Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-27T18:49:28.119Z Has data issue: false hasContentIssue false

Analysis of a counting process associated with a semi-Markov process: number of entries into a subset of state space

Published online by Cambridge University Press:  01 July 2016

Yasushi Masuda
Affiliation:
University of Rochester
Ushio Sumita*
Affiliation:
University of Rochester
*
Postal address: William E. Simon Graduate School of Business Administration, University of Rochester, Rochester, NY 14627, USA.

Abstract

Let N(t) be a finite semi-Markov process on 𝒩 and let X(t) be the associated age process. Of interest is the counting process M(t) for transitions of the semi-Markov process from a subset G of 𝒩 to another subset B where 𝒩 = BG and BG = ∅. By studying the trivariate process Y(t) =[N(t), M(t), X(t)] in its state space, new transform results are derived. By taking M(t) as a marginal process of Y(t), the Laplace transform generating function of M(t) is then obtained. Furthermore, this result is recaptured in the context of first-passage times of the semi-Markov process, providing a simple probabilistic interpretation. The asymptotic behavior of the moments of M(t) as t → ∞ is also discussed. In particular, an asymptotic expansion for E[M(t)] and the limit for Var [M(t)]/t as t → ∞ are given explicitly.

Type
Research Article
Copyright
Copyright © Applied Probability Trust 1987 

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

This paper has been partially supported by the IBM Program of Support for Education in the Management of Information Systems.

The second author was partially supported by the National Science Foundation under grant No. ECS-8404071. Reproduction in whole or in part is permitted for any purpose of the United States Government.

References

[1] Çinlar, E. (1969) Queues with semi-Markovian arrivals. J. Appl. Prob. 4, 365379.CrossRefGoogle Scholar
[2] Çinlar, E. (1969) Markov renewal theory. Adv. Appl. Prob. 1, 123187.CrossRefGoogle Scholar
[3] Çinlar, E. (1975) Markov renewal theory: A survey. Management Sci. 21, 727752.CrossRefGoogle Scholar
[4] Gavish, B. and Sumita, U. (1984) Analysis of channel and disk subsystems in computer systems. Queueing Systems. To appear.Google Scholar
[5] Gertsbakh, I. B. (1984) Asymptotic methods in reliability theory. Adv. Appl. Prob. 16, 147175.CrossRefGoogle Scholar
[6] Hofri, M. (1980) Disk scheduling: FCFS vs. SSTF revised. Comm. ACM 23, 645653.CrossRefGoogle Scholar
[7] Jewell, W. S. (1963) Markov renewal programming, I: Formulation, finite return models. Operat. Res. 11, 938948.CrossRefGoogle Scholar
[8] Jewell, W. S. (1967) Fluctuations of a renewal-reward process. J. Math. Anal. 19, 303329.CrossRefGoogle Scholar
[9] Keilson, J. (1969) On a matrix renewal functions for Markov renewal processes. Ann. Math. Statist. 40, 19011907.CrossRefGoogle Scholar
[10] Masuda, Y., Shanthikumar, J. and Sumita, U. (1986) A general software availability/reliability model: numerical exploration via the matrix Laguerre transform. Stochastic Models 2, 203236.CrossRefGoogle Scholar
[11] Mclean, R. A. and Neuts, M. F. (1967) The integral of a step function defined on a semi-Markov process. SIAM J. Appl. Math. 15, 726737.CrossRefGoogle Scholar
[12] Neuts, M. F. (1977) Some explicit formulas for the steady-state behavior of the queue with semi-Markovian service times. Adv. Appl. Prob. 9, 141157.CrossRefGoogle Scholar
[13] Pyke, R. (1961) Markov renewal processes: definitions and preliminary properties. Ann. Math. Statist. 32, 12311242.CrossRefGoogle Scholar
[14] Pyke, R. (1961) Markov renewal processes with finitely many states. Ann. Math. Statist. 32, 12431259.CrossRefGoogle Scholar
[15] Pyke, R. and Schaufele, R. (1964) Limit theorem for Markov renewal processes. Ann. Math. Statist. 35, 17461764.CrossRefGoogle Scholar
[16] Pyke, R. and Schaufele, R. (1966) The existence and uniqueness of stationary measures for Markov renewal processes. Ann. Math. Statist. 37, 14391462.CrossRefGoogle Scholar
[17] Sumita, U. and Masuda, Y. (1986) Analysis of software availability/reliability under the influence of hardware failures. IEEE Trans. Software Eng. 12, 3241.CrossRefGoogle Scholar
[18] Sumita, U. and Masuda, Y. (1987) An alternative approach to the analysis of finite semi-Markov and related processes. Stochastic Models 3, 6787.Google Scholar