Editing Queueing theory (section)

==== Balance equations ====

The [[steady state]] equations for the birth-and-death process, known as the [[balance equation]]s, are as follows. Here <math>P_n</math> denotes the steady state probability to be in state ''n''.

: <math>\mu_1 P_1 = \lambda_0 P_0</math>
: <math>\lambda_0 P_0 + \mu_2 P_2 = (\lambda_1 + \mu_1) P_1</math>
: <math>\lambda_{n-1} P_{n-1} + \mu_{n+1} P_{n+1} = (\lambda_n + \mu_n) P_n</math>

The first two equations imply
: <math>P_1 = \frac{\lambda_0}{\mu_1} P_0</math>
and
: <math>P_2 = \frac{\lambda_1}{\mu_2} P_1 + \frac{1}{\mu_2} (\mu_1 P_1 - \lambda_0 P_0) = \frac{\lambda_1}{\mu_2} P_1 = \frac{\lambda_1 \lambda_0}{\mu_2 \mu_1} P_0</math>.

By mathematical induction,
: <math>P_n = \frac{\lambda_{n-1} \lambda_{n-2} \cdots \lambda_0}{\mu_n \mu_{n-1} \cdots \mu_1} P_0 = P_0 \prod_{i = 0}^{n-1} \frac{\lambda_i}{\mu_{i+1}}</math>.

The condition <math>\sum_{n = 0}^{\infty} P_n = P_0 + P_0 \sum_{n=1}^\infty \prod_{i=0}^{n-1} \frac{\lambda_i}{\mu_{i+1}} = 1</math> leads to
: <math>P_0 = \frac{1}{1 + \sum_{n=1}^{\infty}\prod_{i=0}^{n-1} \frac{\lambda_i}{\mu_{i+1}} }</math>
which, together with the equation for <math>P_n</math> <math>(n\geq1)</math>, fully describes the required steady state probabilities.