Optimal NT policies for M/G/1 system with a startup and unreliable server

This paper studies the control policies of an M/G/1 queueing system with a startup and unreliable server, in which the length of the vacation period is controlled either by the number of arrivals during the idle period, or by a timer. After all the customers are served in the queue exhaustively, the server immediately takes a vacation and operates two different policies: (i) the server reactivates as soon as the number of arrivals in the queue reaches to a predetermined threshold N or the waiting time of the leading customer reaches T units; and (ii) the server reactivates as soon as the number of arrivals in the queue reaches to a predetermined threshold N or T time units have elapsed since the end of the completion period. If the timer expires or the number of arrivals exceeds the threshold N, then the server reactivates and requires a startup time before providing the service until the system is empty. Furthermore, it is assumed that the server breaks down according to a Poisson process and his repair time has a general distribution. We analyze the system characteristics for each scheme. The total expected cost function per unit time is developed to determine the optimal thresholds of N and T at a minimum cost.     

Note on the article: Maximum entropy analysis of the M/M/1 queueing system with multiple vacations and server breakdowns

Wang et al. [Wang, K. H., Chan, M. C., & Ke, J. C. (2007). Maximum entropy analysis of the M^[x]/M/1 queueing system with multiple vacations and server breakdowns. Computers & Industrial Engineering, 52, 192–202] elaborate on an interesting approach to estimate the equilibrium distribution for the number of customers in the M^[x]/M/1 queueing model with multiple vacations and server breakdowns. Their approach consists of maximizing an entropy function subject to constraints, where the constraints are formed by some known exact results. By a comparison between the exact expression for the expected delay time and an approximate expected delay time based on the maximum entropy estimate, they argue that their maximum entropy estimate is sufficiently accurate for practical purposes. In this note, we show that their maximum entropy estimate is easily rejected by simulation. We propose a minor modification of their maximum entropy method that significantly improves the quality of the estimate.