Analysis of discrete-time queueing systems with negative customers and disasters

Abstract

In this dissertation, discrete-time queues with negative customers or disasters are addressed. The arrival of a negative customer to a queueing system causes one positive customer to be removed if any customer is present. On the other hand, the arrival of a disaster removes all customers in the system. The discrete-time Geo/Geo/1 queue, which has either negative customers or disasters, has recently appeared in the literature. This Geo/Geo/1 queue with this notion is extended into the Geo/G/1 queue and the GI/Geo/1 queue, respectively, in this research. Using the supplementary variable technique, the stationary queue length distribution of the Geo/G/1 queue with negative customers or disasters is obtained. In addition, for the GI/Geo/1 case, the queue length distribution at an arrival epoch is derived via the embedded Markov chain and the time-average queue length distribution is obtained by level-crossing technique and Semi-Markov process. The stationary sojourn time distributions of both the Geo/G/1 queue and the GI/Geo/1 queue are also presented. Furthermore, it is illustrated how the results of the discrete-time queue with negative arrivals are converted into those of the continuous-time queue. Numerical examples also show the effect of the parameter, which is the arrival rate of either negative customers or disasters, on these queueing models.