Sojourn time in a single server queue with threshold service rate control

We study the sojourn time in a queueing system with a single exponential server, serving a Poisson stream of customers in order of arrival. Service is provided at low or high rate, which can be adapted at exponential inspection times. When the number of customers in the system is above a given threshold, the service rate is upgraded to the high rate, and otherwise, it is downgraded to the low rate. The state dependent changes in the service rate make the analysis of the sojourn time a challenging problem, since the sojourn time now also depends on future arrivals. We determine the Laplace transform of the stationary sojourn time and describe a procedure to compute all moments as well. First we analyze the special case of continuous inspection, where the service rate immediately changes once the threshold is crossed. Then we extend the analysis to random inspection times. This extension requires the development of a new methodological tool, that is "matrix generating functions". The power of this tool is that it can also be used to analyze generalizations to phase-type services and inspection times.Comment: 16 pages, 13 figure

Performance evaluation of a discrete-time Geo[X]/G/1 retrial queue with general retrial times

AbstractWe consider a discrete-time Geo[X]/G/1 retrial queue with general retrial times. The system state distribution as well as the orbit size and the system size distributions are obtained in terms of their generating functions. These generating functions yield exact expressions for different performance measures. The present model is proved to have a stochastic decomposition law. Hence, a measure of the proximity between the distributions of the system size in the present model and the corresponding one without retrials is derived. A set of numerical results is presented with a focus on the effect of batch arrivals and general retrial times on the system performance. It appears that it is the mean batch size (and not the batch size distribution) that has the main effect on the system performance. Moreover, increasing the mean batch size is shown to have a noticeable effect on the size of the stability region. Finally, geometric retrial times are shown to have an overall better performance compared with two other distributions