Cooperative Packet Routing using Mutual Information Accumulation

Abstract

We consider the resource allocation problem in cooperative wireless networks
wherein nodes perform mutual information accumulation. We consider a unicast
setting and arbitrary arrival processes at the source node. Source arrivals can
be broken down into numerous packets to better exploit the spatial and temporal
diversity of the routes available in the network. We devise a
linear-program-based algorithm which allocates network resource to meet a
certain transmission objective. Given a network, a source with multiple
arriving packets and a destination, our algorithm generates a policy that
regulates which nodes should participate in transmitting which packets, when
and with what resource. By routing different packets through different nodes
the policy exploits spatial route diversity, and by sequencing packet
transmissions along the same route it exploits temporal route diversity.

Full text

Page 1

1
Cooperative Packet Routing using Mutual Information
Accumulation
Yanpei Liu, Student Member, IEEE, Jing Yang, Member, IEEE, Stark C. Draper, Member, IEEE
Abstract—We consider the resource allocation problem in
cooperative wireless networks wherein nodes perform mutual
information accumulation. We consider a unicast setting and
arbitrary arrival processes at the source node. Source arrivals
can be broken down into numerous packets to better exploit
the spatial and temporal diversity of the routes available in the
network. We devise a linear-program-based algorithm which al-
locates network resource to meet a certain transmission objective.
Given a network, a source with multiple arriving packets and a
destination, our algorithm generates a policy that regulates which
nodes should participate in transmitting which packets, when
and with what resource. By routing different packets through
different nodes the policy exploits spatial route diversity, and by
sequencing packet transmissions along the same route it exploits
temporal route diversity.
Index Terms—Wireless relay networks, cooperative communi-
cation, mutual information accumulation, rateless codes.
I. INTRODUCTION
In multi-hop cooperative relay networks, relay nodes co-
operate with each other to deliver data from network source
to network sink. The benefits of cooperation include im-
provements in energy efficiency, in robustness to fading and
interference, and reductions in the likelihood of the loss
of connectivity [1], [2]. Cooperation arises naturally in the
wireless medium due to its broadcast nature. When two nodes
are communicating, neighboring nodes can listen in on the
transmission and thus are in a position to help. As an example,
nodes further from the source and closer to the destination can
overhear earlier nodes’ transmissions. This gives them a head
start on decoding the message and speeds up message delivery
in contrast to conventional multi-hop transmission.
In order to be able to exploit the broadcast nature of
wireless transmission, we assume nodes are equipped with the
following abilities:
• Nodes can listen in on all transmissions.
• Nodes store all observations and slowly accumulate noisy
observations until they can decode the message.
To simplify our model we assume transmitters operate on
orthogonal channels and thus there is no interference. The
first point simply means nodes continually monitor the full
bandwidth of the system. The second point means that nodes
have large memories and thus can store all observations
This work was presented in part at the 49th annual Allerton Conference on
Communication, Control, and Computing, Monticello IL, September 2011.
The authors are with the Department of Electrical and Computer Engi-
neering, University of Wisconsin, Madison, WI 53706 (E-mail: {yliu73@,
yangjing@ece., sdraper@ece.}wisc.edu).
The authors were supported by the National Science Foundation under grant
CCF-0963834, Air Force Office of Scientific Research under grant FA9550-
09-1-0140 and by a grant from the Wisconsin Alumni Research Foundation.
until they become useful. This is what we mean by mutual
information accumulation. Mutual information accumulation
can be realized through the use of rateless (or “fountain”)
codes [1], [3]–[7].
Enabling nodes with this physical layer (PHY) ability
changes the routing and resource allocation problem from that
traditionally considered (e.g., in backpressure [8]–[10] or in
network coding [11]). Simpler PHY layer modications such
as energy accumulation using space-time or repetition coding
have been considered in [12]–[14]. The difference between
energy accumulation and mutual information accumulation
can be easily understood from the following example [1].
Consider binary signaling over a pair of independent erasure
channels each having erasure probability pefrom two relays
to a single receiver. If the two relays use repetition coding,
corresponding to energy accumulation, then each symbol will
be erased with probability p2
successfully received on average per transmission. Instead, if
we use different codes, the transmissions are independent and
on average 2(1 − pe) novel parity symbols are received per
transmission.
In contrast to previous works that consider the routing of
a single packet [1], [15], in this work we consider general
arrival processes and the routing of multiple packets. This
generalization is important to consider. With multiple packets
on the go there are extra spatial and temporal degrees of
freedom to consider that do not arise in the single packet
case. As we show in this paper, by using our algorithm
designed for multiple packets transmission, we can exploit the
route diversity of the network in both spatial and temporal
aspects. Different packets may be routed along partially (or
wholly) disjoint routes, thereby exploiting spatial or route
diversity. This is a desired feature when packets compete for
resource; it may be desirable to route a group of packets
through one link and the rest through another to balance the
traffic and resource consumption. Depending on the arrival
process different packets may be sent in sequence along the
same route, thereby exploiting temporal diversity. This usually
happens when routes choices are limited or packets arrive at
different times. Essentially, our algorithm exploits available
resource within the network and distributes it efficiently.
The main contribution of our work is the design of a joint
routing and resource allocation algorithm for multiple packets
transmission from the source to the destination in a given net-
work. Our work is a natural extension of the work in [1] where
only single packet transmission is considered. The contribution
of our work is twofold. We first present a formulation of
the routing and resource allocation problem. The formulation
considers various forms of energy and bandwidth constraints
e. Therefore, 1 − p2
esymbols are
arXiv:1108.3097v1 [cs.IT] 15 Aug 2011

Page 2

2
and is expressed in the form of linear program (LP). Second,
we design a centralized algorithm which solves the routing and
resource allocation jointly by solving a sequence of LPs. Each
LP solves for the optimal resource allocation given a route
decision of all packets. The resource allocation result is then
used to update the route decision and the method proceeds
iteratively.
Our work differs from previous cooperative routing and
resource allocation works in the sense that we consider both
PHY layer and routing (rather than just the PHY layer in
simple networks or both in simplified settings). It differs
from the current state-of-the-art cooperative routing algorithms
(e.g., opportunistic multi-hop routing [2], backpressure-based
algorithm [8]–[10], and network-coding-based algorithm [16],
[17]) in that it involves a modified PHY layer with an extra
feature – mutual information accumulation which can be easily
implemented using rateless codes. Meanwhile, there are many
existing works in cooperative routing using this feature. In [3],
mutual information is considered for single relay networks.
In [18], mutual information is also considered, but without
a consideration on resource allocation. Routing and resource
allocation with mutual information accumulation is considered
in [1], [15], but they only consider single packet routing.
To the best of our knowledge, there has been little prior
work investigating routing and resource allocation with mutual
information accumulation for multiple packets.
The rest of the paper is organized as follows. We present
the system model and problem formulation in Sec. II. The
centralized algorithm is developed and discussed in Sec. III.
We provide detailed numerical results in Sec. IV. Discussion
and conclusion are in Sec. V.
II. SYSTEM MODEL
We consider a system with L nodes: the source, always
labeled 1, the destination, always labeled L and L − 2 relay
nodes. Suppose that K large data files (not necessarily of the
same size) arrive at the source at a set of specified times
τ1,τ2,...,τK. Upon arrival each file is subdivided into a
number of packets. In all there are N ≥ K packets. The
indices of the packets are ordered according to the arrival
times of the corresponding data files. We consider each of
these packets as a separate commodity from a routing point
of view. The network’s goal is to deliver N such packets from
the source to the destination under various channel constraints.
Each relay may participate in transmitting a subset of these
packets or remain silent. To simplify the analysis we assume
that the only significant power expenditure for each node lies
in transmission.
A. Problem parameterization
We first introduce the definition of a “decoding event”.
Definition 1: A decoding event occurs when either a node
decodes a packet, or a packet is made available at the source.
We assume a decode-and-forward relaying strategy. We
denote as Tc
c ∈ {1,2,...,N}. In particular, each Tc
arrival time of the corresponding data file ? which packet c is
ithe time at which node i decodes packet c,
1corresponds to the
€
T1
1
€
T1
2
€
T1
3
€
T2
1
€
T3
2
€
T3
3
€
t
€
0
€
T1
€
T2
€
T3
€
T4
€
T8
€
T9
€
t
€
0
€
Δ1
€
Δ2
€
Δ3
€
Δ4
€
Δ9
Fig. 1. A sample decoding order for N = 3, L = 3 and M = NL.
subdivided from, i.e., Tc
of a “decoding order” – a key component in our algorithm.
Definition 2: A decoding order T , containing Tc
Tc
∈
{T1
It should be noted that a decoding order T only specifies the
order of its decoding events. It does not specify when those
events occur (or the value of Tc
We let T denote the decoding order and let M = |T | denote
the number of decoding events inside this decoding order T .
Since a node cannot transmit a packet until it has decoded it,
the order of decoding events puts constraints on the resource
allocated to nodes for each packet. We further note that a
decoding order can contain up to NL decoding events, i.e.,
M ≤ NL. Then the label pair (i,c) of Tc
its position s, s ∈ {1,2,...,M}. In other words, there exists
a unique mapping f which maps (i,c) to s and we further
denote this mapping as Tc
With a decoding order T , we define the “inter-decoding
intervals” for its decoding events.
Definition 3: Given a T , the inter-decoding interval ∆sis
the time interval between Ts, Ts−1∈ T and is defined as
1= τ?. We next present the definition
1 and
Lfor all c
1,...,TN
{1,2,...,N}, is a subset of the set
1,...,TN
L} sorted in increasing order.
i’s).
iuniquely determines
i≡ Tf(i,c)≡ Ts.
∆s= Ts− Ts−1,
where we denote time 0 as T0.
The entire message transmission can be thought of as con-
sisting M intervals. The sth interval is of duration ∆sand is
characterized by the fact that at the end of this ∆s, a decoding
event associated with Tsmust occur. A typical decoding order
is shown in Fig. 1.
We assume the ith node operates at a fixed power spectral
density (PSD) denoted as Pi(joules/sec/Hz), uniform across
its transmission band. We assume the channel between any
pair of nodes is block-fading and frequency non-selective. The
channel gain between node i and node j is denoted as hi,j.
Under these assumptions, the spectral efficiency (bits/sec/Hz)
between node i and node j can be expressed as [1], [19]
?
Ci,j= log2
1 +hi,jPiWi
N0Wi
?
= log2
?
1 +hi,jPi
N0
?
,
(1)
where N0/2 denotes the PSD of the white noise process. We
further denote the time-bandwidth product allocated to node i
to transmit packet c during a given time interval ∆sas Ac
(sec-Hz). Then the information flow from node i to node j
during this interval is Ac
i,s
i,sCi,jbits.

Page 3

3
B. Problem constraints
For a given decoding order we find the resource allocation
minimizing an objective function subject to the following
constraints:
1) ∆s≥ 0 for all s.
2) Ac
3) Arrival process constraint; the source cannot transmit
packet c until its corresponding data file has reached the
source.
4) Decoding constraint; node j must decode packet c at the
associated timing point Ts.
5) Constraint(s) on energy and bandwidth.
Arrival process constraint puts a constraint on value of Tc
(thus the time intervals between them). Each Tc
to the arrival time of the corresponding data file ? which packet
c is subdivided from, i.e., Tc
4) and 5) in more details below.
We assume that nodes use codes that are ideal in the sense
that they fully capture this potential flow, working at the
Shannon limit at any rate. We also assume that distinct packets
can be simultaneously transmitted from any node with no
interference between packets and receivers have perfect knowl-
edge to distinguish them. Nodes are further designed to use
independently generated codes. This design leads to another
assumption that a receiver can combine information flows from
two or more transmitters for each packet without any rate loss.
As noted in [1], the use of independently-generated codes
is crucial for the mutual information accumulation process.
If the transmitters used the same code, the receiver would
get multiple looks at each codeword symbol. This is “energy
accumulation.” By looking at different codes the receiver
accumulates mutual information rather than energy.
With mutual information accumulation, the decoding con-
straint imposes a constraint on some timing points. If Tc
j ?= 1 is in the decoding order, node j must be able to decode
packet c at this time. This constraint is formally expressed as
i,s≥ 0 for all s, i and c.
1’s
1corresponds
1= τ?. We describe constraints
j,
?
i:f(i,c)<f(j,c)
f(j,c)
?
s=f(i,c)+1
Ac
i,sCi,j≥ Bc, for all c,j ?= 1, (2)
where Bcis the size of packet c. Recall that Ci,jis the spectral
efficiency (bits/sec/Hz) of the channel connecting node i to
node j. Equation (2) says that in order for node j to decode
packet c, the total accumulated information at node j for
packet c must exceed Bcbits by the time Tc
nature of existing implementations of rateless codes can be
handled by incorporating an overhead factor (1 + ?) into the
right-hand side of (2). Also, the relation (2) expresses the
constraint that node i can transmit packet c to node j only
after it has decoded packet c.
Constraints on energy and bandwidth can either be system-
wide constraints or be imposed on a node by node basis. We
state various possible constraints in the followings.
1) Per-node bandwidth constraint: If node i is assigned
with bandwidth Wi, its resource allocated during given
j. The non-ideal
∆smust satisfy the following
N
?
c=1
Ac
i,s≤ ∆sWi, for all i,s.
(3)
2) Sum bandwidth constraint: If the total bandwidth WT
is allocated, a sum bandwidth constraint applies across
all nodes during any given ∆s. This can be formally
expressed as
L
?
i=1
N
?
c=1
Ac
i,s≤ ∆sWT, for all s.
(4)
3) Per-node energy constraint: If node i with transmission
power Pi is assigned with energy budget Ei, we can
express the per-node energy constraint as
N
?
c=1
M
?
s=f(i,c)+1
Ac
i,sPi≤ Ei, for all i.
(5)
4) Sum energy constraint: If a sum energy ET is assigned
for all nodes, a sum energy constraint can be applied as
L
?
i=1
N
?
c=1
M
?
s=f(i,c)+1
Ac
i,sPi≤ ET.
(6)
In the next section we develop a centralized algorithm based
on the LP framework.
III. CENTRALIZED ALGORITHM
The LP framework can take many objective functions. If
there is only one data file (thus T1
natural objective function to use is the total transmission time
1,T2
1,...,TN
1
= τ1), one
Tt=
M
?
s=1
∆s.
(7)
Alternatively, if K data files arrive at different times (thus
packets are not available to the source at the same time),
instead of minimizing total transmission time, one might want
to minimize the average transmission time given by
Ta=
1
K
K
?
?=1
T?,
(8)
where T?is the duration data file ? is being transmitted. This
objective function is appropriate in the sense that it does not
penalize the arrival time.
Other linear programming frameworks are also possible.
For example, one may wish to minimize the total energy
expenditure given by
L
?
i=1
N
?
c=1
M
?
s=f(i,c)+1
Ac
i,sPi,
(9)
subject to total transmission time or average transmission time
constraint.

Page 4

4
A. Characteristics of the problem
We now study the properties of routing and resource alloca-
tion under different constraints with total transmission time (7)
as our objective function. First, under the per-node bandwidth
constraint, we have the following lemma.
Lemma 1: Under a given decoding order, suppose ∆sand
Ac
bandwidth constraint (3). Then for each s, there exists a node
i such that the inequality in (3) becomes equality.
Proof: We prove by contradiction. Suppose for some s
inequality is strict for all i, i.e.,
i,sare the solution to the problem under the per-node
N
?
c=1
Ac
i,s< ∆sWi, for all i.
Then we can scale down ∆s, yielding a smaller objective
value. This contradicts the assumption that ∆sis the optimal
solution to the LP.
This proposition suggests that the ∆scan be calculated as
∆s= max
i
?N
c=1Ac
Wi
i,s
.
Under the sum bandwidth constraint, we first have the
following lemma.
Lemma 2: Under a given decoding order, suppose ∆sand
Ac
constraint (4). Then the inequality in (4) must be equality for
all s.
Proof: We prove this by contradiction. Suppose for some
s the equality does not hold, i.e.,
i,sare the solution to the problem under the sum bandwidth
L
?
i=1
N
?
c=1
Ac
i,s= ∆sWs< ∆sWT, for some s.
Then for these s we can scale down the corresponding ∆sby
Ws
WTwhile increase Wsto WT. We therefore obtain a solution
which has smaller objective value.
This lemma leads to the following important theorem.
Theorem 3: Under the sum bandwidth constraint and a
given decoding order, if Pi = P for all i then the solution
that minimizes the objective in (7) also minimizes the sum
energy.
Proof: The total energy expenditure of the entire system
is
L
?
i=1
N
?
L
?
M
?
c=1
M
?
M
?
s=f(i,c)+1
Ac
i,sP
=
i=1
N
?
c=1s=1
Ac
i,sP
=
s=1
∆sWTP
= TtWTP.
The first equality holds because a node cannot transmit a given
packet (resource allocated to it is zero) before it decodes the
packet. The second equality follows from Lemma 2. Since the
objective Tt is proportional to the energy used, minimizing
one minimizes the other.
This theorem tells us that in this setting there is no trade
off between total transmission time and energy. The minimum
transmission time policy is identical to the minimum energy
policy.
Theorem 4: Under the sum bandwidth constraint and a
given decoding order, if the minimum transmission time for
transmitting one packet is T, then the minimum transmission
time for routing N packets of same size is NT.
Proof: Let R denote the optimal set of routes on which
transmitting a packet takes T amount of time. These routes
are equivalent in the sense that transmitting a packet on any
of them takes T amount of time, consuming TWTa amount of
total resource. Then transmitting N packets of the same size
on R can be finished within NT amount of time by allocating
each packetTWT
N
amount of total resource.
We now show that this is the best the system can do.
Suppose otherwise, i.e., not all packets are routed through
R and the total transmission time is smaller than NT. Then
there exists at least one packet traveling on a different set
of equivalent routes R??= R. Since the transmission time
is smaller than NT, each packets traveling on R should be
given more than
N
amount of resource. This implies that
the packet on R?shares less than
whereas achieves a total transmission time less than NT.
This contradicts our assumption that R is the optimal set of
equivalent routes.
This theorem suggests that in sum bandwidth constraint, the
performance may not be improved by dividing a big data file
into smaller packets. We will illustrate this in simulation.
TWT
TWT
N
amount of resource
B. Optimizing the Decoding Order
In this section we show an important theorem that moti-
vates the centralized algorithm. The theorem tells us how to
manipulate the decoding order based on the solution of the LP
problem. Denote as P a LP framework with a chosen objective
and constraints being considered. Given any decoding order of
length M, define
?
to be the solution to the LP with optimal objective value Topt
on this decoding order. We then have the following theorem.
Theorem 5: If ∆m = 0 for some m and we swap the
positions of Tm and Tm−1 in the decoding order, then the
objective value T∗
order satisfies T∗
Proof:We prove the theorem by showing that when
swapped decoding order is used, the original solution x with
optimal objective value Toptis still feasible under new decod-
ing order. To show x is feasible under the new decoding order,
we show the decoding constraint of the swapped decoding
order is satisfied by this solution.
If a decoding event associated with Tm−1has its decoding
constraint satisfied at this time, its decoding constraint is
certainly satisfied at a later time Tm. If a decoding event
x =∆1,...,∆M,A1
1,1,...,AN
L,M
?
optobtained with this swapped decoding
opt≤ Topt.

Page 5

5
associated with Tmhas its decoding constraint satisfied at this
time, it actually has the constraint satisfied at an earlier time
Tm−1since Ac
with optimal objective value Topt is feasible under swapped
decoding order
The idea behind Theorem 5 is based on the following
observation. A solution to the LP with ∆m = 0 indicates
that the event associated with Tm takes place at exactly the
same time with the previous event Tm−1, or actually occurs
before it. Therefore, swapping the position of Tmand Tm−1
typically gives a decrease in the objective value once the LP
is solved under swapped decoding order. In the case that Tc
swapped with Tc
is excluded from the new decoding order by our algorithm.
i,m= 0 for all i and c. Therefore, solution x
iis
Lfor i not equal to L, the decoding event Tc
i
C. Centralized algorithm
We design and implement a centralized resource allocation
and routing algorithm iterating between two sub-problems:
1) For a given decoding order, a resource allocation scheme
is determined by solving a given LP problem P. Solving
the optimization problem with these constraints gives
time allocation ∆s(thus Ts) for all s and the resource
allocation Ac
2) With a given resource allocation scheme, we update the
decoding order.
In the following we discuss the algorithm in more details.
1) Initialization: We first consider the decoding order ini-
tialization as if there were only one packet to be routed.
We design a heuristic algorithm similar to the greedy filling
algorithm in [12] based on network state. For single packet
transmission, the decoding order is essentially the order of
nodes which successfully decode that packet and can come
online to broadcast it. Therefore, with a set of already decoded
nodes, the next node to decode is most likely the one which is
able to benefit the most from the already decoded nodes. We
thus choose the next node to be the one that has maximum sum
spectral efficiency between itself and all previously decoded
nodes. The algorithm pseudocode is given in Algorithm 1.
i,sfor all i, c and s.
Algorithm 1 Initialize the decoding order
1: T ← {T1
2: while length(T ) < L − 1 do
3:
j ← argmaxk/ ∈T ,k?=L
4:
T ← {T ,T1
5: end while
6: T ← {T ,T1
1}
?
i∈TCi,k
j}
L}
In multiple packets case, since we have no initial knowledge
about which nodes should decode which packets and in which
order, we believe it is difficult to design an effective decoding
order initialization algorithm. We therefore initialize the de-
coding order based purely on the one with single transmitted
packet. Suppose in single packet case the following decoding
order is found
T =
?
T1
1,T1
i,...,T1
L
?
,
the decoding order for multi-packet case is simply chosen to
be
?
2) Decoding order updates: Based on the solution to the
LP, the algorithm first searches for all those m’s such that
∆m= 0. Then for those m, it checks for the situation where
Tc
the corresponding Tc
it swaps the order of Tm−1and Tm. If the swapped decoding
events are all the same as the ones in the previous iteration,
and if there was no decoding event dropped in the previous
iteration, the algorithm terminates. We present the sketch of
the algorithm in Algorithm 2.
T =T1
1,...,TN
1,T1
i,...,TN
i,...,TN
L
?
.
iis followed by Tc
Lfor some i ?= L. If it finds one, it drops
iand reruns the LP. If it does not find one,
Algorithm 2 The routing and resource allocation algorithm
1: Choose a LP P
2: Initialize a decoding order T
3: DoAgain ← true
4: while DoAgain do
5:
result ← LP(T , P)
6:
index ← SearchForZeroDeltas(result)
7:
ToDrop
←
tion(index)
8:
if ToDrop ?= 0 then
9:
DoAgain ← true
10:
drop the decoding event
11:
else if drop = 0 and index = index in last iteration and
no event is dropped in last iteration then
12:
DoAgain ← false
13:
break
14:
else
15:
DoAgain ← true
16:
SwapEvents(result)
17:
end if
18: end while
SearchForNodeDecodeAfterDestina-
Because of the exponential number of orderings we expect
the problem of finding the optimal decoding order to be NP-
hard. The sub-optimality of our heuristic algorithm comes
from the fact that the excluded decoding events may actually
be helpful. Without a mechanism to “re-introduce” the ex-
cluded events our algorithm is not expected to achieve global
minimum for all networks.
3) Characteristics of final route: Our algorithm has the
property that different packets may take different paths toward
the destination under the per-node resource constraint. This
exploits the spatial route diversity. For example, the algorithm
may schedule the first 3 packets to pass through a set of relay
nodes and schedule the next 3 packets to pass through yet
another set of relay nodes. The intuition is that instead of
choking an existing route and overloading the already busy
nodes, the algorithm balances the traffic by invoking other
free nodes to fully utilize the network.
4) Challenges: The challenges we face in designing this
multi-packets routing and resource allocation algorithm are
more difficult than the ones the authors encountered in [1].