M/M/1 Interdependent Queuing Model with Vacation and Controllable Arrival Rates

Abstract

Objectives: Rather than working nonstop in the service area, servers take vacations when they have no clients. To determine the probability and features of the queuing system, this study introduces controllable arrival rates and interdependency in the system's service and arrival processes. It also performs a numerical verification of the results. Methods: A recursive method is employed to solve the steady-state probability equations, yielding explicit iterative formulas under the assumption that a single server provides services to all clients. Here, customer arrivals are controlled as either faster or slower, with Poisson assumed by default. Findings: For this model, steady-state solutions and characteristics are derived and explored, and some numerical analysis is carried out using MATLAB. All the probabilities are expressed in terms of , which indicates the system when empty. The movement of the average number of customers in the system and the expected waiting time, and respectively, of the customers in the system is investigated through a graph. and decrease when dependence service rate, and faster arrival rate increase. Additionally, increases and decreases when the slower arrival rate increases. Novelty: Although there have been studies on vacation in queuing theory, this new approach aims to bridge the gap between vacation and interdependency in the arrival and service process, as well as controllable arrival rates. When vacations with predictable arrival rates are utilised advantageously for the benefit of both the server and the client, waiting times may be minimised and the most practical, economical service can be provided. Keywords: Markovian Queuing System, Vacation, Loss and Delay, Finite Capacity, Interdependent Arrival and Service Rates, Varying Arrival Rates, Bivariate Poisson Process