Acta Mathematica Academiae Scientiarum Hungaricae 12. (1961)

1961 / 1-2. szám - Foster, F. G.: Queues with batch arrivals. I

F. 0. FOSTER the end of a service time a unit is present, it departs, otherwise nothing happens, and a fresh service begins. Such batch-size queueing processes do not appear to have been treated explicitly in the literature, although they have obvious applications. They are, however, implicit in the work of Erlang (see [1]) and Wishart [9]. These authors suppose that a service time devoted to one unit is composed of r consecutive phases. If, instead, we think of the unit as composed of r sub­units corresponding to the phases of service, we have the idea of batch arrivals. Justification for the explicit consideration of batch arrivals systems resides in the fact that the results one can obtain are elegant, and a natural generalization of the case of unit arrivals, as treated for example in [5] and [7]. This paper covers much the same ground as [9], but the analysis is different and the results obtained here are in fact new. (Cf. my remarks in the Discussion of [8].) Denote by §(f) the number of units in the system, including the one being served, at the instant t and put §„ = §(t„ — 0) (л = 1,2, ...). The main result of this paper is the determination of the limiting distribution, ft = limP [?»=/]. W—>- CO I am indebted to Dr. L. Takács for suggesting a substantial improvement in my original method of proof. The distribution {pj■} exists and is independent of the initial state of the system if and only if /'pel. The proof of this statement follows the same lines as that for the case r— 1, as given in [2]. The limiting distribution of the waiting time for an arbitrary unit will also be derived. 2. Let {г,,} (л = 1,2, ...) be a sequence of identically distributed inde­pendent random variables with distribution kj = PK =j] (y' = 0, 1,2,...) where CO Then vn is thought of as the number of real or virtual departures during the лш inter-arrival time. CO Put K(z) = ^kjZJ. We note that K(z) — <p{p(\—г)}. We assume rod. j=o Then it follows from Rouché’s theorem that the equation 1 (1) A'(z) = z'- k;= I e-"x ^ dF(x). 0

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