A group of 20 trunks provides a grade of service of 0.01 when offered 12E of traffic. a. How much is the grade of service improved if one extra trunk is added to the group? b. How much does the grade of service deteriorate if one trunk is out of service? c. Give your observation based on the results of a and b.
The correct answer and explanation is:
Let’s analyze the problem step-by-step. We have:
- Original group size: 20 trunks
- Offered traffic: 12 Erlangs (E)
- Grade of Service (GoS) with 20 trunks: 0.01 (blocking probability)
a. Grade of Service improvement if one trunk is added (from 20 to 21 trunks):
To find the new grade of service when one trunk is added, calculate the blocking probability for 21 trunks carrying 12 Erlangs of traffic using the Erlang B formula: B(E,N)=ENN!∑k=0NEkk!B(E, N) = \frac{\frac{E^N}{N!}}{\sum_{k=0}^N \frac{E^k}{k!}}
Where:
- EE is the offered traffic in Erlangs (12)
- NN is the number of trunks
Calculating exact values can be complex manually, but using Erlang B tables or calculators:
- GoS for 20 trunks and 12E = 0.01 (given)
- GoS for 21 trunks and 12E ≈ 0.0065
Improvement = 0.01 – 0.0065 = 0.0035
So, adding one trunk reduces blocking probability from 1% to about 0.65%.
b. Grade of Service deterioration if one trunk is out of service (from 20 to 19 trunks):
Similarly, calculate GoS for 19 trunks:
- GoS for 19 trunks and 12E ≈ 0.015
Deterioration = 0.015 – 0.01 = 0.005
So, losing one trunk increases blocking probability from 1% to about 1.5%.
c. Observation:
The results show that the system’s grade of service is more sensitive to the loss of a trunk than to the gain of one. Removing a trunk (from 20 to 19) causes a 0.5% increase in blocking probability, while adding a trunk (from 20 to 21) reduces blocking probability by only 0.35%. This is because blocking probability decreases non-linearly with the number of trunks, especially near the operational point.
In practical terms, the network experiences a larger negative impact when a trunk fails than the benefit gained from adding an extra trunk. This highlights the importance of system reliability and maintenance to avoid service degradation. Designing systems with some trunk redundancy is beneficial to maintain acceptable quality of service during outages.
Summary:
- Adding one trunk reduces blocking probability by 0.0035 (from 0.01 to 0.0065).
- Losing one trunk increases blocking probability by 0.005 (from 0.01 to 0.015).
- The system is more sensitive to trunk loss than gain, emphasizing reliability importance.