Heavy traffic queue with stochastic arrival and service rates

A new approach is proposed to model heavy traffic queuing behaviour by stochasticizing in various ways the arrival and service rates so as to incorporate the random effect of the environment. This approach yields for the queue size a differential equation with stochastic parameters. The corresponding Fokker-Planck equation for the probability density function of the queue size is employed to analyse the queuing system. In this framework we consider the arrival and service rates to be subject to gaussian, white-noise perturbations and obtain the probability density function of the queue size and the distribution of the busy period. The study is generalized by considering the effect of coloured-noise perturbations in these rates. The effect of the white-noise perturbations on the state-dependent service rate is also investigated.