Queueing Networks With Time-varying Arrival

Queue is widely used in our daily life, production, technology and computer. With the progress of times, queueing network has been applied more widely. Queueing network with time-varying arrival is studied in this paper. The laws of customer arrival and the staffing are important for the queue. Time-varying arrival is mainly studied in this paper.We should study the multi-queue model Gt/Mt/st + GIt first. We has made a series of assumptions, about the feasibility of the staffing function, dynamic constraints, initial conditions, boundary waiting time, smoothness, initial queue density, minimum service rate and the time-varying abandonment. Under the assumptions, we have get a series of performance for the multi-queue, the system service capacity B(t) and its density function b(t,x),the queue density q(t,x), the boundary waiting time ?(t) and the potential waiting time v(t). We have proved the existence and uniqueness of the solution of ?(t) by using the classical Picard-Lindelof theorem. The Lipschitz continuity has been proved about E(t),S(t),Q(t),B(t), A(t).The model studied in this paper is the conversion between underload and overload, so how to control conversion step is also important. Through the queue Gt/M/st + M, We have studied the relationship about the computation time C(?T) and the step size AT, the relationship about the computation time C(?) and switch time ? for fixed time interval, the relationship about the computation time C(T) and time interval T for fixed switch time.The main work of this paper is to study the queueing network model. The fixed point equation (FPE) method and the ordinary differential equation (ODE) method are used to analyze the arrival rate function. For fixed point equation, we can know that it is an monotonic contraction operator by used Banach fixed point theorem. The arrival rate function is obtained by recursive iteration method. Then we can use the performance calculation about the multi-queue Gt/Mt/st + GIt to get the performance for network. For ODE, we consider the multi-dimensional ordinary differential equation. After the arrival rate function of each queue, the performance function of the queueing network is obtained by the multi-queue. Finally the stationary queueing network is studied in this paper.If we have determined the customer arrival rate and the staffing for server, the system can be optimized, and we can create better economic and social benefits.