**Authors:** Mario Lefebvre

**Keywords:** State space, balance equations, queue discipline,
priority, cluster-type multiprocessor computing systems.

### Abstract

The classic M/M/m/n queueing model is modified by allow-
ing a given task to require up to m servers to be performed.
Moreover, the maximum time that a task can wait in the queue
before being executed is a random variable having an exponen-
tial distribution. Both FIFO (First In, First Out) and priority
disciplines are considered. The case when m = 2 is treated: the
state space needed to fully describe the system is given, its size is
calculated and the balance equations are presented when n = 1,2
and 3. The queueing process can be used to model cluster-type
multiprocessor computing systems.

Polytechnique Montréal

Department of Mathematics and Industrial Engineering

2500, chemin de Polytechnique

Montréal (Québec), Canada H3T 1J4

E-mail:

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