The M/G/1Queueing System With Multiple Server Vacations And Replaceable Repair Facility

This paper considers the M/G/1queueing system with multiple server vacations and replaceable repair facility, in which the "repair facility subject to breakdowns" is firstly introduced in our present system. Assume that (Ⅰ) the customer arrival process follows negative exponential distribution, the customer’s service time follows arbitrarily distribution;(Ⅱ) the lifetime of the service station obeys the negative exponential distribution, and the repair time of the failed service station is arbitrarily distributed;(Ⅲ) the multiple vacation policy is adopted by the server;(Ⅳ) the lifetime of the repair facility also obeys negative exponential distribution, and the replacement time of the repair facility follows general distribution, the following performance measures are discussed.Firstly, some queueing characteristics of the system which included,1) the transient and stationary queue-length distributions at arbitrary time epoch;2) the average queue length, the customer’s waiting time and sojourn time in steady state.Secondly, the reliability indices of the service station which included,1) the busy probability of the service station at time t;2) the transient and steady-state availability of the service station;3) the mean number of failures of the service station during the time interval (0,t].Thirdly, some reliability indices of the repair facility which included,1) the transient and stationary unavailability of the repair facility;2) the expected number of replacement during the time interval (0,t].