Performance characteristics of $\mathrm{Geo/Geo/1/}R$ queue with hysteretic load control

Threshold load control is one of the key techniques to prevent overloads in telecommunication networks. Its variants are used for overload detection in signalling system № 7 as well as in next generation networks where session initiation protocol (SIP) is the main signalling protocol. Consideration is given to discrete-time $\mathrm{Geo/Geo/1/}R$ queueing system which is one of the possible mathematical models of the SIP proxy-server. Bi-level hysteretic load control is implemented in the system. The methods that allow one to obtain a stationary joint probability distribution of the number of customers in a system and system's state, stationary waiting and sojourn time distributions, and distribution of first passage times from different system's states are presented. A numerical example based on obtained analytical expressions is given.