Editing Queueing theory (section)

=== Example analysis of an M/M/1 queue ===

Consider a queue with one server and the following characteristics:
* ''<math>\lambda</math>'': the arrival rate (the reciprocal of the expected time between each customer arriving, e.g. 10 customers per second)
* ''<math>\mu</math>'': the reciprocal of the mean service time (the expected number of consecutive service completions per the same unit time, e.g. per 30 seconds)
* ''n'': the parameter characterizing the number of customers in the system
* <math>P_n</math>: the probability of there being ''n'' customers in the system in steady state

Further, let <math>E_n</math> represent the number of times the system enters state ''n'', and <math>L_n</math> represent the number of times the system leaves state ''n''. Then <math>\left\vert E_n - L_n \right\vert \in \{0, 1\}</math> for all ''n''. That is, the number of times the system leaves a state differs by at most 1 from the number of times it enters that state, since it will either return into that state at some time in the future (<math>E_n = L_n</math>) or not (<math>\left\vert E_n - L_n \right\vert = 1</math>).

When the system arrives at a steady state, the arrival rate should be equal to the departure rate.

Thus the balance equations
: <math>\mu P_1 = \lambda P_0</math>
: <math>\lambda P_0 + \mu P_2 = (\lambda + \mu) P_1</math>
: <math>\lambda P_{n-1} + \mu P_{n+1} = (\lambda + \mu) P_n</math>
imply
: <math>P_n = \frac{\lambda}{\mu} P_{n-1},\ n=1,2,\ldots</math>
The fact that <math>P_0 + P_1 + \cdots = 1</math> leads to the [[geometric distribution]] formula
: <math>P_n = (1 - \rho) \rho^n</math>
where <math>\rho = \frac{\lambda}{\mu} < 1</math>.