The system M^θ /G/1/m with two operation modes and hysteretic control of queue

We consider the $mathrm{M^{ heta}/G/1/m}$ queue with twoservice modes (basic and postthreshold) with the distributionfunctions of service time $F(x)$ and $widetilde{F}(x)$respectively. The postthreshold mode is used in conjunctionwith blocking of input flow if at the beginning of service ofthe next customer the number of customers in the system$xi(t)$ satisfies the condition $xi(t)>h_2.$ Return to thebasic mode and stop blocking carried out at the beginning ofservice of the next customer, if $xi(t)le h_1,$ where $h_1leh_2.$ Laplace transforms for distributions of the number ofcustomers in the system during the busy period and for the busytime distribution function are found. The mean duration of thebusy time is found, and formulas for the stationarydistribution of number of customers in the system andstationary characteristics of queue are obtained. The case of$m=infty$ is considered.