Strategic Analysis In M/M/1 Queueing Systems With Working Vacation Or Working Breakdown

Vacation queueing systems are used in many fields,such as computer systems,communication networks,and manufacturing systems.In view of more practical applications,the concepts of working vacation and working breakdown were proposed.The working vacation policy are important for saving costs and increasing returns.In recent years,the strategic analysis of queueing systems has attracted the attention of scholars.This paper considers the strategic analysis for M/M/1 queueing systems with a single working vacation and multiple vacations,and strategic analysis for M/M/1 queueing systems with negative customers and working breakdowns.Arriving customers decide whether to enter the system based on the state of the system and the linear reward-cost structure.First,the M/M/1 queues with a single working vacation and multiple vacations is considered.After the working vacation,the system enters the regular busy period if there are still customers in the system.If there are no customers in the system then it enters vacation period.When the vacation ends,if there is a customer in the system,it enters the regular busy period,otherwise it enters the next vacation.Using Markovian process the steady-state probability in observable case is solved,the equilibrium threshold strategy and social benefits under equilibrium strategy are obtained.The steady-state probability in unobservable case is solved by using the matrix geometry method,the equilibrium strategy,socially optimal strategy,social benefit under equilibrium strategy and socially optimal benefits are derived,and optimal pricing strategy is analyzed.Through numerical examples,we compared the strategy and benefits under different information level of the system.Then,the observable and unobservable cases in M/M/1 queues with negative customers and working breakdowns are consider.Once a negative customer arrives,ordinary customer being served is forced to leave the system and the server come in working breakdown period.The server serves customers in a lower rate at working breakdown time.We investigate the customers’ behavior under different information levels of the system,using Markovian process the steady-state probability in observable case is solved,and equilibrium strategy for the customers are obtained.In unobservable case,by using the probability generating functions,the steady-state distribution and the mean sojourn time of the arriving customers are obtained,and equilibrium strategy,optimal strategy and optimal pricing strategy are derived.Some numerical results are provided to illustrate the effect of the system parameters on equilibrium strategies of the observable case and customers’ sojourn time of the unobservable case.