---
title: "Decision tree; finite source blocking formulas"
subject: "Telecommunications"
chapter: 6
key: "semester_8/telecommunications:decision-tree-finite-sources"
source: https://www.bctengineer.com/q/semester_8/telecommunications/decision-tree-finite-sources
site: BCT Engineering
---

# Decision tree; finite source blocking formulas

Telecommunications · Chapter 6: Telephone Traffic

- Asked in: 2072 Chaitra, 4+4 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-finite-sources
- Whole chapter: https://www.bctengineer.com/md/semester_8/telecommunications/chapter-6

## Question

What are the formulas used in telecommunication traffic engineering decision tree? Describe the blocking formulas used in finite sources.

## Short answer

The decision tree selects formulas by infinite/finite sources, loss/delay, and blocked-call treatment (cleared/held/delayed). For finite sources with cleared calls, Engset gives call congestion B = [C(N-1,n)β^n]/ΣC(N-1,i)β^i; with held calls and N≤n, Binomial blocking = Σ_{k=n}^{N-1}C(N-1,k)a^k(1-a)^{N-1-k}.

## Answer

#### Decision tree of traffic formulas

The right traffic formula depends on three questions: is the number of sources large (infinite) or small (finite)? Is the system a loss system or a delay system? And what happens to a blocked call? A blocked call can be **cleared** (LCC: it is lost and the user goes away), **held** (LCH: the user keeps trying for the holding time) or **delayed** (LCD: it waits in a queue).

```text
                  Traffic formula
                        |
         +--------------+--------------+
    Infinite sources            Finite sources
   (Poisson arrivals)          (N users, N small)
         |                             |
  +------+------+               +------+------+
 LCC   LCH    LCD              LCC   LCH    LCD
  |     |      |                |     |      |
Erlang Poisson Erlang         Engset Binomial Finite
  B            C                            queue
```

| Source model | Blocked calls | Formula used |
|---|---|---|
| Infinite | Cleared (LCC) | Erlang B |
| Infinite | Held (LCH) | Poisson |
| Infinite | Delayed (LCD) | Erlang C |
| Finite | Cleared (LCC) | Engset |
| Finite | Held (LCH) | Binomial |
| Finite | Delayed (LCD) | Finite-source queue (Palm/Molina machine-repair model) |

Main formulas:

- **Erlang B** (A erlang offered to n trunks): $E_{1,n}(A) = \dfrac{A^n/n!}{\sum_{k=0}^{n} A^k/k!}$
- **Erlang C** (probability of delay): $E_{2,n}(A) = \dfrac{\frac{A^n}{n!}\frac{n}{n-A}}{\sum_{k=0}^{n-1}\frac{A^k}{k!} + \frac{A^n}{n!}\frac{n}{n-A}}$
- **Poisson** (LCH): $P = e^{-A}\sum_{k=n}^{\infty} \dfrac{A^k}{k!}$

#### Blocking formulas for finite sources

When the number of sources $N$ is not much larger than the number of servers $n$, the arrival rate falls as more sources become busy. Erlang B then overestimates the blocking, so finite-source formulas are used.

**1. Engset formula (lost calls cleared, N > n).** Let $\beta$ be the offered traffic per idle source. The probability that $k$ servers are busy is

$$
P(k) = \frac{\binom{N}{k}\beta^k}{\sum_{i=0}^{n}\binom{N}{i}\beta^i}, \quad k = 0,1,\dots,n
$$

- **Time congestion** (fraction of time all n servers are busy): $E = P(n)$.
- **Call congestion** (fraction of calls lost): an arriving call comes from an idle source, so it sees only the other $N-1$ sources:

$$
B = \frac{\binom{N-1}{n}\beta^n}{\sum_{i=0}^{n}\binom{N-1}{i}\beta^i}
$$

For finite sources $B < E$, because fewer idle sources exist to make calls when the group is busy.

**2. Binomial formula (lost calls held, or N ≤ n).** Each source is busy with probability $a$, independently of the others, so the number of busy sources follows

$$
P(k) = \binom{N}{k} a^k (1-a)^{N-k}
$$

A call from one source is blocked when $n$ or more of the other $N-1$ sources are already busy, so the blocking probability is

$$
P(\text{blocking}) = \sum_{k=n}^{N-1}\binom{N-1}{k}a^k(1-a)^{N-1-k}
$$

If $N \le n$ the sum is empty and there is no blocking at all.

As $N \to \infty$ with total traffic fixed, Engset tends to Erlang B and the binomial formula tends to the Poisson formula.

## Check yourself

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

1. Finite sources with blocked calls cleared use?
   A. Erlang B
   B. Poisson
   C. Erlang C
   D. Engset
   Answer: D. Engset is the finite LCC formula. (Key point: Finite LCC)

2. In Engset, how do B and E compare?
   A. B < E
   B. B > E
   C. B = E always
   D. B is unrelated
   Answer: A. An arriving call sees only N-1 other sources, fewer being idle when busy. (Key point: Call vs time congestion)

3. Call congestion uses which source count?
   A. N+1
   B. N-1
   C. N
   D. n
   Answer: B. The arriving call is from one of the other N-1 sources. (Key point: Call congestion)

4. If N <= n in the binomial model?
   A. All calls are blocked
   B. Erlang B applies
   C. No blocking occurs
   D. Delay is infinite
   Answer: C. Every source can always find a server. (Key point: Binomial)

## Close variants set in other exams

- [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×

## Related in Telecommunications

- [Decision tree formulas; 20 servers, 10 E](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Adecision-tree-20-servers) — Telecommunications, chapter 6
- [Infinite-source blocking; 40 subscribers](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Ainfinite-source-blocking-40-subscribers) — Telecommunications, chapter 6