Robust control of discrete-time queuing system

We consider the discrete-time queuing system as a control object. Control problem consists in providing a desired system state. The system state is defined as the number of requests in the system. The number of requests in the system is controlled by rejection of arriving requests with some probability. It is assumed that, the probability of arrival and the probability of service completion of request are unknown and may change during the operation of the system. It is also assumed that the system operates under congestion conditions when the probability of arrival greater than the probability service completion of request. The control system should have a type of feedback and should have property of robustness in relation to change of parameters of a control object and the entering flow of requests. The control object as a finite Markov chain is described by the system of Kolmogorov equations p(k + 1) = Ap(k) + (Bp(k))u(k), y(k) = cp(k), where A, B, c are system matrices, y is the average number of requirements in the system, u is control signal. This is a bilinear system with restrictions on the state and control variables. Analysis of this system has shown that it is possible to proceed to a simpler description of the control object in the form of a first-order difference equation with uncertain parameters y(k + 1) = y(k) + au(k) + b. It is proposed to apply the classical proportional-integral feedback to solve the problem of controlling the state of this system. We have investigated the conditions of stability and robustness of the system with control. The results of numerical experiments and simulation confirm the possibility of using the proposed control in queuing systems operating under congestion conditions.

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