---
title: "Traffic intensity, GOS and call duration probability"
subject: "Telecommunications"
chapter: 6
key: "semester_8/telecommunications:traffic-intensity-call-duration-probability"
source: https://www.bctengineer.com/q/semester_8/telecommunications/traffic-intensity-call-duration-probability
site: BCT Engineering
---

# Traffic intensity, GOS and call duration probability

Telecommunications · Chapter 6: Telephone Traffic

- Asked in: 2076 Chaitra, 2+2+2+2 marks
- Video: [Trunking and Grade of Service (GOS) | Traffic Intensity](https://www.youtube.com/watch?v=OGGEmAjMx7I) — ECIT Engineering
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_8/telecommunications/traffic-intensity-call-duration-probability
- Whole chapter: https://www.bctengineer.com/md/semester_8/telecommunications/chapter-6

## Question

Define Traffic Intensity and Grade of Service (GOS). In a telephone system, the average call duration is 2 minutes. A call has already lasted 4 minutes. What is the probability that: a) The call will last at least another 4 minutes? b) The call will end within the next 4 minutes?

## Answer

#### Traffic intensity
Average number of simultaneous calls in a period, in Erlangs: $A = \dfrac{C\,h}{T}$ (C calls, mean holding time h, period T). 1 E = one circuit busy all the time.

#### Grade of Service
Proportion of calls lost (or excessively delayed) in the busy hour: $GoS = \dfrac{\text{calls lost}}{\text{calls offered}}$; smaller is better.

#### Numerical
Call durations are taken as **negative exponential** with mean $h = 2$ min:
$$
P(T > t) = e^{-t/h}
$$
The exponential distribution is **memoryless**: the remaining duration of a call does not depend on how long it has already lasted. So the fact that the call has lasted 4 min does not change the answer.

**(a) Lasts at least another 4 min**
$$
\begin{aligned}
P(T > 8 \mid T > 4) &= P(T > 4) = e^{-4/2} \\
&= e^{-2} = 0.1353
\end{aligned}
$$

**(b) Ends within the next 4 min**
$$
\begin{aligned}
P(T \le 8 \mid T > 4) &= 1 - e^{-2} \\
&= 0.8647
\end{aligned}
$$

**Answer:** (a) **0.1353** (about 13.5%), (b) **0.8647** (about 86.5%).

## Check yourself

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

1. What is traffic intensity?
   A. The total number of calls in an hour
   B. The fraction of calls lost
   C. The longest call duration
   D. The average number of simultaneous calls over a period, in Erlangs
   Answer: D. A = C h / T; 1 E means one circuit busy continuously. (Key point: Traffic intensity)

2. Why does the 4 minutes already elapsed not change the answer?
   A. Exponential holding time is memoryless
   B. Call durations are constant
   C. Calls always end after 8 minutes
   D. Elapsed time is added to the mean
   Answer: A. For the exponential distribution the remaining time is independent of the elapsed time. (Key point: Memoryless property)

3. Probability the call lasts at least another 4 min when mean is 2 min?
   A. e^-4 = 0.0183
   B. e^-2 = 0.1353
   C. 1 - e^-2 = 0.8647
   D. e^-1 = 0.368
   Answer: B. P(T>4) = e^(-4/2) = e^-2. (Key point: Remaining time probability)

4. Probability the call ends within the next 4 min?
   A. 0.1353
   B. 0.5
   C. 0.8647
   D. 0.9817
   Answer: C. It is 1 - e^-2 = 0.8647. (Key point: Complement)

## Related in Telecommunications

- [Traffic intensity, GOS and national numbering](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Atraffic-intensity-gos-numbering) — Telecommunications, chapter 6
- [GOS and blocking probability in a loss system](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Agos-blocking-loss-system) — Telecommunications, chapter 6
- [GOS vs blocking; 600 calls, 2.5 min](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Agos-blocking-600-calls-2-5min) — Telecommunications, chapter 6, asked 6×
- [GOS numerical: 1400 calls, 14 lost](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Agos-busy-hour-1400-calls) — Telecommunications, chapter 6, asked 5×
- [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×
- [Traffic numerical: 600 calls, 40 lost, 120 s](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Agos-600-calls-120s) — Telecommunications, chapter 6, asked 5×

## Related in other subjects

- [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
- [Intensity-duration curve from storm record](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Aintensity-duration-30min-5hr) — Engineering Hydrology, chapter 2
- [Intensity-duration graph from mass curve ordinates](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Aintensity-duration-15-180min) — Engineering Hydrology, chapter 2
- [Hyetograph and intensity-duration curve, 6:00-12:00 storm](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Ahyetograph-intensity-duration-6to12) — Engineering Hydrology, chapter 2