From Single Server Queues to M/G/1

Abstract

Sometimes production and manufacturing are complicated processes. Usually it has to do with limited facilities. For example, in order to produce a car the manufacturer may have to go through a number of working centers: a storage area with raw materials at one end, the quality control inspection at the other end, and a few assembly points in between. Various points along the production process interact. The outflow of one is the inflow of another. A slow machine may reduce the productivity of other working centers along the line which will be starved for more input. However, in order to gain some insight into such systems, we first have to look at single server systems in isolation. When looked at in isolation, each server (or servers who work in tandem) and the demand for its service can be modeled as customers arriving at service stations. If there is too great a delay in the service facility, a waiting line might form. The purposes of this chapter are:

• To see why queues sometimes form;

• To see what the main factors are in determining the length of the queue and the waiting time;

• To show how waiting times and queue lengths are related (Little’s law);

• To define the virtual waiting times and the actual waiting time;

• To derive the mean waiting time (the Khintchine–Pollaczek formula) and the mean time of a busy period for single server queues with a Poisson arrival process.