Some numerical Problems
1. Suppose that a user at a PC at one end of Canada sends a 1-Mbit file to
a remote server on the other end over a data link operating at 64 kbps.
Assume that we are using a fiber optic link with a propagation rate of
the speed of light, approximately 3 * 108 m/sec, and that the distance
is 4800 km.
(a) What is the overall network delay, i.e. time to transmit the file?
(Ignore any processing or queueing delays.)
Introduction: 1-1
Some numerical Problems
dtotal = dpropagation + dtransmission
dpropagation = distance/speed = 4800*103/3*108 = 0.016 seconds
dtransmission = length/rate = 106/64*103 = 15.625 seconds
Introduction: 1-2
Some numerical Problems
2. Suppose you are downloading an MP3 file of 64 MB, the server has a
transmission rate of Rs = 4 Mbps, and you have an access link of Rc = 2
Mbps. What is the throughput ? What is the overall network delay, i.e.
time to transmit the file?
(Ignore any processing or queueing delays.)
Throughput =min(Rs, Rc) = 2Mbps
Time to transmit the file = 64*8/2 = 256 seconds
Introduction: 1-3
Some numerical Problems
3. What is the propagation time if the distance between the two points is
12,000 km? Assume the propagation speed to be 2.4 × 108 m/s in cable.
Propagation time = 12000*1000 / 2.4*108 = 50 ms
Introduction: 1-4
Some numerical Problems
4. What are the propagation time and the transmission time for a 2.5-kbyte
message (an e-mail) if the bandwidth of the network is 1 Gbps? Assume that the
distance between the sender and the receiver is 12,000 km and that light travels
at 2.4 × 108 m/s.
Propagation time = 12000*1000 / 2.4*108 = 50 ms
Transmission time = 2500*8 / 109 = 0.02 ms
Message is short and the bandwidth is high, the dominant factor is the
propagation time, not the transmission time. The transmission time can be
ignored.
Introduction: 1-5
Some numerical Problems
5. What are the propagation time and the transmission time for a 5 MB image if
the bandwidth of the network is 1 Mbps? Assume that the distance between the
sender and the receiver is 12,000 km and that light travels at 2.4 × 108 m/s.
Propagation time = 12000*1000 / 2.4*108 = 50 ms
Transmission time = 5,000,000*8 / 106 = 40 s
Message is long and the bandwidth is less, the dominant factor is the
transmission time, not the propagation time. The propagation time can be
ignored.
Introduction: 1-6
Some numerical Problems
6. In the optical fiber, the core has a refractive index equal to 1.7 and a
cladding of refractive index of 1.2.
a) What is the speed of light inside the core?
b) What is the critical angle at the core-cladding interface?
c) What is the maximum angle θ that the rays leaving the source of light
should make with the axis of the fiber so that total internal reflections
takes place at the core cladding interface?
Introduction: 1-7
Some numerical Problems
a. Refractive index of the core is defined as n1 = C/V (C- speed of
light, and V- velocity of light in core) V = C/n1 = 1.76х108 m/sec.
b. θc =arcsin (1.2/1.7) = 44.9ᵒ (note: at critical angle θ in cladding
will be 90ᵒ)
c.
θ < 90 – θc = 45.1ᵒ.
Introduction: 1-8
Some numerical Problems
7. Ten signals, each requires 4000 Hz, are multiplexed onto a single
channel using FDM. What is the minimum bandwidth required for the
multiplexed channel? (a) without guard band, and (b) with guard bands
are of 400 Hz wide.
a) 40KHz,
b) 43.6 KHz
Introduction: 1-9
Some numerical Problems
8. four channels , two with a bit rate of 200kbps and two with a bit rate
150 kbps are to be multiplexed using multiple slots TDM with no
synchronization bits. Answer the following questions: assume 4 bits are
sent from each of the first 2 sources and 3 bits from each of the second 2
sources.
i. What is the size of a frame in bits?
ii. What is the frame rate?
iii. What is the duration of a frame?
iv. What is the date rate?
Introduction: 1-10
Some numerical Problems
a)
The frame carries 4 bits from each of the first two sources and 3 bits from each of the second
two sources. Frame size = 4 × 2 + 3 × 2 = 14 bits.
b) Each frame carries 4 bit from each 200-kbps source or 3 bits from each 150 kbps. Frame rate =
200,000 / 4 = 150,000 /3 = 50,000 frames/s.
c)
Frame duration = 1 /(frame rate) = 1 /50,000 = 20 μs.
d) Output data rate = (50,000 frames/s) × (14 bits/frame) = 700 kbps. We can also calculate the
output data rate as the sum of input data rates because there are no synchronization bits. Output
data rate = 2 × 200 + 2 × 150 = 700 kbps.
Introduction: 1-11
Some numerical Problems
9. Suppose users share a 10 Mbps link. Also suppose each user requires 200
kbps when transmitting, but each user transmits only 10 percent of the
time. (See the discussion of packet switching versus circuit switching in
Section 1.3.)
a. When circuit switching is used, how many users can be supported?
b. For the remainder of this problem, suppose packet switching is used.
Find the probability that a given user is transmitting. c. Suppose there
are 120 users.
c. Find the probability that at any given time, exactly n users are
transmitting simultaneously. (Hint: Use the binomial distribution.)
d. Find the probability that there are 51 or more users transmitting
simultaneously.
Introduction: 1-12
Some numerical Problems
10. Suppose users share a 2 Mbps link. Also suppose each user
transmits continuously at 1 Mbps when transmitting, but each user
transmits only 20 percent of the time.
a. When circuit switching is used, how many users can be
supported?
b. For the remainder of this problem, suppose packet switching is
used. Why will there be essentially no queuing delay before the
link if two or fewer users transmit at the same time? Why will
there be a queuing delay if three users transmit at the same time?
c. Find the probability that a given user is transmitting.
d. Suppose now there are three users. Find the probability that at
any given time, all three users are transmitting simultaneously.
Find the fraction of time during which the queue grows.
Introduction: 1-13
Some numerical Problems
Introduction: 1-14
Some numerical Problems
11. (a) Suppose N packets arrive simultaneously to a link at which no
packets are currently being transmitted or queued. Each packet is of
length L and the link has transmission rate R. What is the average
queuing delay for the N packets?
(b) Now suppose that N such packets arrive to the link every LN/R
seconds. What is the average queuing delay of a packet?
Introduction: 1-15
Some numerical Problems
a) The queuing delay is 0 for the first transmitted packet, L/R for the
second transmitted packet, and generally, (n-1)L/R for the n th
transmitted packet. Thus, the average delay for the N packets is:
(L/R + 2L/R + ....... + (N-1)L/R)/N = L/(RN) * (1 + 2 + ..... + (N-1)) =
L/(RN) * N(N-1)/2 = LN(N-1)/(2RN) = (N-1)L/(2R) Note that here
we used the well-known fact: 1 + 2 + ....... + N = N(N+1)/2
b) b) It takes LN / R seconds to transmit the N packets. Thus, the
buffer is empty when a each batch of N packets arrive. Thus, the
average delay of a packet across all batches is the average delay
within one batch, i.e., (N-1)L/2R
Introduction: 1-16
Some numerical Problems
12. Suppose Host A wants to send a large file to Host B. The path from
Host A to Host B has three links, of rates R1 = 500 kbps, R2 = 2 Mbps,
and R3 = 1 Mbps.
a. Assuming no other traffic in the network, what is the throughput
for the file transfer?
b. Suppose the file is 4 million bytes. Dividing the file size by the
throughput, roughly how long will it take to transfer the file to Host
B?
c. Repeat (a) and (b), but now with R2 reduced to 100 kbps.
a) 500 kbps b) 64 seconds c) 100kbps; 320 seconds
Introduction: 1-17
Some numerical Problems
Introduction: 1-18
Some numerical Problems
Introduction: 1-19
Some numerical Problems
14. 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 four other packets are waiting to be transmitted. Packets are
transmitted in order of arrival. Suppose all packets are 1,500 bytes and
the link rate is 2.5 Mbps. What is the queuing delay for the packet?
More generally, what is the queuing delay when all packets have length
L, the transmission rate is R, x bits of the currently-being-transmitted
packet have been transmitted, and n packets are already in the queue?
Introduction: 1-20
Some numerical Problems
Introduction: 1-21