Classification of Markov Processes of Matrix M/G/l type with a Tree Structure and its Applications to the MMAP[K]/G[K]/1 Queues

Abstract

This paper studies the classification problem of discrete time and continuous time Markov processes of matrix M/G/l type with a tree structure. It is shown that the Perron-Frobenius eigenvalue of a nonnegative matrix provides information for a complete classification of the Markov process of interest. A computational method is developed to find whether a Markov process of matrix M/G/l type with a tree structure is positive recurrent, null recurrent, or transient. The method is then used to study the impact of the last-come-first-served general preemptive resume (LCFS-GPR) service discipline on the stability of a MAP/PHA queue. Two sufficient conditions are identified for the positive recurrence and transience of the Markov processes of interest, respectively. As an example, the results are used to show that the discrete time (and continuous time) MMAP[K]/G[K]/1queue with a work conserving service discipline is stable if and only if its traffic intensity is less than one, unstable if its traffic intensity is larger than one