article

Work Rates in Closed Queuing Networks with General Independent Servers

Pages 838 - 847

https://doi.org/10.1287/opre.22.4.838

Published: 01 August 1974 Publication History

Abstract

<P>Closed queuing networks are composed of interconnected service stages serving a fixed number of customers. Each stage consists of a queue and identical parallel servers. The path of a customer's progress throughout the network is described by a finite Markov chain over the stage names. Service times are assumed to be mutually independent with arbitrary distribution functions. The work rate for a stage, defined as the long-run time-average amount of service time rendered by the stage, is shown to exist with probability one. Explicit expressions relating the work rates of different stages and explicit expressions for the asymptotic work rates as the number of customers becomes large are derived. Work rates are shown to depend continuously on the service-time distributions.</P>

References

[1]

R. E. Barlow and F. Proschan, Mathematical Theory of Reliability, Wiley, New York, 1965.

Google Scholar

[2]

J. P. Buzen, Queuing Network Models of Multiprogramming, Ph.D. Thesis, Division of Engineering and Applied Science, Harvard University, Cambridge, Massachusetts, 1971.

Google Scholar

[3]

A. Chang and S. S. Lavenberg, "Work Rates in Closed Queuing Networks with General Independent Servers," IBM Research Report RJ 989, San José, California, 1972.

Google Scholar

[4]

K. L. Chung, Markov Chains with Stationary Transition Probabilities, Springer-Verlag, Berlin, 1960.

Google Scholar

[5]

D. R. Cox, "A Use of Complex Probabilities in the Theory of Stochastic Processes," Proc. Camb. Phil. Soc. 51, 313-319 (1955).

Crossref

Google Scholar

[6]

D. P. Gaver, "Probability Models for Multiprogrammed Computer Systems," J. ACM 14, 423-438 (1967).

Digital Library

Google Scholar

[7]

W. J. Gordon and G. F. Newell, "Closed Queuing Systems with Exponential Servers," Opns. Res. 15, 254-265 (1967).

Digital Library

Google Scholar

[8]

E. Koenigsberg, "Cyclic Queues," Opnal Res. Quart. 9, 22-35 (1958).

Crossref

Google Scholar

[9]

M. Loeve, Probability Theory, Van Nostrand, Princeton, New Jersey, 1955.

Google Scholar

[10]

G. S. Shedler, "A Cyclic-Queue Model of a Paging Machine," IBM Research Report RC 2814, Yorktown Heights, N. Y., 1970.

Google Scholar

[11]

W. L. Smith, "Renewal Theory and its Ramifications," J. Roy. Stat. Soc. B 20, 243-301 (1958).

Crossref

Google Scholar

Cited By

View all

Lavenberg SMoeller TSauer C(1979)Concomitant Control Variables Applied to the Regenerative Simulation of Queuing SystemsOperations Research10.1287/opre.27.1.13427:1(134-160)Online publication date: 1-Feb-1979
https://dl.acm.org/doi/10.1287/opre.27.1.134
Asai HLee SHull LChantker ANutt GFranta W(1976)A queuing model with discrete service speedsProceedings of the 4th symposium on Simulation of computer systems10.5555/800244.807311(161-167)Online publication date: 10-Aug-1976
https://dl.acm.org/doi/10.5555/800244.807311
Buzen JBuzen JOckene AChen PFranklin MGreen PMaksin J(1976)Fundamental laws of computer system performanceProceedings of the 1976 ACM SIGMETRICS conference on Computer performance modeling measurement and evaluation10.1145/800200.806196(200-210)Online publication date: 29-Mar-1976
https://dl.acm.org/doi/10.1145/800200.806196
Show More Cited By

Recommendations

Priority retrial queueing model operating in random environment with varying number and reservation of servers

We consider a multi-server queueing system with two types of customers operating in the Markovian random environment. Under the fixed state of the random environment, the arrival flow is described by the marked Markovian arrival process. Type 1 ...
Closed Queueing Networks Under Congestion: Nonbottleneck Independence and Bottleneck Convergence

We analyze the behavior of closed multiclass product-form queueing networks when the number of customers grows to infinity and remains proportionate on each route or class. First, we focus on the stationary behavior and prove the conjecture that the ...
Some Equivalence Results for Load-Independent Exponential Queueing Networks

In this paper we derive a number of results concerning the behavior of closed load-independent exponential queueing networks. It is shown that if the service rate of any station is increased (decreased), then the throughput of the network itself also ...

Comments

Please enable JavaScript to view thecomments powered by Disqus.

Information & Contributors

Information

Published In

Operations Research Volume 22, Issue 4

August 1974

259 pages

ISSN:0030-364X

Issue’s Table of Contents

Publisher

INFORMS

Linthicum, MD, United States

Publication History

Published: 01 August 1974

Qualifiers

Article

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

8
Total Citations
View Citations
0
Total Downloads

Downloads (Last 12 months)0
Downloads (Last 6 weeks)0

Reflects downloads up to 11 Feb 2025

Other Metrics

View Author Metrics

Citations

Cited By

View all

Lavenberg SMoeller TSauer C(1979)Concomitant Control Variables Applied to the Regenerative Simulation of Queuing SystemsOperations Research10.1287/opre.27.1.13427:1(134-160)Online publication date: 1-Feb-1979
https://dl.acm.org/doi/10.1287/opre.27.1.134
Asai HLee SHull LChantker ANutt GFranta W(1976)A queuing model with discrete service speedsProceedings of the 4th symposium on Simulation of computer systems10.5555/800244.807311(161-167)Online publication date: 10-Aug-1976
https://dl.acm.org/doi/10.5555/800244.807311
Buzen JBuzen JOckene AChen PFranklin MGreen PMaksin J(1976)Fundamental laws of computer system performanceProceedings of the 1976 ACM SIGMETRICS conference on Computer performance modeling measurement and evaluation10.1145/800200.806196(200-210)Online publication date: 29-Mar-1976
https://dl.acm.org/doi/10.1145/800200.806196
Gelenbe EPujolle GBuzen JOckene AChen PFranklin MGreen PMaksin J(1976)Probabilistic models of computer systemsProceedings of the 1976 ACM SIGMETRICS conference on Computer performance modeling measurement and evaluation10.1145/800200.806188(118-125)Online publication date: 29-Mar-1976
https://dl.acm.org/doi/10.1145/800200.806188
Asai HLee S(1976)A queuing model with discrete service speedsACM SIGSIM Simulation Digest10.1145/1013610.8073117:4(161-167)Online publication date: 1-Jul-1976
https://dl.acm.org/doi/10.1145/1013610.807311
Buzen J(1976)Fundamental operational laws of computer system performanceActa Informatica10.1007/BF002657697:2(167-182)Online publication date: 1-Jun-1976
https://dl.acm.org/doi/10.1007/BF00265769
Gelenbe EPujolle G(1976)The behaviour of a single queue in a general queueing networkActa Informatica10.1007/BF002657667:2(123-136)Online publication date: 1-Jun-1976
https://dl.acm.org/doi/10.1007/BF00265766
Lavenberg SShedler G(1975)Derivation of Confidence Intervals for Work Rate Estimators in a Closed Queuing NetworkSIAM Journal on Computing10.1137/02040104:2(108-124)Online publication date: 1-Jun-1975
https://dl.acm.org/doi/10.1137/0204010

Abstract

References

Cited By

Recommendations

Priority retrial queueing model operating in random environment with varying number and reservation of servers

Closed Queueing Networks Under Congestion: Nonbottleneck Independence and Bottleneck Convergence

Some Equivalence Results for Load-Independent Exponential Queueing Networks

Comments

Information

Published In

Publisher

Publication History

Qualifiers

Contributors

Other Metrics

Bibliometrics

Article Metrics

Other Metrics

Citations

Cited By

View options

Share

Share this Publication link

Share on social media