AN M/M/1 QUEUE SUBJECT TO DIFFERENTIATED VACATION WITH PARTIAL INTERRUPTION AND CUSTOMER IMPATIENCE

ABSTRACT In this work, we consider an M/M/1 queueing model subject to differentiatedvacations. When the server is idle, he takes a break for a specific duration(termed Type I vacation). Upon completing the vacation, he takes another short break if the server stillfinds an empty system (termed as Type IIvacation). Both types of vacation follow an exponential distribution. As TypeI vacation persists for an extended period, it is reasonable to assume that theservice continues during this vacation by an alternate server at a slower rate,unlike the Type II vacation. Also, Type II being a shorter duration vacation,we assume that the Type II vacation is interrupted when the queue size reaches a predefined threshold value. Further, the arriving customers willwait only for a fixed duration of time, and if the service is incomplete by then,they leave the system permanently. We obtain an exact analytical expressionfor the transient state probabilities using Laplace transform, generatingfunctions, and continued fraction methodology. We also derive some useful measuresof effectiveness like the time-dependent mean and variance andillustrate their variations graphically.