Short note: Finite and infinite source models

Tele-traffic models are classified by the number of traffic sources compared with the number of servers N.

Infinite-source models

• Number of sources is very large (S ≫ N), so the call arrival rate $\lambda$ is constant and independent of busy sources; arrivals are Poisson.
• Offered traffic $A = \lambda h$ is fixed.
• Formulas: Erlang B (lost calls cleared), Poisson (lost calls held), Erlang C (lost calls delayed).
• Erlang B: $B = \dfrac{A^N/N!}{\sum_{k=0}^{N}A^k/k!}$
• Used for junction routes between large exchanges.

Finite-source models

• Small number of sources S (comparable to N). A busy source cannot make a new call, so the arrival rate $(S-n)\gamma$ falls as more servers are busy.
• Formulas: Engset (S > N, lost calls cleared), Bernoulli/binomial (S ≤ N, no loss).
• Engset call congestion: $B = \dfrac{\binom{S-1}{N}\alpha^N}{\sum_{k=0}^{N}\binom{S-1}{k}\alpha^k}$, $\alpha$ = traffic per idle source.
• Used for PBXs, concentrators, small rural exchanges.