Matrix Scheme - in Queueing Theory

In Queueing Theory

A matrix scheme is easily represented as a simple M/M/1 queue within the context of queueing theory In such a system you have a Markovian arrival, Markovian service, and one single server (F. S. Hiller and G. J. Lieberman. Introduction to Operations Research. McGraw-Hill, New York, 1995). In the standard Matrix queue service rates are a function of arrival rates since the time to cycle out of the queue is based on the entry fee into the matrix from arriving members. Also, since members move through the matrix in single file, it is easy to associate the single server.

The basic premise of queueing theory is that when arrival rates equal or exceed service rates overall waiting time within the queue moves towards infinity (Hiller and Lieberman).

The basic formulation includes three formulae. The traffic intensity, ρ, is the average arrival rate (λ) divided by the average service rate (μ):

The mean number of customers in the system (N):

And the total waiting time within the queue (T):

It is possible to see that as arrival rates rise towards service rates the total waiting time (T) and mean number of customers in the system (N) will move towards infinity. Since service time can never exceed the arrival time in the standard matrix, and total waiting time can only be defined if service times exceed arrival times, the only way for the matrix queue to reach stability is for outside income sources to exceed those being entered into the system.