---
title: "Pure loss system and Engset model"
subject: "Telecommunications"
chapter: 6
key: "semester_8/telecommunications:pure-loss-engset-model"
source: https://www.bctengineer.com/q/semester_8/telecommunications/pure-loss-engset-model
site: BCT Engineering
---

# Pure loss system and Engset model

Telecommunications · Chapter 6: Telephone Traffic

- Asked in: 2072 Kartik, 4+6 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_8/telecommunications/pure-loss-engset-model
- Whole chapter: https://www.bctengineer.com/md/semester_8/telecommunications/chapter-6

## Question

What is pure loss system? Explain Engset model.

## Short answer

A pure loss system clears blocked calls immediately. The Engset model handles finite N>n sources, arrival rate (N-k)γ, and state distribution P(k) ∝ C(N,k)β^k. Time congestion E = P(n); call congestion B = E_n(N-1,β) with B<E. Carried traffic Y = A(1-B).

## Answer

#### Pure loss system

A **pure loss system** is a system with $n$ servers and no queue. A call arriving when all $n$ servers are busy is **lost (cleared)** immediately, and the user does not retry. Its quality is measured by the probability of blocking (grade of service). Examples: a trunk group between two exchanges, a PABX with a few outgoing lines, the radio channels of a mobile cell. Erlang B (infinite sources) and Engset (finite sources) are the standard pure loss models.

Two blocking measures are used:

- **Time congestion E:** fraction of time all servers are busy.
- **Call congestion B:** fraction of call attempts that are lost.

#### Engset model

The Engset model is a pure loss model with a **finite number of sources** $N$ greater than the number of servers $n$ ($N > n$). It is used when $N$ is small, e.g. a PABX of 20 extensions sharing 5 trunks.

**Assumptions**

1. $N$ identical, independent sources.
2. An idle source makes calls at rate $\gamma$; a busy source makes no new call. So in state $k$ the arrival rate is $(N-k)\gamma$, which falls as $k$ increases.
3. Holding times are exponential with mean $h = 1/\mu$.
4. Blocked calls are cleared; full availability.

**State diagram**

```text
   N.g      (N-1)g            (N-n+1)g
 0 -----> 1 -----> 2 ... n-1 ---------> n
   <-----   <-----             <-------
    mu       2mu                 n.mu
```

**State probabilities.** The cut equations give

$$
(N-k)\gamma\,P(k) = (k+1)\mu\,P(k+1)
$$

With $\beta = \gamma/\mu$ (offered traffic per idle source):

$$
P(k) = \frac{\binom{N}{k}\beta^k}{\sum_{i=0}^{n}\binom{N}{i}\beta^i}, \qquad 0 \le k \le n
$$

This is a truncated binomial distribution.

**Time congestion**

$$
E = P(n) = \frac{\binom{N}{n}\beta^n}{\sum_{i=0}^{n}\binom{N}{i}\beta^i}
$$

**Call congestion.** An arriving call comes from one of the idle sources, so it sees the system as if there were only $N-1$ sources:

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

**Traffic.** Carried traffic $Y = \sum k P(k)$; offered traffic per source $a = \beta/(1+\beta(1-B))$; offered traffic $A = N a$, and $A(1-B) = Y$.

**Properties**

- $B < E$ for finite sources, because when the group is busy fewer idle sources remain to call.
- As $N \to \infty$ with $N\gamma$ fixed, Engset tends to **Erlang B**.
- If $N \le n$ there is no blocking, and the distribution becomes the full **binomial**.
- Erlang B applied to a small group gives a pessimistic (too high) blocking, so Engset allows fewer trunks for the same GoS.

## Check yourself

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

1. How is the arrival rate in state k in Engset?
   A. N gamma constant
   B. k gamma
   C. gamma/k
   D. (N-k) gamma, falling with k
   Answer: D. Busy sources make no new calls. (Key point: Engset assumption)

2. State probabilities are?
   A. Truncated binomial
   B. Poisson
   C. Geometric
   D. Uniform
   Answer: A. P(k) proportional to C(N,k) beta^k for k up to n. (Key point: Distribution)

3. Why is B < E?
   A. More sources call when busy
   B. Fewer idle sources remain when the group is busy
   C. Holding times shorten
   D. Servers are added
   Answer: B. Arriving calls see N-1 sources. (Key point: Call congestion)

4. As N tends to infinity with N gamma fixed, Engset tends to?
   A. Binomial
   B. Erlang C
   C. Erlang B
   D. Zero blocking
   Answer: C. Finite source becomes infinite source. (Key point: Limit)

## Related in Telecommunications

- [Infinite and pure loss systems; Engset model](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Ainfinite-pure-loss-engset) — Telecommunications, chapter 6
- [Pure loss system and binomial model](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Apure-loss-binomial-model) — Telecommunications, chapter 6