Recent results in performance modelling of finite-source retrial queues with collisions and their applications

The goal of the plenary talk is to give a survey about the latest results about this topic. Different solution methods (tool supported, simulation and asymptotic) are treated depending on the distribution of the arrival, service, retrial, operation, restoration and impatience times. Systems with an unreliable server are also investigated with and without impatience of the customers in the orbit. Systems with two-way communications are analysed as well. Special attention is paid on our latest result, namely the analysis of the steady-state distribution of the waiting time in a finite source M/G/1 retrial queuing system where collisions may happen and the server is unreliable. The failure rates depend on whether the server is busy or idle. An asymptotic method is used when the number sources N tends to infinity, the arrival intensity from the sources, intensity of repeated calls tend to zero while service intensity, breakdown intensity, recovery intensity are fixed. It is proved that the steady-state probability distribution of the number of transitions/retrials of a customer into the orbit is geometric, and the waiting time of a customer is generalized exponentially distributed. The average total service time of a customer is also determined. Prelimit distributions obtained by means of stochastic simulation are compared to the asymptotic ones. Several examples are treated and figures show the accuracy and the area of applicability of the proposed asymptotic method.