Exact and Approximate Solutions for a Class of Infinite Markovian Models
This paper presents an efficient equilibrium solution algorithm for infinite multi-dimensional Markov chains of the quasi-birth-and-death (QBD) type. The algorithm is not based on an iterative approach, so that the exact solution can be computed in a known, finite number of steps. The key step on which the algorithm is based, is the identification of a linear dependence among variables. This dependence is expressed in terms of a matrix whose size is finite. The equilibrium solution of the Markov chain is obtained operating on this finite matrix. An extremely attractive feature of the newly proposed algorithm is that it allows the computation of approximate solutions with any desired degree of accuracy. The solution algorithm, in fact, computes a succession of approximate solutions with growing accuracy, until the exact solution is achieved in a finite number of steps.
Results for a case study show that the proposed algorithm is very efficient and quite accurate, even when providing approximate solutions.
KeywordsMarkov Chain Approximate Solution Arrival Rate Equilibrium Solution Krylov Subspace
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