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Approximations of general discrete time queues by discrete time queues with arrivals modulated by finite chains

Part of: Stochastic processes Operations research and management science Special processes Markov processes

Published online by Cambridge University Press:  01 July 2016

Vinod Sharma
Vinod Sharma*
Affiliation:
Indian Institute of Science
*
*Postal address: Department of Electrical Engineering, Indian Institute of Science, Bangalore 560012, India.
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Abstract

Recently, Asmussen and Koole (Journal of Applied Probability30, pp. 365–372) showed that any discrete or continuous time marked point process can be approximated by a sequence of arrival streams modulated by finite state continuous time Markov chains. If the original process is customer (time) stationary then so are the approximating processes. Also, the moments in the stationary case converge. For discrete marked point processes we construct a sequence of discrete processes modulated by discrete time finite state Markov chains. All the above features of approximating sequences of Asmussen and Koole continue to hold. For discrete arrival sequences (to a queue) which are modulated by a countable state Markov chain we form a different sequence of approximating arrival streams by which, unlike in the Asmussen and Koole case, even the stationary moments of waiting times can be approximated. Explicit constructions for the output process of a queue and the total input process of a discrete time Jackson network with these characteristics are obtained.

Keywords

MARKED POINT PROCESSESDISCRETE TIME QUEUESAPPROXIMATIONS OF POINT PROCESSESCONVERGENCE OF STATIONARY MOMENTS

MSC classification

Primary: 60K25: Queueing theory
Secondary: 60G55: Point processes 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 90B22: Queues and service
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 29 , Issue 4 , December 1997 , pp. 1039 - 1059
Copyright
Copyright © Probability Trust 1997 

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References

[1] Asmussen, S. (1987) Applied Probability and Queues. Wiley, Chichester.Google Scholar
[2] Asmussen, S. and Koole, G. (1993) Marked point process as limits of Markovian arrival streams. J. Appl. Prob. 30, 365–372.CrossRefGoogle Scholar
[3] Borovkov, A. A. (1976) Stochastic Processes in Queuing Theory. Springer, Berlin.CrossRefGoogle Scholar
[4] Borovkov, A. A. (1978) Ergodicity and stability theorems for a class of stochastic equations and their applications. Theory Prob. Appl. 23, 227–247.Google Scholar
[5] Brandt, A., Franken, P. and Lisek, B. (1990) Stationary Stochastic Models. Wiley, Chichester.Google Scholar
[6] Daley, D. J. and Rolski, T. (1992) Finiteness of waiting time moments in general stationary single server queue. Ann. Appl. Prob. 2, 987–1008.Google Scholar
[7] Gibson, D. and Seneta, E. (1987) Augmented truncation of infinite stochastic matrices. J. Appl. Prob. 24, 600–608.Google Scholar
[8] Gut, A. (1988) Stopped Random Walks. Springer, New York.Google Scholar
[9] Hermann, C. (1993) Analysis of discrete time SMAP/D/1/s finite buffer queue with applications in ATM. In INFOCOM'93. Google Scholar
[10] Lucantoni, D. (1991) New results on a single server queue with a batch Markovian arrival process. Stoch. Models 7, 1–46.Google Scholar
[11] Kalashnikov, V. (1994) Topics on Regenerative Processes. CRC Press, Boca Raton, FL.Google Scholar
[12] Neuts, M. F. (1981) Matrix Geometric Solutions in Stochastic Models. Johns Hopkins University Press, Baltimore, MD.Google Scholar
[13] Royden, ?. L. (1989) Real Analysis. Macmillan, Basingstoke.Google Scholar
[14] Sharma, V. (1993) Open queuing networks in discrete time—some limit theorems. Queueing Systems 14, 159–175.Google Scholar
[15] Sharma, V. (1993) Markov modulated queues in discrete time—some limit theorems. IISc-ISTC technical report. Bangalore.Google Scholar
[16] Sharma, V. (1995) Analysis of discrete time queues with applications to ATM based BISDNS. IISc-ISTC technical report. Google Scholar
[17] Takine, T., Suda, T. and Hasegawa, T. (1993) Cell loss and output process analysis of a finite buffer discrete time ATM queuing system with correlated arrivals. In INFOCOM'93. Google Scholar
[18] Wirth, K. D. (1982) On stationary queues with batch arrivals. Inf. Kybernet. 18, 603–619.Google Scholar