On approximating the conditional wait process of general multiserver queueing systems

In view of the extremely difficult task of analyzing G/G/K queueing systems, relatively few general results have been established. Most of the literature dealing with the G/G/K system has been concerned with the heavy traffic situation. Of special note are the bounds given by Kingman[7] on the wait process, the weak convergence theorems established by Iglehart and Whitt[4] and Loulou[8], and the many server approximations due to Newell[9].In this paper a presentation will be made of new formulas constructed to approximate the mean and variance of the conditional wait process. Two key concepts play major roles behind the formulas to be presented. They are: (a) the use of a continuous (diffusion-type) process to approximate the steady-state probabilities of the number in the queue, and (b) the substitution of the original process of interdeparture times by a process of independent variables.In order to assess the quality of the suggested approximation for the mean and variance of the conditional wait process, the approximation will be numerically compared with the results obtained from Kingman's[7] and Kendall's[5] exact formula for the special case of the G/M/K system, and will be tested against point estimate simulations.