Sojourn times in a Markovian queue with working breakdowns and delayed working vacations

The current study considers a Markovian queue, where the server is subject to breakdowns while providing service to customers. During a breakdown period, the server lowers the service rate, rather than completely halting services provision. When there are no customers in the system, the server leaves for a vacation. Upon return from a vacation, if the server finds no customers in the system, the server leaves for another vacation; however, if the server from a vacation to find at least one customer waiting in the queue, the server will serve customers immediately. During a vacation period, the server serves customers at a different service rate and does not stop working. Service times during normal, breakdown and vacation periods follow exponential distributions. For such a system, to find the closed-form solutions for the steady-state probabilities, we employ the spectral expansion method. Several performance measures of the system are developed. We additionally derive the Laplace-Stieltjes transform of the sojourn time of an arbitrary customer and obtain the expected sojourn time. Furthermore, we provide numerical examples to illustrate the effects of various system parameters on performance measures and the expected sojourn time.