Behavior Analysis of an M/M/1 vacation Queue in Random Environment

This study examines an M/M/1 vacation queue in a random environment. This system is considered a classical M/M/1 queueing system in which customers are served one by one in the operative phase i; i = 1, 2, ... , n, and the arrival and service rates are lambda(i) and mu(i.), respectively. The system starts the vacation phase when it is empty and there are no customers to which to provide the service. Furthermore, the system moves to some operating phase i with probability q(i,) i = 1, 2, ... , n at the end of the vacation phase. Based on the generating function technique, we derived the transient probabilities and some performance measures. The steady-state analysis is also discussed. Finally, the effectiveness of the proposed system parameters on the behaviour of the system is illustrated with some numerical examples.