Stationary characteristics of the Mθ/G/1/m system with the threshold functioning strategy

Abstract

The Mθ/G/1 queueing system with the group arrival of customers, switchings of service regimes, and threshold blocking of the flow of customers is considered. The input flow is blocked if, at the instant of the beginning of the service of a successive customer, the number of customers in the system exceeds given threshold level h. If, at instant t of the beginning of the service of the customer, the number of customers in the system satisfies the condition h i < ɛ(t) ≤ h i + 1 \(h_i < \varepsilon (t) \leqslant h_{i + 1} (i = \overline {1,r} )\), then, the service time of this customer is associated with distribution function F i (t). When 1 ≤ ɛ(t) ≤ h = h 1, the service time of the customer is distributed according to law F(t) (the basic service regime). For the case of a single switching (r = 1), the mean duration of the busy period intervals with the absence and presence of the input flow blocking, the probability of customer service, and the stationary characteristics of the queue are determined. The character of the dependences of the mean busy period duration and the probability of service on parameters m and h is investigated. For the case m = ∞, some problems of optimal synthesis of systems with given characteristics and the problem of minimization of the service cost are solved.