Calculating the characteristics of a queue in a synchronous random Markov environment
The steady-state distribution of a two-dimensional Markov chain (queue-size and state of the environment) in a single-server queue with general service time distribution and synchronous Markov environment is calculated. Unlike random environment which changes the state independent of the queue, in this model, changes occur (in Markovian fashion) at some completion service times only. At that the number of service times between changes has geometrical distribution. The authors consider both discrete and continuous time models. The embedded (two-dimensional) Markov chain is introduced at the completion service times, and then the generating function and some properties of the so-called quasi-Toeplitz matrix are applied to obtain an explicit solution in discrete time. At that the matrix analog of the Pollaczek-Khinchin formulae is deduced. The calculation of the stationary distribution of the Markov chain (by the solution of a linear functional matrix equation) is given in detail. To consider the continuous time case, the authors introduce a two-dimensional semi-Markov process whose embedded Markov chain connects closely with the one in discrete time. The steady-state distribution of the sojourn time is found when the arrival rate is constant (independent of the state of environment). In previous works, an approximate solution for the slowly varying random environment has been obtain only.
scientific article; zbMATH DE number 1290213