The system $mathrm{M^{theta}/G/1/m}$ with two operation modes and hysteretic control of queue

We consider the $mathrm{M^{theta}/G/1/m}$ queue with two service modes (basic and postthreshold) with the distribution functions of service time $F(x)$ and $widetilde{F}(x)$ respectively. The postthreshold mode is used in conjunction with blocking of input flow if at the beginning of service of the next customer the number of customers in the system $xi(t)$ satisfies the condition $xi(t)>h_2.$ Return to the basic mode and stop blocking carried out at the beginning of service of the next customer, if $xi(t)le h_1,$ where $h_1le h_2.$ Laplace transforms for distributions of the number of customers in the system on the busy period and for the busy time distribution function are found. The mean duration of the busy time is found, and formulas for the stationary distribution of number of customers in the system and stationary characteristics of queue are obtained. The case of $m=infty$ is considered.