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A unified method to analyze overtake free queueing systems

Part of: Operations research and management science Special processes

Published online by Cambridge University Press:  01 July 2016

Dimitris Bertsimas  and
Georgia Mourtzinou
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.
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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.

Keywords

PERFORMANCE ANALYSISDISTRIBUTIONAL LAWSPRIORITY QUEUESGENERALIZED VACATIONSHEAVY TRAFFICMIXED GENERALIZED ERLANG ARRIVALS

MSC classification

Primary: 60K25: Queueing theory 90B22: Queues and service
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 28 , Issue 2 , June 1996 , pp. 588 - 625
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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