Stationary Characteristics of the GI/M/1 Queue with General Renovation and Feedback

Consideration is given to the queuing system, consisting of one server and a queue of unlimited capacity, with the implemented mechanism of general renovation and feedback. This mechanism, which may be considered as a variant of an active queue management scheme, works as follows. In this short note we show how the main ingredients needed to compute some of the main stationary performance characteristics of the system can be found. The basic methods are transform techniques and methods for the solutions of Volterra integral equations of the second kind with the kernels of convolution type. We concentrate on the stationary distribution of the embedded Markov chain, show how it is related with the stationary distribution of the process, describing the evolution of the total number of customers in the system. Under the two assumptions about the service discipline and the order in which the customers are removed from the queue whenever renovation occurs, we derive expressions for stationary loss probability and the sojourn time distribution in terms of Laplace-Stielties transform.