Study of Retrial Queueing System with Differentiated Vacation, Failure and Repair

This paper presents study of a M/M/1 retrial system incorporating differentiated vacation, failure and repair. Customer arrives according to Poisson process with rate λ. In regular state, if server is busy in serving customers, incoming customers who cannot be served immediately enter an orbit (Pool or virtual queue) of infinite capacity. When server becomes free during regular state, customers waiting in orbit reattempt for service according to classical retrial policy with rate nχ, where n shows orbit or pool size otherwise customer will have to wait for the server to be free. Customers get served with rate μ during regular busy state. Server join complete vacation if server is idle in free regular state, where no service will be provided to customer. If customer arrives during this state, then server transition to the working vacation (WV) state where despite not serving customers completely, now get served with some slow rate ω, (ω˂μ). If no customer remains during WV, server may return to complete vacation. During WV completion instant, if customers are still present in the system, then server resumes regular busy state for serving customers otherwise continue working vacation. Additionally, the server is subject to random breakdowns during its regular busy state. In such cases, it is sent for immediate repair and, upon completion, resumes service in the regular state. By using Probability generating function (PGF) approach, steady state analysis of model, analytical expression of distinct metrics of the system have been derived. The model’s analytical results were further supported by numerical simulations and visualizations implemented using Python, an open-source scientific computing language. The analysis provides insights into how system parameters affect the operational efficiency and quality of service.