Quantiles of response time distribution in fork-join queueing systems with pareto distribution of service times

Abstract

The paper studies a system with separation and parallel service of requests, also called a fork-join queuing system, with a Pareto distribution of service time and various options for the dis-tribution of intervals between arrivals of requests for an incoming flow, namely, the Erlang distribution, exponential distribution, and also a hyperexponential distribution (a mixture of two exponential ones). A new approach to estimating the quantiles of the distribution of a request’s residence time in a fork-join system is proposed. The definition of this characteristic is no less important a task than the more traditional approximation of the mathematical expectation and, accordingly, the moments of a higher order of the system response time, since it gives a broader idea of the required amount of resources for service the requests coming into the system, the mathematical model of which is a system with separation and parallel service. In particular, 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. The approach is based on approximating the system response time distribution by the Fréchet distribution, the parameters of which are determined statistically using the method of moments. The algorithm for finding quantile estimates also includes simulation modeling and an optimization method that can significantly reduce the approximation error of the original formulas. The numerical experiment showed good quality of approximation for high-level response time quantiles; the average relative ap-proximation error in all three cases does not exceed 2%, and the maximum is 5 %.