GI’X/M’Y/1 systems with resident server and generally distributed arrival and service groups

GI’X/M’Y/1 systems with resident server and generally distributed arrival and service groups

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Article ID: iaor19962272
Country: United States
Volume: 9
Issue: 2
Start Page Number: 159
End Page Number: 170
Publication Date: Apr 1996
Journal: Journal of Applied Mathematics and Stochastic Analysis
Authors:
Keywords: bulk queues
Abstract:

Considered are bulk systems of GI/M/1 type in which the server stands by when it is idle, waits for the first group to arrive if the queue is empty, takes customers up to its capacity and is not available when busy. Distributions of arrival group size and server’s capacity are not restricted. The queueing process is analyzed via an augmented imbedded Markov chain. In the general case, the generating function of the steady-state probabilities of the chain is found as a solution of a Riemann boundary value problem. This function is proven to be rational when the generaitng function of the arrival group size is rational, in which case the solution is given in terms of roots of a characteristic equation. A necessary and sufficient condition of ergodicity is proven in the general case. Several special cases are studied in detail: single arrivals, geometric arrivals, bounded arrivals, and an arrival group with a geometric tail.

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