Sojourn times in different modes in the queueing system with batch Poisson arrivals, general service times, and hysteretic policy

Consideration is given to the model of the SIP server modelled as a single-server queueing system with finite capacity R and two batch Poisson flows (priority and non-priority), general service times, and bi-level hysteretic control of arrival rates, which is defined by two numbers: L and H (0 < L < H < R). The hysteretic policy under consideration implies that, at each instant, the system can be in one of the three operation modes: normal, overload, and blocking. The system accepts customers from both flows when it is in normal mode, accepts only priority flow when is it overloaded, and rejects all new customers in the discard modes. It is assumed that the switching between operation modes can occur only at the instant of service completion. The analytical method for computing the distributions (in terms of Laplace-Stieltjes transforms) of system's sojourn times in each operation mode and return times to the normal mode is proposed. Illustative examples are presented.