Finite Source Markovian Single Server Feedback Queue with State Dependent Rates

Abstract

The paper's content is a Finite Source single server queue with a finite waiting line. The arrival process and service time distributions are the Poisson process and negative exponential distribution respectively. The arrival and service rates are state-dependent. In addition, after completion of the service, the customer may decide to leave the system or enter the head of the waiting line or feedback into the waiting line for another service. The decision follows a Bernoulli distribution. This model is mathematically defined using an infinitesimal generator matrix and the matrix is solved using the method group generalized inverse. Using the group generalized inverse matrix, the steady-state probabilities are obtained analytically. Some performance measures are derived. Also, some numerical illustrations are provided by a particular healthcare situation. This research presents an analysis of a finite-source Markovian single-server feedback queue with state-dependent rates, a model that captures the dynamics of a queuing system subject to varying service rates based on its current state. The system comprises a limited number of sources generating entities, a single server processing these entities, and a feedback mechanism allowing entities to re-enter the queue after service completion.