MARKOV INTERACTIVE PROCESSES

Akram A. ELTANNIR

We introduce a new family of Markov processes called “Markov Interactive Processes" (MIP). Such a process consists of several variables or compo­nents.  Jumps occur in one component at a time and the transition rates of each component depend on the current states of the remaining components.  We show that the t-step transition probabilities of a MIP have a product form when the transition rates of the jumps in one component are propor­tional to a function of the states of the remaining components.  This yields a product form stationary distribution as well.  Several standard network models including closed Jackson networks are MIP's with product form distributions.  In addition, we approximate the stationary distributions for three cases of general two-component MIP's that do not have tractable equilibrium distributions.

The first approximation is for an MIP in which one of the components has either very sluggish or very- frequent transitions relatively to the other.  The second approximation is for an MIP in which one component has two states where one of them dominates the other in the sense that this component stays longer in the dominating state.  The third approximation is for a MIP in which one component's transition rates are independent of the state of the other and has many non-dominating states in that their equilibrium probabilities are approximately equal.  We apply our results to develop several new models for queuing systems, which can fit in the framework of MIP's.

 Finally, we study a family of Markov pro­cesses whose state spaces are partitioned into subspaces where one of them is “central" in that transitions from a non-central subspace into another are only possible through one or more central states.  We express the stationary distribution of such a process in terms of the distributions of Markov pro­cesses with the same transition rates of the original process but restricted to the states of a non-central and the central subspaces.