Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-23T19:55:56.324Z Has data issue: false hasContentIssue false
  • English
  • Français

AN ALTERNATING SERVICE PROBLEM

Published online by Cambridge University Press:  31 August 2005

M. Vlasiou  and
I. J. B. F. Adan
M. Vlasiou
Affiliation:
EURANDOM, 5600 MB Eindhoven, The Netherlands, E-mail: [email protected]
I. J. B. F. Adan
Affiliation:
Department of Mathematics & Computer Science, Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands, E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider a system consisting of a server alternating between two service points. At both service points, there is an infinite queue of customers that have to undergo a preparation phase before being served. We are interested in the waiting time of the server. The waiting time of the server satisfies an equation very similar to Lindley's equation for the waiting time in the GI/G/1 queue. We will analyze this Lindley-type equation under the assumptions that the preparation phase follows a phase-type distribution, whereas the service times have a general distribution. If we relax the condition that the server alternates between the service points, then the model turns out to be the machine repair problem. Although the latter is a well-known problem, the distribution of the waiting time of the server has not been studied yet. We derive this distribution under the same setting and we compare the two models numerically. As expected, the waiting time of the server is, on average, smaller in the machine repair problem than in the alternating service system, but they are not stochastically ordered.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 19 , Issue 4 , October 2005 , pp. 409 - 426
Copyright
© 2005 Cambridge University Press

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

REFERENCES

Adan, I.J.B.F. & Zhao, Y. (1996). Analyzing GI/Er/1 queues. Operations Research Letters 19: 183–190.Google Scholar
Asmussen, S. (2003). Applied probability and queues. New York: Springer-Verlag.
Bertsekas, D. & Gallager, R. (1992). Data networks. Englewood Cliffs, NJ: Prentice-Hall.
Cohen, J.W. (1982). The single server queue. Amsterdam: Elsevier Science.
Kelly, F.P. (1979). Reversibility and stochastic networks. Chichester: Wiley.
Kleinrock, L. (1976). Queueing systems, Vol. 2: Computer applications. New York: Wiley.
Lehmann, E. (1955). Ordered families of distributions. Annals of Mathematical Statistics 26: 399–419.Google Scholar
Mann, H.B. & Whitney, D.R. (1947). On a test of whether one of two random variables is stochastically larger than the other. Annals of Mathematical Statistics 18: 50–60.Google Scholar
Park, B.C., Park, J.Y., & Foley, R.D. (2003). Carousel system performance. Journal of Applied Probability 40: 602–612.Google Scholar
Schassberger, R. (1973). Warteschlangen. Wien: Springer-Verlag.
Szekli, R. (1995). Stochastic ordering and dependence in applied probability. New York: Springer-Verlag.CrossRef
Takács, L. (1962). Introduction to the theory of queues. Oxford: Oxford University Press.
Tijms, H.C. (1994). Stochastic models: An algorithmic approach. Chichester: Wiley.
Vlasiou, M., Adan, I.J.B.F., & Wessels, J. (2004). A Lindley-type equation arising from a carousel problem. Journal of Applied Probability 41: 1171–1181.Google Scholar