On some new bivariate negative binomial and gamma distributions with applications to queues, inventories and maintenance

Abstract

Some new theory and applications of certain bivariate non-normal distributions are presented. In particular, new bivariate negative binomial and gamma distributions are discussed and an existing bivariate exponential distribution is applied to single stage and tandem queueing systems (both single server) which have particular kinds of correlated structure.

One new bivariate negative binomial distribution is derived by convoluting an existing bivariate geometric distribution; the probability function has six parameters and admits of positive or negative correlations and linear or nonlinear regressions. Given are the moments to order two and for special cases, the regression function, a recursive formula for the probabilities, a method of moments parameter estimation technique, the likelihood equations, the differentialdifference equations and for maximum likelihood parameter estimates, a necessary relationship for the parameters. Certain results are extended to a dual bivariate gamma distribution. Another bivariate negative binomial distribution, which has four parameters', results by reducing a particular trivariate negative binomial distribution with independent marginals; only positive correlations and linear regressions are possible here. Both bivariate negative binomial distributions are fitted to data and their special features illustrated.

Applications of bivariate distributions to certain aircraft logistical problems are investigated. Primarily, a bivariate negative binomial distribution is fitted to spare parts demand data in two periods and to monthly abort data on either side of a large scale maintenance event and it is shown how the associated sample distributions can be useful in parts inventory control and in investigating the effect of maintenance on an aircraft's performance.

A bivariate exponential distribution is applied to tandem queues to study the effect of correlated exponential service times and to single stage queues to study the effect of correlation between a customer's service time and the inter-arrival interval separating himself and his predecessor. Arrivals to both systems are according to a Poisson process. Simulation is used to show that the mean waiting time is quite sensitive to departures from the traditional assumptions of mutually independent service times for tandem queues and independence of service times and interarrival intervals for single stage queues, especially at higher utilizations. For the cases of infinite interstage storage between two-stage tandem queues and infinite storage before a single stage queue, system performance is increased by positive correlation and impaired by negative correlation. For two-stage queues this change is reversed for zero interstage storage and depends on the value of the utilization rate for the case where interstage storage equals unity. By using spectral analysis techniques and a nonparametric test applied to sample power spectra associated with certain simulated waiting times the effects are shown to be statistically significant. For correlation equal unity and infinite interstage storage results are given for two through twenty-five stages in series.