Equilibrium Analysis And Optimal Control Of Queueing Systems With N-policy

In the past research on queueing systems,most scholars started from the classic view of queueing theory and did not consider the customer's strategies and behaviors.In recent years,more and more scholars have analyzed the queueing system from the perspective of economics.Assuming that customers decide whether to enter the system to accept services at the moment they reach according to the information level of the queueing system(including the queueing length and the status of the server)and other customer' choices,in order to maximize their own benefit.The motivation of service providers or managers is also to maximize their own revenue as much as possible,and the pricing strategy must take into account the behavioral strategies that the customers who arrive at the system will adopt.In this way,a game between two or more par-ties is formed between the customer and the customer or the customer and the service provider or the manager.Over time,the result of the game will converge to a sta-ble strategy combination known as the Nash equilibrium.Since the service rules,cost structure,and other factors between queueing systems can be different,the application of such problems in the company's order production,computer communications,bank-ing services,etc.is very extensive.This article focuses on the individual equilibrium strategies of customers and the optimal control strategies of server in different queueing systems based on N-policy.These strategies can provide the basis of management de-cisions for service providers or managers,and it is of great significance for the efficient management and operation of the enterprise.In this article,we first consider an M/M/1 constant retrial queueing system with reserved idle time under N-policy.We assume that a customer can occupy the serv-er instantaneously when finding the server turned on and idle upon arrival.After the service of the last customer,the server stays idle for some random time.During this period,if a new customer arrives,he obtains service immediately.Otherwise,the sever will shut down for saving energy and be reactivated if the number of waiting customers in retrial orbit reaches a given threshold N(N>1).The probabilities of the server in different states are derived through generating function method.Moreover,we studied the customer's equilibrium joining strategies with the status of the server visible based on the reward-cost structure.Finally,we establish the net profit function of the service provider per unit of time and illustrate the necessity of the reserved idle time's exis-tence from the perspective of the service provider through some numerical examples.It is found the longer the reserved idle time,the greater the server's profit.Next,we study a single server Markovian queueing system with breakdowns and repairs under N-policy.We make an assumption that the server will not work immedi-ately when an arrival occurs unless the number of waiting customers reaches a specified threshold N(N>1).i.e.the arrival of the Nth customer triggers the start of the server.However,a breakdown may occurs when the server works and it will be repaired right away once it breaks down.We assume arrivals and services are not permitted during a repair time.After repaired,the server restarts work and will turn off instantaneously after completing the service of all customers in the system.Similarly,We derive the steady state probabilities of the server in different states.Furthermore,we obtain the customers' equilibrium strategic behavior in the almost fully unobservable case and the fully unobservable case based on the expected sojourn time under different information levels.In addition,we also analyze the equilibrium social benefit per time unit at the corresponding inforamation level.Finally,we performe a sensitivity analysis on some system parameters(N,R,?)and make comparisons of the equilibrium,arrival rates and equilibrium social benefits of the two information levels via numerical examples.