# Integrating Traffic Estimation and Routing Optimization for Multi-Radio Multi-Channel Wireless Mesh Networks

**ABSTRACT** Traffic routing plays a critical role in determining the performance of a wireless mesh network. To investigate the best solution, existing work proposes to formulate the mesh network routing problem as an optimization problem. In this problem formulation, traffic demand is usually implicitly assumed as static and known a priori. Contradictorily, recent studies of wireless network traces show that the traffic demand, even being aggregated at access points, is highly dynamic and hard to estimate. Thus, in order to apply the optimization-based routing solution into practice, one must take into account the dynamic and unpredictable nature of wireless traffic demand. This paper presents an integrated framework for network routing in multi-radio multi-channel wireless mesh networks under dynamic traffic demand. This framework consists of two important components: traffic estimation and routing optimization. By analyzing the traces collected at wireless access points, the traffic estimation component predicts future traffic demand based on its historical value using time-series analysis, and represents the prediction result in two forms - mean value and statistical distribution. The optimal mesh network routing strategies then take these two forms of traffic demand estimations as inputs. In particular, two routing algorithms are proposed based on linear programming which consider the mean value and the statistical distribution of the predicted traffic demands, respectively. The trace-driven simulation study demonstrates that our integrated traffic estimation and routing optimization framework can effectively incorporate traffic dynamics in mesh network routing, where both algorithms outperform the shortest path algorithm in about 80% of the test cases.

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**ABSTRACT:**Most WSN routing protocols select route based on hop count. Efficient routing metric should consider both inter-flow and intra-flow interference. A new minimum route-interference routing metric for multi-radio WMN is proposed. The path which has a minimum total interference can be found by using this metric. Simulation result shows that network performance can be significantly improved with the new routing metric.01/2009; -
##### Conference Paper: Distributed Algorithms for Joint Routing and Frame Aggregation in 802.11n Wireless Mesh Networks

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**ABSTRACT:**A wireless mesh network (WMN) is a special type of wireless ad-hoc network, which consists of mesh clients, mesh routers and gateways to the Internet, organized in a mesh topology. The mesh clients are often laptops, cell phones and other wireless devices. Mesh routers forward traffic between mesh clients and gateways. Despite a number of promising features provided by WMNs, such as low deployment cost, self-healing, etc., the throughput of WMNs is often limited by severe congestion and collisions, and thus cannot satisfy the increasing traffic demands of numerous applications. In this paper, we study how to maximize the throughput of IEEE 802.11n WMNs by joint routing and frame aggregation. Frame aggregation is to aggregate multiple frames into a large frame before transmission, to reduce communication overhead and alleviate collisions. We first show that previous frame aggregation strategies cannot achieve optimal network throughput. We then formulate the joint problem into a linear programming (LP) problem by considering traffic in the network as flow. As most previous algorithms for LP are centralized and difficult to deploy in large-scale WMNs, we propose a distributed algorithm to solve the formulated problem, in which each mesh router determines the amount of traffic flow for its adjacent links based on the traffic information of neighbors and interfering links. However, in realistic 802.11n WMNs, traffic is transmitted in frames instead of flow, and the traffic to different routers needs to be distinguished. Thus, we further provide an algorithm to determine the routing and frame aggregation strategy for each mesh router, using the traffic flow derived from the first algorithm. We have conducted extensive simulations to evaluate the proposed algorithms and the results demonstrate that the network throughput can be significantly improved compared with existing schemes.Parallel & Distributed Processing (IPDPS), 2013 IEEE 27th International Symposium on; 01/2013 - [Show abstract] [Hide abstract]

**ABSTRACT:**Joint channel assignment and routing is an essential yet challenging issue for multi-radio multi-channel wireless mesh networks. Though several works are presented in the existing literature to approach this problem, the key question – how to ensure that the resulting network performance can closely track the optimal solution under high traffic variability without incurring too much overhead, remains unanswered. In this work, we present a new solution called “Robust joint Channel Assignment and Routing with Time partitioning (RCART)” for WMNs. RCART consists of three steps: (1) Time Partitioning and Traffic Characterization, which accomplishes the goal of partitioning time into periodic intervals with consistent properties which can be routed efficiently, (2) Robust Routing, which finds a robust routing scheme that provides an upper bound on the worst-case network performance for traffic demands that fall into a convex region, (3) Channel Assignment, which allocates radios to fixed channels during the time interval identified in step 1 and based on the knowledge of traffic distribution from step 2, using the worst-case congestion ratio as a robustness metric in its objective. Introducing time partitions as an additional control variable in the robust mesh routing RCART solution significantly improves average-case performance. Performance evaluation is conducted for RCART using real traffic demand traces. The results show that our RCART solution significantly outperforms the existing works without time partitioning or with simpler traffic profile models.Ad Hoc Networks 02/2014; 13:210–221. · 1.94 Impact Factor

Page 1

Integrating Traffic Estimation and Routing Optimization

for Multi-Radio Multi-Channel Wireless Mesh Networks

Liang Dai, Yuan Xue, Bin Chang, Yanchuan Cao, Yi Cui

Department of Electrical Engineering and Computer Science

Vanderbilt University

Email: {liang.dai, yuan.xue, bin.chang, yanchuan.cao, yi.cui}@vanderbilt.edu

Abstract—Traffic routing plays a critical role in determining

the performance of a wireless mesh network. To investigate

the best solution, existing work proposes to formulate the

mesh network routing problem as an optimization problem. In

this problem formulation, traffic demand is usually implicitly

assumed as static and known a priori. Contradictorily, recent

studies of wireless network traces show that the traffic demand,

even being aggregated at access points, is highly dynamic and

hard to estimate. Thus, in order to apply the optimization-based

routing solution into practice, one must take into account the

dynamic and unpredictable nature of wireless traffic demand.

This paper presents an integrated framework for wireless mesh

network routing under dynamic traffic demand. This framework

consists of two important components: traffic estimation and

routing optimization. By studying the traces collected at wireless

access points, we present a traffic estimation method which pre-

dicts future traffic demand based on its historical data using time-

series analysis. This method provides not only the mean value

of the future traffic demand estimation but also its statistical

distribution. We further investigate the optimal routing strategies

for wireless mesh network which take these two forms of traffic

demand estimations as inputs. Based on linear programming, we

present two routing algorithms which consider the mean value

and the statistical distribution of the predicted traffic demands,

respectively. The trace-driven simulation study demonstrates

that our integrated traffic estimation and routing optimization

framework can effectively incorporate traffic dynamics in mesh

network routing.

I. INTRODUCTION

Wireless mesh networks have attracted increasing attention

and deployment as a high-performance and low-cost solution

to last-mile broadband Internet access. In a wireless mesh

network, local access points and stationary wireless mesh

routers communicate with each other and form a backbone

structure which forwards the traffic between mobile clients and

the Internet. To alleviate the problem of location-dependent

interference in wireless communication, mesh routers are

usually equipped with multiple radios which enable them to

transmit and receive simultaneously or transmit on multiple

channels simultaneously.

Traffic routing and channel assignment jointly play a critical

role in determining the performance of a wireless mesh

network. Thus it attracts extensive research recently. The pro-

posed approaches usually fall into two ends of the spectrum.

On one end of the spectrum are the heuristic algorithms

(e.g., [1]–[4]). Although many of them are adaptive to the

dynamic environments of wireless networks, these algorithms

lack the theoretical foundation to analyze how well the net-

work performs globally (e.g., whether the traffic shares the

network in a fair fashion).

On the other end of the spectrum, there are theoretical

studies based on optimization methods (e.g., [5], [6]). The

algorithms derived from these optimization formulations can

usually claim analytical properties such as resource utiliza-

tion optimality and throughput fairness. In these optimization

frameworks, traffic demand is usually implicitly assumed as

static and known a priori. Contradictorily, recent studies of

wireless network traces [7] show that the traffic demand, even

being aggregated at access points, is highly dynamic and hard

to estimate. Such observations have significantly challenged

the practicability of the existing optimization-based routing

solutions in wireless mesh networks.

To address this challenge, this paper investigates the optimal

mesh network routing framework which takes into account

the dynamic nature of wireless traffic demand. This routing

framework could work as a part of the joint routing and

channel assignment solution in [5]. To incorporate the traffic

dynamics, the following two components must be seamlessly

integrated into this framework.

• Traffic demand estimation which derives the traffic model

of a wireless mesh network. The model should be de-

pendable at predicting the mean demand at long term,

yet agile at containing often uncertain dynamics at short

term.

• Routing optimization which distributes the traffic along

different routes so that minimum congestion will be

incurred even under dynamic traffic. The routing strategy

should effectively take into account the traffic demand

estimation results.

By studying the traces collected at Dartmouth College

campus wireless network [8], this paper first presents a traffic

prediction method based on time-series analysis. This method

derives future traffic demand based on its historical data. The

mean value of the predicted demand, together with its pre-

diction error distribution, are used in establishing a statistical

model for the traffic demand at a local access point.

This paper further identifies an optimization framework

which integrates the demand prediction into traffic routing.

In particular, two forms of traffic demands are considered as

the inputs for routing optimization, namely the mean value

of the demand prediction and its statistical distribution. We

present two routing algorithms for each form of the traffic

demand estimation respectively. For the first case, based on

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the classical maximum concurrent flow problem, we formu-

late optimal mesh network routing as a linear programming

problem to maximize, among all flows, the minimum scaling

factor of throughput to fixed-value demand (λ) and present a

fast (1 − ǫ)-approximation algorithm (i.e. fixed-demand mesh

network routing (FMR) algorithm) which could accept the

mean value of the demand prediction as the input. For the

second case, in order to incorporate the statistical distribution

of the demand estimation into the problem formulation, we

characterize the traffic demand using a random variable. Now

the scaling factor λ under a given routing solution is also

a random variable. The throughput optimization problem is

then extended to a stochastic optimization problem where the

expected value of the scaling factor λ is maximized. Finally,

based on the design of FMR algorithm, a (1−ǫ)-approximation

algorithm (uncertain-demand mesh network routing (UMR))

is presented for optimal mesh network routing under uncertain

demand.

To evaluate the performance of our algorithms under real-

istic wireless networking environment, we conduct a trace-

driven simulation study. In particular, we derive the traffic

demand for the local access points of our simulated wire-

less mesh network based on the traffic traces collected at

Dartmouth College campus wireless networks. Our simulation

results demonstrate that our integrated traffic estimation and

optimal routing framework could effectively incorporate the

traffic dynamics into the routing optimization of wireless mesh

networks.

The original contributions of this paper are two-fold. Practi-

cally, the integration of traffic estimation and routing optimiza-

tion effectively improves the routing performance of wireless

mesh networks under dynamic and uncertain traffic. The full-

fledged simulation study based on real wireless network traffic

traces provides convincing validation of the practicability of

our solution. Theoretically, upon the classical linear optimiza-

tion algorithm which only accepts the fixed-value demands

as inputs, we extend it into a stochastic optimization solution

capable of serving uncertain demands that are modelled by

their statistical distributions.

The remainder of this paper is organized as follows. Sec. II

presents the system model and solution overview. Sec. III

formulates the mesh network routing problem under fixed-

value traffic demand and uncertain traffic demand and two

fast approximation algorithms (FMR and UMR). Sec. IV

describes the traffic prediction method. We show simulation

results in Sec. V, present related work in Sec. VI and finally

conclude the paper in Sec. VII.

II. SYSTEM MODEL AND SOLUTION OVERVIEW

A. Network and Interference Model

In a multi-hop wireless mesh network, local access points

aggregate and forward the traffic from the mobile clients that

are associated with them. They communicate with each other,

also with the stationary wireless routers to form a multi-

hop wireless backbone network. This wireless mesh backbone

network forwards the user traffic to the gateways which are

connected to the Internet. We use w ∈ W to denote the

set of gateways in the network. In the following discussion,

local access point, gateway and mesh router are collectively

called mesh nodes and denoted by set V (Note that W ⊂ V ).

Further, we assume that node v is equipped with κ(v) radios.

The network could use a set of orthogonal wireless channels

denoted by C. For example, in the IEEE 802.11b standard,

|C| = 3.

In a wireless network, packet transmissions in the same

channel are subject to location-dependent interference. We

assume that all mesh nodes have the uniform transmission

range denoted by RT. Usually the interference range is larger

than its transmission range. We denote the interference range

of a mesh node as RI = (1 + ∆)RT, where ∆ ≥ 0 is

a constant. In this paper, we consider the protocol model

presented in [9]. Let r(u,v) be the distance between u and

v (u,v ∈ V ). In the protocol model, packet transmission

from node u to v on channel c ∈ C is successful, if and

only if (1) the distance between these two nodes r(u,v)

satisfies r(u,v) ≤ RT; (2) any other node w ∈ V within the

interference range of the receiving node v, i.e., r(w,v) ≤ RI,

is not transmitting on the same channel. If node u can transit

to v directly on channel c, they form an edge e(c). We denote

the capacity of this edge as φe(c) which is the maximum data

rate that can be transmitted. Let Ec be the set of all edges

e(c). We say two edges e(c),e′(c) interfere with each other,

if they can not transmit simultaneously based on the protocol

model. Further we define interference set Ie(c) which contains

the edges that interfere with edge e and e itself.

Finally, we introduce a virtual node w∗to represent the

Internet. w∗is connected to each gateway with a virtual edge

e′= (w∗,w),w ∈ W. For simplicity, we assume that the

link capacity in Internet is much larger than the wireless

channel capacity, and thus the bottleneck always appears in

the wireless mesh network. Under this assumption, the virtual

edges could be regarded as having unlimited capacity. Note

that all the virtual links do not interfere with any of the

wireless transmissions.

B. Solution Overview

The performance of a multi-radio multi-channel wireless

mesh network critically depends on the design of three

interdependent components: scheduling, channel assignment,

and routing. Their joint design has been studied in several

existing works [5], [6]. In this paper, we adopt the same

approach as in [5] which formulates this problem as an integer

linear programming problem. To solve this problem, [5] first

solves its LP (linear programming) relaxation and derives the

routing solution based on the necessary conditions of channel

assignment and schedulability. Then the channel assignment

and post processing algorithms are designed to adjust the flows

to yield a feasible solution.

We assume that the system operates synchronously in a

time-slotted mode. The result we obtain will provide an upper

bound for systems using IEEE 802.11 MAC. We further

assume that the traffic between a local access point and

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the Internet could be infinitesimally divided and routed over

multiple paths to multiple gateways achieving the optimal load

balancing and the least congestion.

Formally, let ye(c) be the flow rate on edge e(c) ∈ Ec, y

be the link flow vector, ρe(c) =

channel c over link e, and E(v) be the set of links that is

adjacent to node v. Based on the results presented in [5], the

necessary conditions of channel assignment and scheduling are

summarized in the following claim:

Claim 1 (Necessary Condition of Channel Assignment and

Schedulability). For the multi-channel, multi-radio wireless

mesh network, if a given link flow vector y does not satisfy

the following inequalities:

ye(c)

φe(c)be the utilization of

?

e′∈Ie(c)

?

c∈C

ρe′(c) ≤ γ(∆);∀e(c) ∈ Ec

(1)

?

e(c)∈E(v)

ρe(c) ≤ κ(v);∀v ∈ V

(2)

then y is not schedulable.

In particular, Inequality (1) is the congestion constraint over

an individual channel. γ(∆) is a constant that only depends

on the interference model. Inequality (2) gives the node radio

constraint. Recall that a mesh node v ∈ V has κ(v) radios,

and thus can only support κ(v) simultaneous communications.

The focus of this paper is to investigate the optimal routing

scheme under dynamic traffic based on the above necessary

conditions of channel assignment and schedulability. Once the

flow routes are derived, we simply apply the same method

presented in [5] to adjust the flow routes and scale the flow

rates to yield a feasible routing and channel assignment.

III. OPTIMAL ROUTING

This paper investigates the optimal routing strategy for

wireless mesh backbone network. Thus it only considers the

aggregated traffic among the mesh nodes. In particular, we

regard the virtual node w∗that connects to gateways as the

source of all incoming traffic and the destination of all outgo-

ing traffic of a mesh network. Similarly, the local access points,

which aggregate the client traffic, serve as the sources of all

outgoing traffic and the destinations of incoming traffic. It is

worth noting that although we consider only the aggregated

traffic between gateway access points and local access points

in this paper, our problem formulations and algorithms could

be easily extended to handle inter-mesh-router traffic.

For simplicity, we call the aggregated traffic from a local

access point to the Internet a flow and denote it as f ∈ F,

where F is the set of all aggregated flows. We also denote the

rate of an aggregated flow f ∈ F as xf, and use x = (xf,f ∈

F) to represent the aggregated flow rate vector.

A. Fixed Demand Mesh Network Routing

We first study the formulation of throughput optimization

routing problem in a wireless mesh backbone network under

the fixed traffic demand. We use df to denote the demand

of flow f and d = (df,f ∈ F) to denote the demand

vector consisting of all flow demands. Consider the fairness

constraint that, for each flow f, its throughput being routed

is in proportion to its demand df. Our goal is to maximize

λ (so called scaling factor) where at least λ · df amount of

throughput can be routed for flow f.

We assume an infinitesimally divisible flow model where

the aggregated traffic flow could be routed over multiple paths

and use Pf to denote the set of unicast paths that connect the

source of f and w∗. Let xf(P) be the rate of flow f over

path P ∈ Pf. Obviously the link flow rate ye(c) is given by

ye(c) =?

rates that are routed through paths P passing edge e(c) ∈ Ec.

Based on the necessary conditions of scheduling and channel

assignment in Claim 1 (Eq.(1) and Eq.(2)), we have that

f:P∈Pf&e(c)∈Pxf(P), which is the sum of the flow

?

e′(c)∈Ie(c)

1

φe(c)

?

f:P∈Pf&e′(c)∈P

1

φe(c)

f:P∈Pf&e′(c)∈P

xf(P) ≤ γ(∆);∀e(c) ∈ Ec

(3)

?

c∈C

?

e′(c)∈E(v)

?

xf(P) ≤ κ(v);∀v ∈ V

(4)

Tosimplify

=

?

e′(c)∈E(v),e′(c)∈P

routing with fairness constraint is then formulated as the

following linear programming (LP) problem:

theabove equations,

1

φe′(c)

wedefine

Ae(c)P

?

c∈C

?

e′(c)∈Ie(c),e′(c)∈P

φe′(c). The throughput optimization

and

BvP

=

1

PT:

maximize

subject to

λ

(5)

(6)

?

P∈Pf

?

f∈F

∀e(c) ∈ Ec

?

f∈FP∈Pf

λ ≥ 0,xf(P) ≥ 0,∀f ∈ F,∀P ∈ Pf(9)

xf(P) ≥ λ · df,∀f ∈ F

?

P∈Pf

xf(P)Ae(c)P≤ γ(∆),

(7)

?

xf(P)BvP ≤ κ(v),∀v ∈ V(8)

In this problem, the optimization objective is to maximize

λ, such that at least λ · df units of data can be routed

for each aggregated flow f with demand df. Inequality (6)

enforces fairness by requiring that the comparative ratio of

traffic routed for different flows satisfies the comparative ratio

of their demands. Inequality (7) and (8) come from the nec-

essary conditions of channel assignment and scheduling. This

problem formulation follows the same form as the maximum

concurrent flow problem.

Problem PTcould be solved by a LP-solver such as [10].

To reduce the complexity for practical use, we present a fully

polynomial time approximation algorithm for problem PT,

which finds an ǫ-approximate solution. The key to a fast

approximation algorithm lies on the dual of this problem,

which is formulated as follows. We assign a price µeto each

set Ie(c) for e(c) ∈ Ecand a price µv to each node v ∈ V .

The objective is to minimize the aggregated price for all inter-

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ference sets and all nodes. As the constraint, Inequality (11)

requires that the price?

any path P ∈ Pf for flow f must be at least µf, the price

of flow f. Further, Inequality (12) requires that the weighted

flow price µf over its demand df must be at least 1.

e(c)∈EcAe(c)Pµe+?

v∈VBvPµvof

DT:

minimize

?

e(c)∈Ec

?

e(c)∈Ec

∀f ∈ F,∀P ∈ Pf

?

f∈F

γ(∆) · µe+

?

v∈V

κ(v)µv

(10)

subject to

Ae(c)Pµe+

?

v∈V

BvPµv≥ µf,

(11)

µfdf≥ 1

(12)

FMR: Mesh Network Routing Under Fixed Demand

1

∀e ∈ E, γ ← γ(∆), µe ← β/γ, µv ← β/κ(v)

2

xf(P) ← 0, ∀P ∈ Pf, ∀f ∈ F

3

whileP

4

for ∀f ∈ F do

5

d′

6

whileP

d′

7

P ← lowest priced path in Pf using µe and µv

8

δ ← min{d′

9

d′

10

xf(P) ← xf(P) + δ

11

∀e(c) ∈ Ec s.t. Ae(c)P?= 0, µe ← µe(1+

ǫδAe(c)P/γ)

12

∀v ∈ V s.t. BvP ?= 0, µv ← µv(1 + ǫδBvP/κ(v))

13

end while

14

end for

15

end for

e(c)∈E(c)γ · µe+P

v∈Vκ(v)µv < 1

f← df

e(c)∈E(c)γ · µe+P

f> 0 do

v∈Vκ(v)µv < 1 and

f,mine(c)∈P

γ

Ae(c)P,minv∈V

κ(v)

BvP}

f← d′

f− δ

TABLE I

ROUTING ALGORITHM UNDER FIXED DEMAND

Based on the above dual problem DT, our fast approxima-

tion algorithm is presented in Table I. The algorithm design

follows the idea of [11]. In particular, Line 1 and Line 2

initialize the algorithm. Then for each flow f, we route df

units of data. We do so by finding the lowest priced path in

the path set Pf (Line 7), then filling traffic to this path by

its bottleneck capacity (Lines 8 to 10). Then we update the

prices for the interference sets and the nodes appeared in this

path based on the function defined in Line 11 and Line 12. We

keep filling traffic to flow f in the above fashion until all df

units are routed. This procedure is repeated until the weighted

aggregated price of the interference sets and the nodes exceeds

1 (Line 3).

We formally analyze the properties of our algorithm in the

following theorem. The proofs of the theorems in this paper

are available in the Appendix.

Theorem 1: If β = ((|Ec| + |V |)/(1 − ǫ))−1/ǫ, then the

final flow generated by FMR is at least (1 − 3ǫ) times the

optimal value of P. The running time is O(1

|V |)(2|F|log|F| + |Ec| + |V |) + logU)]) · Tmp, where U is

the length of the longest path in G, and Tmp is the running

ǫ2[log(|Ec| +

time to find the shortest path.

B. Uncertain Demand Mesh Network Routing

Now we proceed to investigate the throughput optimization

routing problem for wireless mesh backbone network when

the aggregated traffic demand is uncertain. We model such

uncertain traffic demand of an aggregated flow f ∈ F using a

random variable Df. We assume that Dffollows the following

discrete probability distribution Pr(Df = di

Df = {d1

non-zero probabilities. Let d = (df,df ∈ Df,f ∈ F) be a

sample traffic demand vector, D be the corresponding random

variable, and D be the sample space. Thus the distribution of

D is given by the joint distribution of these random variables:

Pr(D = d) = Pr(Df= di

Let us consider a traffic routing solution (xf(P),P ∈

Pf,f ∈ F) that satisfies the capacity and node-radio con-

straints (Inequality (7) and (8)). It is obvious that λ is a func-

tion of d: λ(d) = minf∈F{xf

Further let us consider the optimal routing solution under

demand vector d. Such a solution could be easily derived

based on Algorithm I shown in Table I. We denote the optimal

value of λ as λ∗(d). We further define the performance ratio ω

of routing solution (xf(P),P ∈ Pf,f ∈ F) as ω(d) =

Obviously, the performance ratio is also a random variable

under uncertain demand. We denote it as Ω which is a function

of random variable D. Now we extend the wireless mesh

network routing problem to handle such uncertain demand.

Our goal is to maximize the expected value of Ω, which is

given by E(Ω) = Pr(D = d) ×

We abbreviate Pr(D = d) as p(d). It is obvious that

?

optimization routing problem for wireless mesh backbone

network under uncertain traffic demand as follows.

f) = qi

f, where

f,d2

f,...,dm

f} is the set of of values for Df with

f,f ∈ F).

df}, where xf=?

P∈Pfxf(P).

λ(d)

λ∗(d)

λ(d)

λ∗(d)

d∈Dp(d) = 1. Formally, we formulate the throughput

PU:

maximize

?

d∈D

∀d ∈ D,where d = (df,f ∈ F)

?

P∈Pf

?

f∈F

P∈Pf

p(d)λ(d)

λ∗(d)

(13)

subject to

xf(P) ≥ λ(d) · df,∀f ∈ F

(14)

?

xf(P)Ae(c)P≤ γ(∆),∀e(c) ∈ Ec

(15)

(16)

?

f∈F

λ ≥ 0,xf(P) ≥ 0,∀f ∈ F,∀P ∈ Pf

?

P∈Pf

xf(P)BvP ≤ κ(v),∀v ∈ V

(17)

Similar to problem PT, the constraints of PUcome from

the fairness requirement and the wireless mesh network ca-

pacity. In particular, Inequality (14) enforces fairness for

all demand d ∈ D, and Inequality (15) enforces capacity

constraint as Inequality (7) in problem PT.

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Now we consider the dual problem DUof PU. Similar to

DT, the objective of DUis to minimize the aggregated price

for all adjusted interference sets. However, in Inequality (20),

for each sample demand vector d, the aggregated price of all

flows weighted by their demand needs to be larger than its

probability.

DU:

minimize

?

e(c)∈Ec

?

e(c)∈Ec

∀f ∈ F,∀P ∈ Pf

γ(∆) · µe+

?

v∈V

κ(v)µv

(18)

subject to

Ae(c)Pµe+

?

v∈V

BvPµv≥ µf,

(19)

?

f∈F

where d = (df,f ∈ F)

µfdf≥

p(d)

λ∗(d),∀d ∈ D

(20)

UMR: Mesh Network Routing Under Uncertain Demand

1

∀e ∈ E, γ ← γ(∆), µe ← β/γ, µv ← β/κ(v)

2

xf(P) ← 0, ∀P ∈ Pf, ∀f ∈ F

3

loop

4

for ∀f ∈ F do

5

¯P ← lowest priced path in Pf using µe, µv

6

µf ←P

7

end for

8

for ∀d ∈ D do

9

µd ←P

p(d)

10

end for

11

µmin← mind∈Dµd

12

dmin← argmind∈Dµmin

13

if µmin≥ 1

14

return

15

for ∀f ∈ F do

16

d′

f

17

while d′

18

P ← lowest priced path in Pf using µe, µv

19

δ ← min{d′

20

d′

21

xf(P) ← xf(P) + δ

∀e(c) ∈ Ec s.t. Ae(c)P?= 0, µe ← µe(1+

22

ǫδAe(c)P/γ)

23

∀v ∈ V s.t. BvP ?= 0, µv ← µv(1 + ǫδBvP/κ(v))

24

end while

25

end for

26

end loop

e∈EAe(c)¯

Pµe+ Bv¯

Pµv

f∈Fµfdf

λ∗(d)

f← dmin

f> 0 do

f,mine(c)∈P

γ

Ae(c)P,minv∈V

κ(v)

BvP}

f← d′

f− δ

TABLE II

ROUTING ALGORITHM UNDER UNCERTAIN DEMAND

Now we present an approximation algorithm for PU in

Table II. This algorithm (UMR) has the same initialization as

the algorithm for problem PT(FMR). Then we march into the

iteration, in which we find dmin, the demand whose price µmin

is the minimum among others (Lines 4 to 12). If µmin≥ 1,

then the algorithm stops (Lines 13 and 14), since Inequality

(19) and (20) would be satisfied for all demand. Otherwise, we

will increase the price of dminby routing more traffic through

its node pairs. This procedure (Lines 16 to 23) is the same as

what has been described in Lines 4 to 11 of FMR algorithm.

Following the same proving sequence for FMR, we are able

to prove the similar properties with UMR.

Theorem 2: If β = ((|Ec| + |V |)/(1 − ǫ))−1/ǫ, then the

final flow generated by UMR is at least (1 − 3ǫ) times the

optimal value of PU. The running time is O(1

|V |)(2|D||F|log|F| + |Ec| +|V |) +logU)]) ·Tmp, where U

is the length of the longest path in G, Tmpis the running time

to find the shortest path.

ǫ2[log(|Ec| +

IV. TRAFFIC ESTIMATION

In this section, we study the dynamic behavior of aggregated

traffic at local access points. Our goal is to (1) develop a

reliable estimation method that is able to predict the aggre-

gated traffic demand of an access point based on its historical

data, and (2) develop a statistical model to characterize the

prediction results. The estimated traffic demand will serve

as the input of mesh network routing algorithms which are

presented in Sec. III-A and Sec. III-B.

In order to develop such a traffic demand model, we study

the traces collected at the campus wireless LAN network of

Dartmouth College in Spring 2002 [8]. By analyzing the snmp

log from each access point, we derive the dynamic behavior

of the aggregated traffic demand. We argue that the access

points of a wireless LAN serve a similar role and thus exhibit

similar behavior as the local access points of a wireless mesh

network.

To illustrate our analysis procedure, we choose one of the

access points (ResBldg97AP3) as an example. The time series

of its incoming traffic is plotted in Fig. 1. From the figure, we

can easily observe that (1) the traffic demand is non-stationary

over large time scales due to the diurnal and weekly working

cycles; (2) compared with the traffic behavior in the backbone

Internet [13], the traffic at an access point is significantly

bursty due to the insufficient level of multiplexing. The above

observations are consistent with the findings in [7].

0 10 2030 4050 6070

0

2

4

6x 108

Time (#day since 03/25/2002)

Traffic (bypte/hour)

Fig. 1.

2002 - June 9, 11pm, 2002 EST).

Incoming Traffic Time Series of ResBldg97AP3 (March 25, 12am,

The first step of our analysis is to identify and remove

the daily and weekly cyclic patterns in the time series. This

requires us to calculate the weekly/daily cyclic average. For-

mally, let us denote x(t) as the raw traffic series. We estimate

the moving average of this series based on the same time of the

day (note that the weekday and the weekend are differentiated

in averaging): ¯ x(t) =?W

the size of moving window. To eliminate the effect of bursty

traffic, we also filter out the spike traffic during the above

averaging procedure. Fig. 2(a) plots the raw traffic as well as

i=1x(t − 24 × i)/W, where W is

Page 6

6

30 3540 455055 60 65

0

0.5

1

1.5

2

2.5

3x 108

Time (#day since 03/25/2002)

Traffic (byte/hour)

raw

average

(a) Raw Traffic vs. Moving Average Series

303540 4550 55 60 65

-1

0

1

2

3x 108

Time (#day since 03/25/2002)

Adjusted Traffic (byte/hour)

(b) Adjusted Traffic Series

Fig. 2. Traffic Series in 5 weeks

its moving average with W = 5. By removing the cyclic effect

from the raw data, we derive the adjusted traffic series z(t)

as z(t) = x(t) − ¯ x(t).

The adjusted series of the one shown in Fig. 2(a) is given in

Fig. 2(b). This adjusted traffic exhibits short-term (a few hours)

traffic correlations. We model the adjusted traffic series with

an autoregressive process as follows1.

z(t) = β1z(t−1)+β2z(t−2)+...+βKz(t−K)+ǫ (21)

where K is the process order. To apply this model for

prediction, we estimate the parameters of this process. Given

N observations z1,z2,...,zN, the parameters β1, ..., βK are

estimated via least squares by minimizing:

N

?

t=K+1

?z(t) − β1z(t − 1)... − βKz(t − K)?2

(22)

Based on these parameters, we further derive the adjusted

traffic prediction ˆ z(t) as ˆ z(t) = β1z(t − 1) + β2z(t − 2) +

... + βKz(t − K). Fig. 3 illustrates the estimation results

for the adjusted traffic series in Fig. 2(b), where K = 2,

β1= 0.531, β2= 0.469. The figure plots the predicted series

for the adjusted traffic as well as its raw data. In this figure,

the number of observations used for parameter estimation is

N = 60. The fitted traffic series is also plotted for the interval

[720,779] for the purpose of comparison.

We now consider the errors involved in this prediction

process. In particular, we define the adjusted traffic prediction

error as ǫz(t) = z(t)−ˆ z(t). Based on this definition, Fig. 4(a)

plots the cumulative distribution function of the prediction

error of the adjusted traffic series shown in Fig. 3. It is obvious

that the error distribution fits the normal distribution with a

mean close to zero.

Finally, we define traffic prediction ˆ x as follows:

1Ideally, z(t) should have zero mean. In some cases, z(t) has a small mean

value which needs to be removed before fitting an autoregressive process.

720 740760 780800820 840

-4

-2

0

2

4

6

8x 107

Time (#hour since 03/24/2002, 11pm EST)

Adjusted traffic (byte/hour)

raw

fit

predict

fitted part

predicted part

Fig. 3.Adjusted Traffic and Its Prediction

ˆ x(t) = [¯ x(t) + ˆ z(t)]+

(23)

where [x]+= max{0,x}. Fig. 5 plots the predicted traffic

series ˆ x(t) in comparison with the raw traffic. We can see

the predicted traffic closely matches the real(raw) traffic. The

cumulative distribution function of the prediction error ǫx(t),

which is defined as ǫx(t) = x(t)−ˆ x(t), is plotted in Fig. 4(b).

It clearly shows that this distribution also fits the normal

distribution with a near-zero mean.

-3 -2 -101234

x 107

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Adjusted traffic (byte/hour)

F(x)

Empirical CDF

-1 -0.50 0.51 1.5

x 109

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Adjusted traffic (byte/hour)

F(x)

Empirical CDF

(a) Prediction error for adjusted traffic(b) Prediction error for

entire series

Fig. 4. Cumulative Density Function of Prediction Error

303540 45

Time (#day since 03/24/2002)?

5055 6065 70

0

1

2

3

4

5

6x 108

Traffic (byte/hour)

raw

predict

Fig. 5. Raw Traffic vs. Predicted Traffic

To this end, we could consider the estimated traffic demand

at time t as a random variable X(t) which follows the

normal distribution with mean ˆ x(t) and the same variance as

ǫx. To summarize, the presented estimation method provides

two prediction models: mean value and statistical distribution.

These two traffic prediction models will serve as the inputs for

the fixed-demand mesh network routing algorithm (FMR) and

the uncertain-demandmesh network routing algorithm (UMR.

V. SIMULATION STUDY

A. Simulation Setup and Performance Metrics

We evaluate the performance of our algorithms via simula-

tion study. In the simulated wireless mesh network, 60 mesh

Page 7

7

nodes are randomly deployed over a 1000 × 2000m2region.

20 nodes at the edge of this network are selected as the local

access points (LAP) that forward traffic for clients. 4 nodes

in the center of the deploy region are selected as the gateway

access points. The simulated network topology is shown in

Fig. 6. Each mesh node has a transmission range of 250m

and an interference range of 500m, which means ∆ = 2.

The channel capacity φc(e) is the same for all links e and

channels c, which is set as 54 Mbps. In the basic setting, each

mesh nodes are equipped with 3 radio interfaces. And there

are 3 orthogonal channels in the network. Aside from this

basic setting, we have also evaluated the performance of our

algorithms with different configurations of radio and channel

numbers, which we will show in the later part of this section.

0

200

400

600

800

1000

0 500 1000

X Position

1500 2000

Y Position

0

1

2

3

4

5

6

7

8

9

10

11

12

20

13

14

15

16

17

18

19

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

Fig. 6. Mesh Network Topology.

To realistically simulate the traffic demand at each LAP,

we employ the traces collected in the campus wireless LAN

network. The network traces used in this work are collected in

Spring 2002 at Dartmouth College and provided by CRAW-

DAD [8]. By analyzing the snmp log trace at each access point,

we are able to derive its 1108-hour incoming and outgoing

traffic volume since 12:00AM, March 25, 2002 EST. We select

the access points from the Dartmouth campus wireless LAN

and assign their traffic traces to the LAPs in our simulation.

The traffic assignment is given in Table III.

We evaluate and compare different traffic prediction and

routing strategies for this simulated network. In particular, we

consider the following strategies.

• Oracle Routing (OR). In this strategy, the traffic demand

is known a priori. It runs the FMR algorithm (presented

in Tab. I) based on this demand. This solution runs every

hour based on the up-to-date traffic demand from the trace

and returns the optimal set of routes. This ideal strategy

is designed to return the benchmark result, which the rest

of the practical strategies compare to.

• Mean-Value Prediction Routing (MVPR). This strategy

does not know the traffic demand a priori. Instead, it only

predicts the traffic demand based on its historical data. In

particular, it employs the mean value prediction model

and runs the FMR algorithm based on this predicted

demand. This solution also runs every hour to provide

the set of routes for the next hour.

• Statistical-Distribution Prediction Routing (SDPR). Simi-

lar to MVPR, this strategy also relies on traffic prediction.

It predicts not only the mean-value of the traffic demand

in the next hour, but also its distribution. It runs the UMR

algorithm (presented in Tab. II) with the predicted traffic

demand distribution as its input. Since UMR only accepts

discrete probability distribution, we need to discretize the

demand distribution by sampling the following values, the

mean value µ, and values µ−σ, µ+σ, µ−2σ, and µ−2σ.

Since about 95% of all traffic demand values fall within

the range [µ − 2σ,µ + 2σ], we ignore the values which

has a probability smaller than 5%.

• Shortest-Path Routing (SPR). This strategy is agnostic of

traffic demand, and returns fixed routing solution purely

based on the shortest distance (number of hops) from

each mesh node to the gateway. The purpose to evaluate

this strategy is to quantitatively contrast the advantage of

our traffic-predictive routing strategies.

Note that the flows derived from the above routing strategies

will be adjusted by the channel assignment, post processing

and flow scaling algorithms in [5]. We denote the final rate

of flow f along path P as xA

the maximum flow throughput under the fairness constraint

weighted by the traffic demand, which maximizes the scaling

factor λ. However, for performance study, λ is not a suitable

performance metric. First, we are more interested in the

network performance (i.e., congestion) incurred by the given

traffic demand, instead of the achievable throughput. Second,

the absolute value of λ could be misleading, especially when

the actual demand is not the same as the predicted demand

which is being used for routing.

Now we proceed to define the performance metric we use

in the simulation study. First, we scale the achievable flow rate

xA

by its actual traffic demand df:

f=

?

P∈PfxA

f(P). This is

fderived from the routing and channel assignment process

x′

f(P) = xA

f(P) ·df

xA

f

(24)

x′

f(P) is the actual traffic load that is imposed on path

P under our routing and channel assignment scheme. Thus

the traffic being routed within the interference set Ie(c) over

channel c is given by?

f∈F

?

the congestion of an interference set Ie(c) using its utilization

and denote it as θch

e(c) =

θch= maxe(c)∈Ecθch

all the interference sets. We further consider the conges-

tion at a single mesh node incurred by the traffic from all

channels. The congestion of a node v is defined as θrd

P

f∈F

P

κ(v)

. And θrd= maxv∈Vθrd

network congestion θ is defined as θ = max{θrd,θch}.

P∈Pfx′

f(P)Ae(c)P. We define

P

f∈F

P

P∈Pfx′

γ(∆)

f(P)Ae(c)P

. Then

e(c) is the maximum congestion among

v

=

P∈Pfx′

f(P)BvP

v. Finally, the

B. Simulation Results

We experiment with the above routing strategies along the

time range [108,1108], a 1000-hour period excerpted from the

trace2. Note that all the simulation results presented in this

2Note that the beginning part of the trace [0,107] is used as training data,

thus is not included in the simulation result.

Page 8

8

AP 31AP3

22

27AP3

9

34AP5

18

3AP3

23

55AP4

57

21AP2

25

57AP2

5

23AP4

33

62AP3

55

33AP2

19

62AP4

20

62AP2

35

82AP4

53

82AP3

58

94AP1

3

84AP1

42

94AP3

56

90AP2

6

94AP8

27

97AP2

48

Node ID

AP

Node ID

TABLE III

OVERVIEW OF TRAFFIC DEMAND ASSIGNMENT

section are using 108 as the zero point.

0

2

4

6

8

10

0 200 400 600 800

1000

θ

Time (#hour since 03/29/2002, 11am EST)

OR

MVPR

SDPR

SPR

Fig. 7. Overview of All Strategies

We start by presenting the congestion achieved by all strate-

gies (OR, MVPR, SDPR, and SPR) during the entire 1000-hour

simulation period. As seen in Fig. 7, OR constantly achieves

the minimum worst-case congestion among others, due to its

unrealistic capability to know the actual traffic demand. We

note that the burstiness of θ applies to all strategies including

OR. Such observation comes from the burstiness of the traffic

load in the snmp log trace, which is caused by the insufficient

level of traffic multiplexing at wireless local access points.

1

190 200 210 220 230 240 250 260 270 280 290

Time (#hour since 03/29/2002, 11am EST)

2

3

4

5

6

7

θ/θOR

MVPR/OR

SDPR/OR

SPR/OR

1

2

3

4

5

6

7

0 20 40

Time Instance

60 80 100

θ/θOR

MVPR/OR

SDPR/OR

SPR/OR

Fig. 8. (a) Congestion Ratio (

θ

θOR) (b) Sorted View

To filter out the noise caused by traffic burstiness, in

Fig. 8(a), we normalize θ achieved by other strategies by the

same value of OR. Since OR always achieves the minimum θ

among others, this ratio will end up at least 1. Also we take

a close-up look during the hour range [190,290]. Here, the

MVPR and SDPR strategies achieve less than 2 times of the

optimal congestion in most cases, while the SPR strategy can

only achieve 4−7 times of the optimal performance.The above

observations get clearer when we sort out the normalized

congestion ratio for the three strategies in Fig. 8(b). It is

clear that our MVPR and SDPR strategies which integrate the

traffic prediction with the optimal routing outperform the SPR

strategy which is agnostic about the traffic demand. Further,

SDPR achieves lower congestion than MVPR in most of the

time due to more comprehensive representation of the traffic

demand estimation. However, in a few cases (less than 10%

of the time), the worst-case congestion of SDPR is higher

than MVPR. This problem can be mostly attributed to the

inaccuracy of traffic prediction.

0

0.5

1

1.5

2

2.5

0 50 100 150 200 250 300 350 400 450

Interference Set Instances (sorted)

θ

OR

MVPR

SDPR

SPR

ch

Fig. 9. Adjusted Interference Set Sorted By Congestion

Next, we take a closer look at each strategy’s ability to bal-

ance the traffic within the mesh network. In Fig. 9, we unfold

a single time instance at hour 271 and exhibit the congestion

θch

In order to achieve the lowest worst-case congestion, a good

strategy should maximally even out the traffic routed through

all interference sets. Obviously, OR achieves such a balance,

which resulted in the best θ value 0.65. SPR has the highest θ

value as more than 2. The results for MVPR and SDPR are 0.8

and 0.7 respectively. We also observe that the distribution of

θch

e(c)at each interference set Ie(c) resulted from each strategy.

e(c)under the SDPR strategy closely matches the OR strategy.

1

1.5

2

2.5

3

3.5

4

4.5

5

0 10 20 30 40 50 60 70 80 90 100

Time Instances

(a) MVPR

θ/θOR

# of radio=2

# of radio=3

# of radio=4

1?

1.5?

2?

2.5?

3?

3.5?

4?

4.5?

5?

0? 10? 20? 30? 40? 50? 60? 70? 80? 90? 100?

Time Instances?

(c) SDPR

θ? /? θ?OR?

# of radio=2?

# of radio=3?

# of radio=4?

Fig. 10. Impact of Number of Radio Interfaces

In what follows, we alter our simulation configurations to

examine the abilities of different strategies at adapting various

network settings, such as radio interface numbers and channel

numbers. Here, we focus on the traffic prediction strategies,

namely, MVPR and SDPR. Also we plot their performances

by the congestion ratio θ/θORnormalized by the OR routing

results. We first vary the number of radio interfaces from

2 to 4 and study the congestion θ during the time interval

[190,290]. Fig. 10 plots the sorted normalized congestion

of the two strategies. Comparing these two figures, we could

see that the SDPR strategy performs slightly better than the

MVPR strategy. The improvement of both strategies over the

θ

θOR

Page 9

9

OR strategy increases (i.e., normalized congestion decreases)

with the radio number.

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

3.2

1 2 3 4 5

θ/θOR

Channel Numbers

(a) MVPR

# of radio=1

# of radio=2

# of radio=3

# of radio=4

# of radio=5

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

1 2 3 4 5

Channel Numbers

(b) SDPR

# of radio=1

# of radio=2

# of radio=3

# of radio=4

# of radio=5

Fig. 11. Impact of Channel/Radio

Finally, Fig. 11 plots the normalized congestion under dif-

ferent radio and channel numbers at a single time instance 271

for these two strategies. The results show that the improvement

of both strategies over the OR strategy decreases with the

channel number. This is because when the network has more

channels, the algorithms are likely to find more paths and the

prediction error is more likely to be magnified.

VI. RELATED WORK

We evaluate and highlight our original contributions in light

of previous related work.

The problem of wireless mesh network routing, channel

assignment, and the joint solution of these two has been

extensively studied in the existing literature. For example,

routing algorithms are proposed to improve the throughput

for wireless mesh networks via integrating MAC layer infor-

mation [2], such as expected packet transmission time [1],

channel cost metric (CCM) which is the sum of expected

transmission time weighted by the channel utilization [4].

Joint solutions for channel allocation and routing are explored

in [14] using a centralized algorithm and in [3] in a distributed

fashion. These heuristic solutions are designed to adapt to the

dynamic network condition. However, they lack the theoretical

foundation to analyze how well the network performs globally

(e.g., whether the network resource is fully utilized, whether

the flows share the network in a fair fashion) under their

routing schemes.

There are also theoretical studies that formulate these

network planning decisions into optimization problems. For

example, the works of [5], [6] study the optimal solution of

joint channel assignment and routing for maximum through-

put under a multi-commodity flow problem formulation and

solves it via linear programming. The work of [15] presents

bandwidth allocation schemes to achieve maximum throughput

and lexicographical max-min fairness respectively. Further, the

work of [16] presents a rate limiting scheme to enforce the fair-

ness among different local access points. These results provide

valuable analytical insights to the mesh network design under

ideal assumptions such as known static traffic input. However,

they may be unsuitable for practical use under highly dynamic

traffic situation. Different from these existing works, our work

explicitly incorporates traffic behavior analysis and prediction

into the routing optimization, thus better fits the routing need

in the dynamic wireless mesh networks. Distributed algorithms

have presented for joint scheduling and routing in [17],

and for joint channel assignment, scheduling and routing in

[17]. These distributed algorithms only use local information

for traffic routing, thus have the potential to accommodate

dynamic traffic. However, their crucial properties, such as

convergence speed and messaging overhead, are yet to be

evaluated under realistic traffic conditions.

Trace analysis has been used to study the behavior of

wireless networks in many recent works. For example, [7]

statistically characterizes both static flows and roaming flows

in a large campus wireless network. Different from these exist-

ing works, which focus on either user behavior, network flow

or link performances, we provide a framework that integrates

traffic uncertainty model with its performance optimization.

Our work is also related to dynamic traffic engineering [13]

in Internet, which also consider the impact of demand uncer-

tainty in make routing decisions. The major difference between

our work and these existing works lies in the different network

and traffic models of wireless mesh network and Internet.

VII. CONCLUSION

This paper studies the optimal routing strategies for wireless

mesh networks. Different from existing works which implicitly

assume traffic demand as static and known a priori, this

work considers the traffic demand uncertainty. It studies the

dynamic behavior of wireless network traffic, establishes two

prediction models based on time series analysis, and extends

the classical maximum concurrentflow problem with statistical

demand input. Simulation study is conducted based on the

traffic demand from the real wireless network traces. The

results show that our problem formulation and algorithm could

effectively incorporate the traffic demand dynamics.

VIII. APPENDIX

A. Proof for Theorem 1

The proof to Theorem 1 is precluded by a sequence of

lemmas. We first make the following denotations. We use

OPT to represent the optimal solution of both PT and

DT, and OPT′to represent the solution derived from FMR

algorithm.

Lemma 1 : If OPT ≥ 1, scaling the final flow by

log1+ǫ1/β yields a feasible primal solution of value OPT′=

t−1

log1+ǫ1/β, t being the number of phases the algorithm takes

to stop.

Proof: We first make the following denotations. Regard-

ing a set of price assignments µe for e(c) (e(c) ∈ Ec), µv

for v (v ∈ V ), the objective function of DT is Lµ. Let

Pµ(f) be the minimum path of the flow f ∈ F using µeand

µv. µ(Pµ(f)) ??

is the aggregated price of Pµ(f). Each phase contains |F|

iterations, where traffic for each flow in F is routed according

to its demand. In each iteration, the price of an interference set

is updated. We use µ(i)(j)

e

to denote the price of e(c) ∈ Ec,

µ(i)(j)

v

to denote the price of v ∈ V after the jth iteration

e(c)∈EcAe(c)Pµ(f)µe+?

v∈VBvPµ(f)µv

Page 10

10

of the ith phase. Regarding µ(i)(j)

the notation Lµ(i)(j)

µ(Pµ(i)(j)) into µ(P(i)(j)). Based on the price update function

(Line 11 in Tab. I), we have

e

and µ(i)(j)

v

, we simplify

into P(i)(j), and

into L(i)(j), Pµ(i)(j)

L(i)(j)

?

e(c)∈Ec

?

v∈V

L(i)(j−1)+ d(fj)µ(P(i)(j−1))

=µ(i)(j−1)

e

+ ǫ

?

e(c)∈P(i)(j−1)

?

v∈P(i)(j−1)

Ae(c)P(i)(j−1)µ(i)(j−1)

e

d(fj)

procedure applies for any node v. Thus, scaling the flow

by log1+ǫ1/β will yield a feasible solution. Since in each

phase, d(f) units of data are routed for each flow, we have

OPT′=

+µ(i)(j−1)

v

+ ǫ

BvP(i)(j−1)µ(i)(j−1)

v

d(fj)

=

The price assignment at the start of the (i + 1)th phase are

the same as that at the end of the ith phase, i.e., µ(i+1)(0)

µ(i)(|F|)

e

. The price of any interference set e(c) is initialized

as µ(1)(0)

e

= µ(0)(|F|)

e

= β/γ(∆), and the price of any node v

is initialized as µ(1)(0)

v

= µ(0)(|F|)

V

e

=

= β/κ(v). Hence,

L(i)(|F|)≤ L(i)(0)+ ǫ

|F|

?

j=1

d(fj)µ(P(i)(|F|))

since µeand µvare monotonically increasing.

Let us define µ(i)(|F|)=

the objective of DT is to minimize L(i)(|F|), subject to the

constraint that µ(i)(|F|)≥ 1. This constraint can be easily

satisfied if we scale the prices of all inference sets and nodes

by 1/µ(i)(|F|). So DTis equivalent to finding a set of inference

set lengths, such thatL(i)(|F |)

value of DTis OPT ? minµ(i)(|F |)L(i)(|F |)

SinceL(i)(|F|)

?|F|

j=1d(fj)µ(P(i)(|F|)). Then

µ(i)(|F |)is minimized. Thus the optimal

µ(i)(|F |).

µ(i)(|F|)≥ OPT, we have

L(i)(|F|)≤L(0)(|F|)

1 − ǫ

e

ǫ(i−1)

OPT(1−ǫ)

Since L(0)(|F|)= β(|Ec| + |V |), we have

L(i)(|F|)

≤

β(|Ec| + |V |)

(1 − ǫ/OPT)i

β(|Ec| + |V |)

(1 − ǫ/OPT)(1 +

β(|Ec| + |V |)

(1 − ǫ/OPT)e

β(|Ec| + |V |)

1 − ǫ

=

ǫ

OPT − ǫ)i−1

≤

ǫ(i−1)

OPT−ǫ

≤

e

ǫ(i−1)

OP T(1−ǫ)

where the last inequality assumes that OPT ≥ 1. The

algorithm stops at the first phase t for which L(t)(|F|)≥ 1.

Therefore,

1 ≤ L(t)(|F|)≤β(|Ec| + |V |)

1 − ǫ

e

ǫ(t−1)

OPT(1−ǫ)

which implies

OPT

t − 1≤

ǫ

(1 − ǫ)ln

1−ǫ

β(|Ec|+|V |)

(25)

Now consider an interference set e(c). For every γ(∆)

units of flow routed through e(c), we increase its price by

at least a factor (1 + ǫ). Initially, its length is β/γ(∆) and

after t − 1 phases, since L(t)(|F|)< 1, the price of e(c)

satisfies µ(t−1)(|F|)

e

< 1/γ(∆). Therefore the total amount

of flow through e(c) in the first t − 1 phases is strictly less

than log1+ǫ

1/γ(∆)

β/γ(∆)= log1+ǫ1/β times its capacity. The same

t−1

log1+ǫ1/β.

Lemma 2: If OPT ≥ 1, then the final flow scaled by

log1+ǫ1/β has a value at least (1 − 3ǫ) times OPT, when

β = ((|Ec| + |V |)/(1 − ǫ))−1/ǫ.

Proof: By Lemma 1, scaling the final flow by log1+ǫ1/β

yields a feasible solution. Therefore,

OPT

OPT′< log1+ǫ1/β

(26)

Substituting the bound on OPT/(t − 1) from In Equality

(25), we get

OPT

OPT′<

ǫlog1+ǫ1/β

(1 − ǫ)ln

1−ǫ

β(|Ec|+|V |)

=

ǫ

(1 − ǫ)ln(1 + ǫ)

ln1/β

1−ǫ

β(|Ec|+|V |)

ln

When β = ((|Ec|+|V|)/(1−ǫ))−1/ǫ, the above in Equality

becomes

OPT

OPT′

≤ (1 − 3ǫ)

<

ǫ

(1 − ǫ)2ln(1 + ǫ)≤

ǫ

(1 − ǫ)2(ǫ − ǫ2/2)

1

≤ (1 − ǫ)3

Lemma 3: If OPT ≥ 1 and β = ((|Ec| + |V |)/(1 −

ǫ))−1/ǫ, Algorithm I terminates after at most t = 1 +

OPT

ǫ

log1+ǫ

1−ǫ

phases.

Proof: From In Equality (26) and weak-duality, we have

|Ec|+|V |

1 ≤OPT

OPT′< log1+ǫ1/β

Hence, the number of phases t is strictly less than 1 +

OPT log1+ǫ1/β. If β = ((|Ec| + |V |)/(1 − ǫ))−1/ǫ, then

t ≤ 1 +OPT

ǫ

log1+ǫ

1−ǫ

These lemmas require that OPT ≥ 1. The running time

of the algorithm also depends on OPT. Thus we need to

ensure that OPT is at least one and not too large. Let ζibe

the maximum traffic value of flow fi when all other flows

have zero traffic. Let ζ = mini

commodity maximum flows can be routed simultaneously, ζ

is an upper bound on OPT′. On the other hand, routing 1/|F|

fraction of each flow of value ζiis a feasible solution, which

implies that ζ/|F| is a lower bound on OPT. To ensure that

OPT ≥ 1, we can scale the original demands so that ζ/|F|

is at least one. However, by doing so, OPT might be made

as large as |F|, which is also undesirable.

To reduce the dependence on the number of phases on

OPT, we adopt the following technique. If the algorithm does

not stop after T =

ǫlog1+ǫ

|Ec|+|V |

ζi

d(fi). Since at best all single

2

(|Ec|+|V |)

1−ǫ

phases, it means that

Page 11

11

OPT > 2. We then double demands of all commodities, so

that OPT is halved and still at least 1. We then continue the

algorithm, and double demands again if it does not stop after

T phases.

These lemmas require that OPT ≥ 1. The running time

of the algorithm also depends on OPT. Thus we need to

ensure that OPT is at least one and not too large. Let ζf

be the maximum traffic value of flow f when all other flows

have zero traffic. Let ζ = minf

commodity maximum flows can be routed simultaneously, ζ

is an upper bound on OPT′. On the other hand, routing 1/|F|

fraction of each flow of value ζf is a feasible solution, which

implies that ζ/|F| is a lower bound on OPT. To ensure that

OPT ≥ 1, we can scale the original demands so that ζ/|F|

is at least one. However, by doing so, OPT might be made

as large as |F|, which is also undesirable.

To reduce the dependence on the number of phases on

OPT, we adopt the following technique. If the algorithm

does not stop after T =2

ǫlog1+ǫ

that OPT > 2. We then double demands of all commodities,

so that OPT is halved and still at least 1. We then continue

the algorithm, and double demands again if it does not stop

after T phases.

ζf

df. Since at best all single

(|Ec|+|V |)

1−ǫ

phases, it means

Lemma 4: Given ζf for each flow f, the running time of

Algorithm I is O(log(|Ec|+|V |)

ǫ2

Tmp.

Proof: The above demand-doubling procedure is repeated

for at most log|F| times. Thus, the total number of phases is

at most T log|F|. Since each phase contains |F| iterations,

the algorithm runs for at most |F|T log|F| iterations.

Now we observe how many steps are within each iteration.

For each step except for the last step in an iteration, the

algorithm increases the price of some edge inference set or

node by 1 + ǫ. µe has initial value β/γ(∆) and value at

most 1/γ(∆) before the final step of the algorithm. The

same condition applies for nodes v ∈ V . Otherwise, the

stop criterion of the algorithm would have been reached. This

means that the price of an edge inference set or node can

be updated in at most log1+ǫ

Therefore, the algorithm contains at most

|Ec|+|V |

ǫ

log1+ǫ

1−ǫ

≤

mal” steps, and |F|T log|F| ≤2|F|log |F|

steps. Each step contains a minimum overlay spanning tree

operation.

Theorem 1: The total running time of Algorithm I is

O(1

Proof: Computing ζi corresponds to the maximum flow

problem, where fiis the only commodity. The running time of

getting ζiis O(|Ec|+|V |

ǫ2

(logU))·Tmp, where U is the length of

the longest unicast route, and Tmpdenotes the running time to

find the minimum path. Such an operation has to be repeated

for each flow. Also from the result of Lemma 4, we can obtain

the total running time as described by the theorem.

(2|F|log|F| + |Ec| + |V |)) ·

1

β=

1

ǫlog1+ǫ

|Ec|+|V |

1−ǫ

steps.

|Ec|+|V ||Ec|+|V |

ǫ2

log|Ec|+|V |

log|Ec|+|V |

1−ǫ

such “nor-

“final”

ǫ2

1−ǫ

ǫ2[log(|Ec|+|V|)(2|F|log|F|+|Ec|+|V |)+logU)])·Tmp.

B. Proof for Theorem 2

The proof for Theorem 2 follows the same sequence as

the proof to Theorem 1, with minor modification. We start

with Lemma 1. Each phase of the algorithm contains |F|

iterations, where traffic for each flow in F is routed according

to its demand. We reuse the same denotations defined in the

original proof to Lemma 1. We further introduce d(i)as the

demand vector chosen at the ith phase.

Based on the price update function (Line 11 in Tab. I), we

have

L(i)(j)

L(i)(j−1)+ d(fj)µ(P(i)(j−1))λ∗(d(i))

=

p(d(i))

The price assignment at the start of the (i + 1)th phase are

the same as that at the end of the ith phase, i.e., µ(i+1)(0)

µ(i)(|F|)

e

. The price of any interference set Seis initialized as

µ(1)(0)

e

= µ(0)(|F|)

e

= β/γ(∆), µ(1)(0)

Hence,

e

=

v

= µ(0)(|F|)

v

= β/κ(v).

L(i)(|F|)

=L(i)(0)+ ǫ

|F|

?

j=1

d(fj)µ(P(i)(j−1))λ∗(d(i))

p(d(i))

≤L(i)(0)+ ǫ

|F|

?

j=1

d(fj)µ(P(i)(|F|))λ∗(d(i))

p(d(i))

since the edge lengths are monotonically increasing.

Let us define µ(i)(|F|)=

Then the objective of D is to minimize L(i)(|F|), subject to

the constraint that µ(i)(|F|)≥ 1, i.e.,L(i)(|F|)

The rest of the proof follows the same as the original proof

to Lemma 1. The proofs to Lemma 2, 3 remain the same. In

the proof of Lemma 4, the total number of phases is changed

from at most T log|F| to T|D|log|F|. The proof of Theorem

2 follows these results.

?|F|

j=1d(fj)µ(P(i)(|F|))λ∗(d(i))

p(d(i)).

µ(i)(|F|)≥ OPT.

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