Generating function approach for discrete queueing analysis with decomposable arrival and service Markov chains

This paper uses generating function approach with spectral decomposition to analyze discrete queues with arrival and service processes characterized by Markov chain (MC). Both generating function and distribution function of the queue are constructed from vanishing and non-vanishing roots. The vanishing roots are used to obtain linear solutions for the boundary probabilities; each non-vanishing root constructs a geometric term in the queue distribution function. The queue asymptotic behavior is expressed in a simple geometric form, which is determined by the minimum non-vanishing root. A key condition for the success of this approach is that all the eigenvalues of both arrival and service MC generating function matrices are distinct and given in explicit analytic form. In order to express eigenvalues in explicit analytic form, both arrival and service MCs must be a special class of MCs which are decomposable into a set of independent MC elements, and each element has no more than four states. Finding roots then becomes no longer difficult in large systems, since the evaluation of each individual root is well decomposed in a simple convergent form. One can use simple Kronecker product properties to obtain queueing solutions. This paper presents steady state queueing solutions for both arrival and service MCs decomposed in units of heterogeneous two-state MCs.