Asymptotic analysis retrial queueing system with exclusion customers and phase follow-up

In this paper, we consider a retrial queueing system, which has a two-phase serviced and displacement of customers. Frequently such tasks arise in control systems. The input of the system is a simplest request flow with intensity λ. If the received customer finds the server free, it takes it for service in the first phase. The service time in the first phase has an exponential distribution with a parameter μ₁. After successful completion of service in the first phase, the request goes to the second phase of service. Service time in the second phase has an exponential distribution with a parameter μ₂. If the server is busy, then the incoming request pushes the previous request out of service and takes its place, while the pushed out request goes to the orbit. The orbit is conventionally divided into two zones. If the customer was serviced in the first phase, it goes to the first zone of the orbit, but if it was serviced in the second phase, then it comes in the second phase of the orbit. In the orbit zones, the requests are subject to a random delay. The duration of such delays has an exponential distribution with a parameter σ. After a random delay the customer re-accesses the server from an orbit to the re-attempt service. From the first zone of the orbit, the customer is appealed for service to the first phase, from the second zone, one is appealed to the second phase, thus wise, the phase maintenance of customers whose service has been interrupted occurs. The throughput of RQ-system with the displacement of customers and phase follow-up. By throughput we mean the maximum average number of requests that the system can serve per unit of time. In this RQ-system it has the form S = μ₁μ₂ /(μ₁ + μ₂) > 0 , so the stationary mode in such a system exists for any λ < S. Further, the observation of the system is carried out using the method of asymptotic analysis. It was shown that the limit characteristic function of the number of customers in orbit has the form of a characteristic function of the Gaussian distribution. Numerical experiments were carried out. The asymptotic probability distribution of the number of customers in orbit was obtained for the selected parameter values.

Keywords

Authors

References