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JOIN MINIMUM COST QUEUE FOR MULTICLASS CUSTOMERS: STABILITY AND PERFORMANCE BOUNDS

Published online by Cambridge University Press:  01 October 2004

Rahul Tandra ,
N. Hemachandra  and
D. Manjunath
Rahul Tandra
Affiliation:
Department of EECS, University of California, Berkeley, CA, E-mail: tandra@eecs.berkeley.edu
N. Hemachandra
Affiliation:
IE and OR Interdisciplinary Programme, Indian Institute of Technology, Bombay, Powai Mumbai, 400 076 India, E-mail: nh@iitb.ac.in
D. Manjunath
Affiliation:
Department of Electrical Engineering, Indian Institute of Technology, Bombay, Powai Mumbai, 400 076 India, E-mail: dmanju@ee.iitb.ac.in
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Abstract

We consider a system of K parallel queues providing different grades of service through each of the queues and serving a multiclass customer population. Service differentiation is achieved by specifying different join prices to the queues. Customers of class j define a cost function ψij(ci,xi) for taking service from queue i when the join price for queue i is ci and congestion in queue i is xi and join the queue that minimizes ψij(·,·). Such a queuing system will be called the “join minimum cost queue” (JMCQ) and is a generalization of the join shortest queue (JSQ) system. Non-work-conserving (called Paris Metro pricing system) and work-conserving (called the Tirupati system) versions of the JMCQ are analyzed when the cost to an arrival of joining a queue is a convex combination of the join price for that queue and the expected waiting time in that queue at the arrival epoch. Our main results are for a two-queue system.

We obtain stability conditions and performance bounds. To obtain the lower and upper performance bounds, we propose two quasi-birth–death (QBD) processes that are derived from the original systems by suitably truncating the state space. The state space truncation in the non-work-conserving JMCQ follows the method of van Houtum and colleagues. We then show that this method is not applicable to the work-conserving JMCQ and provide sample-path-based proofs to show that the number in each queue is bounded by the number in the corresponding queues of these QBD processes. These sample-path proof techniques might also be of independent interest. We then show that the performance measures like mean queue length and revenue rate of the system are also bounded by the corresponding quantities of these QBD processes. Numerical examples show that these bounds are fairly tight. Finally, we generalize some of these results to systems with more queues.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 18 , Issue 4 , October 2004 , pp. 445 - 472
Copyright
© 2004 Cambridge University Press

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