Skip to main content

Decision tree formulas; 20 servers, 10 E

Short answer

The decision tree uses: infinite sources – Erlang B (cleared), Poisson (held), Erlang C (delayed); finite sources – Engset (cleared), binomial (held), finite queue (delayed). With 20 servers, total 10 E and 3‑min calls: total calls/h = (10×60)/3 = 200 calls/h; per server = 10/20 = 0.5 E, calls per server = (0.5×60)/3 = 10 calls/h.

What are the formulas used in telecommunication traffic engineering decision tree? A group of 20 servers carry a traffic of 10 erlangs. If the average duration of a call is three minutes, calculate the number of calls put through by a single server and the group as a whole in a one-hour period.

Formulas in the traffic engineering decision tree

The choice of formula depends on (1) whether the number of sources is infinite or finite, and (2) how blocked calls are treated: cleared (LCC), held (LCH) or delayed (LCD).

                 Traffic sources
           +-------------+-------------+
       Infinite                     Finite
   +-------+-------+          +-------+-------+
  LCC     LCH     LCD        LCC     LCH     LCD
Erlang B Poisson Erlang C   Engset  Binomial Finite
                                            queue
SourcesBlocked callsFormula
InfiniteClearedErlang B: B=AN/N!∑k=0NAk/k!B = \dfrac{A^N/N!}{\sum_{k=0}^{N}A^k/k!}
InfiniteHeldPoisson: P=e−A∑k=N∞Akk!P = e^{-A}\sum_{k=N}^{\infty}\dfrac{A^k}{k!}
InfiniteDelayedErlang C (probability of delay)
FiniteClearedEngset
FiniteHeldBinomial
FiniteDelayedFinite-queue model

Numerical

Given: N = 20 servers, A = 10 E, h = 3 min, T = 60 min. Using A=C h/TA = C\,h/T, so C=A T/hC = A\,T/h.

Group as a whole:

Cgroup=10×603=200 calls/h\begin{aligned} C_{group} &= \frac{10 \times 60}{3} = 200\ \text{calls/h} \end{aligned}

Single server (traffic shared equally):

A1=1020=0.5 EC1=0.5×603=10 calls/h\begin{aligned} A_1 &= \frac{10}{20} = 0.5\ \text{E} \\ C_1 &= \frac{0.5 \times 60}{3} = 10\ \text{calls/h} \end{aligned}

Check: 20×10=20020 \times 10 = 200.

Answer: each server handles 10 calls/hour; the group handles 200 calls/hour.

✓ Answer checked by Claude Opus 5.5 (high effort)

Related questions

All Telecommunications chapter 6 questions →