [~rlldlN Some Practical Considerations on Multi-Server Queues with Multiple Poisson Arrivals GEORGE P COSMETATOS Imperial College, London [Rc(t'itt'd ~,I.L 197;~F A simple approximate formula is presented and evaluated in this paper for the steady-state average number of customers in the M(~)/G/r queueing system. The derivation of the formula is based on a heuristic argument. Comparisons between approximate results and the ones obtained analyti- cally or by extensive simulation experiments seem to suggest that the degree of approximation achieved is satisfactory for most practical purposes. INTRODUCTION QUEUEING systems where customers arrive in groups and/or get served in batches are quite frequent in practice. However, models of such systems are seldom amenable to analytic methods leading to directly applicable results, and in the absence of approximation tech- niques or solutions the practitioner may have no alternative than to resort to simulation. Yet this approach has its drawbacks: it is time con- suming and not always easy to design, analyse or evaluate. A useful model which does not seem to lend itself to easy mathematical treatment is the one described as MI~/G/r:(~/GD)in Lee's [7] notation. It is assumed in this model that" (a) Customers arrive in groups (batches) of size X, where X is an integer random variable whose probability distribution is Pr{X =xl = c~(1 < x < ~) with mean E(x) and variance var(x). (b) Arrivals of batches of size x form a Poisson process with mean ,;.~,; ihe average rate of arrival of customers is ,:.E(x), where ,:.= ~- ;.x. (c) There are r servers in parallel; each server has an independently and identically distri- buted service-time distribution B(t); the average service time is /~-~ and the coeffi- cient of variation of service times ~'. (d) Arriving Customers join a single queue and are served individually. (e) The system is in statistical equilibrium, i.e. the traffic intensity p = ~.E(x)/r,u < I. Let Lx(r,v) denote the average number of customers in the system (those in the queue and the ones being served). In this paper an approximate formula for Lx(r,v) is developed and the degree of approximation achieved under various conditions investigated. The derivation of the formula is based on a heuris- tic argument whereby a reformulation of the expression for Lx(1,v) is extended to the multi- server queue. From a computational viewpoint, the formula is very simple to apply and the relative percentage errors incurred seem to be quite small. THE M(X)/G/1 QUEUE Saaty [12] derives the following expression for the probability generating function (Pgf) of the number of customers in the system: (1 - p){l - :1K(z) P(z) = (1) K(:) - : where K(z)= B*[,:.- 2C(z)], with B*(s) and C(z) denoting respectively the Laplace-Stieltjes transform of the service-time distribution and the pgf of the variable X. 443