On two Bulk-Service Queueing Models

Abstract

Abstract This paper studies two bulk-service queueing models useful in transportation systems. In both models the interarrival times are i.i.d. r.v.’s each having the general distribution, the service is in bulk, and there is a single server of capacity b. In model A, denoted by GI/M b–X/1, X is a random variable representing the number of units already present with the server at a service instant and if X=m (≤b) then the server takes (b — m) more units or the whole queue, whichever is smaller. Under the assumption that the server does not wait for. the units to arrive if he finds the queue empty, the limiting distribution of the queue size at a prearrival epoch is derived and is then used to obtain limiting queue size distributions at random and post-departure epochs. In model B, denoted by GI/Mb /1, the server is empty at a service ínstant and waits for a unit to arrive if there is none in the queue. Results of model A are used to obtain similar results for model B. First two moments of the distributions are listed. An interesting feature of this study is that results are expressed ín terms of a unique real root of the characteristic equation and are, therefore, easy to compute numerically. Sample listing of the root is given.