Hostname: page-component-7479d7b7d-qlrfm Total loading time: 0 Render date: 2024-07-12T19:30:03.444Z Has data issue: false hasContentIssue false
  • English
  • Français

The single server queue with Poisson input and semi-Markov service times

Published online by Cambridge University Press:  14 July 2016

Marcel F. Neuts
Marcel F. Neuts*
Affiliation:
Purdue University
Get access Rights & Permissions [Opens in a new window]

Abstract

We assume that the successive service times in a single server queue with Poisson arrivals form an m-state semi-Markov process.

The results for the M/G/1 queue are extended to this case. Both the time-dependence and the stationary solutions are discussed.

Type
Research Papers
Information
Journal of Applied Probability , Volume 3 , Issue 1 , June 1966 , pp. 202 - 230
Copyright
Copyright © Sheffield: Applied Probability Trust 

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.)

References

[1] Bellman, R. (1960) Introduction to Matrix Analysis. McGraw-Hill Book Company Inc., New York.Google Scholar
[2] Gaver, D. (1963) A comparison of queue disciplines when service orientation times occur. Naval Res. Logist. Quart. 10, 219–235.CrossRefGoogle Scholar
[3] Loynes, R. M. (1962) A continuous-time treatment of certain queues and infinite dams. J. Aust. Math. Soc. 2, 484–498.CrossRefGoogle Scholar
[4] Loynes, R. M. (1962) Stationary waiting-time distributions for single server queues. Ann. Math. Statist. 33, 1323–1339.Google Scholar
[5] Miller, H. D. (1961) A convexity property in the theory of random variables defined on a finite Markov chain. Ann. Math. Statist. 32, 1260–1270.CrossRefGoogle Scholar
[6] Pyke, R. (1961) Markov renewal processes: Definitions and preliminary properties. Ann. Math. Statist. 32, 1231–1242.Google Scholar
[7] Pyke, R. (1961) Markov renewal processes with finitely many states. Ann. Math. Statist. 32, 1243–1259.Google Scholar
[8] Pyke, R. (1963) Limit theorems for Markov renewal processes. Technical Report No. 6, University of Washington, Dept. of Math. NSF P-11008.Google Scholar
[9] Rosenblatt, M. (1962) Random Processes. Oxford University Press, New York.Google Scholar
[10] Saaty, T. L. (1961) Elements of Queueing Theory. With Applications. McGraw-Hill Book Co., N. Y. Google Scholar
[11] Smith, W. L. (1955) Regenerative stochastic processes. Proc. Roy. Soc. (London) Ser. A. 232, 6–31.Google Scholar
[12] Takács, L. (1960) The transient behavior of a single server queueing process with a Poisson input. Proc. Fourth Berkeley Symposium on Mathematical Statistics and Probability, II, 535–567.Google Scholar
[13] Takács, L. (1960) The time dependence of a single server queue with Poisson input and general service times. Ann. Math. Statist. 33, 1340–1348.CrossRefGoogle Scholar
[14] Takács, L. (1962) Introduction to the Theory of Queues. Oxford University Press, New York.Google Scholar
[15] Wielandt, H. (1950) Unzerlegbare nicht-negative Matrizen. Math. Z. 52, 642–648.CrossRefGoogle Scholar
[16] Zygmund, A. (1951) A remark on characteristic functions. Proc. Second Berkeley Symposium on Mathematical Statistics and Probability.Google Scholar