Analysis of a MAP/PH/1 queueing model with flexible group service, Bernoulli vacation and Bernoulli feedback

In practice, queueing models where services are delivered in groups (or blocks or batches) have shown to be quite useful and these queues have received extensive analysis in the literature. One such group service queueing model with Bernoulli vacation and Bernoulli feedback is provided in this paper. The arrival processes follow Markovian distribution. Customers are served in groups ranging in size from 1 to a fixed constant, let’s say N. A batch’s service time is determined by the phase type distribution that corresponds to each group size. The service time for a group is calculated as the highest of all the customers who make up the group. Customer groups who are dissatisfied with the service will likely receive it again with a probability d and with probability c, the satisfied customer group will exit the system. The group’s feedback is defined here as the highest of the feedbacks made by each of the customers who make up the group. The server may choose to serve the next block of customers with probability p or take a vacation with probability q after providing a service. The matrix geometric method was used to obtain the steady state probabilities, and from there, we generated a few performance measures. We have studied the busy time and we have calculated the distribution of waiting times. Results are illustrated with some graphical representations.