Des and Res Processes and Their Explicit Solutions

Abstract

This paper defines and studies the down entrance state (DES) and the restart entrance state (RES) classes of quasi-skip free (QSF) processes specified in terms of the nonzero structure of the elements of their transition rate matrix Q. A QSF process is a Markov chain with states that can be specified by tuples of the form (m, i), where m ∈ Z represents the "current" level of the state and i ∈ Z + the current phase of the state, and its transition probability matrix Q does not permit one-step transitions to states that are two or more levels away from the current state in one direction of the level variable m. A QSF process is a DES process if and only if one step "down" transitions from a level m can only reach a single state in level m − 1, for all m. A QSF process is a RES process if and only if one step "up" transitions from a level m can only reach a single set of states in the highest level M 2 , largest of all m.We derive explicit solutions and simple truncation bounds for the steady-state probabilities of both DES and RES processes, when in addition Q insures ergodicity. DES and RES processes have applications in many areas of applied probability comprising computer science, queueing theory, inventory theory, reliability, and the theory of branching processes. To motivate their applicability we present explicit solutions for the well-known open problem of the M/Er/n queue with batch arrivals, an inventory model, and a reliability model.

Level product form QSF processes and an analysis of queues with Coxian interarrival distribution

“…Without loss of generality, we may then assume that they are non-negative, that is, m ≤ 0. Then, the infinitesimal generator Q or q -matrix of (see [9] and [2]) takes the form:…”

“…In , we have introduced a new procedure to compute the invariant measure for a class of Markov chains. This procedure is called the successive lumping method.…”

“…This procedure is called the successive lumping method. Furthermore, an explicit solution and bounds for the steady state probabilities for the class of ergodic QSF processes that possess the successive lumpability property have been derived in . Ramaswami and Latouche discuss QBD processes with a special structure, namely where the upward or the downward transitions rates form a row times column matrix.…”

A comparative analysis of the successive lumping and the lattice path counting algorithms

“…[17,16] and pricing models. In this paper we are particularly interested in a comparison of the new successive lumping (SL) methodology developed in [19] with the popular lattice path counting [24] in obtaining rate matrices for queueing models, as in [22] and [21].…”

“…In Section 2 we first define the notation for the QBD processes that we will use throughout the paper. In Section 2 we summarize the results of [19] for the DES processes as they apply to quasi birth and death processes with a down entrance state and the resulting quasi birth and death down entrance state algorithm (QDESA). In Section 2 the QDESA procedure is specialized depending on the structure of the transition rate Q, applicable to the models under investigation in this paper.…”