The distribution of the return time from the set of overload states to the set of normal load states in a system $M|M|1|\langle L,H \rangle |\langle H,R \rangle$ with hysteretic load control

An analytical method for studying the parameters of the hysteretic control, which is implemented as one of the effective solutions to the overload problem in the network of SIP-servers, is suggested. As a mathematical model, the queuing system $M|M|1|\langle L,H \rangle |\langle H,R \rangle$ with two loops hysteretic control was developed, where $H$ is the overload onset threshold, $L$ is the overload abatement threshold, and $R$ is the discard threshold. Two methods of calculating the Laplace–Stieltjes transform of the distribution function of the return time from the set of overload system states to the set of normal load system states were obtained. The first method consists in solving a system of equations with return times for each state of the set of overload system states as unknowns, the second deals with the recurrence for the Laplace–Stieltjes transform of the distribution function of the return time for each state of the set of overload system states as rational fractional expressions. Both methods allow the effective calculations with standard software tools, as shown in the numerical example.