Traffic engineering objectives; Engset formula

Objectives of traffic engineering

1. Dimensioning: find the minimum number of trunks, switch paths, registers or channels that carry the expected busy-hour traffic.
2. Meet the grade of service: keep blocking (loss systems) or delay (delay systems) within the target, e.g. GoS = 0.01.
3. Minimise cost: balance equipment cost against service quality; avoid both over- and under-provision.
4. Traffic measurement and forecasting: measure busy-hour traffic and predict growth for planning.
5. Efficient routing: design direct, tandem and alternate (overflow) routes for high utilisation.
6. Overload control and reliability: keep the network stable during peaks and failures.

Engset formula (finite traffic sources)

The Engset formula gives the loss probability when a finite number of sources S share N servers (S > N) and blocked calls are cleared, e.g. a PBX or small rural exchange. Since busy sources cannot make new calls, the call rate falls as more sources are busy, so Erlang B (which assumes infinite sources) over-estimates loss.

Assumptions: S identical sources; each idle source calls at rate $\gamma$; mean holding time $h = 1/\mu$; $\alpha = \gamma h$ = offered traffic per idle source; full availability; lost calls cleared; equilibrium.

Birth–death model: in state n (n servers busy) arrival rate $=(S-n)\gamma$, departure rate $= n\mu$. Balance:

$$(S-n+1)\gamma P_{n-1} = n\mu P_n \;\Rightarrow\; P_n = \binom{S}{n}\alpha^n P_0$$

Normalising over $n = 0..N$:

$$P_n = \frac{\binom{S}{n}\alpha^n}{\sum_{k=0}^{N}\binom{S}{k}\alpha^k}$$

Time congestion (all N busy):

$$E = \frac{\binom{S}{N}\alpha^N}{\sum_{k=0}^{N}\binom{S}{k}\alpha^k}$$

Call congestion (probability an arriving call is lost) – the Engset loss formula:

$$B = \frac{\binom{S-1}{N}\alpha^N}{\sum_{k=0}^{N}\binom{S-1}{k}\alpha^k}$$

Notes

• $B < E$ for finite S.
• As $S \to \infty$ with $S\alpha = A$, Engset → Erlang B.
• If $S \le N$, no calls are lost (Bernoulli/binomial distribution).