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Calculating exit times for series Jackson networks

Published online by Cambridge University Press:  14 July 2016

William A. Massey
William A. Massey*
Affiliation:
AT&T Bell Laboratories
*
∗Postal address: AT&T Bell Laboratories, 600 Mountain Avenue, Murray Hill, NJ 07974, USA.
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Abstract

We define a new family of special functions that we call lattice Bessel functions. They are indexed by the N-dimensional integer lattice such that they reduce to modified Bessel functions when N = 1, and the exponential function when N = 0. The transition probabilities for an M/M/1 queue going from one state to another before becoming idle (exiting at 0) can be solved in terms of modified Bessel functions. In this paper, we use lattice Bessel functions to solve the analogous problem involving the exit time from the interior of the positive orthant of the N-dimensional lattice for a series Jackson network with N nodes. These special functions allow us to derive asymptotic expansions for the taboo transition probabilities, as well as for the tail of the exit-time distribution.

Keywords

ASYMPTOTIC EXPANSIONSGROUP SYMMETRYBESSEL FUNCTIONSMULTIDIMENSIONAL RANDOM WALKSTRANSIENT BEHAVIOUR
Type
Research Papers
Information
Journal of Applied Probability , Volume 24 , Issue 1 , March 1987 , pp. 226 - 234
Copyright
Copyright © Applied Probability Trust 1987 

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References

[1]Feller, W. (1971) An Introduction to Probability Theory and its Applications, Vol. II. Wiley, New York.Google Scholar
[2]Ledermann, W. and Reuter, G. E. H. (1954) Spectral theory for the differential equations of simple birth and death processes. Phil. Trans. R. Soc. London A 246, 321–369.Google Scholar
[3]Massey, W. A. (1984) An operator analytic approach to the Jackson network. J. Appl. Prob. 21, 379–393.Google Scholar
[4]Massey, W. A. and Wright, P. E. (1986) Asymptotic expansions for lattice Bessel functions. To appear.Google Scholar