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Modelling the flow of coalescing data streams through a processor

Published online by Cambridge University Press:  14 July 2016

V. Anantharam
V. Anantharam*
Affiliation:
University of California, Berkeley
*
∗ Present address: School of Electrical Engineering, Phillips Hall, Cornell University, Ithaca, NY 14853, USA.
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Abstract

In a data processing network, two data streams A and B arrive at a node independently at the same Poisson rate λ. Service at exponential rate µ can take place iff there is at least one of each of A and B present. The output is the combined processed data AB. We consider models of this situation with finite buffers, with infinite buffers and with finite buffers for the excess of each input type over the other. We apply the filtering theory for point process functionals of a Markov chain to study whether the output flow is Poisson in equilibrium. The motivation is to examine, if the output is to subsequently be processed by a queueing system, whether it can be treated as an independent Poisson input to that system.

A result of independent interest is that a subset of transitions of a countable-state Markov process does not yield a Poisson process when counted, if the rate matrix of counted transitions is nilpotent, and we prove a generalization of Pakes' lemma for countable-state Markov chains.

Keywords

FILTERINGMANUFACTURINGOUTPUT PROCESSESQUASIREVERSIBILITYSTATIONARY DISTRIBUTIONS
Type
Research Papers
Information
Journal of Applied Probability , Volume 25 , Issue 1 , March 1988 , pp. 184 - 193
Copyright
Copyright © Applied Probability Trust 1988 

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References

[1] Brémaud, P. (1981) Point Processes and Queues. Springer-Verlag, Berlin.CrossRefGoogle Scholar
[2] Burke, P. J. (1956) The output of a queueing system. Operat. Res. 4, 699–704.CrossRefGoogle Scholar
[3] Brown, T. C. and Pollett, P. K. (1982) Some distributional approximations in Markovian queueing networks. Adv. Appl. Prob. 14, 654–671.CrossRefGoogle Scholar
[5] Kelly, F. (1975) Networks of queues with customers of different types. J. Appl. Prob. 12, 542–554.CrossRefGoogle Scholar
[6] Melamed, B. (1979) On Poisson transition processes in discrete state Markovian systems with applications to queueing theory. Adv. Appl. Prob. 11, 218–239.CrossRefGoogle Scholar
[7] Walrand, J. (1982) On the equivalence of flows in networks of queues. J. Appl. Prob. 19, 195–203.CrossRefGoogle Scholar
[8] Walrand, J. (1981) Filtering formulas and the M/M/1 queue in a quasireversible network. Stochastics 6, 1–22.CrossRefGoogle Scholar
[9] Walrand, J. (1985) Lectures on Queueing Theory (in preparation).Google Scholar
[10] Walrand, J. and Varaiya, P. (1981) Flows in queueing networks: A martingale approach. Math. Operat. Res. 6, 387–404.CrossRefGoogle Scholar