Matrix Geometric Solutions

Abstract

In Example 8.12 we analyzed a scalar state process that was a modification of the M/M/1 queue. In the example, we classified two sets of states: boundary states and repeating states. Transitions between the repeating states had the property that rates from states 2, 3,..., were congruent to rates between states j, j + 1,... for all j ≥ 2. We noted in the example that this implied that the stationary distribution for the repeating portion of the process satisfied a geometric form. In this chapter we generalize this result to vector state processes that also have a repetitive structure. The technique we develop in this chapter to solve for the stationary state probabilities for such vector state Markov processes is called the matrix geometric method. (The theory of matrix geometric solutions was pioneered by Marcel Neuts; see [86] for a full development of the theory.) In much the same way that the repetition of the state transitions for this variation of the M/M/1 queue considered in Example 8.12 implied a geometric solution (with modifications made to account for boundary states), the repetition of the state transitions for vector processes implies a geometric form where scalars are replaced by matrices. We term such Markov processes matrix geometric processes.