## Journal of Applied Probability

- J. Appl. Probab.
- Volume 41, Number 2 (2004), 557-569.

### Infinite- and finite-buffer Markov fluid queues: a unified analysis

#### Abstract

In this paper, we study Markov fluid queues where the net fluid rate to a single-buffer system varies with respect to the state of an underlying continuous-time Markov chain. We present a novel algorithmic approach to solve numerically for the steady-state solution of such queues. Using this approach, both infinite- and finite-buffer cases are studied. We show that the solution of the infinite-buffer case is reduced to the solution of a generalized spectral divide-and-conquer (SDC) problem applied on a certain matrix pencil. Moreover, this SDC problem does not require the individual computation of any eigenvalues and eigenvectors. Via the solution for the SDC problem, a matrix-exponential representation for the steady-state queue-length distribution is obtained. The finite-buffer case, on the other hand, requires a similar but different decomposition, the so-called additive decomposition (AD). Using the AD, we obtain a modified matrix-exponential representation for the steady-state queue-length distribution. The proposed approach for the finite-buffer case is shown not to have the numerical stability problems reported in the literature.

#### Article information

**Source**

J. Appl. Probab. Volume 41, Number 2 (2004), 557-569.

**Dates**

First available in Project Euclid: 26 April 2004

**Permanent link to this document**

http://projecteuclid.org/euclid.jap/1082999086

**Digital Object Identifier**

doi:10.1239/jap/1082999086

**Mathematical Reviews number (MathSciNet)**

MR2052592

**Zentralblatt MATH identifier**

1046.60078

**Subjects**

Primary: 60K25: Queueing theory [See also 68M20, 90B22]

Secondary: 65C40: Computational Markov chains 65F15: Eigenvalues, eigenvectors 68M20: Performance evaluation; queueing; scheduling [See also 60K25, 90Bxx] 47A15: Invariant subspaces [See also 47A46]

**Keywords**

Stochastic fluid model Markov fluid queue performance analysis computer network spectral divide-and-conquer problem generalized Newton iteration

#### Citation

Akar, Nail; Sohraby, Khosrow. Infinite- and finite-buffer Markov fluid queues: a unified analysis. J. Appl. Probab. 41 (2004), no. 2, 557--569. doi:10.1239/jap/1082999086. http://projecteuclid.org/euclid.jap/1082999086.