University of Illinois Spring 2007
ECE 413: Problem Set 7 Due 3/7/07 at beginning of class
System Reliability and CDFs
Assigned reading: Section 4.9, and review Chapters 3 and the rest of Chapter 4.
Noncredit exercises: Chapter 4, problems 17 and 19, theoretical exercises 17,21,22,23, and even
numbered self-test problems and exercises.
1. The reliability of multiple connections
A certain Internet service provider in a midsize city relies on kseparate connections between the
city and neighboring cities, to connect its customers to the Internet. Based on past experience,
management assumes that a given connection will be down on a given day with probability p=
0.001, independently of what happens on other days or with other connections. Total outage is said
to occur if all connections are down on the same day. How large should kbe so that the probability
total outage occurs at least one day in a year is less than or equal to 0.001?
2. Failure rate function for discrete random variables
Suppose an item, such as a light bulb, a sensor, or a satellite, will eventually fail. Suppose the time
of failure, in some appropriate units of time, is modeled as a random variable T, with values in the
set of positive integers. The failure rate function λTis the function λT= (λT(k) : k≥1), defined
by λT(k) = P(T=k|T≥k) if P(T≥k)>0. If P(T≥k) = 0,then λk(T) is not defined.
(a) Give a general expression for λTin terms of the pmf, pT.
(b) Note that if λT(k) is defined for a given value of k, then its value is in the interval [0,1] (this is
true for any conditional probability). What does it mean about the distribution of Tif λT(kf) = 1
for some value kf≥1?
(c) Find λTexplicitly in case pT(k) = 1
nfor 1 ≤k≤n. (Hint: λT(k) is not well defined for
k≥n+ 1 for this example.)
(d) Find the pmf pTin case λT(k) = pfor all k≥1, where pis a constant with 0 < p ≤1. (Hint:
begin by finding pT(1) and pT(2).)
(e) A network consists of two links in series. Let Tbe the time the network fails, which is the first
time that at least one of the links fails. Assume the two links fail independently, with the failure
rate function of link ibeing λi= (λi(k) : k≥1), for i= 0 or i= 1. Express the failure rate
function of the network, λT, in terms of the functions λ1and λ2. (Hint: Make sure λT(k) is in the
interval [0,1] if λ1(k) and λ2(k) are both defined. You should also have λT(k) = 1 if λi(k) = 1 for
i= 1 or i= 2.)
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