On the features of service speed control in fork-join systems with pareto distribution of service time

Abstract

The article studies a fork-join queuing system. It is assumed that the request servicing time has a Pareto distribution, and the input flow is Poisson. With the help of fork-join structures, the processes of functioning of systems using distributed or parallel computing or systems using division of the original task into parts in order to optimize work processes are modeled. One of the most important issues for the owners of such systems is the issue of the optimal number of allocated resources for servicing users. On the one hand, their number should be sufficient for high-quality servicing of input tasks, on the other hand, their cost should not be too high. For the considered system, a mathematical model for determining the optimal cost of its functioning is constructed, which is based on maintaining an optimal balance between the average response time of the system (or the average time a request stays in the system) and reasonable costs of the resources necessary for this. In this case, resources are understood as the intensity or speed of servicing on servers necessary for maintaining the optimal level of system performance. The model uses an approximate expression for estimating the average response time of the system obtained earlier. The cost is a function of the load factor. Finding the optimal cost of operation is actually reduced to determining the minimum of this cost function. The value of the load factor determined in this way, in turn, determines the service intensity, which will be considered optimal. To find the optimal service intensity on the system devices, an equation is obtained that can be solved numerically. A numerical example is considered, and an analysis of the asymptotic behavior of the solution is carried out.