Markov additive process

Abstract: This paper applies the earlier work of Barlow, Rogers and Williams on the Wiener-Hopf factorization of finite Markov chains to a number of questions in the theory of fluid models of queues. Specifically, the invariant distribution for an infinite-buffer model and for a finite-buffer model are derived. The laws of other functionals of the fluid models can be easily derived and compactly expressed in terms of the fundamental Wiener-Hopf factorization.

Abstract: We establish new multidimensional martingales for Markov additive processes and certain modifications of such processes (e.g., such processes with reflecting barriers). These results generalize corresponding one-dimensional martingale results for Levy processes. This martingale is then applied to various storage processes, queues and Brownian motion models.

Abstract: Markov-modulated queueing systems are those in which the primary arrival and service mechanisms are influenced by changes of phase in a secondary Markov process. This influence may be external or internal, and may represent factors such as changes in environment or service interruptions. An important example of such a model arises in packet switching, where the calls generating packets are identified as customers being served at an infinite server system. In this paper we first survey a number of different models for Markov-modulated queueing systems. We then analyze a model in which the workload process and the secondary process together constitute a Markov compound Poisson process. We derive the properties of the waiting time, idle time and busy period, using techniques based on infinitesimal generators. This model was first investigated by G.J.K. Regterschot and J.H.A. de Smit using Wiener-Hopf techniques, their primary interest being the queue-length and waiting time.