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Periodic steady state of loss systems with periodic inputs

Part of: Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Helmut Willie
Helmut Willie*
Affiliation:
Deutsche Telekom
*
∗ Postal address: Deutsche Telekom AG, Research and Technology Center, Postfach 100003, D-64276 Darmstadt, Germany.
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Abstract

The input of a multiserver loss system is assumed to be a periodic random marked point process which has, with probability one, infinitely many construction points. It is shown that, independently of the initial distribution, there exists a unique periodic process modeling the periodic steady-state behaviour of the loss system. In addition, practical sufficient conditions for the existence of enough construction points are derived.

Keywords

Loss systemperiodic random marked point processperiodic steady stateconstruction pointτ-ergodic input

MSC classification

Secondary: 60G55: Point processes
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 30 , Issue 1 , March 1998 , pp. 152 - 166
Copyright
Copyright © Applied Probability Trust 1998 

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References

[1] Asmussen, S. and Rolski, T. (1994). Risk theory in a periodic environment: the Cramer–Lundberg approximation and Lundberg's inequality. Math. Operat. Res. 19, 410–433.Google Scholar
[2] Asmussen, S. and Thorisson, H. (1987). A Markov chain approach to periodic queues. J. Appl. Prob. 24, 215–225.Google Scholar
[3] Bambos, N. and Walrand, J. (1989). On queues with periodic inputs. J. Appl. Prob. 26, 381–389.Google Scholar
[4] Bauer, H. (1972). Probability Theory and Elements of Measure Theory. Holt, Rinehart and Winston, New York.Google Scholar
[5] Borovkov, A.A. (1976). Random Processes in Queueing Theory. Springer, Berlin.Google Scholar
[6] Borovkov, A.A. (1984). Asymptotic Methods in Queueing Theory. Wiley, New York.Google Scholar
[7] Ewdokimowa, G.S. (1974). On the waiting time distribution in the case of a periodically arriving traffic stream. Tech. Cyb. 3, 114–118. (In Russian.).Google Scholar
[8] Fichtenholz, G.M. (1978). Differential- und Integralrechnung II. Deutscher Verlag der Wissenschaften, Berlin.Google Scholar
[9] Franken, P. (1970). Ein Stetigkeitssatz für Verlustsysteme. Operationsforsch. und Math. Statistik II. Akademie, Berlin. pp. 9–23.Google Scholar
[10] Franken, P. and Kalähne, U. (1978). Existence, uniqueness and continuity of stationary distributions for queueing systems without delay. Math. Nach. 86, 97–115.Google Scholar
[11] Franken, P., König, D., Arndt, U. and Schmidt, V. (1982). Queues and Point Processes. Wiley, New York.Google Scholar
[12] Green, L. and Kolesar, P. (1991). The pointwise stationary approximation for queues with nonstationary arrivals. Management Sci. 37, 84–97.Google Scholar
[13] Harrison, J.M. and Lemoine, A.J. (1977). Limit theorems for periodic queues. J. Appl. Prob. 14, 566–576.CrossRefGoogle Scholar
[14] Heyman, D.P. and Whitt, W. (1984). The asymptotic behavior of queues with time-varying arrival rates. J. Appl. Prob. 21, 143–156.CrossRefGoogle Scholar
[15] Lemoine, A.J. (1981). On queues with periodic Poisson inputs. J. Appl. Prob. 18, 889–900.CrossRefGoogle Scholar
[16] Lemoine, A.J. (1989). Waiting time and workload in queues with periodic Poisson input. J. Appl. Prob. 26, 390–397.Google Scholar
[17] Rolski, T. (1987). Approximation of periodic queues. Adv. Appl. Prob. 19, 691–707.CrossRefGoogle Scholar
[18] Rolski, T. (1989). Queues with nonstationary inputs. Queueing Systems 5, 113–130.Google Scholar
[19] Rolski, T. (1992). Approximations of performance characteristics in periodic Poisson queues. In: Queueing and related models. ed. Bhat, U.N. and Basawa, I.V.. Queueing and related models. Clarendon, Oxford. pp. 285–298.Google Scholar
[20] Svoronos, A. and Green, L. (1987). The N-seasons S-servers loss system. Naval Res. Logist. 34, 579–591.Google Scholar