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Superposition of Markov renewal processes and applications

Part of: Special processes

Published online by Cambridge University Press:  01 July 2016

C. Teresa Lam
C. Teresa Lam*
Affiliation:
The University of Michigan
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Abstract

In this paper, we study the superposition of finitely many Markov renewal processes with countable state spaces. We define the S-Markov renewal equations associated with the superposed process. The solutions of the S-Markov renewal equations are derived and the asymptotic behaviors of these solutions are studied. These results are applied to calculate various characteristics of queueing systems with superposition semi-Markovian arrivals, queueing networks with bulk service, system availability, and continuous superposition remaining and current life processes.

Keywords

S-MARKOV RENEWAL EQUATIONSUPERPOSITION MARKOV RENEWAL THEOREMQUEUES WITH SUPERPOSITION SEMI-MARKOVIAN ARRIVALSQUEUEING NETWORKS WITH BULK SERVICESTATE DEPENDENT SERVICE RATESSYSTEM AVAILABILITYCONTINUOUS SUPERPOSITION REMAINING AND CURRENT LIFE PROCESSES

MSC classification

Primary: 60K15: Markov renewal processes, semi-Markov processes
Secondary: 60K10: Applications (reliability, demand theory, etc.) 60K20: Applications of Markov renewal processes (reliability, queueing networks, etc.)
Type
Research Article
Information
Advances in Applied Probability , Volume 25 , Issue 3 , September 1993 , pp. 585 - 606
Copyright
Copyright © Applied Probability Trust 1993 

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Footnotes

Research supported in part by a fellowship from Horace H. Rackham School of Graduate Studies, The University of Michigan.

References

Barbour, A. D. (1982) Generalized semi-Markov schemes and open queueing networks. J. Appl. Prob. 19, 469–474.CrossRefGoogle Scholar
Burman, D. Y. (1981) Insensitivity in queueing systems. Adv. Appl. Prob. 13, 846–859.Google Scholar
Cherry, W. P. (1972) The superposition of two independent Markov renewal processes. Ph.D. Dissertation, Department of Industrial and Operations Engineering, University of Michigan.Google Scholar
Cherry, W. P. and Disney, R. L. (1983) The superposition of two independent Markov renewal processes. Zastos Mat. XVII, 567–602.Google Scholar
ÇInlar, E. (1967) Queues with semi-Markovian arrivals. J. Appl. Prob. 4, 365–379.Google Scholar
ÇInlar, E. (1969a) Markov renewal theory. Adv. Appl. Prob. 1, 123–187.Google Scholar
ÇInlar, E. (1969b) On semi-Markov processes on arbitrary spaces. Proc. Camb. Phil. Soc. 66, 381–392.Google Scholar
ÇInlar, E. (1972) Superposition of point processes. In Stochastic Point Processes, ed. Lewis, P. A. W., pp. 546–606. Wiley-Interscience, New York.Google Scholar
ÇInlar, E. (1975a) Introduction to Stochastic Processes. Prentice-Hall, Englewood Cliffs, NJ.Google Scholar
ÇInlar, E. (1975b) Markov renewal theory: a survey. Management Sci. 21, 727–752.CrossRefGoogle Scholar
Cox, D. R. and Smith, W. L. (1954) On the superposition of renewal processes. Biometrika 41, 91–99.Google Scholar
Harris, C. M. (1967a) Queues with state-dependent stochastic service rates. Operat. Res. 15, 117–130.Google Scholar
Harris, C. M. (1967b) Queues with stochastic service rates. Naval Res. Logist. Quart. 14, 219–230.Google Scholar
Jacod, J. (1971) Théorème de renouvellement et classification pour les chaînes semi-Markoviennes. Ann. Inst. H. Poincaré, B 7, 83–129.Google Scholar
Jacod, J. (1974) Corrections et compléments à l'article: Théorème de renouvellement et classification pour les chaînes semi-Markoviennes. Ann. Inst. H. Poincaré B 10, 201–209.Google Scholar
Kesten, H. (1974) Renewal theory for functionals of a Markov chain with general state space. Ann. Prob. 2, 355–386.Google Scholar
Kshirsagar, A. M. and Becker, M. (1981) Superposition of Markov renewal processes. South African Statist. J. 15, 13–30.Google Scholar
Lam, C. Y. T. (1990) Stationary distributions for the superposition of Markov renewal processes. Technical Report 90-23, Department of Industrial and Operations Engineering, The University of Michigan, Ann Arbor.Google Scholar
Lam, C. Y. T. and Lehoczky, J. P. (1991) Superposition of renewal processes. Adv. Appl. Prob. 23, 64–85.Google Scholar
Lawrance, A. J. (1973) Dependency of intervals between events in superposition process. J. R. Statist. Soc. B 35, 306–315.Google Scholar
Neuts, M. F. (1965) The busy period of queue with batch service. Operat. Res. 13, 815–819.Google Scholar
Neuts, M. F. (1966) Semi-Markov analysis of a bulk queue. Bull. Soc. Math. Belg. 18, 28–42.Google Scholar
Palm, C. (1943-44) Variation in intensity in telephone conversations. Ericsson Technics, 1–89.Google Scholar
Prabhu, N. U. (1965) Queues and Inventories. Wiley, New York.Google Scholar
Pyke, R. (1961a) Markov renewal processes: definitions and preliminary properties. Ann. Math. Statist. 32, 1231–1242.Google Scholar
Pyke, R. (1961b) Markov renewal processes with finitely many states. Ann. Math. Statist. 32, 1243–1259.CrossRefGoogle Scholar
Pyke, R. and Schaufele, R. (1964) Limit theorems for Markov renewal processes. Ann. Math. Statist. 35, 1746–1764.Google Scholar
Pyke, R. and Schaufele, R. (1966) The existence and uniqueness of stationary measures for Markov renewal processes. Ann. Math. Statist. 37, 1439–1462.CrossRefGoogle Scholar
Schassberger, R. (1976) On the equilibrium distribution of a class of finite-stage generalized semi-Markov process. Math. Operat. Res. 1, 395–406.Google Scholar
Schassberger, R. (1977) Insensitivity of steady-state distributions of generalized semi-Markov process–Part I. Ann. Prob. 5, 87–99.CrossRefGoogle Scholar
Schassberger, R. (1978a) Insensitivity of steady-state distributions of generalized semi-Markov processes–Part II. Ann. Prob. 6, 85–93.Google Scholar
Schassberger, R. (1978b) Insensitivity of steady-state distributions of generalized semi-Markov process with speeds. Adv. Appl. Prob. 10, 836–851.Google Scholar
Schassberger, R. (1978c) Insensitivity of stationary probabilities in networks of queues. Adv. Appl. Prob. 10, 906–912.Google Scholar
Smith, W. L. (1955) Regenerative stochastic processes. Proc. R. Soc. A. 232, 6–31.Google Scholar