Martingale relations for the M/GI/1 queue with Markov modulated Poisson input
A single server queueing system with renewal service process and Poisson arrivals modulated by a finite-state Markov process \((y(t),\quad t\geq 0)\) with rate \(\lambda\) (i) when the environment is in state i is considered. The arrival process \((A(t),\quad t\geq 0)\) is a counting process with \(A(0)=0\) such that the process \((\alpha (t),\quad t\geq 0)\) defined by \[ \alpha (t)=A(t)-\int^{t}_{0}\lambda (Y(s^-))ds,\quad t\geq 0, \] is an \({\mathcal F}_ t\)-martingale, where \({\mathcal F}_ t=\sigma \{Y(s),A(s)\), \(0\leq s\leq t\}\). It is shown that martingales are a useful tool for analyzing the dynamic, transient and stationary behaviors of the system. Several exponential martingales which are associated with a chain embedded at service completion epochs in the process describing the joint evolution of the queue-size and the state of the environment are investigated. New conservation laws in the form of a system of linear relations satisfied by the joint distributions and the absolute continuity properties of certain queues are obtained. Some computational problems are discussed.
scientific article; zbMATH DE number 4215115
scientific article