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The superposition of independent discrete Markovian packet streams

Published online by Cambridge University Press:  14 July 2016

Marcel F. Neuts  and
Charles E. M. Pearce
Marcel F. Neuts*
Affiliation:
University of Arizona
Charles E. M. Pearce*
Affiliation:
University of Adelaide
*
∗Postal address: Department of Systems and Industrial Engineering, University of Arizona, Tucson, AZ 85721, USA.
∗∗Postal address: Department of Applied Mathematics, University of Adelaide, Adelaide, SA 5000, Australia.
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Abstract

In a finite, aperiodic, irreducible Markov chain, a visit to some states generates a one; to some others, a zero, while the remaining states are considered silent, in that neither symbol is generated. States during which either a one or a zero is generated are called active states, and sojourns in the set of active states correspond to messages. The output process is called a Markovian packet stream. Informally, a stream is called thin, if the steady-state fraction of time spent in the active states is small and messages are separated by silent periods of long durations. A limit theorem for the superposition of a large number of independent, stochastically identical and appropriately thin Markovian packet streams is obtained. The class of limit processes consists of a family of stationary, integer-valued, discrete-parameter processes of dependent Poisson random variables. Some properties of the limit processes are established.

Keywords

DISCRETE POINT PROCESSESLIMIT PROCESS OF SUPERPOSITION
Type
Research Papers
Information
Journal of Applied Probability , Volume 28 , Issue 1 , March 1991 , pp. 84 - 95
Copyright
Copyright © Applied Probability Trust 1991 

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Footnotes

This research was supported in part by Grants ECS-8803061 from the National Science Foundation and AFOSR-88–0076 from the Air Force Office of Scientific Research.

References

[1] Heffes, H. (1980) A class of data traffic processes – Covariance function characterization and related queueing results. Bell Syst. Tech. J. 59, 897–929.Google Scholar
[2] Mckenzie, E. (1989) Some ARMA models for dependent sequences of Poisson counts. Adv. Appl. Prob. 20, 822–835.Google Scholar
[3] Neuts, M. F. (1981) Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach. The Johns Hopkins University Press, Baltimore.Google Scholar
[4] Neuts, M. F. (1990) On Viterbi's formula for the mean delay in a queue of data packets. Stochastic Models 6, 87–98.Google Scholar
[5] Neuts, M. F. (1990) On the packet stream generated by a random flow of messages of random durations. Stochastic Models 6, 445–470.Google Scholar
[6] Neuts, M. F. (1989) Structured Stochastic Matrices of M/G/1 Type and their Applications. Marcel Dekker, New York.Google Scholar
[7] Ramaswami, V. (1988) Traffic performance modeling for packet communication, whence where and whither? Keynote Address, 3rd Australian Teletraffic Research Seminar, Melbourne.Google Scholar
[8] Ramaswami, V. and Latouche, G. (1988) Modeling packet arrivals from asynchronous input lines. Proc. 12th Internat. Teletraffic Conference, Torino, Italy. Google Scholar
[9] Sriram, K. and Whitt, W. (1986) Characterizing superposition arrival processes in packet multiplexers for voice and data. IEEE J. Select. Areas Commun. Special Issue on Network Performance Evaluation 4, 833–846.CrossRefGoogle Scholar
[10] Viterbi, A. M. (1986) Approximate analysis of time-synchronous packet networks. IEEE J. on Select. Areas Commun. Special Issue on Network Performance Evaluation 4, 879–890.Google Scholar