Homework 2
Due Date: Monday, Nov 28, 2016
Problem 1: Time is slotted and during every slot nodes A and B may choose
any of channels 1, 2, 3 to transmit. If they transmit in the same channel a
collision occurs. At every time slot they change the frequency they are using
(e.g. to not be detected) pseudo-randomly. The changes from channel i to
B
channel j are driven by a 3×3 Markov chain, where pA
ij (or pij ) is the probability
that A (or B) moves to channel j in the next slot, given that it is now using
channel i.
Assuming that node A starts from channel 1, and node B from channel 3,
find the expected time until a collision occurs (i.e. both nodes choose the same
channel) in the following cases:
1/3
(Case 1) PA = 1/3
1/3
1/3
1/3
1/3
0.2
(Case 2) PA = 0.4
0.5
0.6
0.3
0.1
1/3
1/3
1/3 PB = 1/3
1/3
1/3
0.2
0.4
0.3 PB = 0.2
0.4
0.3
1/3
1/3
1/3
0.3
0.5
0.3
1/3
1/3
1/3
0.3
0.3
0.4
Problem 2 You start downloading 2 files in parallel. The time to complete
downloading file 1 is exponentially distributed with parameter λ1 , and the time
to download file 2 is independent from file 1, and exponentially distributed with
parameter λ2 .
1. What is the expected time until at least one file has finished downloading
(whichever finishes first)?
2. What is the probability that file 1 downloads first?
3. What is the expected time until both downloads are finished?
4. Assume you download n files in parallel (all exponential with the same
parameter λ and independent). What is the time until all finish downloading?
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Problem 3: A new malware is out in the Internet! Your goal is to estimate its
spread/damage by time t, assuming it starts at time 0.
• New Internet hosts get infected by this malware according to a Poisson
process with parameter λ, where λ is not known.
• You company has installed a Honeypot security system to detect whether
hosts are infected. Unfortunately there is a delay between when a computer is infected by the malware and the time the Honeypot detects the
damage. Assume that this delay is distributed as Exp(µ).
• Suppose that the Honeypot system has detected N1 (t) infected hosts by
time t. Your boss is worried that, because of the delay, the number of
infected hosts is actually much higher than N1 (t).
• The goal of the problem is to understand how many additional hosts,
N2 (t), are expected to also be infected at time t.
Please, answer the following questions:
1. Suppose that an infection happens at time s, where 0 < s < t. What is
the probability that the infection is detected by time t?
2. Consider an arbitrary infection that happens before time t. What is the
(unconditional) probability, p, that the infection is detected by the Honeypot by time t?
3. How can we use our knowledge of N1 (t) to estimate λ?
4. Use your estimate of to determine the expected value of N2 (t).
[Note: None of the above solutions requires more than a couple lines.]
Problem 4: Exercise 13.5
Problem 5: Exercise 14.6 (Only (a) and (b)!)
Problem 6: Exercise 17.2 (based on material we will cover on Monday, Nov.
21)
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