DISTRIBUTED ROUTING ALGORITHMS FOR WIRELESS MULTIHOP NETWORKS
Alejandro Ribeiro and Georgios B. Giannakis
Dept. of ECE, University of Minnesota
Nikolaos D. Sidiropoulos
Dept. of ECE, Technical University of Crete
ABSTRACT
We introduce distributed algorithms to find rate-optimal routes based on
local knowledge of the pairwise error probability (reliability) matrix. The
distributed algorithms are built by (re)-formulating optimization prob-
lems amenable to application of dual decomposition techniques. Con-
vergence of our algorithms to the optimal routing matrix is guaranteed
under mild conditions. Many rate-optimality criteria of practical inter-
est can be casted in our framework including maximization of: i)worst
user’s rate; ii) weighted sum of rates; iii) product of rates; and iv) relay
network rate. We test robustness of our algorithms to node mobility.
Keywords: Communication systems routing, Wireless networks, Opti-
mization methods, Linear programming, Distributed computing.
1. INTRODUCTION
Capitalizing on the potential energy savings of multihop wireless net-
works requires solving the challenging problem of finding multihop routes
according to properly defined optimality criteria. Existing routing pro-
tocols/algorithms are built on our accumulated knowledge of routing in
wired networks. Consequently, the usual approach is to i) define a com-
munication radius for each node; ii) draw the corresponding connectivity
graph; and iii) utilize network optimization tools to find pertinent routes.
While definitely valuable as a first approach, a graph is not an accu-
rate model of a wireless network [3]. In a recent paper we introduced
a framework to design stochastic routing algorithms/protocols using the
reliability matrix R whose (i, j )-th entry Rij represents the probability
that a packet transmitted from the j -th user Uj is correctly received by
the i-th user Ui [6]; see also Fig. 1.
While offering a more accurate model of the broadcast and unreli-
able wireless channel, the usefulness of a model based on R depends on
the algorithmic complexity of finding optimal routes. Enticingly, many
interesting optimality criteria lead to routing algorithms in the form of
convex optimization problems [6]. Even though this ensures manageable
complexity, it requires R to be available at a central location. This en-
tails: i) a large communication cost to collect R and percolate the optimal
routing matrix; ii) considerable delay to compute the optimal routes; and
iii) lack of resilience to changes in R, a problem particularly important
in mobile scenarios.
Distributed algorithms, whereby nodes iteratively interchange vari-
ables only with one-hop neighbors tackle precisely these problems. The
goal of this paper is to show that the optimal routing problems in [6] can
be solved by an iterative distributed algorithm whereby i) node Uj has
access only to link reliabilities for transmission to and from other nodes
(the j -th row and column of R); ii) Uj interchanges messages only with
one-hop neighbors, defined as the set of terminals with non-zero proba-
bility of decoding Uj ’s packets; and iii) as time progresses Uj computes
its optimal routing probabilities.
1.1. Stochastic routing in wireless multihop networks
Consider a wireless network with J +1 user nodes {Uj }
J+1
j=1
in which
the first J users {Uj }
J
j=1
collaborate in routing packets to the destination
D ≡ UJ+1. The physical and medium access layers are such that if a
packet is transmitted by Uj it is correctly received by Ui with probability
Rij that we arrange in the matrix R. Packets are stochastically routed
according to probabilities Tij arranged in the matrix T. When a user
terminal Uj decides to transmit a packet it selects a random terminal
as the intended destination with Ui chosen with probability Tij . If the
transmission is successful the packet moves to Ui’s queue, if not it is kept
Ŧ1500 Ŧ1000 Ŧ500 0 500 1000 1500
Ŧ1500
Ŧ1000
Ŧ500
0
500
1000
1500
AP
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Fig. 1. Connectivity graph for a network with J = 40 user terminals.
The color index represents the value of Rij .
by Uj that attempts transmission, possibly to a different node, at a later
time. To capture the evolution of packets through the network we define
a matrix K whose elements Kij represent the probability that a packet
moves from Uj ’s queue to Ui’s queue. For i = j a packet moves from
Uj to Ui if and only if it is routed towards Ui and is correctly decoded at
Ui . Since these two events are independent,
Kij = Tij Rij for i = j, K
T
1 = 1, T
T
1 = 1. (1)
The last two constraints are because K and T are stochastic matrices.
Also, let ρ := [ρ1,...,ρj ]
T
denote the vector of packet arrival rates
and α := [α1,...,αJ ]
T
the rate of departures that we constraint by
0 α 1. Defining K0 as the J × J upper left submatrix of K it is
not difficult to see that we can relate ρ and α by [6]
ρ =(I − K0)α (2)
With R available at the AP, we look for routes maximizing a measure of
the arrival rate vector ρ. Letting f (ρ): R
J
→ R be the function used
to compare arrival rate vectors ρ, the optimal routing matrix T
∗
is given
as the solution of the generic optimization problem (symbols and
denote componentwise inequalities between vectors):
(K
∗
, T
∗
) = arg max f [(I − K0)α]
s.t.Kij = Rij Tij for i = j, K
T
1 = 1, T
T
1 = 1
0 α 1. (3)
Finding efficient methods to solve (3) is challenging for general f (ρ).
However, for any f (ρ) that is concave and monotonically non-decreasing
in each component
1
(3) can be transformed to an equivalent convex op-
timization problem. Indeed, [6] establishes that for functions that are
monotonically non-decreasing in each component there exists an opti-
mal solution of (3) with α = 1, thus implying that (3) can be rewritten
as
(K
∗
, T
∗
) = arg max f [(I − K0)1] (4)
s.t.Kij = Rij Tij for i = j, K
T
1 = 1, T
T
1 = 1.
1
We say g(v) is monotonically non-decreasing in each component if for vec-
tors v
1
, v
2
with v
1
j
≤ v
2
j
and v
1
i
= v
2
i
for i = j, g[v
1
] ≤ g[v
2
].
III 517 1424407281/07/$20.00 ©2007 IEEE ICASSP 2007