Study on Bernoulli scheduling, customers priority shift in busy state, intolerance of customers in working vacation, complete vacation in a M/M/1/∞ model.

This study examines Bernoulli scheduling approach, customers(clients) priority shift during server’s busy state and explores customers intolerance during server’s working vacation (WV) state with complete vacation in a M/M/1/∞ queueing model. During the busy state, customers might join the queue with a probability d or leave with a probability s=1-d. Some customers might enter the system but do not join the queue during the busy state are assumed to result from changing priorities or other pressing matters, rather than due to long wait times or frustration. The server enters a working vacation in free time with an exponentially distributed rate θ, providing service at a slower pace. During this phase, customers may become impatient if faced with extended waits, abandoning the queue with an exponentially distributed parameter α if their patience expires. Upon WV completion, the server either resumes a busy state with probability p if any customer present, or join complete vacation with probability m and rate β. If new clients arrive during complete vacation, the server returns to busy state at a rate ψ otherwise availing complete vacation. This paper employs the PGF method to derive equilibrium-state probabilities and assess system measures, including average queue lengths during busy state, WV states and complete vacation, mean sojourn time, and abandonment rate. Additionally, the influence of specific parameters on system metrics is illustrated graphically.