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.