Spectral Expansion Solutions for Markov-Modulated Queues

There are many computer, communication and manufacturing systems which give rise to queueing models where the arrival and/or service mechanisms are influenced by some external processes. In such models, a single unbounded queue evolves in an environment which changes state from time to time. The instantaneous arrival and service rates may depend on the state of the environment and also, to a limited extent, on the number of jobs present. The system state at time t is described by a pair of integer random variables, (It, Jt), where It represents the state of the environment and Jt is the number of jobs present. The variable It takes a finite number of values, numbered 0, 1, . . . , N ; these are also called the environmental phases. The possible values of Jt are 0, 1, . . .. Thus, the system is in state (i, j) when the environment is in phase i and there are j jobs waiting and/or being served. The two-dimensional process X = {(It, Jt) ; t ≥ 0} is assumed to have the Markov property, i.e. given the current phase and number of jobs, the future behaviour of X is independent of its past history. Such a model is referred to as a Markov-modulated queue (see, for example, Prabhu and Zhu [21]). The corresponding state space, {0, 1, . . . , N} × {0, 1, . . .} is known as a lattice strip. A fully general Markov-modulated queue, with arbitrary state-dependent transitions, is not tractable. However, one can consider a sub-class of models which are sufficiently general to be useful, and yet can be solved efficiently. We shall introduce the following restrictions: