### Problems Guide

```Probability
Review
Given
•
•
•
•
N = population
X = sample size
P = probability of success/event
Q = 1-P, probability of failure/non event
Wanted
Probability of X successes in population of N
where member event occurs with probability P
Solution
BiCoef(N,X) PX Q(N-X)
N!
--------------X! (N-X)!
PX Q(N-X)
Example
• What is probability of having 4 heads after
flipping a coin 6 times?
• N=6
• X=4
• P=½
• Q=1–P
6!
30
------- (.5)6 = ----- 0.0156 = 0.234 or 23.4%
4!2!
2
Example
• What is probability of having 7 heads after
flipping a coin 7 times with a trick coin?
• N=6
• X=4
• P = .9
• Q=1–P
7!
------- (.9)7 (.1)0 = (.9)7 = .478
7!0!
Problems
Solver Guide
Similar to
Chapter 1
Problem 12
Packet length:
64 bytes
Transmission rate:
40 Gbps
Current packet
transmission:
50%
Packets in queue:
480 packets
Queuing delay:
_________
A packet switch receives a packet and
determines the outbound link to which
the packet should be forwarded. When
the packet arrives, one other packet is
halfway done being transmitted on this
outbound link and 480 other packets
are waiting to be transmitted. Packets
are transmitted in order of arrival.
Suppose all packets are 64 bytes and
the link is 40 Gbps. What is the
queuing delay for the packet?
Similar to
Chapter 1
Problem 12
Packet length:
L=64 bytes
Transmission rate:
R=40 Gbps
Current packet
transmission:
B=50%
Packets in queue:
Q=480
packets
Queuing delay:
_________
What is the queuing delay for the
packet?
Queing delay =
((L * (Q+1) – B*L) / (R)
= 6.15 usec
Similar to
Chapter 1
Problem 16
Average packets in
buffer:
360 packets
Average queuing delay:
300 msec
Packet transmission rate:
60
packets/sec
Average packet arrival
rate:
_________
Consider a router buffer preceding an
outbound link. In this problem you will
use Little's formula, a famous formula
from queuing theory. Let N denote the
average number of packets in the buffer
plus the packet being transmitted. Let a
denote the rate of packets arriving ".
"
at the link. Let d denote the average total
delay (i.e., the queuing delay plus the
transmission delay) experienced by a
packet. Little's formula is N = a · d.
Suppose that on average, the buffer
contains 360 packets (in addition to the
packet currently being transmitted) and
the average queuing delay is 300 msec
The links transmission rate is 60
packets/sec. Using Little's formula, what
is the average packet arrival rate,
assuming there is no packet loss?
Similar to
Chapter 1
Problem 16
Average packets in
buffer:
N-1 = 360
packets
Average queuing delay:
Q=300 msec
Packet transmission rate:
R=60
packets/sec
Average packet arrival
rate:
a=_________
Little's formula is N = a · d.
Using Little's formula, what is the
average packet arrival rate, assuming
there is no packet loss?
T=1/R
a = N / (Q+T)
= 1137 packets/sec
Similar to
Chapter 1
Problem 24
Data size:
4 Tbytes
Transmission rate:
2.4 Gbps
Transmit time:
_________
Suppose you would like to urgently
deliver 4 Tbytes of data from Boston to
Los Angeles. You have available a 2.4
Gbps dedicated link for data transfer.
Would you prefer to transmit the data
overnight delivery? Explain.
Similar to
Chapter 1
Problem 24
Data size:
4 Tbytes
Transmission rate:
2.4 Gbps
Transmit time:
_________
Suppose you would like to urgently
deliver 4 Tbytes of data from Boston to
Los Angeles. You have available a 2.4
Gbps dedicated link for data transfer.
Would you prefer to transmit the data
overnight delivery? Explain.
4E12*8bits / 2.4E9bps
3.704 hrs to transmit
Similar to
Chapter 1
Problem 25
Distance between
host:
10,000 km
File Size:
6 Tbytes
Propagation speed:
250,000,000 mps
512 kbps
a. bandwidth delay
product:
__________
__________
c. bandwidth delay
product (define):
__________
d. Width of a bit:
__________
e. Width of a bit
(formula):
__________
Suppose two host, A and B, are separated by
10,000 km and are connected by a direct link of R
= 512 kbps. Suppose the propagation speed over
a. Calculate the bandwidth delay product, R· dprop
b. Consider sending a file of 48 Tbits from Host A
to Host B. Suppose the file is sent continuously as
one large message. What is the maximum
number of bits that will be in the link at any
given time?
c. Provide an interpretation of the bandwidth
delay product.
d. What is the width (in meters) of a bit in the
e. Derive a general expression for the width of a
bit in terms of the propagation speed, s, the
transmission rate, R, and the length of the link m.
Similar to
Chapter 1
Problem 25
Distance between
host:
D=10,000 km
File Size:
F=6 Tbytes
Propagation speed:
S=250,000,000
mps
R=512 kbps
a. bandwidth delay
product:
__________
__________
c. bandwidth delay
product (define):
__________
d. Width of a bit:
__________
e. Width of a bit
(formula):
__________
a. Calculate the bandwidth delay
product:
R*D/S = 20.48 kbits
b. Max bits in link: 20.48 kbits
c. bandwidth delay product is the
(max) number of bits that could be in
d. Width of bit:
D/(R*D/S)= S/R = 488.281 m
e. Width of bit = S/R
Similar to
Chapter 1
Problem 26
Distance between
host:
10,000 km
Propagation speed:
250,000,000
mps
R:
__________
Suppose two host, A and B, are
separated by 10,000 km and are
connected by a direct link, R. Suppose
the propagation speed over the link is
2.5·108 meters/sec. For what value of
R is the width of a bit as long as the
Similar to
Chapter 1
Problem 26
Distance between
host:
D=10,000 km
Propagation speed:
S=250,000,000
mps
R:
__________
Suppose two host, A and B, are
separated by 10,000 km and are
connected by a direct link, R. Suppose
the propagation speed over the link is
2.5·108 meters/sec. For what value of R
is the width of a bit as long as the
R= (S / width_of_bit) = S/D = 25 bps
Similar to
Chapter 1
Problem 28
Distance between
host:
100 km
File Size:
350 Gbytes
Propagation speed:
250,000,000
mps
56 kbps
Packet Size:
1.45 kbits
(a) Transmit time
continuous:
____________
(b) Transmit time
segmented:
____________
(c) Compare:
____________
Suppose two host, A and B, are separated by
100 km and are connected by a direct link of
R = 56 kbps and will send a file of 350 Gbytes.
Suppose the propagation speed over the link
is 2.5·108meters/sec.
a. How long does it take to send the file,
assuming it is sent continuously?
b. Suppose not the file is broken into packets
of length 1.45 kbits. Suppose that each packet
is acknowledged by the receiver and the
transmission time of the acknowledgment is
negligible. Finally assume that the sender
cannot send a packet until the preceding one
is acknowledged. How long does it take to
send the file?
c. Compare the results from (a) and (b).
Similar to
Chapter 1
Problem 28
Distance between
host:
D=100 km
File Size:
F=350 Gbytes
Propagation speed:
S=250,000,000
mps
R=56 kbps
Packet Size:
P=1.45 kbits
(a) Transmit time
continuous:
____________
(b) Transmit time
segmented:
____________
(c) Compare:
____________
a. How long does it take to send the file, assuming
it is sent continuously?
Transmission delay + Propagation Delay=
F/R+D/S=
350E9bytes / 56kbps +100km/2.5E8mps=
19.29 months
b. Suppose now the file is broken into packets of
length 1.45 kbits. Suppose that each packet is
acknowledged by the receiver and the
transmission time of the acknowledgment is
negligible. Finally assume that the sender cannot
send a packet until the preceding one is
acknowledged. How long does it take to send the
file?
NumPackets*2*Prop delay + Transmission delay
(F/P)*2D/S+ F/R = 19.29 months + 25.7 min
c. Compare the results from (a) and (b).
Similar to
Chapter 1
Problem 29
Orbit:
36,000 km
Propagation speed:
240,000,000
mps
10 Mbps
(a) propagation delay:
____________
(b) bandwidth delay
product:
____________
(c) size of the photo:
____________
Suppose there is a 10 Mbps microwave
link between a geostationary satellite and
its base station on Earth. Every minute the
satellite takes a digital photo and sends it
to the base station. Assume a propagation
speed of 2.4·108meters/sec.
a. What is the propagation delay of the
b. What is the bandwidth delay product,
R·dprop?
c. Let x denote the size of the photo.
What is the minimum value of x for the
transmitting?
Similar to
Chapter 1
Problem 29
Orbit:
D=36,000 km
Propagation speed:
S=240,000,000
mps
R=10 Mbps
(a) propagation delay:
____________
(b) bandwidth delay
product:
____________
(c) size of the photo:
____________
a. What is the propagation delay of the
D/S=
150 msec
b. What is the bandwidth delay
product, R·dprop?
R*D/S=1.5 Mbits
c. Let x denote the size of the photo.
What is the minimum value of x for the