Transient diffusion approximation for the G/G/m queueing systems

Abstract

The main purpose of wokr is to present a general approach by which fairly complicated multiserver queueing system in transient state could be simply approximated by diffusion processes. In order to approximate the transient behavior of multiserver queueing system we construct diffusion processes with suitable boundary conditions and difusion parameters which are determined by the parameters of queueing system. Using the transition density function of the diffusion process, we derive the apporximate forms of the system size distribution. To find the transition density function of the difusion process, we solve a partial differential equation so called Kolmogorov forward equation or Fokker-Planck equation. The solution of Kolmogorov forward equation is given in the form of Laplac transform with respect to the variable t. We also obtain the stationary distribution of queue size for G/G/m (G/G/m/N) system by letting t $\rightarrow$ $\infty$ in transient solution. We derive the distribution of the first passage time of diffusion process from the transition density function of the diffusion process. Using the first passage time of diffusion process, we obtain the approximations for the distributions of the first overflow time in G/G/m/N-1 system, maximun queue length up to time t and k-busty period in G/G/m system. The transient approximation for the distribution of the number of customers at each stages in the two stages cyclic system is also abtained. For the lack of exact transient solution of queueing stsyem with multiple server, we check the accuracy of approximation by comparing approximation results with simulation experiments. The numerical results show that the diffusion approximation is very accuate for all traffic cases.