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Convexity results for single-server queues and for multiserver queues with constant service times

Published online by Cambridge University Press:  14 July 2016

Arie Harel
Arie Harel*
Affiliation:
Rutgers University
*
∗Postal address: Graduate School of Management, Rutgers University, 92 New Street. Newark, NJ 07102, USA.
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Abstract

We show that the waiting time in queue and the sojourn time of every customer in the G/G/1 and G/D/c queue are jointly convex in mean interarrival time and mean service time, and also jointly convex in mean interarrival time and service rate. Counterexamples show that this need not be the case, for the GI/GI/c queue or for the D/GI/c queue, for c ≧ 2. Also, we show that the average number of customers in the M/D/c queue is jointly convex in arrival and service rates.

These results are surprising in light of the negative result for the GI/GI/2 queue (Weber (1983)).

Keywords

GI/G/1 QUEUED/G/c QUEUEM/D/c QUEUEMEAN INTERARRIVAL TIMEMEAN SERVICE TIMEJOINTLY CONVEX
Type
Short Communications
Information
Journal of Applied Probability , Volume 27 , Issue 2 , June 1990 , pp. 465 - 468
Copyright
Copyright © Applied Probability Trust 1990 

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References

Harel, A. (1990) Convexity properties of the Erlang loss formula. Operat. Res. To appear.Google Scholar
Rolfe, A. J. (1971) A note on the marginal allocation in multi-server facilities. Management Sci. 17, 656–658.Google Scholar
Tu, H. Y. and Kumin, H. (1983) A convexity result for a class of GI/G/1 queueing system. Operat. Res. 31, 948–950.Google Scholar
Weber, R. R. (1983) A note on waiting times in single server queues. Operat. Res. 31, 950–951.Google Scholar