Probabilistic characteristics of an M 2 θ /G/1/m queue with two-loop hysteretic control of the service time and arrival rate

Abstract

An M2θ/G/1/m queueing system with the batch arrival of customers is considered. In the system, a threshold mechanism with two hysteretic loops is applied to control the service time and input flow intensity. The system receives two independent flows of customers. One of the flows is blocked in the overload mode. Operation modes are switched at the instants when the service of customers is completed. Complete blocking of the input flow starts when the queue length reaches number m. An approach based on Korolyuk’s potential method is proposed for determining the probabilistic characteristics of the system. Laplace transforms of the distributions of the number of customers in the system during the busy period and of the distribution function of the busy period are found. The mean duration of the busy period is determined. Formulas are derived for the stationary distribution of the number of customers in the system, for the probability of service, and for the stationary characteristics of a queue. The results obtained are verified with the use of a simulation model developed with the help of the GPSS World tools.