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Busy periods in fluid queues with multiple emptying input states

Published online by Cambridge University Press:  14 July 2016

A. J. Field  and
P. G. Harrison
A. J. Field*
Affiliation:
Imperial College London
P. G. Harrison*
Affiliation:
Imperial College London
*
∗Postal address: Department of Computing, Imperial College London, South Kensington Campus, London SW7 2AZ, UK.
∗Postal address: Department of Computing, Imperial College London, South Kensington Campus, London SW7 2AZ, UK.
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Abstract

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A semi-numerical method is derived to compute the Laplace transform of the equilibrium busy period probability density function in a fluid queue with constant output rate when the buffer is nonempty. The input process is controlled by a continuous-time semi-Markov chain (CTSMC) with n states such that in each state the input rate is constant. The holding time in states with net positive output rate - so-called emptying states - is assumed to be an exponentially distributed random variable, whereas in states with net positive input rate - so-called filling states - it may have an arbitrary probability distribution. The result is demonstrated by applying it to various systems, including fluid queues with two on-off input sources. The latter exercise in part shows consistency with prior results but also solves the problem in the case where there are two emptying states. Numerical results are presented for selected examples which expose discontinuities in the busy period distribution when the number of emptying states changes, e.g. as a result of increasing the fluid arrival rate in one or more states of the controlling CTSMC.

Keywords

Fluid queuebusy periodCTSMCdiscontinuity

MSC classification

Secondary: 68U99: None of the above, but in this section 90C99: None of the above, but in this section
Type
Research Article
Information
Journal of Applied Probability , Volume 47 , Issue 2 , June 2010 , pp. 474 - 497
Copyright
Copyright © Applied Probability Trust 2010 

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