Numerical Analysis of the Shortest Queue Problem for a Time-Scalable Queuing System with a Small Parameter on a Non-Uniform Grid Scheme

Abstract

In this paper, we use numerical methods using an uneven Shishkin grid to analyze the queue lengths of a time-scalable queuing system ( ) for which there is a Poisson incoming flow of requests with an intensity and service time , where is the service intensity parameter. The queuing system implements the service discipline in such a way that a random selection of any servers is provided for each incoming request and among them servers that have the shortest queue are selected. The dynamics of such a queuing system can be described using the shortest queue function , which can be found by solving a system of differential equations of infinite order, which can be obtained using an approach that involves the use of Markov chains. For this system of differential equations of infinite order, which can be attributed to a system of differential equations of the Tikhonov type, a singularly perturbed Cauchy problem with a small parameter is formulated. For the system of differential equations of this Cauchy problem, a truncation procedure is used, which allows us to obtain a singularly perturbed truncated Cauchy problem for a system of differential equations of finite order of Tikhonov type. For the numerical analysis of solutions to the truncated Cauchy problem, a non-uniform high-order Shishkin grid scheme is used. This scheme demonstrates good convergence of solutions to the singularly perturbed truncated Cauchy problem when the small parameter tends to zero. The results of numerical analysis of solutions to the truncated Cauchy problem show that a time-scalable queuing system can cope with an intensive incoming flow of applications.

Keywords

• numerical methods in queuing theory
• non-uniform grid scheme
• systems of differential equations of infinite order
• small parameter
• counting markov chains
• time-scalable queuing systems