A Distribution for the Service Model

Abstract

In this paper, we propose a distribution that describes a specific system. The system has a heavy traffic, a fast service and the service rate depends on state of the system. This distribution we call the Maximum-Conway-Maxwell-Poisson-exponential distribution, denoted by MAXCOMPE distribution. The MAXCOMPE distribution is obtained by compound distributions in which we use the zero truncated Conway-Maxwell-Poisson distribution and the exponential distribution. This distribution has adjustment mechanism in order to re-establish the equilibrium of the system when the traffic flow increases and that is described by variations of the pressure parameter. Because of this, the MAXCOMPE distribution contains sub-models, such as, the Maximum-geometric-exponential distribution, the Maximum-Poisson-exponential distribution and the Maximum-Bernoulli-exponential distribution. The properties of the proposed distribution are discussed, including formal proof of its density function and explicit algebraic formulas for their reliability function and moments. The parameter estimation is based on the usual maximum likelihood method. Simulated and real data are shown to illustrate the applicability of the model. A DISTRIBUTION FOR THE SERVICE MODELsystem and avoid congestion. The increase in service rate and the opening the new service channels are the adjustment mechanisms. Some examples of systems with this behavior are nonstop toll electronic, the traffic flow and the access on website.We propose the Maximum-Conway-Maxwell-Poisson-exponential distribution, denoted by MAXCOMPE distribution to describe this system. Moreover, we are particularly interested on the maximum inter-arrival time or maximum service time. The methodology used to obtain this distribution is the compound distributions. We compose the zero truncated Conway-Maxwell-Poisson distribution and the exponential distribution.The MAXCOMPE distribution contains sub-models that describe the variations of the system, such as, the Maximum-geometric-exponential distribution, denoted by MAXGE distribution, the Maximum-Poisson-exponential distribution, denoted by MAXPE distribution and Maximum-Bernoulli-exponential distribution, denoted by MAXBE distribution.The compound distributions has been extensively studied. We mention the following articles: Barreto-Souza et al. (2011) introduced the Weibull-geometric distribution which generalizes the exponential-geometric distribution proposed by Adamidis et al. (2005) and Cancho et al. (2011) proposed the Poisson-exponential distribution with two-parameters, Cordeiro et al. (2012) introduced a new three parameters distribution called exponential-Conway-Maxwell Poisson distribution, denoted by ECOMP distribution.We are interested in systems which the service rate depends on state of the system. We would like to mention some articles with that characteristic. Kimura (1991) provided two distributions with an approximation for the mean waiting time in a G I/G/s queue. This approximation are weighted combinations of the exact mean waitin...