The Waiting Time For Queues With Variable Servers

This paper mainly introduces the waiting time for customers queue and servers queue with variable severs under heavy-traffic. The servers have their own changing rules. The essence of the queueing system is to study the efficiency of the service and receiving service between the servers and the customers. The key point of the queueing system is its optimal design and operation. In the infinite waiting space, the optimization of customer and severs can make them win-win. In order to reduce customer abandonment rate, we apply of a new sever if the customers who are waiting for service more than the limited value. The waiting time of the customers will relatively be reduced if the new server’s opening.In this paper, we mainly study the number of servers is increased when the initial system is empty and the customer’s waiting space to be infinite. For a service system, whether the servers opening or closing are depending on the number of the customers waiting for service. In this paper, through the establishment of queueing model under variable servers,we set the first come first served rule and new opened conditions and change limit under the model. If the number of customers waiting for service no less than M at time t, we apply to open another server. But the total amount of the servers are not more than n.We never change the rule whether the new server open or not. Customers may leave the service facility only by completing service. This paper mainly study the waiting time of customers queue and servers queue.This paper concludes the functional weak law of large numbers and the functional central limit theorem of the stochastic process under the variable servers, and we proved the theorem by the continuous mapping theorem. In this paper we also study the correlation properties of the waiting time of customers queue and servers queue. We proved the differential of the boundary waiting time of customers queue. The existence and uniqueness of the boundary waiting time of customers queue is proved by the existence and uniqueness of the solution of the ordinary differential equations theorem. The virtual waiting time of customers queue with right-continuous and has limits from the left is been constructed. Thispaper proved the relationship between the boundary waiting time and the virtual waiting time. This paper proved the differential and the existence and the uniqueness of the virtual waiting time by the existence and uniqueness of the solution of the ordinary differential equations theorem and the relationship between the boundary waiting time and the virtual waiting time. For the waiting time of the servers, we proved it’s continuity by recursive.Finally we study the waiting time of customers queueing and servers queueing in the model of M/M/n_t.