## Journal of Applied Probability

- J. Appl. Probab.
- Volume 47, Number 2 (2010), 474-497.

### Busy periods in fluid queues with multiple emptying input states

A. J. Field and P. G. Harrison

#### Abstract

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.

#### Article information

**Source**

J. Appl. Probab. Volume 47, Number 2 (2010), 474-497.

**Dates**

First available in Project Euclid: 17 June 2010

**Permanent link to this document**

https://projecteuclid.org/euclid.jap/1276784904

**Digital Object Identifier**

doi:10.1239/jap/1276784904

**Mathematical Reviews number (MathSciNet)**

MR2668501

**Zentralblatt MATH identifier**

1192.68808

**Subjects**

Primary: 68U99: None of the above, but in this section

Secondary: 90C99: None of the above, but in this section

**Keywords**

Fluid queue busy period CTSMC discontinuity

#### Citation

Field, A. J.; Harrison, P. G. Busy periods in fluid queues with multiple emptying input states. J. Appl. Probab. 47 (2010), no. 2, 474--497. doi:10.1239/jap/1276784904. https://projecteuclid.org/euclid.jap/1276784904