Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-17T15:19:55.492Z Has data issue: false hasContentIssue false

On regenerative and ergodic properties of the k-server queue with non-stationary Poisson arrivals

Published online by Cambridge University Press:  14 July 2016

Hermann Thorisson*
Affiliation:
Chalmers University of Technology and the University of Göteborg
*
Postal address: Department of Mathematics, Chalmers University of Technology and University of Göteborg, S 412 96 Göteborg, Sweden.

Abstract

We consider the stable k-server queue with non-stationary Poisson arrivals and i.i.d. service times and show that the non-time-homogeneous Markov process Zt = (the queue length and residual service times at time t) can be subordinated to a stable time-homogeneous regenerative process. As an application we show that if the system starts from given conditions at time s then the distribution of Zt stabilizes (but depends on t) as s tends backwards to –∞. Also moment and stochastic domination results are established for the delay and recurrence times of the regenerative process leading to results on uniform rates of convergence.

Type
Research Papers
Copyright
Copyright © Applied Probability Trust 1985 

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

Supported by the Swedish Natural Science Research Council and by the Icelandic Science Foundation.

References

Borovkov, A. A. (1978) Ergodicity and stability theorems for a class of stochastic equations and their applications. Theory Prob. Appl. 23, 227247.Google Scholar
ÇInlar, E. (1975) Introduction to Stochastic Processes. Prentice-Hall, Englewood Cliffs, NJ.Google Scholar
Kalähne, U. (1976) Existence, uniqueness and some invariance properties of stationary distributions for general single server queues. Math. Operationsforsch. Statist. 7, 557575.Google Scholar
Kolmogorov, A. N. (1936) Zur Theorie der Markoffschen Ketten. Math. Ann. 112, 155160.Google Scholar
Loynes, R. M. (1962) The stability of a queue with non-independent interarrival and service time. Proc. Camb. Phil. Soc. 58, 494520.Google Scholar
Miyazawa, M. (1977) Time and customer processes in queues with stationary inputs. J. Appl. Prob. 14, 349357.Google Scholar
Miyazawa, M. (1979) A formal approach to queuing processes in the steady state and their applications. J. Appl. Prob. 16, 332346.Google Scholar
Thorisson, H. (1983) The coupling of regenerative processes. Adv. Appl. Prob. 15, 531561.Google Scholar
Thorisson, H. (1984) Backward limits of non-time-homogeneous regenerative processes. Preprint, Dept. of Math., Göteborg.Google Scholar
Thorisson, H. (1985a) The queue GI/G/1: Finite moments of the cycle variables and uniform rates of convergence. Stoch. Proc. Appl. 19, 8599.Google Scholar
Thorisson, H. (1985b) The queue GI/GI/k: Finite moments of the cycle variables and uniform rates of convergence. Stochastic Models 2.Google Scholar
Tuominen, P. and Tweedie, R. L. (1979) Exponential ergodicity in Markovian queuing and dam models. J. Appl. Prob. 16, 867880.Google Scholar