The system M2θ/G/1/m with threshold control of the arrival rate and service time

We consider a M2θ/G/1/m queueing system with arrival of customer batches, which uses a threshold control mechanism of the service time and arrival rate. The system receives two independent flows of customers, one of which is blocked in an overload mode (under the condition that the number of customers in the system exceeds a given threshold value h). Full blocking of the input flow is carried out from the moment when the queue length reaches the number m until the beginning of the service of the first customer, for which the number of customers in the system does not exceed h. From the beginning of the service of the first customer during the excess of number of customers in the system of h until the completion of full blocking the time of service of customer is distributed under the law of F(x) (an increased service rate is used). Rest of the time the system applies the normal service rate with the distribution function F(x) of service time. Laplace transforms for the distributions of the number of customers in the system during the busy period and for the distribution function of the busy period are found. The average duration of the busy period is obtained. Formulas for the stationary distribution of the number of customers in the system, for the probability of service and for the stationary characteristics of the system are established. The obtained results are verified with the help of a simulation model constructed with the assistance of GPSS World tools.