NUMERICAL ANALYSIS OF SHORTEST QUEUE PROBLEM FOR TIME-SCALE QUEUEING SYSTEM WITH A SMALL PARAMETER

In this paper we apply numerical methods for analysis of the timescale queueing systems (TSQS) evolution dynamics under the the assumption that the number of single-services tends to infinity. We suppose that TSQS implements a service discipline so that for each incoming request is provided a random selection a server from random selected m-set servers that has the s-th shortest queue size. The evolution dynamics TSQS can be describe using the function that can be found by solving a system of differential equations infinite degree. We formulate the singularly perturbed Cauchy problem for this system of differential equations with a small parameter. We use the truncation procedure for this singularly perturbed Cauchy problem and formulate the finite order system of differential equations. We apply a high-order non-uniform grid scheme for numerical solving of the truncated Cauchy problem. We use different sets of small parameters for time-scaling processes analysis for TSQS. The grid scheme demonstrates good convergence of solutions of the singularly perturbed Cauchy problem when a small parameter tend to zero. The results of the numerical simulation show that this TSQS can hold with a high incoming flow of requests.