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May Examination Period 2025/2026
ECS660U Modelling and Performance Duration: 3 hours
Answer ALL questions
The total duration of this exam is three hours. This includes time for downloading the paper,
formatting and uploading your answers.
You MUST submit your answers before the exam end time.
You must follow the online exam guidelines and instructions on the EECS exam access and
submission page.
This is an open-book exam. You may use lecture notes and any module materials made available
to you (online or physical). You must not use other online resources.
YOU MUST COMPLETE THE EXAM ON YOUR OWN, WITHOUT CONSULTING OTHERS.
Examiners:
Dr Ma
Dr Masaracchia
Queen Mary University of London, 2026
Page 2 ECS660U (2026)
Question 1
Figure 1.1 shows an unweighted and directed graph.
Figure 1.1
a) Derive an adjacency matrix for the graph and use the matrix to find the in- and out degree of each node. Identify the source and sink nodes in the graph.
[5 marks]
b) Calculate the average shortest path in the graph. Ignore any unreachable pairs.
[3 marks]
Now, assuming the graph in Figure 1.1 is undirected, answer the following questions:
c) Calculate the local clustering coefficient for each node. What is the global clustering
coefficient of the graph List all the triplets.
[7 marks]
d) You are tasked to use the Failure Propagation Model to simulate the spread of a failure
in the graph (undirected version of the network in Figure 1.1).
i) Assuming the model parameter β is set to 0.3, and the initial failure occurs at Node 2.
Explain, with an aid of a network diagram, why Node 8 will be immune to the initial
failure.
ii) If β is now set to 0.24, how the propagation of the initial failure would be different
Illustrate step-by-step how the initial failure would spread in this case.
[10 marks]
3
4
5
6
7 1
2 8
ECS660U (2026) Page 3
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Question 2
Analyses were performed on an unknown network, and results are shown in Figure 2.1.
Figure 2.1
a) Refer to Panel A and answer the following questions:
i) What conclusions can you draw from the degree distribution of the network
ii) The degree distribution fluctuates and does not form a smooth line. Is it what you
expected Justify your answer.
iii) What network model would you use to generate pseudo networks that would exhibit
comparable properties. Justify your answer.
[10 marks]
b) Refer to Panel B and answer the following questions:
i) Describe the analysis that have been performed which have led to the results shown
in Figure 2.1B.
ii) What conclusions can you draw from Figure 2.1B
iii) The effectiveness of the analysis used here has been questioned. Explain why this
might be case and suggest how you would modify the analysis to ensure statistical
robustness.
[15 marks]
B A
Degree k
P(k)
10 100 1000
100
10-1
10-2
10-3
10-4
K0

10 100 1000
1000
100
10
Empirical
Null
Page 4 ECS660U (2026)
Question 3
a) Suppose that you are rolling a 12-face dice 120 times, and you observe the following
results:
5 times the face 1
8 times the face 2
9 times the face 3
10 times the face 4
12 times the face 5
7 times the 6
15 times the 7
6 times the 8
11 times the 9
14 times the 10
13 times the 11
10 times the 12
i) Calculate the relative frequency of each face.
ii) What is the relationship between the relative frequency and the probability of an event
iii) Assuming that the dice has been rolled an infinite number of times, and that you
observe the following probabilities for each face shown in Table 3.1.
Face Probability
1 0.05
2 0.08
3 0.07
4 0.10
5 0.12
6 0.09
7 0.15
8 0.06
9 0.11
10 0.09
11 0.04
12 0.04
Table 3.1
State whether the dice is fair or not and justify your answer.
[5 marks]
This question continues over the page…
ECS660U (2026) Page 5
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b) You are provided with the function illustrated in the Figure 3.1.
Figure 3.1
i) Explain why this can represent a Probability Mass Function.
ii) Calculate the mean value and the standard deviation.
iii) What is the probability to observe a value less than or equal to 200 Assume the
events are independent.
[5 marks]
c) Suppose you are tasked to characterise the reliability of a generic system. You are
provided with N = 10 identical replicas of this system, and the failure times (in months)
for each copy are:
Copy 1 = 10.4 months
Copy 2 = 9.2 months
Copy 3 = 8.8 months
Copy 4 = 7.6 months
Copy 5 = 13 months
Copy 6 = 14 months
Copy 7 = 6.1 months
Copy 8 = 8.7 months
Copy 9 = 12 months
Copy 10 = 10.2 months
i) Based on these observations, provide the estimation of the failure rate of the
considered system.
ii) Which type of distribution is typically used to model the reliability of a system Provide
the corresponding probability density function.
iii) What is the probability that the considered system will still work after 2 years
[5 marks]
This question continues over the page…
100 150 200 250 300 350 400 450
0.05
0.15
0.25
P(x)
x
Page 6 ECS660U (2026)
d) Suppose that you are requested to evaluate the reliability of the system shown in
Figure 3.2:
Figure 3.2
Each sub-system has the following Mean Time Between Failure (MTBF):
System Name MTBF
SS1 3 months
SS2 6 months
SS3 18 months
i) What is the probability that the resulting system will still work after 1 year
ii) What is the improvement on reliability of the system if all the subsystems are
connected in parallel
[5 marks]
e) You have a system consisting of three parallel sub-systems with identical failure rates.
i) Calculate the MTBF in months assuming that the R(18 months) = 0.5.
ii) What is the reliability of the whole system after 2.5 years
[5 marks]
SS1
SS3
SS2
INPUT OUTPUT
ECS660U (2026) Page 7
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Question 4
a) Consider a system with two possible states, GREEN and RED. Suppose the transition
probabilities between these states are known and remain constant over time. The
system is modelled as a Discrete-Time Discrete-Space (DTDS) process, where the
transition probabilities at a generic time step ! are:
P(RED | GREEN) = p
P(GREEN | GREEN) = 1-p
P(GREEN | RED) = q
P(RED | RED) =1-q
i) Sketch the Markov chain diagram for the considered system.
ii) Determine the steady-state probability of each state.
iii) Assuming p = q = 0.2, which state is the system most likely to be in
[5 marks]
b) Given a discrete M/D/1 queue model, and the probability of having an empty queue,
a.k.a. P(0), is equal to 0.4.
i) What is the arrival rate λ in this queue system
ii) If the service rate is μ = 0.75, calculate the mean waiting time E(w) and the mean
elements in the queue.
[5 marks]
c) You are simulating a communication system in order to evaluate its performance in terms
of packet delay. You run a set of N=15 simulations, and obtain the following values for
the delays in milliseconds after each simulation reaches the steady state:
10, 5, 10, 15, 12, 100, 8, 25, 11, 4, 8, 26, 50, 2
i) Evaluate the estimated mean and 90% Confidence Intervals for these delay results.
You will need to use the t-distribution table.
ii) What do Confidence Intervals tell us
[5 marks]
This question continues over the page…
Page 8 ECS660U (2026)
d) Let us consider a local population of Ntot = N1 + N2 + N3 phones generating calls from A
to B in accordance with the following information:
N1= 100 phones generating calls with a rate of 0.01 per hour
N2= 150 phones generating calls with a rate of 0.02 per hour
N3= 200 phones generating calls with a rate of 0.03 per hour
Each call last 5 mins
i) Calculate the total applied traffic.
ii) What is the Grade of Service (GoS) if these calls are managed by an exchange with 3
lines Do not use the Poisson approximation.
iii) Using the B-Erlang table, provide the value of serving lines N the operator need in
order to guarantee GoS ≈ 0.01.
[5 marks]
A communication system with variable Bit Error Rate (BER) can be well described
using a DTDS Markow chain shown in Figure 4.1, with P1 = 0.3; P2 = 0.1 P3 = 0.35.
Figure 4.1
If you measure over a long enough period, what would you expect to find the overall
mean BER to be
[5 marks]
End of Paper
Low Error
Rate
(BER =0.001)
Medium
Error Rate
(BER =0.01)
High Error
Rate
(BER =0.1)
1-P1 P2
1-P3 1- P2
P1 P3