Simulation Of Queue Systems

based on

Analytical Solution for Time-dependent M/M/1

with

Application to Motorway Traffic

Mathematical solutions have been derived for many queue systems under the assumption of equilibrium or steady-state. These results are very useful for determining properties of a system averaged over long time periods.

However, we are often interested in queue behaviour over short time periods such as a rush hour. This is non-equilibrium queue theory and is much harder mathematically. see Cox & Smith, Queues, DCU library reference 519.82/COX, Chapter III).

Nevertheless, it is possible to establish some results analytically. In particular, Cox & Smith derive a solution for the non-equilibrium case of a single-server queue with random arrivals and exponential service times (denoted in queue theory as M/M/1).

The proposed project is to implement the above solution in a software class, hiding the mathematical details that users of the class do not need to know. Key input parameters are the arrival and service rates while the fundamental output, for a given time t; is an array of values of p_n(t) the probability that there is n units in the system at time t.

Having constructed and tested the class, it should then be possible to simulate various scenarios by creating several objects of the class whose inputs and outputs feed into each other. For example, one might think of doing a crude simulation of motorway traffic in this way.