Analysis of a single server queue in a multi-phase random environment with working vacations and customers' impatience

In this paper, we analyze an M/M/1 queueing system under both single and multiple working vacation policies, multi-phase random environment, waiting server, balking and reneging. When the system is in operative phase j = 1, 2, ... , K, customers are served one by one. Whenever the system becomes empty, the server waits a random amount of time before taking a vacation, causing the system to move to working vacation phase 0 at which new arrivals are served at a lower rate. Using the probability generating function method, we obtain the distribution for the steady-state probabilities of the system. Then, we derive important performance measures of the queueing system. Finally, some numerical examples are illustrated to show the impact of system parameters on performance measures of the queueing system.