Impatient Extent Related To The System State M/m/1 Vacation Queuing System

In qucueing theory,the customers whose waiting time is limited are called impatient customers.The queue system models with impatient customers can be used to simulate and resolve many practical problems,for example,the transportation of data packets in highway networks,real-time computation,automatic production,the process of industrial packting,the storage of perishable foods and so on;while vacation queue systems are refered to the ones that the servers can't work at some time(there are many kinds of reasons causing service interuption).As they are widely used,queue systems with vacations are always a hot problem for recent years,and great advances have been acheived.In this paper,the M/M/1 queue system with expernatial multi-vacation is studied,in which the customers' longest waiting time befor receiving service is dependent on the state of the system,that is to say,the customers' longest waiting time is an expernatial variable with parameterμ₀ when the queue system is in vacation period,and with parameterμ₁ when the queue system is in busy period,so are the M/M/1 queue system with N policy expernatial multi-vacation and the M/M/1 queue system with expernatial single-vacation.Under a certain condition,the probability generating function of the steady queue lenghth distribution and queue lenghth mean are given for the three models.