# Traffic management

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### 4.4.1 Queue Lengths

This subsection provides the key formulae related to queue lengths in an [M/M/1] queuing system. The derivations of these formulae are provided in Commentary 3.

The key formulae are as follows.

The probability of the queue being empty – that is, having no units in service and no units waiting in a queue to be serviced – is:

 $P_{0}=1 - \rho$ 4.2

The probability that there are $$n$$ units in the system, $$n \ge0$$ , including the unit in service (if any), is:

 $P_{n}=(1 -\rho ) \rho ^{n}$ 4.3

The expected number in the system is:

 $E(n)=\frac{\rho }{1-\rho }=\frac{r}{s - r}$ 4.4

The probability of there being more than N items in the system is:

 $\Pr (n>N)=\rho ^{N+1}$ 4.5

The mean queue length, excluding the unit being serviced, is:

 $E(m)=\frac{\rho ^{2} }{1 - \rho } = \frac{\rho }{1 - \rho } - \rho$ 4.6

Thus,

 $E(m) = E(n). \rho = E(n) -\rho$ 4.7

Given that $$\rho <1$$ , E(m) is not (as might be expected) equal to E(n) – 1. This is because there is a finite probability that the system is empty, in which case n = m = 0.

Finally, the variance of the number of units in the system is:

 σ2(n) = $$\sum _{n=0}^{\infty }n^{2}$$ Pn – [E(n)]2 = $$\frac{\rho }{(1-\rho )^{2} }$$ 4.8