Hostname: page-component-7479d7b7d-fwgfc Total loading time: 0 Render date: 2024-07-11T14:12:16.041Z Has data issue: false hasContentIssue false
  • English
  • Français

Stochastic bounds for heterogeneous-server queues with Erlang service times

Published online by Cambridge University Press:  14 July 2016

Oliver S. Yu
Oliver S. Yu*
Affiliation:
Stanford Research Institute
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper establishes stochastic bounds for the phasal departure times of a heterogeneous-server queue with a recurrent input and Erlang service times. The multi-server queue is bounded by a simple GI/E/1 queue. When the shape parameters of the Erlang service-time distributions of different servers are the same, these relations yield two-sided bounds for customer waiting times and the queue length, which can in turn be used with known results for single-server queues to obtain characterizations of steady-state distributions and heavy-traffic approximations.

Keywords

QUEUESERVICE SYSTEMCONGESTION THEORYMULTI-SERVER QUEUEHETEROGENEOUS-SERVER QUEUEQUEUE WITH ERLANG SERVICE TIMESSTOCHASTIC ORDERING OF QUEUESSTOCHASTIC BOUNDING RELATIONS FOR QUEUESEXISTENCE THEOREMS FOR QUEUESLIMIT THEOREMS FOR QUEUESAPPROXIMATIONS FOR QUEUESQUEUES IN HEAVY TRAFFIC
Type
Research Papers
Information
Journal of Applied Probability , Volume 11 , Issue 4 , December 1974 , pp. 785 - 796
Copyright
Copyright © Applied Probability Trust 1974 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Research partially supported at Stanford University by the Office of Naval Research, the National Science Foundation and Stanford Research Institute.

References

[1]Brumelle, S. L. (1971) Some inequalities for parallel-server queues. Operat. Res. 19, 402–413.Google Scholar
[2]Cohen, J. W. (1969) The Single Server Queue. North-Holland, Amsterdam.Google Scholar
[3]Cox, D. R. (1962) Renewal Theory. John Wiley, New York.Google Scholar
[4]Hillier, F. S. and Lo, F. D. (1971) Tables for multiple-server queueing systems involving Erlang distributions. Technical Report No. 14, Operations Research Department, Stanford University.Google Scholar
[5]Iglehart, D. and Whitt, W. (1970a) Multiple channel queues in heavy traffic, I. Adv. Appl. Prob. 2, 150–177.Google Scholar
[6]Iglehart, D. and Whitt, W. (1970b) Multiple channel queues in heavy traffic, II: sequences, networks, and batches. Adv. Appl. Prob. 2, 355–369.Google Scholar
[7]Jacobs, D. R. Jr. and Schach, S. (1972) Stochastic order of relationships between GI/G/k systems. Ann. Math. Statist. 43, 1623–1633.Google Scholar
[8]Kiefer, J. and Wolfowitz, J. (1955) On the theory of queues with many servers. Trans. Amer. Math. Soc. 78, 1–18.Google Scholar
[9]Kingman, J. F. C. (1961) The single server queue in heavy traffic. Proc. Camb. Phil. Soc. 57, 902–904.Google Scholar
[10]Kingman, J. F. C. (1964) The heavy traffic approximation in the theory of queues. Proc. of the Symp. on Congestion Theory. (Ed. Smith, W. and Wilkinson, W.) The University of North Carolina Press, Chapel Hill. 137–159.Google Scholar
[11]Little, J. D. C. (1961) A proof of the queueing formula: L = λW. Operat. Res. 9, 383–387.Google Scholar
[12] Queueing standardization conference report. (1972) OR/SA Today 2, No. 2, 10.Google Scholar
[13]Stidham, S. Jr. (1970) On the optimality of single-server queueing systems. Operat. Res. 18, 708–732.Google Scholar
[14]Whitt, W. (1974) Exponential heavy traffic approximations for multi-server queues. Ann. Prob. 2, 000–000.Google Scholar