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The N/G/1 queue and its detailed analysis

Published online by Cambridge University Press:  01 July 2016

V. Ramaswami
V. Ramaswami*
Affiliation:
Drexel University
*
∗Postal address: Department of Mathematical Sciences, Drexel University, Philadelphia, PA 19104, U.S.A.
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Abstract

We discuss a single-server queue whose input is the versatile Markovian point process recently introduced by Neuts [22] herein to be called the N-process. Special cases of the N-process discussed earlier in the literature include a number of complex models such as the Markov-modulated Poisson process, the superposition of a Poisson process and a phase-type renewal process, etc. This queueing model has great appeal in its applicability to real world situations especially such as those involving inhibition or stimulation of arrivals by certain renewals. The paper presents formulas in forms which are computationally tractable and provides a unified treatment of many models which were discussed earlier by several authors and which turn out to be special cases. Among the topics discussed are busy-period characteristics, queue-length distributions, moments of the queue length and virtual waiting time. We draw particular attention to our generalization of the Pollaczek–Khinchin formula for the Laplace–Stieltjes transform of the virtual waiting time of the M/G/1 queue to the present model and the resulting Volterra system of integral equations. The analysis presented here serves as an example of the power of Markov renewal theory.

Keywords

COMPUTATIONAL PROBABILITYSINGLE-SERVER QUEUEPHASE-TYPE DISTRIBUTIONSN-PROCESSQUEUE LENGTHWAITING TIME
Type
Research Article
Information
Advances in Applied Probability , Volume 12 , Issue 1 , March 1980 , pp. 222 - 261
Copyright
Copyright © Applied Probability Trust 1980 

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Footnotes

This paper is based on Part I of a Ph.D. dissertation submitted to Purdue University. This research was supported by AFOSR-72-2350C at the Department of Statistics, Purdue University and AFOSR-77-3236 at the Department of Statistics and Computer Science, University of Delaware. A detailed version of the paper in technical report form may be obtained by writing to the author.

References

1. Carson, C. C. (1975) Computational Methods for Single-Server Queues with Interarrival and Service Time Distributions of Phase Type. , Dept. of Statistics, Purdue University.Google Scholar
2. Çinlar, E. (1969) Markov renewal theory. Adv. Appl. Prob. 1, 123–187.CrossRefGoogle Scholar
3. Cox, D. R. (1955) A use of complex probabilities in the theory of stochastic processes. Proc. Camb. Phil. Soc. 51, 313–319.Google Scholar
4. Cox, D. R. (1962) Renewal Theory. Methuen, London.Google Scholar
5. Erlang, A. K. (1917) Solution of some problems in the theory of probabilities of significance in automatic telephone exchanges. Post. Office Elect. Engineers' J. 10, 189–197.Google Scholar
6. Gantmacher, F. R. (1959) The Theory of Matrices, Vol. 2, Chelsea, New York.Google Scholar
7. Heffes, H. (1973) Analysis of first-come, first-served queueing systems with peaked inputs. Bell System Tech. J. 7, 1215–1228.CrossRefGoogle Scholar
8. Hunter, J. J. (1969) On the moments of Markov renewal processes. Adv. Appl. Prob. 1, 188–210.CrossRefGoogle Scholar
9. Kapadia, A. S. (1973) A k-server queue with phase input and service distribution. Operat. Res. 21, 623–628.Google Scholar
10. Kuczura, A. (1972) Queues with mixed renewal and Poisson inputs. Bell System Tech. J. 51, 1305–1326.Google Scholar
11. Lucantoni, D. (1978) Numerical Methods for a Class of Markov Chains Arising in Queueing Theory. , Dept. of Statistics and Computing Science, University of Delaware.Google Scholar
12. Marcus, M. and Minc, H. (1964) A Survey of Matrix Theory and Matrix Inequalities. Academic Press, New York.Google Scholar
13. Naor, P. and Yechiali, U. (1971) Queueing problems with heterogeneous arrivals and service. Operat. Res. 19, 722–734.Google Scholar
14. Neuts, M. F. (1975) Probability distributions of phase type. In Liber Amicorum Professor Emeritus H. Florin, Department of Mathematics, University of Louvain, 173–206.Google Scholar
15. Neuts, M. F. (1975) Computational uses of the method of phases in the theory of queues. Computers Math. Appl. 1, 151–166.Google Scholar
16. Neuts, M. F. (1975) Computational problems related to the Galton–Watson process. Proceedings of Actuarial Research Conference, Brown University.Google Scholar
17. Neuts, M. F. (1976) Algorithms for the waiting time distributions under various queue disciplines in the M/G/1 queue with service time distribution of phase type. Management Sci. Google Scholar
18. Neuts, M. F. (1976) Moment formulas for the Markov renewal branching process. Adv. Appl. Prob. 8, 690–711.Google Scholar
19. Neuts, M. F. (1976) Some explicit formulas for the steady-state behaviour of the queue with semi-Markovian service times. Adv. Appl. Prob. 9, 141–157.Google Scholar
20. Neuts, M. F. (1978) Renewal processes of phase type. Naval. Res. Log. Quart. 25, 445–454.Google Scholar
21. Neuts, M. F. (1978) The M/M1/1 queue with randomly varying arrival and service rates. Opsearch 15, 139–157.Google Scholar
22. Neuts, M. F. (1979) A versatile Markovian point process. J. Appl. Prob. 16, 764–779.CrossRefGoogle Scholar
23. Purdue, P. (1974) The M/M/1 queue in a Markovian environment. Operat. Res. 22, 562–569.CrossRefGoogle Scholar
24. Pyke, R. (1961) Markov renewal processes: definitions and preliminaries. Ann. Math. Statist. 32, 1231–1242.Google Scholar
25. Pyke, R. (1961) Markov renewal processes with finitely many states. Ann. Math. Statist. 32, 1243–1259.CrossRefGoogle Scholar
26. Takács, L. (1962) Introduction to the Theory of Queues. Oxford University Press, New York.Google Scholar
27. Wachter, P. (1973) Solving Certain Systems of Homogeneous Equations with Special Reference to Markov Chains. , Dept. of Mathematics, McGill University.Google Scholar
28. Yechiali, U. (1973) A queueing-type birth and death process defined on a continuous-time Markov chain. Operat. Res. 21, 604–609.Google Scholar