M/m/1 Queuing System Under Variable Environmental Studies

This paper studies the M / M /1 queueing system in two environments changed mutually. The service station works in environment A and environment B,the service rates of the service station in different environment areμ1 andμ2. The duration of the environment A is a non-negative random variables withαexponential distribution .The duration of the environment B is a non-negative random variables, withβexponential distribution. Taking further assumptions, there is only one service station, and the capacity of the system is infinite. The customer first arrived, first be serviced. The environment transformation process,customer arrival process,service process are independent from each other. Using quasi-birth-and-death process,generating function and Laplace transform, we discuss the following questions:1. The law of the changing environment which the service station stay in, and get the transient probability in environment A and B at the moment t from different initial environment, and also get the steady-state probability.2. The steady-state queue-length distribution of the system, and obtain the probability generating function, the steady-state queue-length distribution. Furthermore, we obtain the expected number of customers in the system.3. The steady-state waiting queue-length distribution of the system, and obtain both the expressions of the steady-state waiting queue-length distribution and the average waiting length.4. The sojourn time and waiting time of customers, get the expressions of their LS transform. Moreover, we obtain the average sojourn time and waiting time.5. The special case, that is, non-changing environment, and obtain the corresponding results for the classic M / M /1 queueing system.