The M_t/M/1 queueing system with impatient customers and multiple vacation

This paper presents a queueing system that considers impatient customers and multiple vacation periods. This study uses the M_t/M/1 model and assumes that customers’ impatience is related to the server’s vacation. The system being analyzed exhibits exponential service times and arrival processes that follow a periodic (sinusoidal) Poisson distribution. The system under consideration has 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. The time-dependent probability generating functions and their associated steady-state solutions have been derived explicitly. The pointwise stationary approximation is utilized to calculate long-term average effectiveness sizes performance metrics. 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, we perform a sensitivity analysis using simulations to evaluate the parameters.