## Modelling the Length of a Queue

The length of a queue at timecan be modelled by a simple birth and death process (people joining and leaving the queue respectively).

Letthen the general Kolmogorov equations are

whereFor the simple queuesfor allandfor allThis gives

for

This is a difficult problem and since how a queue behaves in the long term is of more interest, we analyse the steady state solutionand all the parameters are constant. The Kolmogorov equations become

The last equation giveswhich inductively saysfor

henceforwith

The sum of any probability distribution is 1 by definition henceso that

Thenfor