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Tandem Conveyor Queues

Published online by Cambridge University Press:  27 July 2009

F. Baccelli ,
E.G. Coffman Jr  and
E.N. Gilbert
F. Baccelli
Affiliation:
International de Recherche en Informatique et en Automatique 06565 Valbonne Cedex, France
E.G. Coffman Jr
Affiliation:
AT&T Bell Laboratories Murray Hill, New Jersey 07974
E.N. Gilbert
Affiliation:
AT&T Bell Laboratories Murray Hill, New Jersey 07974
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Abstract

This paper analyzes a queueing system in which a constant-speed conveyor brings new items for service and carries away served items. The conveyor is a sequence of cells each able to hold at most one item. At each integer time, a new cell appears at the queue's input position. This cell holds an item requiring service with probability a, holds a passerby requiring no service with probability b, and is empty with probability (1– a – b). Service times are integers synchronized with the arrival of cells at the input, and they are geometrically distributed with parameter μ. Items requiring service are placed in an unbounded queue to await service. Served items are put in a second unbounded queue to await replacement on the conveyor in cells at the input position. Two models are considered. In one, a served item can only be placed into a cell that was empty on arrival; in the other, the served item can be placed into a cell that was either empty or contained an item requiring service (in the latter case unloading and loading at the input position can take place in the same time unit). The stationary joint distribution of the numbers of items in the two queues is studied for both models. It is verified that, in general, this distribution does not have a product form. Explicit results are worked out for special cases, e.g., when b = 0, and when all service times are one time unit (μ = 1). It is shown how the analysis of the general problem can be reduced to the solution of a Riemann boundary-value problem.

Type
Articles
Information
Probability in the Engineering and Informational Sciences , Volume 3 , Issue 4 , October 1989 , pp. 517 - 536
Copyright
Copyright © Cambridge University Press 1989

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