Joint stationary distribution in the ${\mathrm{GI}/M/n/\infty}$ queue with general renovation

Multiserver queuing system with a finite number of identical servers and one queue of unlimited capacity is being considered. Customers enter the system one by one in accordance with a recurrent flow. Service times are exponentially distributed with the same parameter. General renovation mechanism is implemented in the system: a customer, whose service has been completed, upon leaving the system removes a random number of other customers from the queue according to a given probability distribution. The method is proposed for finding the joint stationary distribution of the total number of customers in the system and the time elapsed since the last arrival. Expressions (in terms of transforms) for the calculation of the transient joint distribution are presented.