Correlated Bivariate Sequences for Queueing and Reliability Applications

Abstract

We derive bivariate exponential, gamma, Coxian or hyperexponential distributions. To obtain a positive correlation, we define a linear relation between the variates X and Y of the form Y = aX + Z where a is a positive constant and Z is independent of X. By fixing the marginal distributions of X and Y, we characterize the distribution of Z. To obtain negative correlations, we define X = aP + V and Y = bQ + W where P and Q are exponential antithetic random variables. Our bivariate models are useful in introducing dependence between the interarrivals and service times in a queueing model and in the failure process in multicomponent systems. The primary advantage of our model in the context of queueing analysis is that it remains mathematically tractable because the Laplace Transform of the joint distribution is a rational function, that is a ratio of polynomials. Further, the variates can be very easily generated for computer simulation. These models can also be used for the study of transmission controlled queueing networks.