Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-08T17:37:05.739Z Has data issue: false hasContentIssue false
  • English
  • Français

Superposition of renewal processes

Published online by Cambridge University Press:  01 July 2016

C. Y. Teresalam  and
John P. Lehoczky
C. Y. Teresalam*
Affiliation:
University of Michigan, Ann Arbor
John P. Lehoczky*
Affiliation:
Carnegie-Mellon University
*
∗Postal address: Department of Industrial and Operations Engineering, University of Michigan, Ann Arbor, MI 48109, USA.
∗∗Postal address: Department of Statistics, Carnegie-Mellon University, Pittsburgh, PA 15213, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper extends the asymptotic results for ordinary renewal processes to the superposition of independent renewal processes. In particular, the ordinary renewal functions, renewal equations, and the key renewal theorem are extended to the superposition of independent renewal processes. We fix the number of renewal processes, p, and study the asymptotic behavior of the superposition process when time, t, is large. The key superposition renewal theorem is applied to the study of queueing systems.

Keywords

S-RENEWAL FUNCTIONSS-RENEWAL EQUATIONSKEY SUPERPOSITION RENEWAL THEOREMQUEUEING SYSTEMS
Type
Research Article
Information
Advances in Applied Probability , Volume 23 , Issue 1 , March 1991 , pp. 64 - 85
Copyright
Copyright © Applied Probability Trust 1991 

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

Research of both authors supported in part by a grant from the National Science Foundation DMS-87-02537.

References

Albin, S. L. (1982) On Poisson approximations for superposition arrival processes in queues. Management Sci. 28, 126–137.CrossRefGoogle Scholar
Albin, S. L. (1984) Approximating a point process by a renewal process, II: Superposition arrival processes to queues. Operat. Res. 32, 1133–1162.CrossRefGoogle 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. (1973) Some properties of a process occurring in the superposition of two independent renewal processes. Proc. XX International Meeting, The Institute of Management Sciences , ed. Schifler, E., Tel Aviv, Israel, 517–520.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. (1972) Superposition of point processes. In Stochastic Point Processes , ed. Lewis, P. A. W., Wiley-Interscience, New York, 546–606.Google Scholar
Çinlar, E. (1975) Introduction to Stochastic Processes. Prentice-Hall, Englewood Cliffs, NJ.Google Scholar
Cox, D. R. (1962) Renewal Theory. Wiley, New York.Google Scholar
Cox, D. R. and Smith, W. L. (1954) On the superposition of renewal processes. Biometrika 41, 91–99.CrossRefGoogle Scholar
Disney, R. L. (1975) Random flow in queueing networks. A.I.I.E. Trans. 7, 268–288.Google Scholar
Feller, W. (1971) An Introduction to Probability Theory and Its Applications , Vol. 2, 2nd edn. Wiley, New York.Google Scholar
Karlin, S. and Taylor, H. M. (1975) A First Course in Stochastic Processes , 2nd edn. Academic Press, New York.Google Scholar
Kshirsagar, A. M. and Becker, M. (1981) Superposition of Markov renewal processes. S. Afr. Statist. J. 15, 13–30.Google Scholar
Lawrance, A. J. (1973) Dependency of intervals between events in superposition process. J. R. Statist. Soc. B 35, 306–315.Google Scholar
Newell, G. F. (1984) Approximations for superposition arrival processes in queues. Management Sci. 30, 623–632.CrossRefGoogle Scholar
Ross, S. M. (1983) Stochastic Processes. Wiley, New York.Google Scholar
Whitt, W. (1982) Approximating a point process by a renewal process, I: Two basic methods. Operat. Res. 30, 125–147.CrossRefGoogle Scholar