Approximate Analysis of M/G/c, GI/G/1, and GI/G/c Systems Using Mixture Distributions

There are exact solutions for M/G/1 system and good approximations for M/G/c, GI/G/1, and GI/G/c systems when certain conditions are satisfied. However, various approximation formulas have also been proposed in the literature. In the M/G/c system, the arrival process follows a Poisson distribution, while the service-time distribution is non-exponential with (c) servers. For M/G/1 , the Pollaczek–Khintchine formula provides an exact solution. However, this formula requires the service-time distribution to follow a theoretical distribution and its variance to be known. The results obtained from the M/G/1 system are commonly used as lower and upper bounds based on the coefficient of variation, and approximation formulas have been developed by extending this approach to M/G/c, GI/G/1, and GI/G/c systems. However, these approximation formulas have generally been developed and applied for cases in which the data fit a theoretical distribution. When the data do not fit any theoretical distribution, instead of directly using an empirical distribution, it may be tested whether the dataset follows a mixture distribution. Mixture distributions can be used for modeling heterogeneous populations consisting of several unobserved homogeneous subpopulations.

In this study, an approximate solution is proposed for M/G/c, GI/G/1, and GI/G/c systems when the general distributions consist of pure mixture distributions. The rationale behind the proposed approximation is based on the ability to compute the variance of certain pure mixture distributions. The approximation formula has been adapted for two-component Gaussian, Gamma, and Beta mixture distributions. To evaluate the effectiveness of the proposed approximation, M/G/c, GI/G/1, and GI/G/c systems were simulated using random numbers generated from mixture distributions. The results obtained from the proposed approximation formula were found to be very close to the simulation results.