Multi-server queueing systems have been important models for evaluating the performance of a variety of service systems in the fields of computer/ communications, transportation, manufacturing and so on. However, due to analytical difficulty, very little is known for multi-server queueing systems. To deal with this difficulty, we need approximations, which may require heuristics depending very much on intuition and creativity.
In this study, we consider the M/G/c and the M/G/c/c+r queueing systems. Applying the method of system approximations, we provide approximate solutions for the queue-length distribution, the waiting time distribution, the mean waiting time and the delay probability in the M/G/c queue. We also provide approximate solutions for the queue-length distribution, the mean queue length and the loss probability in the M/G/c/c+r queue. In addition, the approximation for the loss probability in the M/G/c/c+r queue is applied to the buffer design problem to obtain a simple approximation for the optimal buffer capacity. The buffer design problem is to determine the smallest buffer capacity such that the proportion of lost customers is below an acceptable level.
