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A unified method to analyze overtake free queueing systems

Published online by Cambridge University Press:  01 July 2016

Dimitris Bertsimas*
Affiliation:
Massachusetts Institute of Technology
Georgia Mourtzinou*
Affiliation:
Massachusetts Institute of Technology
*
Postal address: Sloan School of Management and Operations Research Center, MIT, Cambridge, MA 02139, USA.
∗∗ Postal address: Operations Research Center, MIT, Cambridge, MA 02139, USA.

Abstract

In this paper we demonstrate that the distributional laws that relate the number of customers in the system (queue), L(Q) and the time a customer spends in the system (queue), S(W) under the first-in-first-out (FIFO) discipline are special cases of the H = λG law and lead to a complete solution for the distributions of L, Q, S, W for queueing systems which satisfy distributional laws for both L and Q (overtake free systems). Moreover, in such systems the derivation of the distributions of L, Q, S, W can be done in a unified way. Consequences of the distributional laws include a generalization of PASTA to queueing systems with arbitrary renewal arrivals under heavy traffic conditions, a generalization of the Pollaczek–Khinchine formula to the G//G/1 queue, an extension of the Fuhrmann and Cooper decomposition for queues with generalized vacations under mixed generalized Erlang renewal arrivals, approximate results for the distributions of L, S in a GI/G/∞ queue, and exact results for the distributions of L, Q, S, W in priority queues with mixed generalized Erlang renewal arrivals.

Type
General Applied Probability
Copyright
Copyright © Applied Probability Trust 1996 

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Footnotes

The research of D. Bertsimas was partially supported by a Presidential Young Investigator Award DDM-9158118 with matching funds from Draper Laboratory. The research of both authors was partially supported by the National Science Foundation under grant DDM-9014751.

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