Download ECE 313 (Section G)

University of Illinois
Spring 2017
ECE 313 (Section G)
In-Class Activity 5 Solution - Wednesday, Apr 5th
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Question 1 (75 points):
Consider the following storage system composed of two subsystems S1 and S2. S1 is the
primary system composed of three disks A, B, and C, where the failure of any of them
will cause the failure of the subsystem S1. S2 is composed of a disk D which acts as a
backup (cold standby) of the subsystem S1 and will be powered on only after the primary
subsystem fails.
Assume that A, B, and C are identical disks and the switching circuit S is perfect. We
model the lifetime of the disks A, B, C, and D with three independent random variables
X1, X2, X3 and X4. Assume X1, X2 and X3 are exponentially distributed with parameter 𝜆.
Part A (25 points) – Find the reliability function and failure rate (𝜆1) of the primary
subsystem (S1).
Reliability function for each of disk A, B, and C: e−𝜆t
From reliability point of view, A, B and C are in series: R S1 = R A R B R C = e−3𝜆t
(15 points)
Therefore, failure rate 𝜆1 = 3𝜆
(10 points)
Part B (25 points) – If X4 is exponentially distributed with parameter 4𝜆 . What
distribution best models the time to failure of the whole system? Use the results of part A
to derive the reliability function and instantaneous failure rate of the system in terms of 𝜆.
The time to failure is sum of two independent sequential phases with different
parameters. Therefore, a 2-stage Hypo-Exponential with 𝜆1 = 3𝜆 and 𝜆2 = 4𝜆
(10 points)
𝑅(𝑡) = 1 − 𝐹(𝑡) =
𝜆2
𝜆1
𝑒 −𝜆1 𝑡 −
𝑒 −𝜆2 𝑡 = 4e−3𝜆𝑡 − 3𝑒 −4𝜆𝑡
𝜆2 − 𝜆1
𝜆2 − 𝜆1
(7 points)
𝑓(𝑡)
Hence, the instantaneous failure rate ℎ(𝑡) = 𝑅(𝑡) =
12𝜆(e−3𝜆𝑡 −𝑒 −4𝜆𝑡 )
4e−3𝜆𝑡 −3𝑒 −4𝜆𝑡
(8 points)
Part C (25 points) – If X4 is exponentially distributed with parameter 3𝜆 . What
distribution best models the time to failure of the whole system? Derive the reliability
function and instantaneous failure rate of the system in terms of 𝜆.
With 𝜆2 = 3𝜆, we have a 2-stage Erlang Distribution with 𝜆 = 3𝜆
Therefore, 𝑅(𝑡) = 1 − 𝐹(𝑡) = e−3𝜆t (1 + 3𝜆𝑡) for t > 0
(9𝜆2 𝑡)𝑒 −3𝜆𝑡
Then, h(t) = (1+3𝜆𝑡)𝑒 −3𝜆𝑡 =
9𝜆2 𝑡
1+3𝜆𝑡
for t > 0
(10 points)
(7 points)
(8 points)
Question 2 (25 points):
Consider a communication network in which packets are being sent from node N1 to node
N2. The packet can take one of two paths that exist between N1 and N2. The time to
transfer is exponentially distributed with the parameter dependent on the path i.e. 𝜆1 for
path 1 and 𝜆2 for path 2. The path to be taken is decided by tossing a coin with
probability of heads 𝑝. The packet takes path 1 if heads turns up and path 2 if tails turns
up. What is the distribution of the time of transfer of a packet from N1 to N2? Derive the
density function of the distribution.
In this problem, the process consists of alternate phases (alternate paths between N1 and
N2), i.e. during any single experiment the process experiences one and only one of the
many alternate phases (a packet can take only one of the two paths). Both these paths
have an exponential distribution. Therefore, it is a two-phase hyper-exponential
distribution. (15 points)
The pdf is given by:
𝑓(𝑡) = 𝑝 𝜆1 𝑒 −𝜆1 𝑡 + (1 − 𝑝) 𝜆2 𝑒 −𝜆2 𝑡
(10 points)
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