Network Coding: Theory and Applications - F. Fitzek, M. Medard, D. Lucani, M. Pedersen
Asymmetric Alice and Bob
In this exercise the traffic flows between Alice and Bob are asymmetric, specifically, the load at Alice shall
be larger or equal than the load at Bob. The task is to give analytical expressions for the throughput for
the case of not using network coding and the case of using network coding, see Figure 1.
.TA→B
.TA→B
.A
. R→A
T
. A→R
T
.R.
. B→R
T
. R→B
T
.A
.B
.TA→R
..
R
.TB→R
.B
.TR→
.TB→A
.TB→A
a) without coding
b) with coding
Figure 1: Data throughput notation in the Alice and Bob topology.
The following notation is used throughout this exercise:
Symbol
Description
i, j
Xi
Ti→j
Ti→
T→j
C
Ti→ /T→i
Denotes a node in the scenario, i, j = A, R, B
The generated unit less load Xi = 0 . . . 1 at node i.
The throughput on the link or path from node i to node j.
The combined throughput on all links from node i.
The combined throughput on all links into node j.
The measured total WiFi capacity in bits/second on the shared medium.
loss rate at a node i
Exercise 1: Throughput without Network Coding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The task of this exercise is to estimate the throughputs of the given Alice and Bob topology and to fill
out the following table:
case
Packets dropped at
loss rate at R
TA→B
TB→A
XB > 1/3 ?
(a)
A,B,R
... ? ... ? ...
(b)
(c) (d)
A,R
R
–
Note that for each case there is a condition, for example, the condition for case (a) is XB > 1/3. If the
condition is true, the throughputs are as given in the respective column. If the condition is not true,
the condition in the next column is checked, and so on. Recall that by definition XA ≥ XB . The loss
rate (for a node not performing coding) is defined as Ti→ /T→i . Note that the loss rate at a node allows
to compute the individual outgoing flows. This means, for instance, that the data throughput from R
to B can be computed as
TR→B = TA→R · (TR→ /T→R ).
Go through the following cases and estimate the respective throughputs TA→B and TB→A . As input
variables XA or XB shall be used.
(a) Packets are dropped at A, R, and B
(b) Packets are dropped at A and R
(c) Packets are only dropped at R
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Network Coding: Theory and Applications - F. Fitzek, M. Medard, D. Lucani, M. Pedersen
(d) Packets are not dropped
Exercise 2: Throughput with Network Coding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Now we look at the same topology and asymmetric loads using network coding. Fill out the following
table regarding the cases (a), (b), and (c):
case
Packets dropped at
TA→B
TB→A
XB > 1/3 ?
(a)
A,B
... ?
(b)
A
(c)
–
(a) Packets are dropped at A and B
(b) Packets are only dropped at A
Explain why at R there are no packets dropped in this case.
(c) Packets are not dropped
Exercise 3: Plot of an Example Scenario with given loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The ratio between the loads of A and B is given as XA = 2.5 · XB , and the capacity as C = 7.2M bits/s
(you may also use a unit less medium load instead). The throughput for with and without network
coding shall be compared for the flows in (a) and (b).
(a) Construct the plot for TA→B !
(b) Construct the plot for TB→A !
You can use GNUPLOT for generating the plots. A conditional plot can be generated as follows:
plot condition ? plot_action : alternative_plot_action
There can also be multiple conditions:
plot condition_1 ? plot_action_1 : condition_2 ? \
plot_action_2 : alternative_plot_action_2
A simple example is given as follows:
plot x>0.5 ? 2*x+3 : x<0.8 ? 3*x : 2
Bidirectional Cross
An elementary topology where network coding can provide interesting benefits is the bidirectional
cross, where we distinguish between three cases:
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Network Coding: Theory and Applications - F. Fitzek, M. Medard, D. Lucani, M. Pedersen
B
A
B
A
B
A
R
R
R
C
a) pure relaying
D
D
C
b) network coding
C
D
c) network coding with overhearing
Specifically, the cases are (a) pure relaying, where packets are not combined, (b) network coding, where
two packets are combined, and (c) network coding with overhearing, where the relay combines four
packets.
Exercise 4: Activities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Give the activities for each node in the topology for cases (a), (b), and (c). The possible activities
for each node are idle, receive, and send. The following tables are intended as a help to construct the
respective activity plots.
(a) Pure Relaying
Enter the activities in the table below !
(b) Network Coding
Enter the activities in the table below !
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Network Coding: Theory and Applications - F. Fitzek, M. Medard, D. Lucani, M. Pedersen
(c) Network Coding with Overhearing
Enter the activities in the table below !
Exercise 5: Throughput . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Give the total throughput for the three cases (a), (b), and (c), including the plot and the analytical
expressions.
Intra session Coding
Consider a source that will send 3 data packets (P1 , P2 , P3 ) with n bits each to a destination using RLNC
with GF
∑(2). A coded packet CP is generated by a linear combination of the original data packets, i.e.,
CP = i=1,2,3 ci Pi , where ci ’s are the coding coefficients.
Exercise 6: Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
What is the field size and the generation size in this problem?
Exercise 7: Probability of a Packet being Innovative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Assuming that the destination has no coded packets at the beginning, what is the probability of a linear
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Network Coding: Theory and Applications - F. Fitzek, M. Medard, D. Lucani, M. Pedersen
combination not being useful? Use the following table that represents the coding coefficients ci of each
original packet Pi .
c1
0
0
0
0
1
1
1
1
c2
0
0
1
1
0
0
1
1
c3
0
1
0
1
0
1
0
1
Check if Useful
Assuming that two coded packets have arrived to the destination with coding coefficients (0, 1, 1) and
(1, 1, 0), calculate again the probability of a new incoming coded packet being useful to the receiver.
Use the following table for help.
c1
0
0
0
0
1
1
1
1
c2
0
0
1
1
0
0
1
1
c3
0
1
0
1
0
1
0
1
Check if Useful
Effect of Imperfect Overhearing
Consider an X topology, with source (Si ) and destination (Di ) for each flow i as well as a relay (R).
For our analysis, let us consider R only XORs packets from the two flows. Assume an independent loss
probability for each link as ei,j , where i, j ∈ {R, S1 , S2 , D1 , D2 }.
D2 X Topology S1 R D1 S2 Let us consider that we have WiFi’s MAC protocol and that it will guarantee delivery of packets in
the links from Si to R. This will be done by retransmission of a data packet until R gets that packet.
Note that receiver Di must overhear transmissions from Sj , j ̸= i in order to be able to recover the
information in the XORed packet.
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Network Coding: Theory and Applications - F. Fitzek, M. Medard, D. Lucani, M. Pedersen
To simplify the problem, consider that the links between the relay R and the two destinations have no
losses, i.e., eR,D1 = eR,D2 = 0. Therefore, both destinations will receive the XORed data packet.
Exercise 8: Probability of Losing a Packet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Considering that a destination will only receive a valid packet if both the XORed packet and the
overheard packet are received, calculate the probability of packet loss at the receiver as a consequence
of the overhearing error.
Hint: the transmissions from sources to relay are modelled as a Geometric distribution with success
probability 1 − eSi ,R .
Plot the loss probability for destination D1 for the case of eS2 ,D1 = eS2 ,R = e.
Exercise 9: Numer of Packet Transmissions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Provide an expression of the mean number of transmissions performed by the system to convey one data
packet to each destination.
Effect of Recoding
Consider an line network , with source (S) and destination (D) and a relay (R). Assume that two data
packets will be sent.
Exercise 10: No Recoding at Relay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
For the case of a relay that immediately forwards (no recoding), determine how many packets are
received at D after 4 transmissions under the loss pattern described in the figure (X is a loss). Is
there any difference if the source sends coded packets or if it only sends packets in Round Robin, i.e.,
P 1 , P2 , P1 , P2 .
S Loss In link S-­‐
>R Loss in link R-­‐>D t1 X
t2 X t3 X t4 X D R Pkts at R Pkts at D Exercise 11: Recoding at Relay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Now consider the case of a relay that stores packets coming from the source in its buffer and always
sends coded packets to the destination. Coded packets are generated at the relay by combining packets
in its buffer. Is the destination able to recover the two data packets?
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Network Coding: Theory and Applications - F. Fitzek, M. Medard, D. Lucani, M. Pedersen
S Loss In link S-­‐
>R Loss in link R-­‐>D t1 X
t2 X t3 X t4 X D R Pkts at R Pkts at D Page 7