Analysis of the M_t/M/1 Queueing System with Impatient Customers and Single Vacation

We consider an M_t/M/1 queueing system with impatient customers and a single vacation, assuming the customers’ impatience is due to the server’s vacation. In the context of non-stationary sinusoidal modeling, this paper introduces systems with exponential service times and periodic (sinusoidal) Poisson arrival processes. We studied a novel analysis of an M_t/M/1 model including simultaneous vacations and impatient customers alongside the relative amplitude changes. In addition, the pointwise stationary approximation has been computed by integrating over time the formula for the stationary performance measure with the arrival rate that applies at each point in time. The time-dependent probability generating functions and the corresponding steady-state results have been obtained explicitly. We focus on five performance measures: the expected number of customers waiting in the queue during vacation, the expected customer waiting time in the queue during vacation, the probability of the server being busy, the probability of the server being on vacation and the probability of customers’ impatience. Finally, to evaluate the performance measure of queue length, we have conducted a sensitivity analysis by running a simulation for a specific set of parameters.