Equilibrium Analysis In The M/M/1 Queue With Breakdowns

With the wide application of random service systems,the economic utility of customers in service systems has become a hot topic at home and abroad in recent years.Combining game theory,analyzing the performance indicators of the queuing system and the influence of various parameters on them can better optimize the management of the service system and obtain more economic benefits.Based on this,and considering that the service system is not completely reliable in reality,this paper analyses the strategic behavior of customers in queuing system with breakdowns.Firstly,the equilibrium strategies of customers in M/M/1 queuing system with two types of breakdowns are considered.The system does not admit a new arrival once a breakdown happens,and there may exist two independent types of breakdowns:(1)partial breakdown:the server continues to serve the customers on spot at a low rate and is repaired when the system is empty;(2)full breakdown: the server stagnates service and is repaired immediately.By using Markov process and solving difference equations,the steady-state probabilities of the system are obtained,and the equilibrium joining strategies of the customers and the average social benefits of the system in the fully observable case and the almost unobservable case are discussed respectively.Secondly,the equilibrium strategies of customers in the queuing system with two types of parallel customers and partial breakdown are analyzed.There are two types of customers with independent arrival process.When a partial breakdown occurs,the server continues to serve the customers on spot at a low rate and doesn't admit a new arrival until normal work resumes.By solving the difference equations and analyzing the equilibrium of the system,the equilibrium joining strategies of the two types of customers and the average social benefits of the system in the fully observable case and the almost observable case are given respectively.Finally,the equilibrium behavior of customers in queuing systems with setup times and partial breakdown are studied.In the queuing system,the reactivated server must go through a period of setup time to reach the normal working state and a failure can occur at any time during the normal service.When a partial breakdown occurs,the server continues to serve the customers on spot at a low rate and does not admit a new arrival.Once the system becomes empty,an exponential repair time starts.Through steady-state analysis and equilibrium analysis,the equilibrium strategies of the customers and the average social benefits of the system in the fully observable case and the almost unobservable case are discussed respectively.