Multi-server Queueing System $$MAP/M/N^{(r)}/\infty $$ Operating in Random Environment

Abstract

A multi-server queueing system with an infinite buffer operating in the Markovian random environment is analyzed. Under the fixed state of the random environment, the arrival flow is described by the Markovian arrival process. Customers in the buffer may be impatient and leave the system. The number of available servers, the arrival process, the rate of customers’ service, and the impatience intensity depend on the state of the random environment. Behavior of the system is described by the multi-dimensional asymptotically quasi-Toeplitz Markov chain. The ergodicity condition is derived. Algorithm for computation of the stationary distribution is provided. The main performance measures of the system are calculated.