An unreliable Geo/G/1 queue with startup and closedown times under randomized finite vacations

A Bernoulli-schedule randomized vacation policy Geo/G/1 queueing system of an unreliable server with startup time and closedown time is studied. The vacation time, repair time, startup time and closedown time are generally distributed. After one basic vacation the server may take a finite vacation policy with Bernoulli successful probability p up to J − 1 vacations. Based on the generating function and supplementary variable technique, the analytical expressions for the steady-state distributions of system size at various states and server state are obtained. Also by the concept of generalized service time, the steady-state length distributions of various state periods are derived. Furthermore, the customer's waiting time in the system is also obtained.