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

# Infinite and pure loss systems; Engset model

Telecommunications · Chapter 6: Telephone Traffic

- Asked in: 2080 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/infinite-pure-loss-engset
- Whole chapter: https://www.bctengineer.com/md/semester_8/telecommunications/chapter-6

## Question

What is infinite system and pure loss system? Describe the tele-traffic Engset model.

## Short answer

An infinite-source system has constant arrival rate as sources are many. A pure loss system clears blocked calls with no queue. The Engset model is a pure loss system with finite sources S>N; arrival rate falls as more sources are busy. State probabilities use binomial coefficients; time congestion is P_N, and call congestion B uses (S-1 choose N), giving B < E.

## Answer

#### Infinite (source) system
An **infinite-source system** is one in which the number of traffic sources (subscribers) is very large compared with the number of servers. The call arrival rate therefore does **not** depend on how many sources are already busy: arrivals are Poisson with a constant mean rate $\lambda$. Example: a junction route between two large exchanges. Erlang B, Erlang C and Poisson formulas assume infinite sources.

#### Pure loss system
A **pure loss (lost-calls-cleared) system** has N servers and **no waiting room**. A call that finds all N servers busy is rejected (lost) and disappears; it is not queued and is assumed not to retry. Its quality is measured by the probability of loss (GoS). In Kendall notation, M/M/N/N. Example: a group of trunks with no queue; Erlang B formula.

#### Engset model
The **Engset model** is a **pure loss system with a finite number of sources** S (S > N), such as a PBX with 40 extensions sharing 8 outgoing lines. As more sources become busy, fewer idle sources remain to make calls, so the arrival rate falls.

**Assumptions**
1. S identical sources, N servers, full availability, $S > N$.
2. Each **idle** source makes calls at rate $\gamma$; mean holding time $h = 1/\mu$. Offered traffic per idle source $\alpha = \gamma/\mu = \gamma h$.
3. Blocked calls are cleared; statistical equilibrium.

**State diagram (birth–death)**
```text
   S*g     (S-1)*g          (S-N+1)*g
 0 ----> 1 ------> 2 ... N-1 ---------> N
   <----   <------     <---------------
    mu      2*mu              N*mu
```
Arrival rate in state n is $(S-n)\gamma$; departure rate is $n\mu$.

**State probabilities** (from cut equations $(S-n+1)\gamma P_{n-1} = n\mu P_n$):
$$
P_n = \frac{\binom{S}{n}\alpha^n}{\sum_{k=0}^{N}\binom{S}{k}\alpha^k},\quad n = 0,1,\dots,N
$$

**Time congestion** (fraction of time all servers busy):
$$
E = P_N = \frac{\binom{S}{N}\alpha^N}{\sum_{k=0}^{N}\binom{S}{k}\alpha^k}
$$

**Call congestion** (Engset loss formula, probability that a call attempt is lost), which is what the caller experiences:
$$
B = \frac{\binom{S-1}{N}\alpha^N}{\sum_{k=0}^{N}\binom{S-1}{k}\alpha^k}
$$
Here $B < E$, because when all servers are busy fewer sources are free to call.

**Limits:** if $S \to \infty$ with $S\alpha = A$ constant, both formulas tend to Erlang B. If $S \le N$ there is no loss and the busy-server distribution is binomial (Bernoulli).

**Use:** dimensioning PBX trunks, small rural exchanges, concentrators and cell sectors with few users, where Erlang B would over-estimate blocking.

## Check yourself

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

1. Why is the arrival rate constant in an infinite-source system?
   A. Sources are very many compared with servers, so busy sources hardly matter
   B. Calls are queued
   C. Holding time is fixed
   D. Servers are unlimited
   Answer: A. With very many sources, arrivals do not depend on how many are busy. (Key point: Infinite source)

2. In a pure loss system a call that finds all servers busy:
   A. Waits in a queue
   B. Is rejected and does not retry
   C. Retries after a delay
   D. Is held until a server frees
   Answer: B. Lost-calls-cleared: no waiting room. (Key point: Pure loss)

3. In the Engset model, arrival rate in state n is:
   A. λ
   B. (S - n)γ
   C. nμ
   D. Sγ always
   Answer: B. Only idle sources (S - n) generate calls. (Key point: Engset arrival rate)

4. Why is Engset call congestion B less than time congestion E?
   A. Holding times are shorter
   B. When all servers are busy fewer sources are free to call
   C. Calls are queued
   D. Servers are more reliable
   Answer: B. Arrivals see fewer idle sources in the all-busy state. (Key point: Call vs time congestion)

## Related in Telecommunications

- [Pure loss system and Engset model](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Apure-loss-engset-model) — 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
- [Traffic engineering objectives; Engset formula](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Atraffic-objectives-engset) — Telecommunications, chapter 6

## Related in other subjects

- [Uniaxial, biaxial and pure shear stress systems](https://www.bctengineer.com/question.md?key=semester_4%2Fstrength_of_materials_mechanical%3Auniaxial-biaxial-pure-shear-systems) — Strength of Materials, chapter 6