In this report we use Network Utility Maximization (NUM) Framework to study the
joint assignment of rate and reliability among sources in multi-hop wireless network.
Formulating the above problem in the NUM framework can lead to natural functional
decomposition into layers and distributed algorithms. The optimality is preserved
through messages passing between layers and local messages exchanging among
nodes. The framework reasons (joint) cross layer design. Heuristic layering turns out
to be solving a NUM optimization problem. This is so called “Layering as optimization
decomposition”. Especially, our problem leads to congestion control at transport
layer and the adaptive coding at the physical layer. The messages passing between
layers are congestion prices (Lagrange multipliers). Updating is performed
distributively using local messages.
Each source s has a utility function 𝑈𝑠 (𝑥𝑠 , 𝑅𝑠 ), where 𝑥𝑠 is an information data rate
and 𝑅𝑠 is the reliability of source s. We assume that the utility function is a
continuous, increasing, and concave function of 𝑥𝑠 and 𝑅𝑠 .
Each source s has its information data rate bounded between a minimum and a
maximum: 𝑥𝑠𝑚𝑖𝑛 and 𝑥𝑠𝑚𝑎𝑥 , and has a minimum reliability requirement 𝑅𝑠min .The
reliability of source s is defined as
𝑅𝑠 = 1 − 𝑝 𝑠
𝑡𝑙,𝑠 = 𝑥𝑠 /𝑟𝑙,𝑠
𝑝𝑙,𝑠 = 𝐸𝑙 (𝑟𝑙,𝑠 )
𝑝 𝑠 = 1 − ∏ (1 − 𝑝𝑙,𝑠 ) = 1 − ∏ (1 − 𝐸𝑙 (𝑟𝑙,𝑠 ))
𝑙∈𝐿(𝑠)
𝑙∈𝐿(𝑠)
Assume 𝑝𝑙,𝑠 ≪ 1
𝑝 𝑠 ≈ ∑ 𝑝𝑙,𝑠 = ∑ 𝐸𝑙 (𝑟𝑙,𝑠 )
𝑙∈𝐿(𝑠)
𝑙∈𝐿(𝑠)
𝑅𝑠 ≈ 1 − ∑ 𝐸𝑙 (𝑟𝑙,𝑠 )
𝑙∈𝐿(𝑠)
𝑠
𝑝 is the end-to-end error probability
𝑟𝑙,𝑠 is the code rate of source s at link l
𝑥𝑠 is the information data rate of source s
𝑡𝑙,𝑠 is the transmission data rate of source s at link l
𝑝𝑙,𝑠 is the error probability of source s at link l
𝐸𝑙 (𝑟𝑙,𝑠 ) is an increasing function of 𝑟𝑙,𝑠 reflecting the rate-reliability trade-off
𝐿(𝑠) is the set of links used by source s
∑ 𝑡𝑙,𝑠 = ∑
𝑠∈𝑆(𝑙)
𝑠∈𝑆(𝑙)
𝑥𝑠
≤ 𝐶𝑙𝑚𝑎𝑥 , ∀𝑙
𝑟𝑙,𝑠
𝑆(𝑙) is the set of sources using link l
𝐶𝑙𝑚𝑎𝑥 is the capacity of link l
Assume 𝑟𝑙,𝑠 = 𝑟𝑙 , ∀𝑠 ∈ 𝑆(𝑙), ∀𝑙
maximize ∑𝑠 𝑈𝑠 (𝑥𝑠 , 𝑅𝑠 )
subject to 𝑅𝑠 ≤ 1 − ∑𝑙𝜖𝐿(𝑠) 𝐸𝑙 (𝑟𝑙 ), ∀𝑠
𝑥𝑠
∑
≤ 𝐶𝑙𝑚𝑎𝑥 , ∀𝑙
𝑟𝑙
𝑠∈𝑆(𝑙)
𝑥𝑠𝑚𝑖𝑛 ≤ 𝑥𝑠 ≤ 𝑥𝑠𝑚𝑎𝑥 , ∀𝑠
𝑅𝑠𝑚𝑖𝑛 ≤ 𝑅𝑠 ≤ 1, ∀𝑠
0 ≤ 𝑟𝑙 ≤ 1, ∀𝑙
𝐿(𝐱, 𝐑, 𝐫, 𝛌, 𝛍) = ∑ 𝑈𝑠 (𝑥𝑠 , 𝑅𝑠 ) + ∑ 𝜆𝑙 (𝑟𝑙 𝐶𝑙𝑚𝑎𝑥 − ∑ 𝑥𝑠 )
𝑠
𝑙
𝑠∈𝑆(𝑙)
+ ∑ 𝜇𝑠 (1 − ∑ 𝐸𝑙 (𝑟𝑙 ) − 𝑅𝑠 )
𝑠
𝑙𝜖𝐿(𝑠)
= ∑ (𝑈𝑠 (𝑥𝑠 , 𝑅𝑠 ) − ∑ 𝜆𝑙 𝑥𝑠 − 𝜇𝑠 𝑅𝑠 )
𝑠
𝑙𝜖𝐿(𝑠)
+ ∑ (𝜆𝑙 𝑟𝑙 𝐶𝑙𝑚𝑎𝑥 − ∑ 𝜇𝑠 𝐸𝑙 (𝑟𝑙 )) + ∑ 𝜇𝑠
𝑙
𝑠∈𝑆(𝑙)
𝑠
= ∑(𝑈𝑠 (𝑥𝑠 , 𝑅𝑠 ) − 𝜆𝑠 𝑥𝑠 − 𝜇𝑠 𝑅𝑠 ) + ∑(𝜆𝑙 𝑟𝑙 𝐶𝑙𝑚𝑎𝑥 − 𝜇 𝑙 𝐸𝑙 (𝑟𝑙 ))
𝑠
𝑙
+ ∑ 𝜇𝑠
𝑠
𝜆𝑠 = ∑𝑙𝜖𝐿(𝑠) 𝜆𝑙 and 𝜇 𝑙 = ∑𝑠∈𝑆(𝑙) 𝜇𝑠
𝜆𝑙 : the Lagrange multiplier on link l with an interpretation of “congestion price”, the
price per unit rate to use link l
𝜇𝑠 : the Lagrange multiplier on source s with an interpretation of “reliability price”,
the price per unit reliability that the source s must pay to the network
𝜆𝑠 : with an interpretation of “end-to-end congestion price” on source s
𝜇 𝑙 : with an interpretation of “aggregate reliability price” paid by sources using link l
The Lagrange dual function is
𝑄(𝛌, 𝛍) =
max
𝐱 min ≼ 𝐱 ≼ 𝐱 max
𝐑min ≼ 𝐑 ≼ 𝟏
𝟎≼𝐫≼𝟏
𝐿(𝐱, 𝐑, 𝐫, 𝛌, 𝛍)
The dual problem is
minimize 𝑄(𝛌, 𝛍)
subject to 𝛌 ≽ 𝟎
𝛍≽𝟎
Since the Lagrangian is separable, this maximization of the Lagrangian over (𝐱, 𝐑, 𝐫)
can be conducted in parallel at each source s
maximize 𝑈𝑠 (𝑥𝑠 , 𝑅𝑠 ) − 𝜆𝑠 𝑥𝑠 − 𝜇𝑠 𝑅𝑠
subject to 𝑥𝑠𝑚𝑖𝑛 ≤ 𝑥𝑠 ≤ 𝑥𝑠𝑚𝑎𝑥
𝑅𝑠𝑚𝑖𝑛 ≤ 𝑅𝑠 ≤ 1
and on each link l
maximize 𝜆𝑙 𝑟𝑙 𝐶𝑙𝑚𝑎𝑥 − 𝜇 𝑙 𝐸𝑙 (𝑟𝑙 )
subject to 0 ≤ 𝑟𝑙 ≤ 1
dual problem can be solved by using the gradient projection algorithm as
+
𝜆𝑙 (𝑡 + 1) = [𝜆𝑙 (𝑡) − 𝛽(𝑡)(𝑟𝑙 (𝑡)𝐶𝑙𝑚𝑎𝑥 − ∑ 𝑥𝑠 (𝑡))] , ∀𝑙
𝑠∈𝑆(𝑙)
+
𝜇𝑠 (𝑡 + 1) = [𝜇𝑠 (𝑡) − 𝛽(𝑡)(1 − ∑ 𝐸𝑙 (𝑟𝑙 (𝑡)) − 𝑅𝑠 (𝑡))] , ∀𝑠
𝑙𝜖𝐿(𝑠)
𝛽(𝑡) is the step size.
Algorithm 1 for Integrated Dynamic Reliability Policy
Source problem and reliability price update at source s:
 Source problem
maximize 𝑈𝑠 (𝑥𝑠 , 𝑅𝑠 ) − 𝜆𝑠 (𝑡)𝑥𝑠 − 𝜇𝑠 (𝑡)𝑅𝑠
subject to 𝑥𝑠𝑚𝑖𝑛 ≤ 𝑥𝑠 ≤ 𝑥𝑠𝑚𝑎𝑥
𝑅𝑠𝑚𝑖𝑛 ≤ 𝑅𝑠 ≤ 1

Price update
𝜇𝑠 (𝑡 + 1) = [𝜇𝑠 (𝑡) − 𝛽(𝑡)(𝑅 𝑠 (𝑡) − 𝑅𝑠 (𝑡))]+
where 𝑅 𝑠 (𝑡) = 1 − ∑𝑙𝜖𝐿(𝑠) 𝐸𝑙 (𝑟𝑙 (𝑡)) is the end-to-end reliability at iteration t.
Link problem and congestion price update at link l:
 Link problem
maximize 𝜆𝑙 (𝑡)𝑟𝑙 𝐶𝑙𝑚𝑎𝑥 − 𝜇 𝑙 (𝑡)𝐸𝑙 (𝑟𝑙 )
subject to 0 ≤ 𝑟𝑙 ≤ 1
 Price update
𝜆𝑙 (𝑡 + 1) = [𝜆𝑙 (𝑡) − 𝛽(𝑡)(𝑟𝑙 (𝑡)𝐶𝑙𝑚𝑎𝑥 − 𝑥 𝑙 (𝑡))]+
where 𝑥 𝑙 (𝑡) = ∑𝑠∈𝑆(𝑙) 𝑥𝑠 (𝑡) is the aggregate information rate on link l at iteration t.
In Algorithm 1, to solve problem (8), source s needs to know 𝜆𝑠 (𝑡), the sum of 𝜆𝑙 (𝑡)’s
of links that are along its path 𝐿(𝑠).This can be obtained by notification from the
links, e.g., through the presence or timing of acknowledgment packets in TCP. To
carry out price update (9), the source needs to know the sum of error probabilities of
the links that are along its path [i.e., its own reliability that are offered by the
network 𝑅 𝑠 (𝑡)].This can be obtained by the notification from the destination
that can measure its end-to-end reliability. To solve link problem (10), each link needs
to know 𝜇 𝑙 (𝑡), the sum of 𝜇𝑠 (𝑡)’s from sources 𝑠 ∈ 𝑆(𝑙) using this link. This can be
obtained by the notification from these sources. To carry out price update (11), the
link needs to know 𝑥 𝑙 (𝑡) , the aggregate information data rate of the sources that
are using it. This can be measured by the link itself.
a linear topology consisting of four links and eight users
min(1−𝛼)
𝑈𝑠 (𝑥𝑠 , 𝑅𝑠 ) = 𝑎𝑠
𝑥𝑠1−𝛼 − 𝑥𝑠
max(1−𝛼)
𝑥𝑠
𝛼 = 1.1, 𝑥𝑖𝑚𝑖𝑛 = 0.1 (
Mb
s
and 𝑅𝑖𝑚𝑖𝑛 = 0.9
1
𝑝𝑙 = exp(−𝑁(1 − 𝑟𝑙 ))
2
min(1−𝛼)
+ (1 − 𝑎𝑠 )
min(1−𝛼)
− 𝑥𝑠
Mb
) , 𝑥𝑖𝑚𝑎𝑥 = 2 (
s
𝑅𝑠1−𝛼 − 𝑅𝑠
max(1−𝛼)
𝑅𝑠
Mb
) , 𝐶𝑙𝑚𝑎𝑥 = 2 (
s
min(1−𝛼)
− 𝑅𝑠
) , 𝑅𝑖𝑚𝑎𝑥 = 1,
maximize ∑𝑠 𝑈𝑠 (𝑥𝑠 , 𝑅𝑠 )
subject to 𝑅𝑠 ≤ 1 − ∑𝑙𝜖𝐿(𝑠) 𝐸𝑙 (𝑟𝑙,𝑠 ), ∀𝑠
𝑥𝑠
∑
≤ 𝐶𝑙𝑚𝑎𝑥 , ∀𝑙
𝑟𝑙,𝑠
𝑠∈𝑆(𝑙)
𝑥𝑠𝑚𝑖𝑛 ≤ 𝑥𝑠 ≤ 𝑥𝑠𝑚𝑎𝑥 , ∀𝑠
𝑅𝑠𝑚𝑖𝑛 ≤ 𝑅𝑠 ≤ 1, ∀𝑠
0 ≤ 𝑟𝑙,𝑠 ≤ 1, ∀𝑙, 𝑠 ∈ 𝑆(𝑙)
By introducing the auxiliary variables 𝑐𝑙,𝑠 , which can be interpreted as the allocated
transmission capacity to source s at link l,
maximize ∑𝑠 𝑈𝑠 (𝑥𝑠 , 𝑅𝑠 )
subject to 𝑅𝑠 ≤ 1 − ∑𝑙𝜖𝐿(𝑠) 𝐸𝑙 (𝑟𝑙,𝑠 ), ∀𝑠
𝑥𝑠
≤ 𝑐𝑙,𝑠 , ∀𝑙, 𝑠 ∈ 𝑆(𝑙)
𝑟𝑙,𝑠
∑ 𝑐𝑙,𝑠 ≤ 𝐶𝑙𝑚𝑎𝑥 , ∀𝑙
𝑠∈𝑆(𝑙)
𝑥𝑠𝑚𝑖𝑛 ≤ 𝑥𝑠 ≤ 𝑥𝑠𝑚𝑎𝑥 , ∀𝑠
𝑅𝑠𝑚𝑖𝑛 ≤ 𝑅𝑠 ≤ 1, ∀𝑠
0 ≤ 𝑟𝑙,𝑠 ≤ 1, ∀𝑙, 𝑠 ∈ 𝑆(𝑙)
0 ≤ 𝑐𝑙,𝑠 ≤ 𝐶𝑙𝑚𝑎𝑥 , ∀𝑙, 𝑠 ∈ 𝑆(𝑙)
We introduce another layer, i.e., link layer (𝑐𝑙,𝑠 ), by using a vertical decomposition of
the optimization problem.
′
It is a GP, let 𝑥𝑠′ = log 𝑥𝑠 , 𝑈𝑠 (𝑒 𝑥𝑠 , 𝑅𝑠 ) = 𝑈𝑠′ (𝑥𝑠′ , 𝑅𝑠 )
maximize ∑𝑠 𝑈𝑠′ (𝑥𝑠′ , 𝑅𝑠 )
subject to 𝑅𝑠 ≤ 1 − ∑𝑙𝜖𝐿(𝑠) 𝐸𝑙 (𝑟𝑙,𝑠 ), ∀𝑠
𝑥𝑠′ − log 𝑟𝑙,𝑠 ≤ log 𝑐𝑙,𝑠 , ∀𝑙, 𝑠 ∈ 𝑆(𝑙)
∑ 𝑐𝑙,𝑠 ≤ 𝐶𝑙𝑚𝑎𝑥 , ∀𝑙
𝑠∈𝑆(𝑙)
𝑥′𝑚𝑖𝑛
≤ 𝑥′𝑠 ≤ 𝑥′𝑚𝑎𝑥
, ∀𝑠
𝑠
𝑠
𝑚𝑖𝑛
𝑅𝑠 ≤ 𝑅𝑠 ≤ 1, ∀𝑠
0 ≤ 𝑟𝑙,𝑠 ≤ 1, ∀𝑙, 𝑠 ∈ 𝑆(𝑙)
0 ≤ 𝑐𝑙,𝑠 ≤ 𝐶𝑙𝑚𝑎𝑥 , ∀𝑙, 𝑠 ∈ 𝑆(𝑙)
𝐿(𝐱 ′ , 𝐑, 𝐫, 𝐜, 𝛌, 𝛍)
= ∑ 𝑈′𝑠 (𝑥′𝑠 , 𝑅𝑠 ) + ∑ 𝜇𝑠 (1 − ∑ 𝐸𝑙 (𝑟𝑙,𝑠 ) − 𝑅𝑠 )
𝑠
𝑠
𝑙𝜖𝐿(𝑠)
+ ∑ ∑ 𝜆𝑙,𝑠 (log 𝑐𝑙,𝑠 + log 𝑟𝑙,𝑠 − 𝑥𝑠′ )
𝑙
𝑠∈𝑆(𝑙)
= ∑ (𝑈′𝑠 (𝑥′𝑠 , 𝑅𝑠 ) − ∑ 𝜆𝑙,𝑠 𝑥′𝑠 − 𝜇𝑠 𝑅𝑠 )
𝑠
𝑙𝜖𝐿(𝑠)
+ ∑ ( ∑ (𝜆𝑙,𝑠 ( log 𝑐𝑙,𝑠 + log 𝑟𝑙,𝑠 ) − 𝜇𝑠 𝐸𝑙 (𝑟𝑙,𝑠 ))) + ∑ 𝜇𝑠
𝑙
𝑠∈𝑆(𝑙)
𝑠
The Lagrange dual function is
𝑄(𝛌, 𝛍) =
𝐱′min
max
≼ 𝐱′ ≼ 𝐱′max
𝐿(𝐱 ′ , 𝐑, 𝐫, 𝐜, 𝛌, 𝛍)
𝐑min ≼ 𝐑 ≼ 𝟏
𝟎≼𝐫≼𝟏
𝐜∈C
𝐶 = {𝑐𝑙,𝑠 | ∑ 𝑐𝑙,𝑠 ≤ 𝐶𝑙𝑚𝑎𝑥 , ∀𝑙, 0 ≤ 𝑐𝑙,𝑠 ≤ 𝐶𝑙𝑚𝑎𝑥 , ∀𝑙, 𝑠 ∈ 𝑆(𝑙)}
𝑠∈𝑆(𝑙)
The dual problem is
minimize 𝑄(𝛌, 𝛍)
subject to 𝛌 ≽ 𝟎
𝛍≽𝟎
Algorithm 2 for Differentiated Dynamic Reliability Policy
Source problem and reliability price update at source s:
 Source problem
maximize 𝑈𝑠 (𝑥′𝑠 , 𝑅𝑠 ) − 𝜆𝑠 (𝑡)𝑥′𝑠 − 𝜇𝑠 (𝑡)𝑅𝑠
subject to 𝑥𝑠′𝑚𝑖𝑛 ≤ 𝑥′𝑠 ≤ 𝑥′𝑚𝑎𝑥
𝑠
𝑅𝑠𝑚𝑖𝑛 ≤ 𝑅𝑠 ≤ 1
 Price update
𝜇𝑠 (𝑡 + 1) = [𝜇𝑠 (𝑡) − 𝛽(𝑡)(𝑅 𝑠 (𝑡) − 𝑅𝑠 (𝑡))]+
where 𝑅 𝑠 (𝑡) = 1 − ∑𝑙𝜖𝐿(𝑠) 𝐸𝑙 (𝑟𝑙,𝑠 (𝑡)) is the end-to-end reliability at iteration t.
Link problem and congestion price update at link l:

Link problems
Link-layer problem
maximize ∑𝑠∈𝑆(𝑙) 𝜆𝑙,𝑠 (𝑡) log 𝑐𝑙,𝑠
subject to ∑𝑠∈𝑆(𝑙) 𝑐𝑙,𝑠 ≤ 𝐶𝑙𝑚𝑎𝑥 ; 0 ≤ 𝑐𝑙,𝑠 ≤ 𝐶𝑙𝑚𝑎𝑥 , 𝑠 ∈ 𝑆(𝑙)
Physical-layer problem for source s, 𝑠 ∈ 𝑆(𝑙)
maximize 𝜆𝑙,𝑠 (𝑡)log 𝑟𝑙𝑠 − 𝜇𝑠 (𝑡)𝐸𝑙 (𝑟𝑙,𝑠 )
subject to 0 ≤ 𝑟𝑙,𝑠 ≤ 1

Price update
+
𝜆𝑙,𝑠 (𝑡 + 1) = [𝜆𝑙,𝑠 (𝑡) − 𝛽(𝑡)(log 𝑐𝑙,𝑠 (𝑡) + log 𝑟𝑙,𝑠 (𝑡) − 𝑥𝑠′ (𝑡))] , 𝑠 ∈ 𝑆(𝑙)
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