IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 32, NO. 2, FEBRUARY 2014 345

Sustainability Analysis and ResourceManagement for Wireless Mesh Networks with

Renewable Energy SuppliesLin X. Cai Member, IEEE, Yongkang Liu, Tom H. Luan Student Member, IEEE,

Xuemin (Sherman) Shen Fellow, IEEE, Jon W. Mark Life Fellow, IEEE, and H. Vincent Poor Fellow, IEEE

Abstract—There is a growing interest in the use of renewableenergy sources to power wireless networks in order to mitigatethe detrimental effects of conventional energy production orto enable deployment in off-grid locations. However, renewableenergy sources, such as solar and wind, are by nature unstablein their availability and capacity. The dynamics of energy supplyhence impose new challenges for network planning and resourcemanagement. In this paper, the sustainable performance of awireless mesh network powered by renewable energy sourcesis studied. To address the intermittently available capacity ofthe energy supply, adaptive resource management and admissioncontrol schemes are proposed. Specifically, the goal is to maximizethe energy sustainability of the network, or equivalently, tominimize the failure probability that the mesh access points(APs) deplete their energy and go out of service due to theunreliable energy supply. To this end, the energy buffer of a meshAP is modeled as a G/G/1(/N) queue with arbitrary patternsof energy charging and discharging. Diffusion approximation isapplied to analyze the transient evolution of the queue lengthand the energy depletion duration. Based on the analysis, anadaptive resource management scheme is proposed to balancetraffic loads across the mesh network according to the energyadequacy at different mesh APs. A distributed admission controlstrategy to guarantee high resource utilization and to improveenergy sustainability is presented. By considering the first andsecond order statistics of the energy charging and dischargingprocesses at each mesh AP, it is demonstrated that the proposedschemes outperform some existing state-of-the-art solutions.

Index Terms—Energy sustainability, resource management,wireless mesh networks, renewable energy supply.

I. INTRODUCTION

THE EXPLOSIVELY growing demand for ubiquitousbroadband wireless access has led to a significant in-

crease in energy consumption by wireless communicationnetworks. To counter this increase, future generations ofwireless networks are expected to make use of renewableenergy sources, e.g., wind, solar, tides, etc., to fulfill the ever-increasing user demand, while reducing the detrimental effects

Manuscript received April 13, 2012; revised October 12, 2012.L. X. Cai and H. V. Poor are with the Department of Electrical Engi-

neering, Princeton University, Princeton, NJ, 08544, USA (e-mail: linlin-cai@gmail.com; poor@princeton.edu).

Y. Liu, T. H. Luan, X. Shen, and J. W. Mark are with the De-partment of Electrical and Computer Engineering, University of Wa-terloo, Waterloo, ON N2L 3G1, Canada (e-mail: yongkang.liu.phd@gmail.com; hluan@bbcr.uwaterloo.ca; xshen@bbcr.uwaterloo.ca; jwmark@bbcr.uwaterloo.ca).

Digital Object Identifier 10.1109/JSAC.2014.141214.

of conventional energy production. However, unlike traditionalenergy supplied from the electricity grid, renewable energysources are intrinsically dynamic with unstable availability andtime varying capacity. For example, a wind turbine usuallyprovides intermittent power which depends on how windythe weather is. Although solar panels can provide relativelycontinuous power supply, the energy supply varies across thetime of a day and the season of the year, and is influenced byatmospheric conditions and geography. As a result, when re-newable energy is deployed to power wireless communicationnetworks, its dynamic and unreliable nature will affect theavailability and efficiency of communications, and thereforewill make energy-sustainable network design a necessity.

Improving energy efficiency has long been a fundamentalresearch issue in wireless communications, mainly becauseof the limited battery power of mobile terminals and/or theincreasing cost of the energy from the electricity grid. Intraditional systems powered by batteries, the energy is alimited resource but it is stable during the battery lifetime.The electricity grid generally provides continuous power ondemand with no stringent usage limit; however, this power isprimary generated from limited and non-sustainable resources,such as coal, natural gas, and petroleum. In contrast, renewableenergy sources are sustainable in the long term but are unstableand intermittently available in the short term. As a result, thefundamental design criterion and the main performance metrichave shifted from energy efficiency to energy sustainability ina network powered by renewable energy [1]. While many ex-isting works focus on energy efficiency, energy sustainabilityhas not been well explored and deserves further investigation.Thus motivated, we first develop a mathematical model tostudy the “energy sustainability” performance of wirelessdevices theoretically and, based on this analysis, we furtherdimension the resource management and admission controlstrategies to improve the sustainable network performanceunder an energy sustainability constraint.

The possibility and advantages of deploying a sustainableenergy powered wireless system are reported in [2]. In par-ticular, it is shown that solar or wind powered access points(APs) provide a cost-effective solution in wireless local areanetworks (WLANs), especially for APs installed in off-gridlocations. Since the publication of that study, resource man-agement for sustainable wireless networks has been studied inmultiple contexts. In [1] and [3], the AP placement problem

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346 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 32, NO. 2, FEBRUARY 2014

has been re-visited in the context of sustainable WLAN meshnetworks, in which a minimal number of APs with renewableenergy sources are deployed in an area such that the qualityof service (QoS) requirements of users can be fulfilled. In [4],by comparing sustainability performance of various routingalgorithms, it is shown that traditional routing strategieswithout considering energy charging capability achieve poorperformance; it is crucial to adapt routing decisions to thetime varying energy supplies. However, without an accurateanalytical model for energy supply dynamics, how to effec-tively make routing decisions in an energy sustainable wirelessnetwork remains an open issue. Moreover, most existing worksuse residual energy or the mean energy charging rate forresource allocation, by either assuming the energy chargingrate is known a priori or using an oversimplified model, e.g.,that the charging rate is uniformly distributed. In reality energycharging is known to be a complicated and dynamic processdue to the restricted energy charging capabilities of hardwareand diverse charging environments which are usually locationdependent. Coupled with the dynamic end host demands onmultimedia services, e.g., bursty media streaming and dataapplications, it is thus crucial to fully understand the impactof dynamic energy availability on resource management andtraffic routing at distributed nodes to provision satisfactory,robust, and sustainable performance of a renewable energypowered mesh network.

In this paper, we develop an analytical model to evaluatethe instantaneous volume of buffered energy at mesh APswith dynamic energy charging and discharging. By exploringthe transient behavior of the energy buffer, adaptive andoptimal energy resource management schemes are proposedto maximize the network-wide energy sustainability such thatthe probability of mesh APs depleting their energy and goingout of service is minimized. Our main contributions are three-fold:

• Analytical Model: A generic analytical framework tostudy the transient evolution of the energy buffer ispresented. Specifically, we model the energy buffer asa G/G/1/∞ and G/G/1/N queue with the generalenergy charging and discharging processes in infiniteand finite energy buffer cases, respectively. Based on thefirst two statistical moments, i.e., the mean and variance,of the energy charging and discharging intervals, weapply the diffusion approximation to obtain closed-formdistributions of the transient (energy) queue length andthe energy depletion duration.

• Adaptive Resource Management: Based on the developedanalytical framework, we propose an adaptive resourcemanagement scheme to assure the energy sustainabilityof the network. In the proposal, traffic flows are distribu-tively scheduled on multi-hop paths towards the minimalenergy depletion probability of mesh APs. The proposedscheme is adaptive to the residual energy level at meshAPs and the dynamic traffic demands of flows.

• Distributed Admission Control: A distributed admissioncontrol strategy is deployed at mesh APs to strike abalance between high resource utilization and energysustainability.

The remainder of the paper is organized as follows. Relatedworks are discussed in Section II. The system model is pre-sented in Section III and an analytical framework is developedto study the transient buffer evolution in Section IV. Based onthe buffer analysis, an adaptive resource management schemeis proposed in Section V. Simulation results are given inSection VI, followed by concluding remarks in Section VII.

II. RELATED WORK

Energy efficiency is a fundamental research issue in wire-less communication networks [5], [6], [7], [8]. An extensivebody of research has been devoted to resource managementfor energy efficient communication and networking, rangingfrom network capacity planning and topology control [9],[10], traffic scheduling and routing [11], [12], and adaptivesleep control of mobile devices [13], [14], to energy efficientcommunication and cooperation [15], [16], and optimal powermanagement [17], [18], etc. In these works, the energy supplyis typically considered to be a fixed yet limited resource.

With recent advances in energy technologies, sustainablewireless networks with renewable energy sources have beenemerging. The development of prototypes of sustainable sen-sors with solar energy harvesting capability was the first issueto be studied in this context. In [19] and [20], experimentalresults show that such prototypes can enable near-perpetualoperation of a sensor node. The solar/wind powered APhas been recognized as a cost-effective alternative to thetraditional AP in WLAN mesh networks, especially when afixed power supply is not available. Resource allocation andenergy management in sustainable mesh networks have alsobeen studied in multiple contexts. Sayegh et al. in [2] considera hybrid solar/wind powered WLAN mesh network and arguethat a hybrid solution provides the optimal cost configuration.Farbod and Todd in [21] describe a solar battery configurationmethodology based on the mean offered capacity profile, andpropose an outage control algorithm to improve the nodeoutage performance. Cai et al. in [1] study the dynamic char-acteristics of sustainable energy sources and elaborate on thefundamental design criterion for a sustainable mesh network.Under the green network paradigm incorporating renewableenergy, network deployment and resource management issueshave been re-visited. Different routing schemes have been ex-amined and compared in the context of sustainable networks,aiming to distribute the traffic loads evenly over the networkto improve the network sustainability. For example, it is shownin [4] that traditional routing strategies that do not considerthe variable nature of renewable energy supplies achieve poorperformance, and it is crucial to adapt the routing decisionsto the time varying power supply conditions in a sustainablenetwork environment. However, how to accurately captureand model the power conditions remains an open researchissue. Most existing works on sustainable communicationsuse the residual energy, a fixed energy charging rate, or anoversimplified energy charging model, e.g., that the chargingrate is uniformly distributed, for resource allocation. As theavailable energy is inherently dynamic due to variations inboth energy charging and discharging processes, it is essentialto characterize the variations in the analytical model of energyconditions. In addition, previous routing algorithms mainly

CAI et al.: SUSTAINABILITY ANALYSIS AND RESOURCE MANAGEMENT FOR WIRELESS MESH NETWORKS WITH RENEWABLE ENERGY SUPPLIES 347

focus on routing metric design and/or route discovery issues.To provide satisfactory sustainable network performance, bothload balancing over various paths and effective admissioncontrol should be incorporated into one resource managementframework.

To tackle this challenging issue, we model the energy bufferas a G/G/1/∞ and G/G/1/N queue in infinite and finite batterycapacity cases, respectively. Based on the first and secondorder statistics of the charging and discharging processes, weapply diffusion approximation to analyze the distributions ofthe queue length and the first passage time at which a node’senergy depletes. As an efficient and accurate approach forstudying transient behavior of queueing systems [22], [23],[24], diffusion approximations have been widely applied inadaptive buffer management [25]. To the best of our knowl-edge, this is the first work to apply a diffusion approximationto analyze the energy sustainability performance of networkdevices, and to use the derived closed-form distributions foradaptive resource management in a sustainable wireless meshnetwork.

III. SYSTEM MODEL

A. Energy Model

We consider a sustainable wireless mesh network as shownin Fig. 1. Each mesh AP is equipped with a battery thatcan be charged repeatedly via a renewable energy supply. LetRi(0) denote the initial battery energy of node i and Ai(t)denote the amount of energy charged over the time interval[t − 1, t] at node i. Note that due to differing environments,e.g., different solar radiation intensities or wind speeds atdifferent geographical locations, the energy charging capacityof each mesh AP varies uniquely over time. Hence, we modelthe energy charging process at a node as a continuous-timestochastic process with an arbitrary but stable distribution, andcount the charged energy units as the arrival events of thequeue. The mean and variance of the inter-charging intervalsare denoted by μa and va, respectively. μa and va can beestimated to be an exponentially weighted moving averageor other estimation approaches. The charged energy can bestored in the batteries. Denote by Ri(t) the residual energyof node i at time t, by Rmax

i the maximum energy storageor the battery capacity of node i, and by Rmin

i ≥ 0 theminimal residual energy level based on battery life and safetyconsiderations. In other words, the energy level of node i iswithin the range [Rmin

i , Rmaxi ]. Without loss of generality,

we simplify the model by considering 0 ≤ Ri(t) ≤ Ni, whereNi = Rmax

i − Rmini which reflects the maximum number of

energy units that a node can use. In the following, we studythe two scenarios of an infinite energy buffer capacity, i.e.,Ni → ∞ and a limited energy buffer capacity, i.e., Ni isfinite.

The energy consumption includes the energy used forreceiving a packet, processing it, and forwarding it en routeto the destination. Receiving and processing energy can beconsidered to be a constant, er, while the transmission energyshould be adjusted to ensure a desired bit error rate at thereceiver. As the signal energy attenuates over a wirelesschannel [26], [27], the energy path loss is usually characterized

by dni,j , where di,j is the distance between nodes i and j,and n is the path loss exponent. For a given signal to noiseratio (SNR) requirement, the minimal energy required fortransmitting one bit over link i ⇀ j should be proportional topath loss, i.e.,

eti,j ∝ dni,j . (1)

The total energy consumption of node i during the timeinterval [t− 1, t] is given by

Si(t) =∑j

∑k

pi,j,k(t)(er + eti,j ), (2)

where pi,j,k(t) is the traffic demand of flow k over link i ⇀ jduring [t − 1, t]. The residual energy of node i at time t canthus be given as

Ri(t) = min{max{Ri(t− 1)+Ai(t)−Si(t), 0}, Ni

}. (3)

We model the traffic arrivals at each AP as a Poisson process.The mean and variance of the energy inter-discharge intervalare denoted as μs and vs, respectively. The values of μs andvs can be estimated via standard estimation approaches.

B. Network Model

We consider a distributed wireless mesh network consistingof multiple stationary mesh APs with sustainable energysupplies, as shown in Fig. 1. Each mesh AP manages alocal WLAN with multiple wireless users, while serving asa mesh router that forwards traffic from other mesh APsor WLANs. One typical application scenario is a home oroffice WLAN in which an AP with sustainable energy supplycan be installed on the roof to harvest energy from theenvironment, e.g., sun, wind, etc. Users of the WLAN maycommunicate with local or remote users. Therefore, eachmesh AP not only bridges traffic inside its coverage, e.g.,within the house or an office, but also routes traffic fromother APs toward the destination over the mesh backbone.Both intra- and inter-WLAN traffic demands are consideredin the system model. Using orthogonal frequency divisionmultiplexing, mesh APs can select orthogonal channels fordata transmissions and receptions, and therefore there is nointerference among intra- and inter-WLAN communications.As the first step, we assume that the bandwidth is sufficientand that the sustainable energy supply is the performancebottleneck. This is acceptable because bandwidth is typicallyan ample resource with advanced wireless technologies whileenergy is more limited in the current setting. Future researchon joint bandwidth and energy management is required tostudy the impact of limited bandwidth on the network per-formance.

We construct a directed graph G = (V,E) for the meshbackbone network such that mesh APs are represented byvertices (v ∈ V ) and wireless links between APs are rep-resented by edges (e ∈ E). Each vertex is associated with aweight wv which represents the energy sustainability level ofthe node, i.e., how likely the node is to drop out by depletingall its energy to serve the traffic flows traversing it. Eachedge, e.g., edge i ⇀ j from node i to node j, is associatedwith a transmission energy, eti⇀j , which is a function of the

348 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 32, NO. 2, FEBRUARY 2014

Mesh Backbone ConnectionLocal Area NetworkMobile User Green AP

Fig. 1. WLAN Mesh Network

link distance. Notice that the energy consumed for relayingtraffic is not only determined by the transmission energy ofthe forwarding links, but also the traffic demands for the relay.

IV. TRANSIENT QUEUEING ANALYSIS OF ENERGYBUFFER

In this section, we develop a general framework for energybuffer analysis. Specifically, we apply a diffusion approxima-tion to study the transient evolution of an energy buffer inboth infinite and finite battery storage cases of a mesh AP.Initially, a mesh AP has energy R(0) = x0.1 Energy harvestedfrom the environment is stored in the energy buffer until thebattery capacity N is reached, and energy is discharged fromthe buffer while serving traffic demands. The evolution ofthe energy buffer is shown in Fig. 2. Due to the dynamicsin the charging and discharging process, a mesh AP maydeplete its energy buffer when R(t) reaches 0. In this case, theAP is considered temporarily unreachable and all associatedlinks are broken. Therefore, a fundamental problem arisingis how to properly distribute the traffic demands across thenetwork according to the energy levels of APs such that theprobability that a link is broken or an AP becomes unavailableis minimized.

A. Queue Model of Infinite Energy Buffer

We first consider the situation in which the battery capacityis sufficiently large such that all charged energy can be storedin the battery. Therefore, the energy buffer can be modeled asa G/G/1/∞ queue, with means and variances of inter-arrivals(inter-departures) denoted as μa (μs) and va (vs), respectively.

We use the diffusion approximation or Brownian motionapproximation approach to analyze the G/G/1/∞ queue [22].

1For notational simplicity, we neglect the node index in the notation inSection III, e.g., the subscript i in Ri(t).

Time t

EnergyBuffer

N

D

X0

Fig. 2. Evolution of an energy buffer

The discrete buffer size R(t) can be approximated by acontinuous process X(t) such that its incremental change overa small interval dt is normally distributed with mean βdt andvariance αdt,

dX(t) = X(t+ dt)−X(t) = βdt+G(t)√αdt, (4)

where G(t) is a white Gaussian process with zero mean andunit variance. β and α are the mean and variance of the changein X(t), respectively, which are also referred to as drift anddiffusion coefficients defined by

β = 1/μa − 1/μs (5)

andα = va/μ

3a + vs/μ

3s. (6)

Given the initial energy level at X(t = 0) = x0, theconditional probability density function (p.d.f.) of X(t), i.e.,

CAI et al.: SUSTAINABILITY ANALYSIS AND RESOURCE MANAGEMENT FOR WIRELESS MESH NETWORKS WITH RENEWABLE ENERGY SUPPLIES 349

the energy buffer size at time t (t > 0),

f(x, t;x0)dx = Pr(x ≤ X(t) < x+ dx|X(0) = x0), (7)

satisfies the forward diffusion equation

∂f(x, t;x0)

∂t=

α

2

∂2f(x, t;x0)

∂x2− β

∂f(x, t;x0)

∂x, (8)

under the boundary condition

X(t) ≥ 0, t > 0. (9)

By applying the method of images [24], we can use (8) and(9) to obtain

f(x, t;x0) =∂

∂x

{Φ

(x− x0 − βt√

αt

)(10)

− exp

{2βx

α

}Φ

(−x+ x0 + βt√

αt

)},

where Φ(x) is the standard normal integral defined as

Φ(x) =

∫ x

−∞

1√2π

exp(−1

2z2)dz. (11)

Denote by D(x0) the energy depletion duration given theinitial condition x0, i.e., the duration from X(0) = x0 untilthe moment when the AP depletes its energy,

D(x0) = inf(t ≥ 0|X(t) = 0, X(0) = x0). (12)

D(x0) is also referred to as the first passage time of X(t)from x0 > 0 to 0.

The density function of D can be obtained from the diffu-sion equation with the absorbing barrier at the origin, whichis equal to

fD(t;x0) = limx→0

{α

2

∂f(x, t;x0)

∂x− βf(x, t;x0)

}. (13)

Solving the partial differential equation (13), we obtain theconditional p.d.f. of D as

fD(t;x0) =x0√2παt3

exp

{− (x0 + βt)2

2αt

}. (14)

The moment generation function of D is given by

f∗D(s;x0) =

∫ ∞

0

e−stfD(t;x0)dt (15)

= exp{−x0

α(β +

√β2 + 2αs)

}.

Note that when s → 0, (15) gives the probability P(0;x0)of reaching the absorbing barrier starting from x0 [22],

P(0;x0) = lims→0

f∗D(s;x0)

=

{1, for β < 0

exp(− 2x0β

α

), otherwise. (16)

Eq. (16) indicates that the energy buffer depletes with proba-bility 1 when the energy replenish rate is lower than or equal tothe energy discharging rate, i.e., 1/μa ≤ 1/μs. On the otherhand, even if the mean energy charging rate is larger thanthe mean energy discharging rate, it is still possible that theAP depletes it energy due to the variance of energy charging

and discharging processes. In the case 1/μa > 1/μs, theenergy depletion probability is dependent on the initial energylevel x0, and the mean and variance of energy charging anddischarging processes.

Differentiation of (15) w.r.t. s gives the mean and varianceof D for β = 0,

E[D;x0] = x0/|β|, (17)V [D;x0] = x0α/|β|3. (18)

B. Queue Model of Finite Energy Buffer

In practice, a mesh AP will have a limited battery capacity.In this case, the energy buffer can be modeled as a G/G/1/Nqueue.

Similar to the infinite buffer case, given the initial conditionand the buffer size N , the conditional p.d.f. of the buffer sizesatisfies the forward diffusion equation,

∂f(x, t;x0, N)

∂t(19)

=α

2

∂2f(x, t;x0, N)

∂x2− β

∂f(x, t;x0, N)

∂x+μaP0(t;x0, N)δ(x− 1)

+μsPN (t;x0, N)δ(x−N + 1),

where P0(t) and PN (t) are the probability mass functions atthe boundaries x = 0 and x = N at time t, given the initialvalue x0, respectively:

P0(t;x0, N) = Pr[X(t) = 0|X(0) = x0] (20)PN (t;x0, N) = Pr[X(t) = N |X(0) = x0]. (21)

When X(t) reaches one of the boundaries x = 0 or x = N ,it stays there for a random interval, referred to as a holdingtime, with mean 1/μa and 1/μs, respectively. The probabilitymass functions P0(t;x0, N) and PN (t;x0, N) should alsosatisfy the following equations:

dP0(t;x0, N)

dt= −μaP0(t;x0, N) (22)

+ limx→0

[α

2

∂f(x, t;x0, N)

∂x− βf(x, t;x0, N)

]and

dPN (t;x0, N)

dt= −μNPN (t;x0, N) (23)

+ limx→N

[−α

2

∂f(x, t;x0, N)

∂x+ βf(x, t;x0, N)

],

subject to the initial condition

f(x, t;x0, N) = δ(x− x0) 0 < x < ∞, (24)

and boundary conditions

limx→0

f(x, t;x0, N) = 0 t > 0 (25)

limx→N

f(x, t;x0, N) = 0 t > 0. (26)

By applying a doubly infinite system of images [22], wecan obtain the transient solution of the probability function

f(x, t;x0, N) =1√2παt

∞∑n=−∞

(A−B) , t > 0, (27)

350 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 32, NO. 2, FEBRUARY 2014

where⎧⎨⎩

A = exp(

β2nNα − (x−x0−2nN−βt)2

2αt

),

B = exp(

β(−2x0−2nN)α − (x+x0+2nN−βt)2

2αt

).

Since∑

i Pi = 1, we have∫ N

0

f(x, t;x0, N)dx+ P0(t;x0, N) + PN (t;x0, N) = 1.

(28)

Similar to (13), the density function of the first passage timein the finite buffer case satisfies

fD(t;x0, N) = limx→0

[α

2

∂f(x, t;x0, N)

∂x− βf(x, t;x0, N)

].

(29)

We derive the Laplace transform of the density functionfD(t;x0, N) with a finite energy capacity N as

f∗D(s;x0, N) = exp

{−β

αx0

}sinh[C(N − x0)]−D

sinh(CN)−D(30)

where {C =

√2αs+β2

α ,D = μa

μa+s exp{β/α} sinh[C(N − 1)].

The inversion of f∗D(s;x0, N) can be approximated by

fD(t;x0, N) (31)

=1

2texp

{E

2

}(f∗D

(E

2t;x0, N

))

+1

texp

{E

2

} ∞∑k=1

(−1)k(f∗D

(E + 2kπi

2t;x0, N

)),

where E is the approximation error function such that the erroris bounded by exp{−E}

1−exp{−E} [28]. For instance, by setting E =

5 ln(10), the approximation error is smaller than 0.99 · 10−5.

Differentiation of (30) w.r.t. s gives the first moment of Das

E[D;x0, N ] (32)

=

{−x0

β + (μa +1β )

exp{− 2βα (N−x0)}−exp{− 2β

α N}1−exp{− 2β

α } , β = 0,

x0(μa +2N−x0−1

α ), β = 0.

Solving Eqs. (19)-(23) at the stationary state whenlimt→∞ P0(t;x0, N) = P0, limt→∞ PN (t;x0, N) = PN , andlimt→∞ f(x, t;x0, N) = f(x;x0, N),

α

2

∂2f(x, t;x0, N)

∂x2− β

∂f(x, t;x0, N)

∂x(33)

= −μ0P0δ(x− 1)− μNPN δ(x−N + 1),

limx→0

[α

2

∂p(x, t;x0, N)

∂x− βp(x, t;x0, N)

]= μ0P0, (34)

limx→N

[α

2

∂p(x, t;x0, N)

∂x− βp(x, t;x0, N)

]= −μNPN ,

(35)

we can obtain the steady state probability function of the

energy buffer length [29]

f(x;x0, N) (36)

=

⎧⎪⎨⎪⎩

−μaP0

β [1− erx], 0 ≤ x ≤ 1

−μaP0

β [e−r − 1]erx, 1 ≤ x ≤ N − 1

−μaP0

β [er(x−N) − 1]er(N−1), N − 1 ≤ x ≤ N,

where r = (2β)/α.

The probability that an AP depletes its energy in the finitebuffer case is given by

P(0;x0, N) (37)

=

{(1 + μs

μaer(N−1) + μs

μa−μs[1− er(N−1)])−1, β = 0,

12 (1 +

N−1v2a/μ

2a+v2

s/μ2s)−1, β = 0.

Eq. (37) indicates that in the finite energy buffer case, due tothe variance of the charging and discharging processes, theenergy buffer is depleted with a certain probability, whichis jointly determined by the first and second moments ofthe sojourn times at the boundary conditions, regardless ofwhether β ≥ 0 or β < 0. The depletion probability decreaseswith increased battery capacity N or energy charging rate. Alarger variance in either charging or discharging results in ahigher energy depletion probability P(0;x0, N).

V. ADAPTIVE RESOURCE MANAGEMENT

In this section, we present an energy-aware adaptive re-source management framework based on the transient energybuffers of mesh APs. To improve network sustainability, weaim to minimize the probability that mesh APs deplete theirenergy in serving traffic demands. Based on the transientenergy level, energy charging capability, and existing trafficdemands at each AP, a path selection metric is designedto distribute traffic flows over diverse relay paths acrossthe network. A distributed admission control strategy is alsopresented to further guarantee the energy sustainability of thenetwork.

A. Relay Path Selection

In a WLAN mesh network depicted in Fig. 1, users’traffic may be relayed over multiple hops until reaching thedestination. To ensure the energy sustainability and networkconnectivity, traffic flows should be scheduled over relaypaths such that the APs along the paths have the minimalprobability of being out of service, i.e., depleting their energyand becoming temporarily unavailable. To track the dynamicsof the traffic demands and charging capability, the schedulingshould be updated periodically and when new events occur(e.g., a new flow joins in the network or traffic demands ofexisting flows change).

Based on the analysis in Section IV, the relay path selectionperforms as follows: a source user first broadcasts a requestthat includes the destination and the estimated first and secondorder statistics of the energy consumption of the traffic flow.When the destination AP receives the request, it first calcu-lates the probability of energy depletion, P(0;x0) in (16) orP(0;x0, N) in (37), based on its current energy level and the

CAI et al.: SUSTAINABILITY ANALYSIS AND RESOURCE MANAGEMENT FOR WIRELESS MESH NETWORKS WITH RENEWABLE ENERGY SUPPLIES 351

accumulated traffic demands. The destination AP updates itsweight as

wv = P(0;x0) or P(0;x0, N). (38)

The destination then attaches its weight in the reply messageback to the source user. Upon receiving the reply message,each mesh AP also updates its weight accordingly. By com-puting its own P(0;x0) with the accumulated load energyconsumption over each link, the source AP can make adecision about the data forwarding path. For example, thesource AP may select the path with the minimal sum ofthe energy depletion probability (MEDP) along the path, i.e.,the path with Min

∑wv , to ensure the overall sustainable

performance of the network; or the AP may prefer to select thepath along which the maximum value of wv is minimized, i.e.,the path with Min Max wv , to ensure that the least sustainableAP can sustain the traffic demands longer. The process ofweight updates in these cases is the same as the shortest pathalgorithm, with the metric of the number of hops replaced bythe path selection metric, i.e.,

∑wv or Max wv .

Notice that in the infinite energy capacity case, P(0;x0)may increase to 1 when β ≤ 0, which implies that the AP willeventually deplete its energy by relaying this flow. Therefore,to differentiate their weights for path selection, the AP needsto further evaluate whether its current energy level can sustainthe flow demand within a finite duration. Denote the survivaltime of a traffic flow as T , i.e., the flow is expected to survivein the network in the following T time slots. The AP thencomputes the probability that it depletes its energy before Texpires, which is given by

FD(T ;x0) = Pr(D ≤ T ) =

∫ T

0

fD(t;x0)dt. (39)

In the infinite buffer case, the destination AP updates itsweight according to the flow request as

wv = P(0;x0) + I(β ≤ 0)FD(T ;x0), (40)

where the indicator I(·) equals to 1 if condition (·) is true and0 otherwise. In a network of queues, the buffer of the relayingAP may absorb the traffic variance in some degree and theoutput traffic characteristics may vary. If the estimated trafficdemand changes during time duration T , APs should updatethe traffic parameters in the remaining time of duration T , andrepeat the aforementioned path selection process. A mesh APmay also need to retransmit a packet after a random period ifits next hop mesh AP is currently out of service, which maychange the energy consumption statistics of the ongoing flow.In the case that P(0;x0) is large, energy consumption statisticsmay vary hop by hop, and mesh APs need to update the energyconsumption statistics and recalculate P(0;x0). However, byminimizing the energy depletion probability and ensuring asufficiently large depletion duration, the probability that anAP is out of service is reduced and becomes negligible.

B. Admission Control

Due to the limited network resources in a wireless network,admission control plays a critical role in providing satisfactoryquality of service to the existing users. Generally, admission

control tackles the trade-off between resource utilization andquality of service provisioning. For instance, more usersadmitted to the network can exploit more network resources toachieve a higher network throughput, but they may also use upthe network resources faster, e.g., the residual energy of meshAPs depletes quickly, and some APs may be out of servicewhich causes long service delays and jitter. Therefore, aneffective admission control strategy is necessary to assure highresource utilization under the energy sustainability constraintand guaranteed user performance.

We provision guaranteed service to admitted users by en-suring a sufficiently large energy depletion duration, e.g., thatD is larger than the longest survival time of traffic flows D̂,in a stochastic manner as

Pr(D ≤ D̂) =

∫ D̂

0

fD(t;x0)dt < ε, (41)

where the parameter 0 < ε � 1 is an adjustable parameterwhich reflects the energy sustainability level. A smaller εimplies a stricter energy sustainability constraint for admittinga new flow. As such, according to the estimated flow statisticsin the request, each mesh AP verifies its availability to relay,i.e., a mesh AP responds only when its energy sustainabilitycondition satisfies (41). If the source AP cannot establish avalid relay path to the destination from the received responsemessages, indicating that one or more APs’ energy supplycannot sustain the traffic demands of the flow, the source APwill reject the flow request from the end user. By upper bound-ing the energy depletion probability, satisfactory sustainablenetwork performance can be achieved.

VI. SIMULATION RESULTS

In this section, we validate the energy buffer analysis andevaluate the performance of the proposed resource manage-ment schemes via extensive simulations, based on a discretetime event-driven simulator coded in C++.

A. Simulation Setup

In this section, we simulate a WLAN mesh network withten APs and multiple mobile users, as depicted in Fig. 1.The distances between adjacent APs are randomly selectedin [R0, 3R0] where R0 is the communication radius of anAP. Ten groups of users are randomly distributed in thecoverage of APs. The energy charging intervals of an APare randomly selected from ta = {1, 2, 3, 4} (in unit of timeslots) with a given probability distribution pta . For example,if pta = {0.3, 0.3, 0.2, 0.2}, the mean and variance of theduration of the charging intervals are μa = 2.3 and va = 1.21,respectively. The energy discharging of an AP depends onthe traffic load demands of users. In each simulation run, anumber of flows are sequentially generated between any twousers in the mesh network. Unless otherwise specified, theinter-packet arrivals of a flow are exponentially distributedwith mean μs = 14. The energy consumption per packet atan AP includes both reception and transmission. For intra-WLAN traffic, i.e., the transmission between the AP and anyuser within its coverage radius R0, the energy consumptionof a packet is set as a constant e0 = 1 energy unit. For

352 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 32, NO. 2, FEBRUARY 2014

0 20 40 60 80 1000

0.2

0.4

0.6

0.8

1

Initial energy, x0 (energy units)

Ene

rgy

depl

etio

n pr

obab

ility

, P0

anasim

Fig. 3. P(0; x0) (Infinite buffer, µa = 2.3, va = 1.21, µs = 2.33,vs = 5.44)

0 5 10 15 20 25 300

0.1

0.2

0.3

0.4

0.5

Energy buffer capacity, N (energy units)

Ene

rgy

depl

etio

n pr

obab

ility

, P0

anasim

Fig. 4. P(0; x0, N) (Finite buffer, µa = 2.3, va = 1.21, µs = 2.33,vs = 5.44)

inter-WLAN traffic, i.e., the transmissions between meshAPs, the energy consumption of one packet is dependenton the link distance R between the APs, which is given bymax{1, (R/R0)

n}e0 where n is selected from {1, 2, 3, 4}. Weignore the energy consumption due to the signaling exchange,which is relatively negligible compared to the traffic demandsin the mesh backbone. We repeat each experiment 10 times,and each experiment contains 1000 runs with different randomseeds. The average results are calculated with 95% confidence.

B. Energy Buffer

We first examine the energy depletion probability in bothinfinite and finite buffer cases. To study the relationshipbetween the charging capability and traffic load consumption,we gradually add six flows to transmit over a link of distanceR0, and collect the buffer depletion statistics. For six flows,the energy arrival rate is larger than the departure rate, andwe have β > 0. To illustrate the buffer evolution, these flowsare not associated with a survival time. To compute P (0;x0)in the simulation, we collect the number of runs on whichthe energy buffer of node A reaches 0 when the simulationruns 6000 time slots, and divide it by the total number runsof 1000 and plot the results in Fig. 3. It can be seen fromFig. 3 that P(0;x0) decreases with the initial energy x0. Aswe only conduct the simulation with limited duration (6000time slots), the simulation results are conservative and thusslightly lower than the analytical results (which converge astime goes to infinity). In the finite buffer case, it can be seenin Fig. 4 that the energy depletion probability decreases whenthe energy buffer capacity N increases.

Fig. 5 plots the cumulative distribution function (CDF) ofthe energy depletion duration for the infinite energy buffercases. To effectively obtain the CDF of D, we add six moreflows to transmit over a link distance R0 and calculate thefirst passage time. Notice that for 12 flows, the energy arrivalrate is lower than the departure rate indicating that we havea stable energy buffer with β < 0 and the energy buffer willeventually deplete. As shown in Fig. 5, for a smaller x0, anAP is more likely to deplete its energy in the near future, andthus the CDF curve shifts to the left. Fig. 6 shows the CDFof the energy depletion duration for the finite energy buffer

cases. The initial energy buffer is full, i.e., x0 = N . It canbe seen that with a large energy buffer capacity and initialbuffer size, an AP is less likely to deplete its energy soon,and therefore the CDF shifts to the right.

C. Sustainable Network Performance

In this section, we evaluate the sustainable network perfor-mance in terms of the network lifetime, which is defined asthe maximal duration that all APs are available until one of theAPs depletes its energy. We consider a heterogeneous networkwhere green APs deployed at different locations have diversecharging capabilities by selecting different probability vectorspta for different APs.

We compare the proposed MEDP using the path selectionmetric of

∑wv and Max wv with two other schemes, namely,

the minimum energy (ME) scheme [30], and the minimumpath recovery time (MPRT) scheme [4]. In ME, a relay pathis selected such that the total energy consumed along the pathis minimized. Unlike ME, the MPRT algorithm selects thepath with the minimum cumulative recovery time such thatthe total consumed energy can be recovered in the shortestduration. Thus, MPRT is more likely to select a path with ahigher charging rate.

The three path selection schemes are compared in Fig. 7 andFig. 8. Without considering the energy charging capability inthe path selection, the sustainable performance of ME is muchlower than those of MPRT and MEDP. The proposed MEDPoutperforms MPRT since MPRT considers only the chargingcapability of mesh APs, neglecting the traffic demands andvariations in both charging and discharging processes. Forthe proposed MEDP, the use of the metric

∑wv favors the

overall network sustainability while that of Max wv ensuresthe worst case sustainability performance. There is no obviousdifference for the use of the two metrics in the MEDP. Itcan also be observed that the network lifetime increases withthe initial energy buffer x0 in Fig. 7 and the energy buffercapacity N in Fig. 8. By studying the transient queue lengthdistribution for a given initial buffer x0 or the buffer capacityN , our proposed MEDP scheme can significantly extend thenetwork lifetime in both finite buffer and infinite buffer cases.

CAI et al.: SUSTAINABILITY ANALYSIS AND RESOURCE MANAGEMENT FOR WIRELESS MESH NETWORKS WITH RENEWABLE ENERGY SUPPLIES 353

0 50 100 150 200 250 3000

0.2

0.4

0.6

0.8

1

Energy depletion duration, D (time units)

CD

F

ana, x0 = 10

sim, x0 = 10

ana, x0 = 20

sim, x0 = 20

ana, x0 = 40

sim, x0 = 40

Fig. 5. CDF of D (Infinite buffer, µa = 2.3, va = 1.21,µs = 1.16,vs = 1.36, )

0 50 100 150 2000

0.2

0.4

0.6

0.8

1

Energy depletion duration, D (time units)

CD

F

ana: N = 5sim: N = 5ana: N = 10sim: N = 10

Fig. 6. CDF of D (Finite buffer, µa = 2.3, va = 1.21, µs = 2.33,vs = 5.44)

4 5 6 7 8 9 100

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

Number of flows

Net

wor

k lif

e tim

e (t

ime

slot

s)

MEDP(sum), x0 = 20

MEDP(sum), x0 = 100

MEDP(MinMax), x0 = 20

MEDP(MinMax), x0 = 100

ME, x0 = 20

ME, x0 = 100

MPRT, x0 = 20

MPRT, x0 = 100

Fig. 7. Path selection comparison (Infinite buffer)

4 5 6 7 8 9 100

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

Number of flows

Net

wor

k lif

e tim

e (t

ime

slot

s)

MEDP(sum), N = 20MEDP(sum), N = 100MEDP(MinMax), N = 20MEDP(MinMax), N = 100ME, N = 20ME, N = 100MPRT, N = 20MPRT, N = 100

Fig. 8. Path selection comparison (Finite buffer)

10 15 20 25 30 35 400

1000

2000

3000

4000

Number of flows

Net

wor

k lif

e tim

e (t

ime

slot

s)

No CACCAC, ε = 0.1CAC, ε = 0.2CAC, ε = 0.3

Fig. 9. Network life time w/wo CAC (Infinite buffer)

0 10 20 30 400

2

4

6

8

10

Number of flows

Num

ber

of a

dmitt

ed fl

ows

ε = 0.1ε = 0.2ε = 0.3

Fig. 10. Number of admitted flows (Infinite buffer)

Fig. 9 shows the comparison of the network lifetime withand without CAC schemes applied. We use the metric of

∑wv

for MEDP path selection. With more flows joining the net-work, the increased traffic loads degrade the network lifetimesignificantly when no CAC is applied. In this simulation, thefirst fix flows are always admitted to the network, but some

following requests are rejected due to the fact that the energyconditions of some APs cannot sustain their traffic demands.The number of admitted flows under different values of ε isshown in Fig. 10. A smaller ε represents a stricter admissioncondition and thus limits the number of admitted flows andthe network throughput. It is seen that by increasing ε from

354 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 32, NO. 2, FEBRUARY 2014

0.1 to 0.2, one more flow can be admitted. However, addingone more flow is likely to drain the energy of some APs andreduce the network lifetime significantly, as shown in Fig. 9.Therefore, with a smaller ε, each AP is less likely to depleteits energy and be out of service, which improves the overallnetwork connectivity and service provisioning to the existingflows.

VII. CONCLUSION

In this paper, we have developed a generic analytical modelto study the energy sustainability of mesh APs poweredby renewable energy sources, by characterizing the transientevolution of an energy buffer of a mesh AP. Based on theclosed-form solutions of energy buffer analysis, i.e., the energydepletion probability and energy depletion duration, we havefurther proposed an adaptive resource management frameworkfor relay path selection and admission control. By mitigatingthe energy depletion probability and ensuring a large energydepletion duration of mesh APs, the sustainable networkperformance can be significantly improved.

For our future work, we plan to consider bandwidth con-straints in this framework and jointly design the bandwidthallocation and energy management in a sustainable energypowered mesh network. We also plan to investigate how onecan deploy a minimal number of mesh APs to ensure that thecharged energy can sustain the traffic demands of users willalso be under investigation.

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Lin X. Cai received the M.A.Sc. and Ph.D. degreesin electrical and computer engineering from theUniversity of Waterloo, Ontario, Canada, in 2005and 2010, respectively. She was working as a post-doctoral research fellow in electrical engineeringdepartment at Princeton University in 2011. Cur-rently, she is a senior engineer at the US wirelessR&D center in Huawei Technologies Inc. Her re-search interests include green communication andnetworking, resource management for broadbandmultimedia networks, and cross-layer optimization

and QoS provisioning.

CAI et al.: SUSTAINABILITY ANALYSIS AND RESOURCE MANAGEMENT FOR WIRELESS MESH NETWORKS WITH RENEWABLE ENERGY SUPPLIES 355

Yongkang Liu is working toward a Ph.D. degreewith the Department of Electrical and ComputerEngineering, University of Waterloo, Canada. Heis currently a research assistant with the Broad-band Communications Research (BBCR) Group,University of Waterloo. He received the Best PaperAward from IEEE Global Communications Confer-ence (Globecom) 2011, Houston, USA. His generalresearch interests include protocol analysis and re-source management in wireless communications andnetworking, with special interest in spectrum and

energy efficient wireless communication networks.

Tom H. Luan received the B.E. degree in Xi’anJiaotong University, China in 2004, the M.Phil.degree in the Hong Kong University of Science andTechnology, Kowloon, Hong Kong in 2007, and thePh.D. degree at the University of Waterloo, ON,Canada in 2012. His current research interests focuson wired and wireless multimedia streaming, QoSrouting in multihop wireless networks, peer-to-peerstreaming and vehicular network design.

Xuemin (Sherman) Shen (IEEE M’97-SM’02-F’09) received the B.Sc.(1982) degree from DalianMaritime University (China) and the M.Sc. (1987)and Ph.D. degrees (1990) from Rutgers University,New Jersey (USA), all in electrical engineering.

He is a Professor and University Research Chair,Department of Electrical and Computer Engineering,University of Waterloo, Canada. He was the Asso-ciate Chair for Graduate Studies from 2004 to 2008.Dr. Shen’s research focuses on resource managementin interconnected wireless/wired networks, wireless

network security, wireless body area networks, vehicular ad hoc and sensornetworks. He is a co-author/editor of six books, and has published manypapers and book chapters in wireless communications and networks, controland filtering. Dr. Shen served as the Technical Program Committee Chair forIEEE VTC’10 Fall, the Symposia Chair for IEEE ICC’10, the Tutorial Chairfor IEEE VTC’11 Spring and IEEE ICC’08, the Technical Program Commit-tee Chair for IEEE Globecom’07, the General Co-Chair for Chinacom’07 andQShine’06, the Chair for IEEE Communications Society Technical Committeeon Wireless Communications, and P2P Communications and Networking.He also serves/served as the Editor-in-Chief for IEEE Network, Peer-to-PeerNetworking and Application, and IET Communications; a Founding AreaEditor for IEEE Trans. Wireless Communications; an Associate Editor forIEEE Trans. Vehicular Technology, Computer Networks, and ACM/WirelessNetworks; and the Guest Editor for IEEE JSAC, IEEE Wireless Communi-cations, IEEE Communications Magazine, and ACM Mobile Networks andApplications.

Dr. Shen is a registered Professional Engineer of Ontario, Canada, anIEEE Fellow, a Fellow of the Canadian Academy of Engineering, a Fellowof Engineering Institute of Canada, and a Distinguished Lecturer of IEEEVehicular Technology Society and Communications Society.

Jon W. Mark (M’62-SM’80-F’88-LF’03) receivedthe B.A.Sc. degree from the University of Torontoin 1962, and the M.Eng. and Ph.D. degrees fromMcMaster University in 1968 and 1970, respectively,all in electrical engineering. In September 1970 hejoined the Department of Electrical and ComputerEngineering, University of Waterloo, Waterloo, On-tario, where he is currently a Distinguished ProfessorEmeritus. He served as the Department Chairmanduring the period July 1984-June 1990. In 1996 heestablished the Centre for Wireless Communications

(CWC) at the University of Waterloo and is currently serving as its foundingDirector.

His current research interests are in broadband wireless communications,resource and mobility management, and cross domain interworking. Herecently co-authored two texts, Wireless Communications and Networking,Prentice-Hall 2003 and Multimedia Services in Wireless Internet, John Wiley& Sons, 2009. He also co-authored a book entitled Wireless BroadbandNetworks, John Wiley & Sons, 2009 and co-edited The Best of the Best:Fifty Years of Communications and Networking Research, the Institute ofElectrical and Electronics Engineers, Inc. and John Wiley & Sons, 2007.Dr. Mark is a Life Fellow of IEEE and a Fellow of Canadian Academy ofEngineering. He is the recipient of the 2000 Canadian Award for Telecommu-nications Research and the 2000 Award of Merit of the Education Foundationof the Federation of Chinese Canadian Professionals, an editor of IEEETrans. Communications (1983-1990), a member of the Inter-Society SteeringCommittee of the IEEE/ACM Transactions on Networking (1992-2003), amember of the IEEE Communications Society Awards Committee (1995-1998), an editor of Wireless Networks (1993-2004), and an associate editorof Telecommunication Systems (1994-2004). Since 2008, he has been amember of the Advisory Board, John Wiley series on Advanced Texts inCommunications and Networking.

H. Vincent Poor (IEEE S’72-M’77-SM’82-F’87)received the Ph.D. degree in EECS from PrincetonUniversity in 1977. From 1977 until 1990, he wason the faculty of the University of Illinois at Urbana-Champaign. Since 1990 he has been on the facultyat Princeton, where he is the Michael Henry StraterUniversity Professor of Electrical Engineering andDean of the School of Engineering and AppliedScience. Dr. Poor’s research interests are in the areasof stochastic analysis, statistical signal processingand information theory, and their applications in

wireless networks and related fields including social networks and smart grid.Among his publications in these areas are the recent books Smart GridCommunications and Networking (Cambridge University Press, 2012) andPrinciples of Cognitive Radio (Cambridge University Press, 2013).

Dr. Poor is a member of the National Academy of Engineering and theNational Academy of Sciences, a Fellow of the American Academy ofArts and Sciences, and an International Fellow of the Royal Academy ofEngineering (U. K). He is also a Fellow of the Institute of MathematicalStatistics, the Optical Society of America, and other organizations. In 1990,he served as President of the IEEE Information Theory Society, and in2004-07 he served as the Editor-in-Chief of the IEEE Trans. InformationTheory. He received a Guggenheim Fellowship in 2002, the IEEE EducationMedal in 2005, and the Marconi and Armstrong Awards of the IEEECommunications Society in 2007 and 2009, respectively. Recent recognitionof his work includes the 2010 IET Ambrose Fleming Medal for Achievementin Communications, the 2011 IEEE Eric E. Sumner Award, and honorarydoctorates from Aalborg University, the Hong Kong University of Scienceand Technology, and the University of Edinburgh.