---
title: "Decision tree formulas; 20 servers, 10 E"
subject: "Telecommunications"
chapter: 6
key: "semester_8/telecommunications:decision-tree-20-servers"
source: https://www.bctengineer.com/q/semester_8/telecommunications/decision-tree-20-servers
site: BCT Engineering
---

# Decision tree formulas; 20 servers, 10 E

Telecommunications · Chapter 6: Telephone Traffic

- Asked in: 2079 Chaitra, 3+5 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_8/telecommunications/decision-tree-20-servers
- Whole chapter: https://www.bctengineer.com/md/semester_8/telecommunications/chapter-6

## Question

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.

## 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.

## Answer

#### 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).

```text
                 Traffic sources
           +-------------+-------------+
       Infinite                     Finite
   +-------+-------+          +-------+-------+
  LCC     LCH     LCD        LCC     LCH     LCD
Erlang B Poisson Erlang C   Engset  Binomial Finite
                                            queue
```

| Sources | Blocked calls | Formula |
|---|---|---|
| Infinite | Cleared | Erlang B: $B = \dfrac{A^N/N!}{\sum_{k=0}^{N}A^k/k!}$ |
| Infinite | Held | Poisson: $P = e^{-A}\sum_{k=N}^{\infty}\dfrac{A^k}{k!}$ |
| Infinite | Delayed | Erlang C (probability of delay) |
| Finite | Cleared | Engset |
| Finite | Held | Binomial |
| Finite | Delayed | Finite-queue model |

#### Numerical
**Given:** N = 20 servers, A = 10 E, h = 3 min, T = 60 min. Using $A = C\,h/T$, so $C = A\,T/h$.

Group as a whole:
$$
\begin{aligned}
C_{group} &= \frac{10 \times 60}{3} = 200\ \text{calls/h}
\end{aligned}
$$

Single server (traffic shared equally):
$$
\begin{aligned}
A_1 &= \frac{10}{20} = 0.5\ \text{E} \\
C_1 &= \frac{0.5 \times 60}{3} = 10\ \text{calls/h}
\end{aligned}
$$

Check: $20 \times 10 = 200$.

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

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Which two factors decide the formula in the traffic decision tree?
   A. Number of sources (infinite/finite) and treatment of blocked calls
   B. Holding time and busy hour
   C. Number of trunks and GoS
   D. Arrival and service rates
   Answer: A. Sources infinite or finite, and blocked calls cleared, held or delayed. (Key point: Decision tree factors)

2. Infinite sources with blocked calls cleared uses:
   A. Poisson
   B. Erlang B
   C. Engset
   D. Binomial
   Answer: B. Infinite/cleared is Erlang B; infinite/held is Poisson; finite/cleared is Engset. (Key point: Formula choice)

3. 20 servers carry 10 E with h = 3 min. Calls per hour for the group?
   A. 100
   B. 200
   C. 300
   D. 600
   Answer: B. C = 10 × 60/3 = 200. (Key point: Group calls)

4. Calls per hour for a single server?
   A. 5
   B. 10
   C. 20
   D. 30
   Answer: B. 0.5 E each: 0.5 × 60/3 = 10; check 20 × 10 = 200. (Key point: Single server)

## Related in Telecommunications

- [Calls handled by 30 servers carrying 15 E](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Aservers-15e-calls-per-hour) — Telecommunications, chapter 6, asked 3×
- [Calls handled by 25 servers carrying 5 E](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Aservers-25-5e-calls) — Telecommunications, chapter 6, asked 3×
- [Decision tree; infinite source blocking formulas](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Adecision-tree-infinite-sources) — Telecommunications, chapter 6, asked 2×
- [Decision tree; finite source blocking formulas](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Adecision-tree-finite-sources) — Telecommunications, chapter 6, asked 2×
- [Traffic intensity, GOS and call duration probability](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Atraffic-intensity-call-duration-probability) — Telecommunications, chapter 6
- [Traffic numerical: 800 calls, 50 lost](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Agos-800-calls-50-lost) — Telecommunications, chapter 6, asked 6×

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