Hostname: page-component-7dd5485656-pnlb5 Total loading time: 0.001 Render date: 2025-10-30T23:08:37.683Z Has data issue: false hasContentIssue false
  • English
  • Français

On up- and downcrossings

Published online by Cambridge University Press:  14 July 2016

J. W. Cohen
J. W. Cohen*
Affiliation:
Mathematical Institute, University of Utrecht
Get access Rights & Permissions [Opens in a new window]

Abstract

For the sample functions of the stationary virtual waiting-time process vt of the GI/G/1 queueing system some properties of the number of up- and downcrossings of level v by the vt -process during a busy cycle are investigated. It turns out that the simple fact that this number of upcrossings is equal to that of downcrossings leads in a rather easy way to basic relations for the waiting-time distributions. This approach to the study of the vt -process of the GI/G/1 system seems to be applicable to many other types of stochastic processes. As another example of this approach the infinite dam with non-constant release rate is briefly discussed.

Keywords

QUEUEING SYSTEMSUPCROSSINGSDOWNCROSSINGSINFINITE DAM

Information

Type
Short Communications
Information
Journal of Applied Probability , Volume 14 , Issue 2 , June 1977 , pp. 405 - 410
Copyright
Copyright © Applied Probability Trust 1977 

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.)

Article purchase

Temporarily unavailable

References

[1] Cohen, J. W. (1969) The Single Server Queue. North-Holland, Amsterdam.Google Scholar
[2] Cohen, J. W. (1976) On regenerative processes in queueing theory. Lecture Notes in Economics and Mathematics 121, Springer-Verlag, Berlin.Google Scholar
[3] 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 time. Sankhya A 25, 91–100.Google Scholar
[4] Takács, L. (1962) Introduction to the Theory of Queues. Oxford University Press, New York.Google Scholar
[5] Gaver, D. P. and Miller, R. G. (1962) Limiting distributions for some storage problems. In Studies in Applied Probability and Management Science, ed. Arrow, K. J., Karlin, S. and Scarf, H., Stanford University Press.Google Scholar
[6] Çinlar, E. and Pinsky, M. (1972) On dams with additive input and a general release rule. J. Appl. Prob. 9, 422–429.CrossRefGoogle Scholar
[7] Rubinovitch, M. and Cohen, J. W. (1976) On level crossings and cycles in the classical dam. Report, Mathematical Institute, University of Utrecht.Google Scholar
[8] Brill, P. H. and Posner, M. J. M. (1975) Levelcrossings in point processes applied to queues: single server case. Report 75–009, Department of Industrial Engineering, University of Toronto.Google Scholar