Page 1

Wireless Mesh Network Routing Under

Uncertain Demands

Jonathan Wellons, Liang Dai, Bin Chang and Yuan Xue

Abstract Traffic routing plays a critical role in determining the performance of a

wireless mesh network. Recent research results usually fall into two ends of the

spectrum. On one end are the heuristic routing algorithms, which are highly adap-

tive to the dynamic environments of wireless networks yet lack the analytical prop-

erties of how well the network performs globally. On the other end are the opti-

mal routing algorithms that are derived from the optimization problem formulation

of mesh network routing. They can usually claim analytical properties such as re-

source utilization optimality and throughput fairness. However, traffic demand is

usually implicitly assumed as static and known a priori in these problem formu-

lations. In contrast, recent studies of wireless network traces show that the traffic

demand, even being aggregated at access points, is highly dynamic and hard to es-

timate. Thus, in order to apply the optimization-based routing solution in practice,

one must take into account the dynamic and uncertain nature of wireless traffic de-

mand. There are two basic approaches to address the traffic uncertainty in optimal

mesh network routing: (1) predictive routing which infers the traffic demand with

maximumpossibilitybasedinits historyandoptimizestheroutingstrategybasedon

the predicted traffic demand and (2) oblivious routing which considers all the pos-

sible traffic demands and selects the routing strategy where the worst-case network

performance could be optimized. This chapter provides an overview of the optimal

routing strategies for wireless mesh networks with a focus on the above two strate-

gies that explicitly consider the traffic uncertainty. It also identifies the key factors

that affect the performanceof each routing strategy and providesguidelines towards

the strategy selection in mesh network routing under uncertain traffic demands.

Vanderbilt University e-mail: : {jonathan.wellons, liang.dai, bin.chang, yuan.xue}@vanderbilt.edu

1

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2 J. Wellons, L. Dai, B. Chang and Y. Xue

1 Introduction

Wireless mesh networks (e.g., [5, 4]) which now offer a rapid and inexpensive so-

lution to last-mile broadband Internet access, are attracting ever greater attention

and widespread deployment. A wireless mesh network is composed of local access

points and wireless mesh routers which form an organic backbone structure which

forwards traffic between mobile clients and the Internet.

Traffic routing plays a critical role in determining the performance of a wire-

less mesh network. Thus it has attracted extensive recent research. The key chal-

lenges come from the scarce wireless channel resource, high dynamic link quality,

and the uncertain traffic demands. The proposed approaches address these chal-

lenges in different ways. On one end of the spectrum are the heuristic algorithms

(e.g., [11, 8, 17, 13]). Although many of them are adaptive to the dynamic environ-

ments of wireless networks, these algorithms lack the analytical properties of how

well the network performs globally (e.g., whether the scarce channel resource is

shared in an optimal and fair fashion). On the other end of the spectrum, there are

theoretical studies that formulate mesh network routing as optimization problems

(e.g., [6, 18]). The routing algorithms derived from these optimization formulations

can usually claim analytical properties such as resource utilization optimality and

throughput fairness. In these optimization frameworks, traffic demand is usually

implicitly assumed as static and known a priori. Recent studies of wireless network

traces [16], however, show that the traffic demand, even being aggregated at access

points, is highly dynamic and hard to estimate. Such observationshave significantly

challenged the practicability of the existing optimization-based routing solutions in

wireless mesh networks.

There are two basic approachesto address the traffic uncertaintyin optimal mesh

network routing:

• Predictive routing [10, 9], which infers the traffic demand with maximum prob-

ability based in its history and optimizes the routing strategy based on the pre-

dicted traffic demand. Underlying predictive routing is the assumption that past

behavior is a good indicator of the future.

• Oblivious routing [20], which makes no assumption on traffic demand and con-

siders all the possible traffic demands and selects the routing strategy where the

worst-case network performance is optimized.

This chapter provides an overview of optimal routing algorithms for wireless

mesh networks: optimal routing under fixed demand, predictive routing, and obliv-

ious routing. It focuses on the latter two routing strategies and show how they ex-

plicitly consider the traffic uncertainty in their problem formulation and algorithm

design.Inthis chapter,we also identifythekeyfactors thataffectthe performanceof

each routing strategy and provide guidelines towards the strategy selection in mesh

network routing under uncertain traffic demands.

The remainder of this chapter is organized as follows. Sec. 3 presents the net-

work and system model. Sec. 4 reviews the backgroundknowledge,i.e., the optimal

routing algorithm under fixed demand. Sec. 5 presents the predictive mesh network

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Wireless Mesh Network Routing Under Uncertain Demands3

routing strategy. Sec. 6 presents the oblivious mesh network routing formulation

and algorithm. Sec. 7 evaluates and compares the performance of different routing

strategies. Sec. 8 and Sec. 9 provide thoughts for practitioners and the directions for

future research. Finally, Sec. 10 concludes the chapter.

2 Terminology and Definitions

Some of the most important terms in this chapter are defined below.

Transmission Range The upper limit of the distance between two routers such

that messages are intelligible between them.

Interference Range The upper limit of the distance between two routers such

that their messages collide when sent on the same channel at the same time. The

interference range is always at least as large as the transmission range.

Wireless Mesh Network A set of interconnected nodes which aggregate and

route traffic to and from end users and the Internet. A WMN is often two-tiered,

consisting of a backbone with hard links to the Internet and a mesh of routers that

extend the reach of the network. WMNs can be used to extend the last-mile of Inter-

net access in rural areas or provide a cost-effective solution in urban areas.

Interference Set

which it interferes with. One or both of the endpoints for an edge in the interference

set of e is within the interference range of one of e’s endpoints.

The Interference Set of an edge e is the set of other edges

Channel

broadcast on.

A section of the radio spectrum that a router can be configured to

Radio A component of a wireless router which can broadcast on one channel.

Routers often have several radios.

Routing

which all traffic flows between any two nodes.

A full set of pathways in a graph which specify the exact routes by

Wireless Routing

information to accommodate channels. Such information may include channel as-

signments and scheduling

A routing in a wireless network, which includes additional

Oblivious Routing A routing system that takes no account of the actual de-

mands on a network. Note that such a routing is static even as demandchangesover

time.

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4 J. Wellons, L. Dai, B. Chang and Y. Xue

Predictive Routing A routing system that attempts to extrapolate future rout-

ing demands and plan the network flows accordingly. This can perform well, if the

traffic is regular in which case it approaches the optimal routing, or it can perform

poorly if subsequent demands are near the worst-case for the routing chosen by the

algorithm.

3 Model

3.1 Network and Interference Model

Internet

gateway access point

clients

mesh node

local access point

Backbone Network

aggregated ?

flow

Fig. 1 Illustration of Wireless Mesh Network

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

traffic for the mobile clients which are associated with them. They communicate

with each other and with the stationary wireless routers to form a multi-hop back-

bone network, which forwards the user traffic to the Internet gateways. Fig. 1 shows

an example of wireless mesh network. We use w ∈ W to denote the set of gateways

in the network and s ∈ S to denote the set of local access points that generate traf-

fic in the network. Local access points, gateways and mesh routers are collectively

called mesh nodes and denoted by the set V .

In a wireless network, packet transmissions are subject to location-dependent

interference. Here we consider the protocol model presented in [12]. We assume

that all mesh nodes have the uniform transmission range denoted by RT. Usually

the interference range is larger than its transmission range, which is denoted as

RI = (1 + ∆)RT, where ∆ ≥ 0 is a constant. For simplicity, in this chapter we

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Wireless Mesh Network Routing Under Uncertain Demands5

Interference

Range

Transmission

Range

u

b

w

x

v

c

a

Fig. 2 Transmission and Interference Range

assume that each node is equipped with one radio interface which operates on the

same wireless channel as others. Let r(u,v) be the distance between two nodes u

and v (u,v ∈ V ). In the protocol model, packet transmission from node u to v 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 x ∈ V within the interference range of the

receiving node v, i.e., r(x,v) ≤ RI, is not transmitting. If node u can transmit to

v directly, they form an edge e = (u,v). As an example shown in Fig.2, nodes

w,x,v are within the transmission range of node u, thus they can transmit the node

u directly. At the same time, nodes w,v,x,b,c are all within the interference range

of node u, which means the signal from node u could be heard by any node of

w,v,x,b,c, and vice versa. Thus they must be silenced, if they are not the intended

sender, when u is receiving a packet.

u

v

b

a

y

x

Fig. 3 Interference Set

We assume that the maximum data rate that can be transmitted along an edge is

the same for all edges, and denote it as c (also called the channel capacity).Let E be

the set of all edges. We say two edges e,e′interfere with each other, if they can not

transmitsimultaneouslybasedonthe protocolmodel.Furtherwe defineinterference

set I(e) which contains the edges that interfere with edge e and e itself. Fig. 3 is an

illustration of the interference set of edge (u,v). The circles are the interference

ranges of node u and v, and the union of these two circles is the interference range

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6 J. Wellons, L. Dai, B. Chang and Y. Xue

of edge (u,v). So the interference set I(u,v) of edge (u,v) includes (u,v), (a,b),

(v,b), (v,a), (a,u), (x,u) and (x,y).

Internet

gateway access point

clients

mesh node

local access point

Backbone Network

aggregated ?

flow

w*

Fig. 4 Illustration of Virtual Node

Finally, we introduce a virtual node w∗to represent the Internet, as shown in

Fig. 4. w∗is connected to each gateway with a virtual edge e∗= (w∗,w),w ∈ W.

Further, let E′= E ∪ {e∗} and V′= V ∪ {w∗}. For simplicity, we assume that

the link capacity in the 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, and

they do not interfere with any of the wireless transmissions.

3.2 Traffic Model and Schedulability

This chapter studies the routing strategies for wireless mesh backbone networks.

Thus it only considers the aggregatedtraffic between the local access points and the

Internet gateways. Here we call the aggregatedtraffic in (or out) a local access point

a flow and denote it as f ∈ F, where F is the set of all aggregated flows. All flows

will take w∗as their source (or destination). We denote the traffic demand of flow f

as dfand use vector d = (df,f ∈ F) to denote the demand vector consisting of all

flow demands.

Now we proceed to study the constraint of the flow rates. Let y = (y(e),e ∈ E)

denote the edge rate vector, where y(e) is the aggregated flow rate along e. Edge

rate vector y is said to be schedulable, if there exists a stable schedule that ensures

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Wireless Mesh Network Routing Under Uncertain Demands7

every packet transmission with a bounded delay. Essentially, the constraint of the

flow rates is defined by the schedulable region of the edge rate vector y.

The edge rate schedulability problem has been studied in several existing works,

which lead to different models [14, 15, 21]. In this chapter, we adopt the model

in [15], which is also extended in [6] for multi-radio, multi-channel mesh network.

In particular, [15] presents a sufficient condition under which an edge scheduling

algorithm is given to achieve stability with bounded and fast approximation of an

ideal schedule. [6] presents a scheme that can adjusts the flow routes and scale the

flow rates to yield a feasible routingand channelassignment.Based on these results,

we have the following claim as a sufficient condition for schedulability.

Claim 1. (Sufficient Condition of Schedulability) The edge rate vector y is

schedulable if the following condition is satisfied:

∀e ∈ E,

?

e′∈I(e)

y(e′) ≤ c

(1)

4 Background

This section provides the background of optimal mesh network routing, introduces

its problem formulations, and reviews its algorithm under fixed traffic demand.

Theexistingworks onoptimalmultihopwireless networkrouting[6,18, 14] usu-

ally formulate it as a throughput optimization problem which maximizes the flow

throughput, while satisfying the fairness constraints. In this formulation, traffic de-

mand is fixed and reflected as the flow weight in the fairness constraints. Recall that

f ∈ F is the aggregated traffic flow between the local access points and the virtual

gateway (i.e., Internet) and d = (df,f ∈ F) is the demand vector consisting of all

flow demands. Consider the fairness constraint that, for each flow f, its throughput

beingroutedis in proportionto its demanddf.The goalofthroughputmaximization

routing is to maximize λ (so called scaling factor) where at least λ · dfamount of

throughput can be routed for flow f.

To balance the traffic load, flow f could be routed over multiple paths, let Pf

be the set of unicast paths that could route flow f, and xf(P) be the rate of flow f

over path P ∈ Pf. Obviously the aggregatedflow rate yealong edge e ∈ E is given

by ye=?

through paths P passing edge e. Based on the sufficient condition of schedulability

in Claim 1 (Eq.(1)), we have that

f:P∈Pf&e∈Pxf(P), which is the sum of the flow rates that are routed

?

e′∈Ie

?

f:P∈Pf&e′∈P

xf(P) ≤ c

(2)

To simplify the above equation, we define AeP = |Ie∩ P| as the number of

wireless links path P passes in the interference set Ie. The throughput optimization

routing with fairness constraint is then formulated as the following linear program-

ming (LP) problem:

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8 J. Wellons, L. Dai, B. Chang and Y. Xue

PT: maximize λ

subject to

(3)

(4)

?

P∈Pf

xf(P) ≥ λ · df,∀f ∈ F

?

f∈F

?

P∈Pf

xf(P)AeP ≤ c,∀e ∈ E

(5)

λ ≥ 0,xf(P) ≥ 0,∀f ∈ F,∀P ∈ Pf

(6)

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 (4) enforces fairness by requiring that the comparative ratio of traffic

routedfor differentflows satisfies the comparativeratio oftheir demands.Inequality

(5) enforces the capacity constraint by requiring the traffic aggregation of all flows

passing wireless link e ∈ E satisfy the sufficient condition of schedulability. This

problem formulation follows the classical maximum concurrent flow problem.

While theabovethroughputmaximizationroutingproblemformulationis widely

used in designing optimal mesh network routing strategies under known demands,

it is not suitable to study the routing performance under dynamic and uncertain

traffic demand. Here we consider a formulation based on another routing perfor-

mance metric – network congestion (or utilization). In the Internet, link utilization

is commonly used for traffic engineering [19], whose objective is to minimize the

utilization at the most congested link under given traffic demand. However, link uti-

lization can not be straightforwardly applied to multihop wireless networks, such

as mesh backbone network, as a metric of routing performance due to the location-

dependentinterference.In what follows, we define the network congestion based on

the utilization of the interference set as the routing performance metric and outline

the relation between the formulation of the throughput optimization problem and

the congestion minimization problem.

Let x′

that?

is then given by?

f∈F

?

ence set Ieis definedas its utilization(i.e., the ratio betweenits load andthe channel

capacity) and denote it as θe:

f(P) be the rate of flow f on path P under traffic demand df. It is obvious

P∈Pfx′

P∈Pfx′

f(P) = df. The traffic being routed within the interference set Ie

f(P)AeP. Formally,the congestionof an interfer-

θe=

?

f∈F

?

P∈Pfx′

c

f(P)AeP

(7)

Further,the network congestionis definedas the maximumcongestionamongall

the interference sets, i.e.,

θ = max

e∈Eθe

(8)

The network congestion minimization routing problem is then formulatedas fol-

lows:

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Wireless Mesh Network Routing Under Uncertain Demands9

PC: minimize θ

subject to

(9)

(10)

?

P∈Pf

x′

f(P) ≥ df,∀f ∈ F

?

f∈F

θ ≥ 0,x′

?

P∈Pf

x′

f(P)AeP ≤ c · θ,∀e ∈ E

(11)

f(P) ≥ 0,∀f ∈ F,∀P ∈ Pf

(12)

To reveal the relation between PTand PC, we let θ =

Problem PCis then transformed to:

1

λand x′

f(p) =

xf(p)

λ

.

P′

C: minimize1

λ

(13)

subject to1

λ

?

P∈Pf

xf(P) ≥ df,∀f ∈ F

(14)

1

λ

?

f∈F

?

P∈Pf

x′

f(P)AeP ≤ c · θ,∀e ∈ E

(15)

λ ≥ 0,x′

f(P) ≥ 0,∀f ∈ F,∀P ∈ Pf

(16)

which is obviously equivalent to the throughput optimization problem PT.

If the demand vector d is known,both problemPTand PCcould be solved by a

LP-solversuchas [2, 3]. To reducethe complexityforpractical use, thework of [10]

also presents 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. First we assign a

price µeto each set Sefor e ∈ E. The objective is to minimize the aggregated price

for all interference sets. As the constraint, Inequality (18) requires that the price

?

f. Further, Inequality (19) requires that the weighted flow price µfover its demand

dfmust be at least 1.

e∈EAePµeof any path P ∈ Pffor flow f must be at least µf, the price of flow

DT: minimize

?

e∈E

?

e∈E

?

f∈F

c · µe

(17)

subject to

AePµe≥ µf,∀f ∈ F,∀P ∈ Pf

(18)

µfdf≥ 1

(19)

Based on the above dual problem DT, the fast approximation algorithm is pre-

sented in Table 1. The properties of this algorithm are shown as follows.

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10 J. Wellons, L. Dai, B. Chang and Y. Xue

Mesh Network Routing Under Fixed Demand

1

∀e ∈ E, µe← β/c

2

xf(P) ← 0, ∀P ∈ Pf, ∀f ∈ F

3

whileP

4

for ∀f ∈ F do

5

d′

f← df

6

whileP

7

P ← lowest priced path in Pfusing µe

8

δ ← min{d′

f,mine∈PAeP}

9

d′

f← d′

10

xf(P) ← xf(P) + δ

11

∀e s.t. AeP?= 0, µe← µe(1 + ǫδAeP)

12

end while

13

end for

14

end for

e∈Ec · µe< 1

e∈Ec · µe < 1 and d′

f> 0 do

f− δ

Table 1 Routing Algorithm Under Fixed Demand

Property: If β = (|E|/(1 − ǫ))−1/ǫ, then the final flow generated by FMR is at

least(1−3ǫ)timestheoptimalvalueofP.TherunningtimeisO(1

|E|) + logU)]) · Tmp, where U is the length of the longest path in G, and Tmpis

the running time to find the shortest path.

ǫ2[log|E|(2|F|log|F|+

5 Predictive Mesh Network Routing

The predictive mesh network routing is based on a two-tier framework as shown in

Fig. 5, which integrates traffic modelling and routing optimization.

traffic modeling

traffic estimation

network routing

traffic ?

demand?

estimation

network ?

performance?

feedback

?

model

model?

parameter

wireless?

network?

traces

network optimization

Fig. 5 Integrated Framework of Traffic Modelling and Network Optimization.

• Trafficmodellingderivesthetrafficmodelofawirelessmeshnetwork.Themodel

should be dependable at characterizing the long term traffic demand, yet agile at

containing the uncertain traffic dynamics in the short term. The traffic modelling

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Wireless Mesh Network Routing Under Uncertain Demands11

component needs to produce traffic demand estimations as inputs to the network

optimization component.

• Routing optimization determines the routing strategy which distributes the traffic

along different routes so that minimum congestion will be incurred even under

dynamic traffic. To achieve this goal, the routing optimization decision should

effectivelytake into account the traffic demand estimation results from the traffic

modelling component.

5.1 Traffic Prediction

First we study the dynamic behavior of aggregated traffic at access points. Our goal

is to (1) develop a reliable prediction method that is able to estimate the aggregated

traffic demand of an access point based on its historical data, and (2) develop a

statistical model to characterize the prediction errors. The predicted traffic demand

will serve as the input of predictive mesh network routing algorithms which will be

presented in the next section.

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 [1]. By

analyzing the snmp log from each access point, we derive the dynamic behavior of

aggregated traffic demand. 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. 6. From the figure, we can easily observe that (1) the traffic

demandis non-stationaryoverlarge time scales due to the diurnaland weeklywork-

ing cycles; (2) compared with the traffic behavior in the backbone Internet [19], the

traffic at an access point is significantly bursty due to the insufficient level of multi-

plexing. The above observations are consistent with the findings in [16].

0 10 2030 40 5060 70

0?

2?

4?

6?x 10?8

Time (#day since 03/24/2002)

Traffic (bypte/hour)

Fig. 6 Incoming Traffic Time Series of ResBldg97AP3 (March 24, 11pm, 2002 - June 9, 10pm,

2002).

Thefirst step ofouranalysisis to identifyandremovethe dailyandweeklycyclic

patterns in the time series. This requires us to calculate the weekly/daily cyclic av-

erage. Formally, 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 same day of the week, i.e.,

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12 J. Wellons, L. Dai, B. Chang and Y. Xue

30? 35?40? 45?50? 55?60? 65?

0?

0.5?

1?

1.5?

2?

2.5?

3?x 10?8?

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

Traffic (byte/hour)?

?

?

raw?

average?

(a) Raw Traffic vs. Moving Average Series

30354045 50 5560 65

-1

0

1

2

3x 108

Time (#day since 03/24/2002)

(b) Adjusted Traffic Series

Adjusted Traffic (byte/hour)

Fig. 7 Traffic Series in 5 weeks

¯ x(t) =

W

?

i=1

x(t − 24 × 7 × i)/W

(20)

where W is 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. 7(a) plots

the raw traffic as well as its moving average with W = 5. By removing the cyclic

effect from the raw data, we derive the adjusted traffic series y(t) as follows.

y(t) = x(t) − ¯ x(t)

(21)

The adjusted series of the one shown in Fig. 7(a) is given in Fig. 7(b). This

adjusted traffic exhibits short-term (a few hours) traffic correlations. We model the

adjusted traffic series with an autoregressive process as follows1.

y(t) = β1y(t − 1) + β2y(t − 2) + ... + βKy(t − K) + ǫ

(22)

where K is the process order. To apply this model for prediction, we estimate the

parameters of this process. Given N observations y1,y2,...,yN, the parameters β1,

..., βKare estimated via least squares by minimizing:

N

?

t=K+1

?y(t) − β1y(t − 1)... − βKy(t − K)?2

(23)

1Ideally, y(t) should have zero mean. In some cases, y(t) has a small mean value which needs to

be removed before fitting an autoregressive process.

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Wireless Mesh Network Routing Under Uncertain Demands 13

Based on these parameters, we further derive the adjusted traffic prediction ˆ y(t) as

follows:

ˆ y(t) = β1y(t − 1) + β2y(t − 2) + ... + βKy(t − K)

(24)

Fig. 8 illustrates the estimation results for the adjusted traffic series in Fig. 7(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] hour 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 follows.

ǫy(t) = y(t) − ˆ y(t)

(25)

Based on this definition, Fig. 9(a) plots the cumulative distribution function of the

prediction error of the adjusted traffic series shown in Fig. 8. It is obvious that the

error distribution follows normal distribution with a mean close to zero.

720740 760780 800 820840

-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. 8 Adjusted Traffic and Its Prediction

Finally, we define traffic prediction ˆ x as follows:

ˆ x(t) = [¯ x(t) + ˆ y]+

(26)

Fig. 10 plots the predicted traffic series ˆ x(t) in comparison with the raw traffic.

We cansee thepredictedtrafficclosely matchesthereal(raw)traffic.Thecumulative

distributionfunctionofthepredictionerrorǫx(t), whichis definedas ǫx(t) = x(t)−

ˆ x(t), is plottedin Fig.9(b).It clearlyshowsthatthis distributionalso followsnormal

distribution with a near-zero mean.

Thus we could consider the traffic demand at time t as a random variable X(t)

which follows normal distribution with mean ˆ x(t) and the same variance as ǫx.

Fig. 11 shows an exampledistributionof the predictedtraffic demandof #976hour.

To summarize, the presented prediction method provides two prediction mod-

els: mean value and statistical distribution. These two traffic prediction models will

serve as the inputs for predictive routing algorithms which are presented in the next

section.

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14 J. Wellons, L. Dai, B. Chang and Y. Xue

-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 -0.500.51 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. 9 Cumulative Density Function of Prediction Error

3035 40 45

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

5055 606570

0

1

2

3

4

5

6x 108

Traffic (byte/hour)

raw

predict

Fig. 10 Raw Traffic vs. Predicted Traffic

-101234

x 108

0

1

2

3

4

5

6x 10-9

Traffic Prediction (byte/hour)

?

Probability Density Function

Prediction < 0 ?

will be cut-off

Fig. 11 Predicted Traffic Distribution

5.2 Predictive Routing Optimization

There are two predictive routing algorithms [9] – one takes the mean value of the

predicted traffic demand as input, the other takes the statistical distribution of the

predicted traffic demand as input.

The mean-value predictive routing algorithm is a natural integration of the opti-

mal routing algorithm under fixed demand (Sec. 4), where the traffic demand dfat

time t takes the mean value of the predicted traffic demand ˆ x(t). In what follows,

we will focus on the statistical-distribution predictive routing.

Page 15

Wireless Mesh Network Routing Under Uncertain Demands 15

First we model the traffic demand of an aggregated flow f ∈ F using a random

variable Df, which follows the following discrete probability distribution

Pr(Df= di

f) = qi

f

(27)

where Df = {d1

bilities. 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. We further assume

that the demand from different access points are independent from each other. Thus

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

follows.

f,d2

f,...,dm

f} is the set of of values for Df with non-zero proba-

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

f,f ∈ F) = Πf∈Fqi

f

(28)

Let us consider a traffic routing solution (xf(P),P ∈ Pf,f ∈ F) that satisfies

the capacity constraint (Inequality (5)). It is obvious that λ is a function of d:

λ(d) = min

f∈F{xf

df}

(29)

where xf =?

under demandvector d. Such a solution could be easily derivedbased on Algorithm

I shown in Table 1. 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 follows:

P∈Pfxf(P). Further let us consider the optimal routing solution

ω(d) =

λ(d)

λ∗(d)

Obviously, the performance ratio is also a random variable under uncertain de-

mand. We denote it as Ω. Ω is a function of randomvariable 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 as follows.

E(Ω) = Pr(D = d) ×λ(d)

λ∗(d)

(30)

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

mally, we formulate the throughputoptimization routing problem for wireless mesh

backbone network under uncertain traffic demand as follows.

d∈Dp(d) = 1. For-

Page 16

16J. Wellons, L. Dai, B. Chang and Y. Xue

PU: maximize

?

d∈D

p(d)λ(d)

λ∗(d)

(31)

subject to ∀d ∈ D,where d = (df,f ∈ F)

?

P∈Pf

xf(P) ≥ λ(d) · df,∀f ∈ F

(32)

?

f∈F

?

P∈Pf

xf(P)AeP ≤ c,∀e ∈ E

(33)

λ ≥ 0,xf(P) ≥ 0,∀f ∈ F,∀P ∈ Pf

(34)

Similar to problem PT, the constraints of PUcome from the fairness require-

ment and the wireless mesh network capacity.In particular,Inequality (32) enforces

fairness for all demand d ∈ D, and Inequality (33) enforces capacity constraint as

Inequality (5) in problem PT.

Now we consider the dual problem DUof PU. Similar to DT, the objective

of DUis to minimize the aggregated price for all interference sets. However, in

Inequality (37), 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∈E

?

e∈E

c · µe

(35)

subject to

AePµe≥ µf,∀f ∈ F,∀P ∈ Pf

(36)

?

f∈F

where d = (df,f ∈ F)

µfdf≥

p(d)

λ∗(d),∀d ∈ D

(37)

Now we present an approximation algorithm for PUin Table 2. Note that since

the channel capacity c will not affect the final result of the algorithm, we simply

omit it here. In the work of [9], we are able to prove the following properties with

this algorithm.

Property: If β = (|E|/(1 − ǫ))−1/ǫ, then the final flow generated by the above

algorithm is at least (1 − 3ǫ) times the optimal value of PU. The running time is

O(1

of the longest path in G, Tmpis the running time to find the shortest path, and Tfmr

is the running time of the optimal routing algorithm under a fixed demand.

ǫ2[log|E|(2|D||Tfmr||F|log|F|+|E|)+logU)])·Tmp, where U is the length

6 Oblivious Mesh Network Routing

In contrast to the predictive routing which establishes traffic models based on time-

series analysisandoptimizestowardsthetrafficdemandswithmaximumpossibility,

Page 17

Wireless Mesh Network Routing Under Uncertain Demands 17

Mesh Network Routing Under Uncertain Demand

1

∀e ∈ E, µe← β

2

xf(P) ← 0, ∀P ∈ Pf, ∀f ∈ F

3

loop

4

for ∀f ∈ F do

5

¯P ← lowest priced path in Pfusing µe

6

µf←P

7

end for

8

for ∀d ∈ D do

9

µd←P

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 Pfusing µe

19

δ ← min{d′

f,mine∈P

20

d′

f← d′

21

xf(P) ← xf(P) + δ

22

∀e s.t. AeP?= 0, µe← µe(1 + ǫδAeP×λ∗(dmin)

23

end while

24

end for

25

end loop

e∈EAe¯

Pµe

f∈Fµfdf

λ∗(d)

p(d)

f← dmin

f> 0 do

1

AeP}

f− δ

p(dmin))

Table 2 Routing Algorithm Under Uncertain Demand

oblivious routing makes no assumptions on the traffic model, rather it considers

all traffic demand possibilities and optimizes towards the worst-case scenario. To

formally study the oblivious routing strategy, we need a performance metric that

could characterize the worst-case congestion under all possible traffic demand.

(1) Routing:

First, let’s examine the formal description of routing, which specifies how traffic

of each flow is distributed across the network. In the previous formulation (PC),

a routing is characterized through the traffic load distribution along different paths

(i.e., x′

flow. When we have to consider all possible traffic demands, it becomes infeasible.

In fact, a routing strategy could be modeled independently of the traffic demand,

which is the core of the oblivious routing problem formulation.

Formally, we define a routing by the fraction of each flow that is routed along

each edge e ∈ E′. We use φf(e) to denote the fraction of demand of flow f that

is routed on the edge e ∈ E′. Thus, a routing could be specified by the set φ =

{φf(e),f ∈ F,e ∈ E′}. Recall that the demand of flow f ∈ F is denoted by

df. Therefore, the amount of traffic demand of f that needs to be routed over e in

routing φ, denoted by y′

f(P)). This description of a routing depends on the traffic demand of each

f(e), is given as follows:

Page 18

18J. Wellons, L. Dai, B. Chang and Y. Xue

y′

f(e) = df· φf(e)

(38)

Thus the congestion θeof an interference set I(e) is given by

θe=

?

e′∈I(e)

?

f∈F

y′

f(e′)

c

=

?

e′∈I(e)

?

f∈F

df· φf(e′)

c

(39)

We further use θ(φ,d) = maxe∈Eθe(φ,d) to denote the network congestion

under a certain routing φ and traffic demand vector d.

(2) Oblivious Performance Ratio:

Now we proceed to study the performance metric that could characterize a

“good” routing solution under all possible traffic demands. We start with the opti-

mal routing φopt(d) for a certain demand vector d, which would give the minimum

congestion under this demand, i.e.,

θopt(d) = min

φ

θ(φ,d)

(40)

Now we define the performance ratio γ(φ,d) of a given routing φ on a given

demand vector d as the ratio between the network congestion under the routing φ

and the minimum congestion under the optimal routing, i.e.,

γ(φ,d) =θ(φ,d)

θopt(d)

(41)

Performance ratio γ measures how far φ is from being optimal on the demand

d. Now we extend the definition of performance ratio to handle uncertain traffic

demand. Let D be a set of traffic demand vectors. Then the performance ratio of a

routing φ on D is defined as the worst-case performance ratio for all demands in

D, i.e.,

γ(φ,D) = max

d∈Dγ(φ,d)

(42)

A routing φoptis optimal for the traffic demand set D if and only if

φopt= argmin

φ

γ(φ,D)

(43)

which means φoptminimizes the performance ratio under the worst-case sce-

nario. When the set D includes all possible demand vectors d, we refer to the per-

formance ratio as the oblivious performance ratio. The oblivious performance ratio

is the worst performance ratio a routing obtains with respect to all possible demand

vectors. To study the optimal routing strategy under uncertain traffic demand, we

are interested in the optimal oblivious routing problem which finds the routing that

minimizes the oblivious performance ratio. We call this minimumvalue the optimal

oblivious performance ratio.

It is worth noting that the performance ratio γ is invariant to scaling. Thus

to simplify the problem, we only consider traffic demand vectors D that satisfy

θopt(d) = 1, instead of considering all possible traffic vectors. In this case,

Page 19

Wireless Mesh Network Routing Under Uncertain Demands 19

γ(φ,D) = max

d∈Dθ(φ,d)

(44)

Thus the goal of oblivious routing is given by

min

φ

max

θopt(d)=1θ(φ,d)

(45)

(3) Flow Conservation

Traffic into and out of a mesh node must be conserved. In PC, a path represen-

tation of the routing is being used (x′

conservation. Here, since we use an edge representation of the routing (φf(e)), the

flow conservation has to be explicitly formulated. In particular, for the node v ∈ V′

that only relays for flow f (i.e.,neither sourceor destination),we have the following

relations:

f(P)), which implicitly formulates the flow

∀f ∈ F,

?

e=(u,v)

y′

f(e) −

?

e=(v,u)

y′

f(e) = 0

(46)

if v is a relay of f

If v is the source node of flow f, then we have

∀f ∈ F,

?

e=(u,v)

y′

f(e) −

?

e=(v,u)

y′

f(e) = −df

(47)

if v is the source node of f

Summarizingthe abovediscussions, the obliviousmesh network routingproblem

is formulated as follows.

PO:

minimize θ

(48)

(49)

subject to

?

e′∈I(e)

?

f∈F

y′

f(e′)

c

≤ θ,∀e ∈ E

(50)

?

e=(u,v)

∀f ∈ F,∀v ∈ V′,if v is a relay of f

?

e=(u,v)

e=(v,u)

∀f ∈ F,∀v ∈ V′,if v is the source node of f

∀f ∈ F,∀e ∈ E,y′

f(e) = df· φf(e) ≥ 0

θ ≥ 0,∀d with θopt(d) = 1

y′

f(e) −

?

e=(v,u)

y′

f(e) = 0

(51)

y′

f(e) −

?

y′

f(e) = −df

(52)

(53)

(54)

Page 20

20J. Wellons, L. Dai, B. Chang and Y. Xue

Different from PC, the oblivious mesh routing problem POcannot be solved

directly, because it is taken over all demand vectors, and θopt(d) is an embedded

maximization in the minimization problem.

Here we use a similar method as in [7], which provides a LP formulation of the

obliviousroutingproblem.The keyinsightis to lookat the dualproblemofthe slave

LPs of the original oblivious routing problem. So let’s first examine the following

slave problem which computes the slave demands for each possible edge.

max

?

f∈F

df· φf(e)

c

(55)

subject to φf(e) is a routing(56)

In the dual formulation, we first introduce interference set weights πe(e′) for

every pair of interference sets e,e′. πe(e′) corresponding to the fraction of the flow

on the interference set Iethat is routed over the interference set Ie′. Each π variable

canbethoughtofas a weightedmultiplierin anLP dualformulation.Therearethree

essential properties shown in Theorem 1. The proofs of these properties follow the

similar idea of [7], which is provided in our technical report[20] due to the space

constraint.

Property: Weights πe(e′),e,e′∈ E on a routing φf(e) with oblivious ratio ≤ θ

have the following properties,

P1?

P2 ∀ paths h1,h2,...hkand each flow f

φf(e) ≤ c ·?p

P3 ∀ interference sets Ie,I′

Because the interference must be minimized, each interference set of the path

is weighted according to how many other interference sets in the path it interferes

with. The number of such paths between any two nodes grows exponentially with

the size of the network.In order to retain polynomialsolvability,we may encodethe

shortest interference path requirement (in P2) in such a way that we only need as

many such variables and constraints as there are pairs of interference sets. Thus we

introduce pe(f) as the length of the shortest path flow f according to interference

set weights πe(e′) (for all e′∈ E). This definition is equivalent to the following

form:

∀e ∈ E,∀f ∈ F,

∀e′= (v,u), where v is the destination of f,

and u is the destination of flow f′

πe(e′) + pe(f) − pe(f′) ≥ 0

e′c · πe(e′) ≤ θ,∀e ∈ E

k=1πe( interference-set-of (hk))

e,πe(e′) ≥ 0

Summarizing the above discussions, the dual problem is given as follows:

Page 21

Wireless Mesh Network Routing Under Uncertain Demands 21

DO:

minimize θ

(57)

∀e,e′∈ E :

?

e

c · πe(e′) ≤ θ

∀e ∈ E,∀f ∈ F :

?

e′∈I(e)

∀e ∈ E,∀f ∈ F,

∀e′= (v,u), where v is the destination of f,

and u is the destination of flow f′

πe(e′) + pe(f) − pe(f′) ≥ 0

∀e,e′∈ E,πe(e′) ≥ 0

∀e ∈ E,∀f ∈ F : pe(f) ≥ 0

(58)

φf(e′)/c ≤ pe(f)

(59)

(60)

In the dual problem, Eq. (58) can be explained by property P1. Property P2 and

the shortest interference set paths account for Eq. (60), and finally property P3 ap-

pears at Eq. (60). The dual problem is a single polynomial-size LP instance, which

can be solved with any LP solver. Our choice of LP solver was lp solve [3], an open

source Mixed Integer Linear Programming (MILP) solver.

7 Simulation Study

In this section, we simulate the predictive and oblivious routing strategies over a

variety of mesh network setups. Our goal is to evaluate and compare their perfor-

mance and identify the key factors that impact the performance. Two other routing

strategies, namely oracle routing and shortest-path routing, are used as the baseline

strategies for comparison. We describe the routing strategies that are evaluated in

the simulation study as follows.

• Oracle Routing (OR). In this strategy, the traffic demand is known a priori. It

runs every hour based on the up-to-date traffic demand and returns the optimal

set of routes. As a result, no other routing strategies can outperform OR, and we

used it to provide a performance upper bound.

• Shortest-Path Routing (SPR). This strategy is agnostic of traffic demand, and re-

turns a fixed routing solution purely based on the shortest distance (number of

hops) from each mesh node to the gateway. Many mesh network routing heuris-

tics resemblethe shortest-pathroutingstrategy.We evaluatethis strategyto quan-

titatively contrast the advantage of routing strategies which explicitly consider

traffic uncertainty.

Page 22

22J. Wellons, L. Dai, B. Chang and Y. Xue

• Predictive Routing (PR). This strategy attempts to adjust to changing the traf-

fic demand. Future demand is estimated based on the historical data every hour

based on the traffic prediction method presented in Sec. 5.

• ObliviousRouting(OBR).Thisstrategyisoblivioustothetrafficdemands.Itcon-

siders all possible traffic demands that may be imposed on the network and finds

a routing that optimizes the worst-case congestion using the algorithm presented

in Sec. 6.

It is worth noting that the SPR and OBR will compute the traffic routes only

once and use them during the entire simulation time, while the OR and PR need to

compute and update the routes every hour.

To realistically simulate the traffic demand at each LAP, we employ the traces

collectedinthecampuswirelessLANnetwork.Thenetworktracesusedinthiswork

arecollectedinSpring2002at DartmouthCollegeandprovidedbyCRAWDAD [1].

By analyzing the snmp log trace at each access point, we are able to derive its 1108-

hour incoming and outgoing traffic volume beginning 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 3 in one of the random topologies as shown in Fig. 12.

0

200

400

600

800

1000

0 200 400 600 800 1000

Y Position

X Position

01

2

26

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

27

28

59

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

60

61

62

63

Fig. 12 Mesh Network Topology

AP 31AP3 34AP5 55AP4 57AP2 62AP3 62AP4 82AP4 94AP1 94AP3 94AP8

2218575 55 Node ID20 533 5627

Table 3 Traffic Assignment from Trace File

Page 23

Wireless Mesh Network Routing Under Uncertain Demands23

Weexperimentwiththeaboveroutingstrategiesalongthetimerange[108,1108],

a 1000-hour period excerpted from the trace2. Note that all the simulation results

presented in this section use 108 as the zero point.

0?

5?

10?

15?

20?

0? 200? 400? 600?

Time (#hour since 03/31/2002, 11pm EST)?

800? 1000? 1200? 1400? 1600?

θ?

OR?

PR?

SPR?

OBR?

Fig. 13 Overview of All Strategies

We start by presenting the congestion achieved by all strategies during the entire

1000-hour simulation period. As seen in Fig. 13, OR constantly achieves the mini-

mum 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. This 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?

620?

1.2?

1.4?

1.6?

1.8?

2?

2.2?

2.4?

2.6?

2.8?

640? 660? 680? 700? 720?

θ? /? θ?OR?

Time (#hour since 03/31/2002, 11pm EST)?

PR/OR?

SPR/OR?

OBR/OR?

1?

1.2?

1.4?

1.6?

1.8?

2?

2.2?

2.4?

2.6?

2.8?

0? 20? 40?

Time Instance?

60? 80? 100?

θ? /? θ?OR?

PR/OR?

SPR/OR?

OBR/OR?

Fig. 14 (a) Congestion Ratio (

θ

θOR) (b) Sorted View

To filter out the noise caused by traffic burstiness, in Fig. 14(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, PR, SPR, and OBR achieve less than 2

timestheoptimalcongestioninmostcases.Theaboveobservationsgetclearerwhen

we sort out the normalized congestion ratio for the three strategies in Fig. 14(b). It

is clear that both PR and OBR which integrate the traffic prediction with the opti-

mal routingoutperformthe SPR strategy which is agnostic about the traffic demand.

2Note that the beginning of the trace [0,107] is used as training data, thus it is not included in the

simulation result.

Page 24

24 J. Wellons, L. Dai, B. Chang and Y. Xue

Further, PR achieves lower congestion than OBR for many time points due to more

comprehensive representation of the traffic demand estimation. However, in other

cases (less than 10% of the time), the worst-case congestion of PR is substantially

higher than OBR. This problem can be mostly attributed to the fundamental inaccu-

racy of traffic prediction which is highly sensitive to the traffic’s erratic behavior.

We also investigate the performance of PR and OBR in a representative random

topologies with Internet gateways near the perimeter. For a more complete picture,

wealsoinvestigatecases with2gatewaysandwith4gateways.Eachofthesetopolo-

gies has a total of 64 nodes, including 10 access points receivingtraffic from mobile

clients, the gateways, and the remaining nodes forwarding traffic on behalf of the

Internet and the access points. The points are distributed at random over a simula-

tion square 1000m on the side, with an interference range of 155m. For simplicity,

the transmission range is equal to the interference range.

In both the 4-gateway and the 2-gateway scenarios, we run PR, OBR and SPR

using the demand data from the Dartmouth trace. Fig. 15 plots the congestion ratios

of PR over SPR and OBR over SPR. In both pictures, OBR and PR outperform SPR

in more than 50% of the cases. During the time when they are inferior to SPR, the

worst-case ratio is bounded by 2. Also when we increase the number of gateways

from 2 to 4, both ratios decrease. Obviously, SPR takes advantage of this topology

change, due to the fact that more gateways will diversify the shortest paths from

access points to nearest gateways, and also shrink the lengths of the paths.

0?

0.2?

0.4?

0.6?

0.8?

1?

0.6? 0.8? 1? 1.2? 1.4? 1.6? 1.8?

θ?PR?/? θ?SPR?

2? 2.2?

Percentage?

#gateway 2?

#gateway 4?

0

0.2

0.4

0.6

0.8

1

0.6 0.8 1 1.2 1.4 1.6 1.8

Percentage

θOBR/θSPR

#gateway 2

#gateway 4

Fig. 15 (a) CDF of Congestion Ratio (θPR

θSPR) (b) CDF of Congestion Ratio (θOBR

θSPR)

0.6?

0.7?

0.8?

0.9?

1?

1.1?

1.2?

1.3?

1.4?

100? 200? 300? 400? 500? 600?

θ?PR?/? θ?OBR?

Time (#hour since 03/31/2002, 11pm EST)?

0?

0.2?

0.4?

0.6?

0.8?

1?

0.6? 0.7? 0.8? 0.9? 1? 1.1? 1.2? 1.3? 1.4?

Percentage?

θ?PR?/? θ?OBR?

#hour 201-300?

#hour 301-400?

#hour 501-600?

Fig. 16 (a) Congestion Ratio (

θPR

θOBR) (b) CDF of Congestion Ratio

Page 25

Wireless Mesh Network Routing Under Uncertain Demands25

8 Thoughts for Practitioners

To summarize this chapter, we provide some thoughts for practitioners:

• Currently, most mesh network routing algorithms and protocols are heuristic-

based. Though adaptive to the dynamic environments of wireless networks, they

lack the analytical properties of how well the network performs globally. Thus

they may lead to sub-optimal resource utilization or unfairness in the network.

The optimal mesh routing algorithms that are derived from optimization formu-

lations can usually claim analytical properties such as resource utilization opti-

mality and throughput fairness. However, they usually have strong assumptions

on static and known traffic demand, which have been shown to be unrealistic by

the studies of wireless network traces [16]. Thus there is a critical need to inves-

tigate the optimal mesh network routing strategies that can accommodate traffic

uncertainty.

• Predictive routing and oblivious routing are two optimal routing strategies that

address the traffic uncertainty in mesh network routing. Their designs, however,

are based on different principles. (1) predictive routing infers the traffic demand

with maximum possibility based in its history and optimizes the routing strat-

egy based on the predicted traffic demand. Underlying predictive routing is the

assumptionthat past behavioris a goodindicatorof the future.(2)oblivious rout-

ing, which makes no assumption on traffic demand and considers all the possible

trafficdemands.In particular,obliviousroutingselects the routingstrategywhere

the worst-case network performance is optimized. For a given mesh network, it

is important to know which routing strategy would provide a better performance.

• Through the simulation study, we find predictive routing performs better un-

der consistent traffic demand compared to highly variable demand. Furthermore,

oblivious routing, being a stateless routing, is unaffected by the traffic behav-

ior. The performance of both algorithms is sensitive to demand and topology,

suggesting that the optimal choice for deployment should be based on local pa-

rameters.

9 Directions for Future Research

This chapter studies optimal routing strategies for wireless mesh networks with at-

tention to traffic demand uncertainty over time and provable robustness. Two ap-

proaches are reviewed and discussed in this chapter. Here we outline several possi-

ble directions for future research.

• Traffic Modelling and Estimation. The predictive routing strategy is sensitive to

traffic dynamics and the prediction accuracy. To obtain a higher prediction accu-

racy, the future research needs to develop appropriate traffic models which can

be integratedwith networkoptimizationformulations.The key probleminvolved

is how to parameterize the traffic models in order to represent its structure with

Page 26

26 J. Wellons, L. Dai, B. Chang and Y. Xue

small number of parameter values that can be estimated from the data. Based on

the traffic model, traffic estimation needs to develop reliable estimation methods

that determine the values of the parameters that provide robust and high accurate

traffic estimate.

• To incorporate traffic uncertainty and dynamics, and integrate different traffic

models, future research should explore the full spectrum of research outlined

in Fig. 17 from two directions. One side of the spectrum starts with the fixed-

demand network optimization, where the traffic demand is known as a fixed

single-value scalar; then it extends to handle the scenarios where the traffic de-

mand is represented using a random variable with statistical distribution. The

other side of the spectrum starts with the oblivious optimization problem where

the traffic demand is completely unknown, where it can be refined to handle the

cases where the range of the traffic demand is known.

fixed demand

uncertain demand?

(statistical distribution)

no knowledge of demand?

(oblivious)

knowledge of ?

demand range

Fig. 17 Research Space for Route Optimization.

10 Conclusions

This chapter studies the optimal routing strategies for wireless mesh networks. Dif-

ferent from existing works which implicitly assume traffic demand as static and

known a priori, this chapter considers the traffic demanduncertainty.It presents two

approaches to address the traffic uncertainty in optimal mesh network routing: (1)

predictive routing which infers the traffic demand with maximum possibility based

in its history and optimizes the routing strategy based on the predicted traffic de-

mand and (2) oblivious routing which considers all the possible traffic demands

and selects the routing strategy where the worst-case network performancecould be

optimized. It also identifies the key factors that affect the performance of each rout-

ing strategy and provides guidelines towards the strategy selection in mesh network

routing under uncertain traffic demands.

11 Exercises

Answers in italics

1. Explain the factors that must be taken into account when deploying a Wireless

Mesh Network.

Many answers are possible, but common answers may include

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Wireless Mesh Network Routing Under Uncertain Demands 27

• hardware choice, number of supported channels

• deployment locations, interference and reflective obstacles

• protocol choice and configuration

2. What is the intuition behind the differing strengths of oblivious and predictive

routing?

Predictive Routing performs best when traffic history provides an accurate view

of future demands. If the traffic is perfectly predictable, it will converge on the

optimal routing. However, if it is not predictable, predictive routing makes no al-

lowance for errors and can suffer severe worst case behavior. By contrast, obliv-

ious routing performs with bounded performance regardless of the demands and

thus elasticity has no direct impact on its performance.

3. What affect does traffic aggregation at network endpoints have on the elasticity

of the demand?

Elasticity will be reduced because the sporadic and random bursts will tend to

cancel out. In fact, the variance in demand tends to the inverse square root of the

number of simultaneous clients.

4. Name some of the key factors that make traffic more predictable. Name some

that make it less predictable.

More predictable: Corporate access with regular hours Automated Web Queries

Less predictable: Streaming video and large downloads that heavily tax network

connections alternating with ordinary web browsing or inactivity.

5. Using Figure 12, how many neighbors would node 55 have if the transmission

range was 100m? Suppose it was 200m? In general, with randomly distributed

nodes, what is the asymptotic relationship begin degree and transmission range?

Your answer should be a simple polynomial, such as “linear”.

The only nodes that appear to be within 100m of node 55 are 51 and 22. Within

200m, there is 22, 21, 57, 18, 51, 42, and 60. In general, we should expect the

numberofnodeswithinrangetobeproportionaltothesquareofthetransmission

range.

6. Suppose nodes A, B, and C in a WMN aggregatetraffic as in the followingtable

of load demand.

Time

1 100 200

2 200 200

3 400 100

4 400 400

ABC

1

3

8

1

A) Which of nodes A, B and C would you say has the most erratic demand?

Which is least erratic? Formalize your intuition.

B) During which of the 3 time intervals shown does demand change the most?

Justify your answer.

This problem has several answers and illustrates the difficulty in quantifying de-

mandvariability. Taken in isolation,B has the largest absoluteshift from moment

to moment (500). However, C has the largest fractional change: 200 % + 167 %

Page 28

28J. Wellons, L. Dai, B. Chang and Y. Xue

+ 88 % = 455 %. By contrast, A has only 100 % + 100 % + 0 % = 200 % and B

has 0 % + 50 % + 300 % = 350 %.

7. Consider the graph topology shown below with the corresponding demand ma-

trix. Assume the rows are the demand sources and the columns are the destina-

tions.

D

??

A

??

??C

B

?????????

???

?

?

?

?

?

?

2

6

6

6

6

4

A B C D

A − 0 1 0

B 0 − 1 0

C 0 0 − 0

D 0 0 0 −

3

7

7

7

7

5

Fig. 18 A) A Simple Topology and B) Demands on this Topology

There are many ways to route this traffic. Calculate the shortest path routing and

also the routing that minimizes the maximal congestion.

Answer:

D

2/3

??

A

1/3

??

2/3??C

?

?

?

?

?

?

B

1/3

?????????

2/3

???

D

0

??

A

0

??

1

??C

B

0

?????????

1

?

???

?

?

?

?

?

Fig. 19 Optimal and Shortest Path routings

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