Stationary sojourn times in $\mathrm{MAP}/\mathrm{PH}/1/r$ queue with bi-level hysteretic control of arrivals

This paper reports some new results concerning the analysis of the time-related stationary characteristics of a finite-capacity queueing system operating in a random environment with the bi-level hysteretic control of arrivals. The topic of the paper is motivated by the overload problem in networks of SIP (session initiation protocol) servers and the viewpoint that multilevel hysteretic control of arrivals in SIP servers can be used to mitigate signalling network congestion. The considered mathematical model of SIP server is the single server queueing system with Markovian arrival processes (MAP), PH (phase-type) service, and bi-level hysteretic control policy. According to this policy, a system may be in one of the three operation modes: normal, overload, or blocking. The switching between modes occurs at instants whenever the total number of customers in the system changes. The analytical method for the computation of the stationary sojourn times in different operation modes (in terms of Laplace–Stieltjes transforms (LST)), which utilizes the knowledge about the presence of hysteretic loops, is given. It is also applicable in the case when, in addition to the sojourn times, one needs to account for the number of lost customers.