Research On M/M/2 Queueing System With Threshold And Vacation Policies

In the artificial service system, the service rates tend to be different, and the number of customer visits the service system often fluctuates over time, so the using intensity of the system servers is not uniform. Aiming at this problem, this paper studies an M/M/2 queuing system of two different servers with a threshold policy, task allocation and synchronous multiple vacations. Further more, we study M/M/2/K queuing system with a threshold policy and asynchronous multiple vacations. The main contents are as follows:Firstly, this paper studies an M/M/2 queuing system with two heterogeneous servers. We analysis three different task assignment policies: dispatching the tasks according to probabilities and two different threshold policies. We obtain steady state probabilities and balance indexes of the system in different policies by the methods of iteration and matrix-geometric solution. By a numerical example, we calculate the minimum expected queue length and sojourn time of different policies, respectively, we obtain the assignment probabilities and the optimal threshold values.Secondly, we study an M/M/2 queuing system with a threshold and synchronous multiple vacations, where the service rates of the two servers are not identical. In the busy period, the faster server provides service, while the slower server serves the customer only when the number of waiting customers is not less than m, or it is always idle. Using the method of iteration and probability generating function, we have derived the explicit expressions of the stationary probabilities and performance measures. At last, a cost model is presented to analyze the effect of all parameters on the minimum cost and the optimal threshold.At last, we study an M/M/2/K queuing system with a threshold and asynchronous multiple vacations, where the service rates of the two servers are not identical. In this queuing system, the first server will take a vacation if he or she finds no waiting customer in the system at a service completion instant, while the second server will take a vacation if he or she finds the number of customers in the system is less than the threshold m at a service completion instant. Using the matrix analytic method, we obtain the stationary probability vectors and performance measures. A cost model is presented to analyze the effect of the system parameters on the optimal cost and the optimal threshold of the system.