Hostname: page-component-848d4c4894-tn8tq Total loading time: 0 Render date: 2024-06-20T10:51:59.706Z Has data issue: false hasContentIssue false
  • English
  • Français

A rate-conservative principle for stationary piecewise Markov processes

Published online by Cambridge University Press:  01 July 2016

Tomasz Rolski
Tomasz Rolski*
Affiliation:
Wrocław University
Get access Rights & Permissions [Opens in a new window]

Abstract

A stationary process yt, t ∈ R1 is considered which is Markov between points of changeover from a stationary point process, and at these points it changes over according to a distribution dependent only on the value of yt just before the change is analysed. An explicit form of a rate-conservative principle is stated, and its relationship with formulae relating the distribution of the process at an instant t and the distribution at a point of changeover is shown. The theory is applied to discrete state processes and to processes which are generalizations of the Takács processes, and is also applied in the theory of G/M/l and G/G/k queues to obtain relations between distributions of a process and some process imbedded in it.

Keywords

PIECEWISE MARKOV PROCESSRATE CONSERVATIVE PRINCIPLEIMBEDDED PROCESSG + M/G/1 QUEUEG/M/1 QUEUEG/G/k QUEUE
Type
Research Article
Information
Advances in Applied Probability , Volume 10 , Issue 2 , June 1978 , pp. 392 - 410
Copyright
Copyright © Applied Probability Trust 1978 

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

Barlow, R. E. and Proschan, F. (1964) Comparison of replacement policies, and renewal theory implications. Ann. Math. Statist. 35, 577–589.CrossRefGoogle Scholar
Chung, K. L. (1967) Markov Chains with Stationary Transition Probabilities. Springer-Verlag, Berlin.Google Scholar
Cinlar, E. (1967) Queues with semi-Markovian arrivals. J. Appl. Prob. 4, 365–379.Google Scholar
Cohen, J. W. (1976) On regenerative processes in queueing theory. Lecture Notes in Economics and Mathematical Systems, 121, Springer-Verlag, Berlin.Google Scholar
Daley, D. J. and Vere-Jones, D. (1972) A summary of the theory of point processes. In Stochastic Point Processes, Theory and Applications, ed. Lewis, P. A. W. Wiley-Interscience, New York, 299–383.Google Scholar
Franken, P. (1975) Steady state probabilities for queueing systems at different points in time (in Russian) Izv. Akad. Nauk SSSR Techn. Kibernet. 1, 101–107.Google Scholar
Jankiewicz, M. and Rolski, T. (1977) Piecewise Markov processes on a general state space. Zast. Mat. 15, 421–435.Google Scholar
Keilson, J. (1963) The first passage time density for homogeneous skip-free walks on the continuum. Ann. Math. Statist. 34, 1003–1011.CrossRefGoogle Scholar
Kerstan, J., Matthes, K. and Mecke, J. (1974) Unbegrenzt teilbare Punktprozesse. Akademie-Verlag, Berlin.Google Scholar
Klimov, G. P. (1978) Queueing Processes (in Polish). To appear.Google Scholar
König, D., Rolski, T., Schmidt, V. and Stoyan, D. (1978) Stochastic processes with imbedded marked-point process (pmp) and their application in queueing theory. Math. Operationsforch. Statist. 9, To appear.Google Scholar
Kopociński, B. (1976) On the virtual waiting time in the GI/G/s queue. Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 24, 779–782.Google Scholar
Kopociński, B. and Rolski, T. (1977) A note on the virtual waiting time in the GI/G/s queue. Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 25, 1279–1280.Google Scholar
Kuczura, A. (1973) Piecewise Markov processes. SIAM J. Appl. Math. 24, 169–181.CrossRefGoogle Scholar
Miyazawa, M. (1976) Conservation laws in queueing theory and their application to the finiteness of moments. Research Report on Information Sciences B-28, Tokyo Institute of Technology.Google Scholar
Morris, R. and Wolman, E. (1961) A note on ‘statistical equilibrium’. Opns. Res. 9, 751–753.Google Scholar
Rolski, T. (1977) A relation between imbedded Markov chains in piecewise Markov processes. Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 25, 185–193.Google Scholar
Stoyan, D. (1977) Qualitative Eigenschaften und Abschätzungen stochastischer Modelle. Akademie-Verlag, Berlin.Google Scholar
Takács, L. (1963) The limiting distribution of the virtual waiting time and the queue size for a single-server queue with recurrent input and general service times. Sankhyā A25, 91–100.Google Scholar