University Problem Set #8 ECE 313
of Illinois Page 1 of 2 Spring 2002
Assigned: Wednesday, March 6
Due: Wednesday, March 13
Reading: Ross, Chapter 4, Section 9; and Chapter 5
Noncredit Exercises: Chapter 5: Problems 1-8; Theoretical Exercises: 1, 8
Note: In honor of Engineering Open House, this Problem Set has only four
problems on it…
Problems:
1. [“… It’s not your father’s Oldsmobile…”] A system works if and only if all of its M
components (numbered 1 through M) work. Each component fails (independently) with
probability p. Consider two possible means of obtaining a more reliable system. We can
replicate each component N times as shown in the graph model on the left. Or, we can
replicate the entire system N times as shown in the graph model on the right. In either
case, the result is called a replicated system. Note that both methods use the same number
N of each component .
N-ary replication of each component N-ary replication of entire system
1.1
1.2
1.3
1.N
2.1
2.2
2.3
2.N
M.1
M.2
M.3
M.N
System.1
System.2
System.3
System.N
A concrete example of the question we wish to consider is:
Which of the following two methods provides more reliable transportation?
•a single gigantic car with N engines, N transmissions, N brakes, … etc. that works (i.e.
provides us with transportation) as long as at least one of its engines and at least one of its
transmissions, and at least one of its brakes … works
•N separate ordinary cars that fail as soon as any one of their components fail, but which
together provide us with transportation as long as at least one car is in working condition.
(a) For each model, find the probability of replicated system failure in terms of p, N and M.
(b) Suppose that M = 5 and p = 0.2. If it is desired that the replicated system failure probability
be less than 0.001, what should N be in each case?
(c) Repeat part (b) assuming that M = 1000.
2. [“Reach out and touch someone”] MiddleEast Bell, a division of Psingular Corp., has built
a telephone network as shown below. Terrorists attack each of the seven links. The
attacks may be considered to be independent events, and the attack on a link succeeds in
severing the link with probability p. If a link is severed, switches automatically re-route
calls so as to avoid the failed link (if possible).
(a) What is the probability of being able to call from ORIAC to SUCSAMAD?
(b) Given that it is possible to call from ORIAC to SUCSAMAD, what is the conditional
probability that the ZEUS to NAMMA link is in working condition?
(c) The link capacities (i.e., the numbers of telephone calls that the links can carry (in either
direction)) are as marked on the diagram. Let X denote the number of calls that can be
made from ORIAC to SUCSAMAD. Find the pmf and the expected value of X.
ORIAC
ZEUS
NAMMA
NONABEL
SUCSAMAD
VIVA LET
100 100
20
20
50
50
100
