Construction of New Copulas with Queueing Application

TL;DR

This paper introduces new copulas constructed via bounds and perturbations, enabling better modeling of dependencies in queueing systems, with explicit formulas and applications to service time analysis.

Contribution

It develops new bound and perturbed copulas that encompass a wide dependency range and are computationally simple, with explicit measures and queueing applications.

Findings

Derived dependency measures for the new copulas.

Applied copulas to queueing system analysis.

Provided explicit formulas for distribution and expectations.

Abstract

In this paper, we construct a bound copula, which can reach both Frechet's lower and upper bounds for perfect positive and negative dependence cases. Since it covers a wide range of dependency and simple for computational purposes, it can be very useful. We then develop a new perturbed copula using the lower and upper bounds of Frechet copula and show that it satisfies all properties of a copula. In some cases, it is very difficult to get results such as distribution functions and the expected values in explicit form by using copulas such as Archemedes, Guassian, $t$-copula. Thus, we can use these new copulas. For both copulas, we derive the strength of measures of the dependency such as Spearman's rho, Kendall's tau, Blomqvist's beta and Gini's gamma, and the coefficients of the tail dependency. As an application, we use the bound copula to analyze the dependency between two service times to evaluate the mean waiting time and the mean service time when customers launch two replicas of each task on two parallel servers using the cancel-on-finish policy. We assume that the inter-arrival time is exponential and the service time is general.