On Regularity of Solution to Diffusion Approximation of GI/G/1 Queueing System

Regularity of non-stationary and stationary solutions to the diffusion approximation of the GI/G/1 queueing system under the elementary return boundary condition are discussed in this paper. Some boundedness of the solutions are also verified by using maximum principle. Introduction We discuss explicit non-stationary and stationary solutions to an initial boundary value problem of a linear partial differential equation of parabolic type, used in the elementary return boundary formulation of diffusion approximation to the GI/G/1 queueing system. It has been one of open problems in the literature [18]. Diffusion approximation is one of the most useful methods for tracing the temporal behavior of queueing systems. It describes the probability distribution function of the customer number in the system or virtual waiting time of a customer at each time, which is formulated by an initial boundary value problem of a linear partial differential equation of parabolic type. It is especially efficient for the GI/G/1 queueing system, where the inter-arrival times are independent and identically distributed random variables, customers are served in order of arrival, the service times of customers are independent and identically distributed random variables, and the inter-arrival and service times form independent sequences. The justification of this approach was provided by Kleinrock [12]. Even though the queue length is assumed to be infinite, the customer number in the system and virtual waiting time take non-negative values. Therefore, we have to consider the problem on the interval R+ ≡ (0,∞) . As a result, some boundary conditions at x = 0 and x → ∞ are necessary. In general, there exist two formulations of the diffusion approximation of the GI/G/1 system according to the form of boundary conditions: the reflecting barrier and elementary return formulations. The former formulation models the sample path of the object to be reflected instantaneously