---
title: "Numerical: trunked channels and blocking (Erlang B)"
subject: "Wireless Communication"
chapter: 2
key: "semester_7/wireless_communication:trunking-five-channel-blocking"
source: https://www.bctengineer.com/q/semester_7/wireless_communication/trunking-five-channel-blocking
site: BCT Engineering
---

# Numerical: trunked channels and blocking (Erlang B)

Wireless Communication · Chapter 2: Cellular Mobile Communication Concepts

- Asked in: 2077 Chaitra, 2+2+2+3 marks
- Video: [Measurement of Telephone Traffic, Erlang B and Erlang C formulas examples](https://www.youtube.com/watch?v=BRKnfS9CGTQ) — Dr Warutumo, PhD
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_7/wireless_communication/trunking-five-channel-blocking
- Whole chapter: https://www.bctengineer.com/md/semester_7/wireless_communication/chapter-2

## Question

A mobile radio system where each user averages three calls per hour and each call lasting an average of 5 minutes.
a) What is traffic intensity for each user?
b) Find the number of users that could use the system with 1% blocking if only one channel is available.
c) Find the number of users that could use the system with 1% blocking if five trunked channels are available.
d) If the number of users you found in (c) is suddenly doubled, what is the new blocking probability of the five channel trunked mobile radio system? Justify whether the performance is acceptable or not.

Traffic (A) in erlangs for B%:

| No. of Trunks (N) | 0.1% | 0.2% | 0.5% | 1% | 1.2% | 3% | 5% | 7% |
|---|---|---|---|---|---|---|---|---|
| 1 | 0.001 | 0.002 | 0.005 | 0.010 | 0.012 | 0.031 | 0.053 | 0.075 |
| 2 | 0.046 | 0.065 | 0.105 | 0.153 | 0.168 | 0.282 | 0.381 | 0.470 |
| 3 | 0.194 | 0.249 | 0.349 | 0.455 | 0.489 | 0.715 | 0.899 | 1.06 |
| 4 | 0.439 | 0.535 | 0.701 | 0.869 | 0.922 | 1.26 | 1.52 | 1.75 |
| 5 | 0.762 | 0.900 | 1.13 | 1.36 | 1.43 | 1.88 | 2.22 | 2.50 |

## Short answer

a) Traffic per user is 0.25 Erlang. b) With one channel at 1% blocking, offered traffic is 0.010 E, supporting 0.04 user, effectively no user. c) With five channels, A=1.36 E, supporting 5.44, about 5 users. d) Doubling to 10 users gives A=2.5 E and about 7% blocking, which is unacceptable.

## Answer

**Given:** λ = 3 calls/hour, H = 5 min, blocked calls cleared (Erlang B).

#### a) Traffic intensity per user
```text
Au = λH = 3 × 5/60 = 0.25 Erlang
```

#### b) One channel, 1 % blocking
From the table, N = 1, B = 1 %: A = 0.010 Erlang.
```text
U = A / Au = 0.010 / 0.25 = 0.04 user
```
Fewer than one user can be supported: even a single user (0.25 E) would cause far more than 1 % blocking. So effectively **no user (only 0.04 user)** can be served at 1 % GOS with one channel.

#### c) Five trunked channels, 1 % blocking
From the table, N = 5, B = 1 %: A = 1.36 Erlangs.
```text
U = 1.36 / 0.25 = 5.44 ≈ 5 users
```

#### d) Number of users doubled
```text
U = 2 × 5 = 10 users
A = 10 × 0.25 = 2.5 Erlangs
```
From the table, N = 5 and A = 2.50 corresponds to **B = 7 %**. (Erlang B formula gives 6.97 %.)

If 5.44 users are doubled (10.88 users, A = 2.72 E), blocking is about 8.7 %.

**Is it acceptable?** No. The blocking probability rises from 1 % to about 7 %, i.e. 7 of every 100 call attempts fail in the busy hour. This is well above the usual design GOS of 1–2 %, so service quality is unacceptable. More channels are needed (from the Erlang B table, about 7 channels would be needed for 2.5 E at 1 %).

| Case | Channels | A (E) | Users | Blocking |
|---|---|---|---|---|
| b | 1 | 0.010 | 0.04 | 1 % |
| c | 5 | 1.36 | 5 | 1 % |
| d | 5 | 2.50 | 10 | ≈ 7 % |

Note: trunking 5 channels supports 5.44 users compared with 5 × 0.04 = 0.2 users on 5 separate single channels, showing the trunking gain.

## Check yourself

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

1. With 3 calls/hour of 5 minutes each, traffic per user is:
   A. 15 Erlang
   B. 0.25 Erlang
   C. 0.6 Erlang
   D. 1.5 Erlang
   Answer: B. Au = 3 x 5/60 = 0.25 E. (Key point: Traffic per user)

2. With 5 channels at 1% GOS the table gives 1.36 E. The number of users is:
   A. about 1
   B. about 340
   C. about 0.04
   D. about 5
   Answer: D. 1.36/0.25 = 5.44, so about 5 users. (Key point: Users on 5 channels)

3. If the users double to 10 (A = 2.5 E) with 5 channels, the blocking is about:
   A. 7%
   B. 1%
   C. 2%
   D. 0.5%
   Answer: A. The table gives 7% for 5 channels at 2.5 E. (Key point: Doubled users)

4. Is a 7% blocking probability acceptable?
   A. Yes, since 1% is only a guideline
   B. Yes, because blocking below 10% is fine
   C. No, it is far above the typical 1-2% design GOS; more channels are needed
   D. No, because blocking should be exactly zero
   Answer: C. About 7 channels would be needed for 2.5 E at 1%. (Key point: Acceptability)

## Related in Wireless Communication

- [Numerical: users supported and market penetration](https://www.bctengineer.com/question.md?key=semester_7%2Fwireless_communication%3Amarket-penetration-erlang) — Wireless Communication, chapter 2, asked 2×
- [Numerical: 12-cell system capacity at 2% GOS](https://www.bctengineer.com/question.md?key=semester_7%2Fwireless_communication%3Acellular-capacity-12-cell-system) — Wireless Communication, chapter 2
- [Numerical: 32-cell system area and channels](https://www.bctengineer.com/question.md?key=semester_7%2Fwireless_communication%3Acellular-32-cells-capacity) — Wireless Communication, chapter 2
- [Numerical: trunking capacity loss with sectoring](https://www.bctengineer.com/question.md?key=semester_7%2Fwireless_communication%3Atrunking-loss-sectoring) — Wireless Communication, chapter 2
- [GSM frame structure and channels during a call](https://www.bctengineer.com/question.md?key=semester_7%2Fwireless_communication%3Agsm-frame-call-channels) — Wireless Communication, chapter 8
- [Numerical: N-TDMA simultaneous users](https://www.bctengineer.com/question.md?key=semester_7%2Fwireless_communication%3Antdma-simultaneous-users) — Wireless Communication, chapter 7

## Related in other subjects

- [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×
- [Device holding time and calls per hour](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Adevice-holding-time-calls) — Telecommunications, chapter 6
- [Traffic numerical: 1000 calls, 5 lost](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Atraffic-1000-calls-5-lost) — Telecommunications, chapter 6, asked 6×
- [Traffic numerical: 1200 calls, 6 lost](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Atraffic-1200-calls-6-lost) — Telecommunications, chapter 6, asked 6×