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