exam2019Jan.pdf
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Prof. dr. ir. P. Van Mieghem 25 Jan. 2019, 13.30-16.30
Examination Performance Analysis (IN4-341)
Question 1: Derive the formula for the virtual waiting time (also called system time) in an
M/M/1 queue under FIFO service discipline in the steady state. Give su¢ cient arguments and
explain the steps in your derivation clearly.
Question 2: Consider a continuous-time Markovian SIS epidemic process on a graph with
N nodes. The eigenvalues of the adjacency matrix A are ordered as
1
2
N
. The
infection rate and the curing rate are the same for each node and link in the graph. This
SIS process features a phase transition in the e¤ective infection rate =
. Demonstrate (and
explain!) that there is a phase transition and relate that critical -regime to properties of the
graph.
Hint: start from
dE[X
i
(t)]
dt
= E
h
X
i
(t) + (1 X
i
(t))
P
N
k=1
a
ki
X
k
(t)
i
for Pr [X
i
(t) = 1] =
E [X
i
(t)], where X
i
(t) is the infection state of node i at time t, and deduce a linear matrix
di¤erential equation as upper bound.
Question 3: (a) Given that a Poisson event has occurred in the interval [0; T ], its occurrence
time is uniformly distributed over [0; T ]. Demonstrate and explain.
(b) What is the di¤erence between a transient state and a null recurrent state in a Markov
process? Is it possible to have a null recurrent state in a …nite state Markov process?
Question 4: The number of viewers watching TV from 19:00 to 20:00 on a certain day
follows a Poisson distribution. The average number of TV viewers within this time interval is
1000. Among the TV channel list, each viewer has a probablity of 10% to choose the “Discovery”
TV channel. Viewers of “Discovery”are independent from each other.
(a) Prove that the number of viewers of “Discovery”follows a Poisson distribution.
(b) What is the probablity that there are fewer than four viewers of “Discovery”from 19:00
to 20:00 on that certain day?
Question 5: An ISP wants to have an idea of the end-to-end delay D along a path in the
Internet. Assume the delay T incurred in each link is an i.i.d. exponential random variable
with mean 1=. The random variable D =
P
H
N
j=1
T
j
, where T
j
is the delay in each constituent
link of the path and H
N
is the hopcount.
(a) Consider a speci…c path with a constant hopcount H
N
= k. What is the generating
function of the end-to-end delay
P
k
j=1
T
j
?
(b) The ISP learned from Performance Analysis that the hopcount H
N
of an arbitrary path
in the Internet is not a constant but approximately follows the generating function
'
H
N
(z) =
N
N 1
(N + z)
N! (z + 1)
1
N
Assume that the hopcount of a path is independent of the delay incurred in a link. Compute
the generating function of the end-to-end delay D =
P
H
N
j=1
T
j
in this case. Hint: Use the law
of total probability on the pgf of D.
1
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