Prioritized Retrial Queueing Systems with Randomized Push-Out Mechanism

Abstract

The article is focused on a single-channel preemptive queuing system. Two stationary Poisson flows of customers are incoming to the system. The first flow has an absolute priority over the second one: a new high-priority customer from the first flow displaces a low-priority one from the service channel and takes its place. The capacity of the system is limited to k customers. There is a probabilistic push-out mechanism in the system: if a new high-priority customer finds that all the places in the queue are occupied, then it has the right to displace one low-priority customer from the queue with probability a. Both types of customers have the same exponentially distributed service times. Customers who failed to enter the system due to the limited size of the queue, as well as those expelled from the queue or service channel when the push-out mechanism is triggered, are not lost immediately, but they are sent to a special part of the system called the orbit and designed to store repeated customers. In orbit, there are two separate unlimited queues, consisting of low-priority and high-priority repeated customers, respectively. If there are no free places in the system, new customers with a probability q are added to the corresponding orbital queue. The waiting time of repeated customers in orbit is distributed according to an exponential law. The parameter of this law may differ for different types of customers. After waiting in orbit, secondary customers try to re-enter the system. The probabilistic characteristics of the described queuing system are calculated by the method of generating functions, previously proposed by the authors for calculating a similar system without repeated customers. This method allows finding the main probabilistic characteristics of distributions for both types of customers. Particular attention is paid to the study of the dependence of the loss probabilities for both types of customers on the parameters of the system, primarily on the push-out probability a, the capacity of the system k, and the probability of repeated circulation (probability of persistence) q. It is shown that the effect of blocking the system and the effect of the linear law of customers’ losses, previously identified in similar problems without repeated customers, remain valid even in the presence of secondary repeated customers. The theoretical results are proved by numerical calculations. The blocking area for the second type of customers was calculated along with the area of linear loss law for both types of customers. We studied the influence of the probability of repeated circulation q on the shape of these areas and on the dependence of the loss probabilities for both types of customers on the push-out probability a.