Strategy And Optimization Analysis Of Repairable Queueing Model With Setup Times

The phenomenon of queueing is ubiquitous and exists in all aspects of life,such as banking services,call centers,communication networks,etc.Originally derived from the study of actual phenomena,queueing theory is a widely applied discipline.In the historical development of queueing theory,the study of retrial,vacation,and repairable queues has continued to deepen and occupy a pivotal place.Therefore,the focus of this paper will be on retrial,vacation,and repairable queues.The optimal joining strategy of customers and the system optimization problem will be considered in two types of repairable queues with different vacation policies.We first consider the equilibrium strategy and social optimization problem for an M/M/1queueing system with negative customers and setup time.When a negative customer arrives,he causes the server to malfunction and forces the receiving customer to leave the system.When the service of the last customer in the system is completed,the server is closed immediately.When a new customer arrives,the server undergoes a random setup time to serve the customer.Based on the linear reward-cost structure,we obtain the equilibrium joining probabilities of customers in both the almost unobservable and fully unobservable cases.The overall optimal joining probabilities are obtained using a genetic algorithm.Numerical examples are used to demonstrate the sensitivity of the optimal joining probability with respect to parameters in two cases,and to compare the optimal social welfare of two cases.In subsequent work,a bi-objective optimization problem is considered to minimize both the cost and the customer sojourn time,and applied to computer systems.The Pareto optimal solution set is found by the second generation non-dominated sorting genetic algorithm(NSGA-Ⅱ).A regression model is established to test the relationship between minimum cost and sojourn time obtained from the Pareto optimal solution set.Secondly,we consider the M/M/1 repairable retrial queueing system with reserved time and setup time under the -policy,and analyze the customer equilibrium strategy and social optimization problem under the almost unobservable and fully unobservable cases.In particular,specific expressions for the equilibrium strategies are derived for the almost unobservable queue.In the treatment of social optimization problem,the genetic algorithm is used to find the optimal solution.Numerical examples are used to demonstrate the sensitivity of the optimal arrival rate with respect to parameters in two cases,and to compare the optimal social welfare of two cases.Finally,we consider the bi-objective optimization problem for call centers.The NSGA-Ⅱ algorithm is used to find the Pareto optimal solution set that minimizes both the cost and sojourn time,and a regression model is established to discuss their relationship.