Abstract—Providing high data rates with minimum energyconsumption is a crucial challenge for next generation wirelessnetworks. There are few papers in the literature which combinethese two issues. This paper focuses on multi-hop wireless meshnetworks using a MAC layer based on Spatial Time Division Mul-tiple Access (S-TDMA). We develop an optimization frameworkbased on linear programming to study the relationship betweenthroughput and energy consumption. Our contributions are two-fold. First, we formulate and solve, using column generation, a newMILP to compute offline energy-throughput tradeoff curve. Weuse a physical interference model where the nodes can performcontinuous power control and can use a discrete set of data rates.Second, we highlight network engineering insights. We show, vianumerical results, that power control and multi-rate functional-ities allow optimal throughput to be reached, with lower energyconsumption, using a mix of single hop and multi-hop routes.
Index Terms—Mesh networks, throughput, energy consump-tion, scheduling, S-TDMA, energy-capacity tradeoff.
I. INTRODUCTION
PROVIDING users with high data rates, irrespective of theirlocation, is a challenge for next generation cellular net-
works, like 3 GPP LTE-Advanced and WIMAX. In this paper,we consider a managed wireless mesh network (WMN) orga-nized in a tiered architecture: i) clients are connected to MeshRouters (MR) and ii) a multi-hop wireless backhaul topologyinterconnects the MRs with the core network (Fig. 1). TheMRs aggregate the uplink traffic generated by mobile clientsand forward it through multi-hop communications to dedicatedMRs, which are denoted gateways that bridge the backhaulnetwork to the core network. Similarly, downlink traffic goesfrom the gateways to the MRs, then to the clients. We assumethat mobile-to-MR and MR-to-MR traffic use independent fre-quencies. This work examines the backhaul network and doesnot take into account the users’ requests, rather, their flowsaggregated by the MRs. Optimizing the capacity of multi-hopwireless networks, defined as the maximum achievable total
Manuscript received May 7, 2013; revised October 10, 2013, March 20,2014, and July 14, 2014; accepted October 14, 2014. Date of publicationOctober 24, 2014; date of current version February 6, 2015. The associateeditor coordinating the review of this paper and approving it for publication wasG. Xing.
C. Rosenberg is with the Department of Electrical and Computer Engineer-ing, University of Waterloo, Waterloo, ON N2L 3G1, Canada (e-mail: [email protected]).
Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TWC.2014.2364815
Fig. 1. Wireless mesh network architecture: mesh routers collect the trafficfrom clients (mobile or static) and forward it to the core network.
throughput in the network topology under a fairness criteria, hasbeen a focus of research since the seminal work of Gupta andKumar [1]. In addition, minimizing the energy expenditure andelectromagnetic pollution of such infrastructures are also a so-cioeconomic challenge [2], [3]. Much work in the literature hasstudied how to maximize the capacity or how to minimize theenergy consumption; however the work was done under strongassumptions and tradeoffs between achievable throughputs andenergy have received very little attention.
The first contribution of this work is to develop a flexi-ble optimization framework, based on linear programming, tostudy multi-hop mesh networks. Several similar optimizationtools have been proposed in the literature [4]–[6]. The maincontribution of this framework is the modeling of continuouspower control which provides fine-tuning of transmit power.The following features are also added by our work:
1) The routing is formulated as an edge-path multi-commodity flow. The routing, scheduling, and powerallocation problems are jointly solved by a column gener-ation algorithm. By computing a restricted set of decisionvariables, this algorithm solves reasonable size instanceswith a detailed modeling of the links.
2) The modeling of links relies on two concepts. Logicallinks efficiently represent routing over origin-destinationpairs. The physical link, described by the parameters ofthe radio transmission, is used for physical layer issues.This combination of link models allows us to have atractable formulation while using a detailed Signal-to-Noise-and-Interference-Ratio (SINR) interference model.
This framework is used to compute, offline, an optimalsystem setting of the backhaul network to minimize the energyconsumption (resp. to maximize the capacity) under somenetwork capacity (resp. energy consumption) requirements.
1132 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 14, NO. 2, FEBRUARY 2015
A system setting corresponds to configuration parameters foroperating the backhaul network, such as routing paths andscheduling, including the transmission power and rate assignedto each transmission. The impact of these mechanisms on theperformances of the network, as well as the energy-throughputtradeoffs, are investigated in depth.
Our second contribution is to provide practical engineeringinsights into WMN.
Our numerical results highlight that:• Combining continuous power control and multi-rate func-
tionalities allow the optimal achievable throughput to bereached with significantly lower energy consumption; insuch tradeoffs some nodes actually use several combina-tions of power and rate at different times.
• The ratio of uplink over downlink traffic demands doesnot have a significant impact on the network capacity andenergy consumption tradeoffs.
• In the case of fixed transmission power, single-hop com-munications are more energy efficient than multi-hopones; in the case of continuous power control, it is theopposite.
• The clique area around the gateway plays a critical role inthe energy-throughput tradeoff. The predominance of theclique in the capacity determination of a WMN has alreadybeen discussed in the literature. We obtain similar resultsconcerning the energy-throughput tradeoff.
The rest of the paper is organized as follows. Section IIreviews related work. Section III gives the problem statementand the network model. Then, in Section IV, we present ourframework based on linear programming and column genera-tion. Section V studies the energy-capacity tradeoff and demon-strates the benefits of continuous power control. In Section VI,we provide practical engineering insights into WMN. Finally,we conclude the paper in Section VII.
II. RELATED WORK
There exists a vast amount of literature devoted to improvingthe capacity of WMN and to minimizing energy consumption,even if these two areas are considered separately. To increasethe throughput provided to nodes, several studies investigatedTDMA scheduling techniques, i.e., to identify sets of links thatcan be simultaneously activated [4], [7]. Molle et al. [4] studythe problem of routing and scheduling in IEEE 802.11 basednetworks. An optimization framework is provided for deter-mining optimal routing and scheduling needed by the traffic inthe network, considering a binary interference model and fixedtransmission power. In a practical system, transmission poweris an important tunable parameter to provide reliable and energyefficient communications: higher transmission power increasesthe SINR at the receiver to enable successful reception on alink, while lower transmission power mitigates interferences toother simultaneously utilized links. The joint problem of powercontrol and scheduling link transmissions in wireless networksto optimize performance objectives (throughput, delay, energy)has received much attention in recent years [5], [8]–[10]. In[5], a joint scheduling, routing, and power control strategy isproposed. The authors develop a computational tool using col-
umn generation to maximize the minimum throughput amongall flows. They highlight the usefulness of power control on theperformance of multi-hop wireless networks. In this work, thepower control is restricted to a small set of power levels. In[8], the problem of finding a minimum-length schedule thatsatisfies a set of specified traffic demands is addressed. It isshown that power control improves the spatial reuse, whichleads to further improvements on the schedule length, comparedto a fixed transmission power. Because scheduling with powercontrol using a SINR model is NP-hard [7], [11], several papershave proposed heuristic algorithms to minimize the schedulelength with and without power control [7], [12].
The optimization of energy consumption has also beenextensively addressed in the literature. Typically, the energyexpenditure in a node is linear with the transmission power[13]. From an energy efficiency standpoint, the most effectivesolution is to put the wireless nodes in sleep mode [14].To produce an effective energy-efficient network, [15] pro-posed an optimization framework which allows for jointlycomputing a planning and energy management solution forWMN. The authors showed that the highest energy savings areachieved when network planning and management are handledsimultaneously.
To the best of our knowledge, only a few papers investigatedcapacity and energy consumption jointly for WMN. Gorce et al.[16] studied energy, latency, and capacity tradeoff existing inmulti-hop ad-hoc wireless networks. The authors assume alinear topology with a simple energy model. They proposedan analytical study that does not take into account a realisticinterference model. The relation between energy minimiza-tion and throughput maximization for a 802.11 WLAN isanalyzed in [17]. In [6], an optimization problems to studythe max-min node lifetime and the max-min throughput of amulti-hop wireless network is formulated. The authors showedthat the optimal tradeoffs between throughput and lifetimeare usually not obtained at the minimum power that enablesnetwork connectivity. A multi-criteria optimization approachis proposed in [18] to study the relationship between energyconsumption and throughput of multi-hop wireless networks.The authors tried to characterize and compute the Paretofront between these two issues using simple model and strongassumptions. In particular, they do not take into account the in-terferences among simultaneous transmissions and power con-trol. Also the scheduling and the unfairness problems are notinvestigated.
Lopez-Peres et al. [19] investigated the problem of the jointallocation of Modulation and Coding Scheme (MCS), resourceblocks, and power assignment to users in LTE cellular systems,while minimizing the overall power consumption. To achievethis objective, the authors break down the problem into twoloops based on a linear program and a metaheuristic algorithm.They showed that to provide a minimum bit rate per user, it isbetter to use more resource blocks with lower MCS and lesstransmission power, than it is to use few resource blocks withhigher MCS, but more power.
The lack of literature on both the capacity and energyconsumption has lead to this in-depth study to investigate thetradeoff between them using a continuous power control.
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III. ASSUMPTIONS AND PROBLEM DEFINITION
A. Assumptions and Network Properties
In this work, we consider a synchronized multi-hop singlechannel WMN where the MAC layer is based on S-TDMA. Weare especially interested in broadband cellular networks like3GPP LTE-Advanced and WIMAX. We assume a repeatingtransmission schedule of duration T (frame) that contains afixed number of time-slots (whose lengths take on continuousvalues) allocated to nodes to transmit their traffic. To increasenetwork efficiency, S-TDMA allows links with sufficient spatialseparation to transmit simultaneously. One of our objectivesis to compute an optimal S-TDMA scheduling that providesmaximum throughput and efficient energy consumption.
We assume that the channel gains are quasi time-invariant.Under the assumption of quasi-static traffic and quasi time-invariant channel gains, it is reasonable to consider a staticnetwork. Each node is equipped with an omni-directional an-tenna. Its transmit power can be adjusted continuously at eachtransmission. Network capacity can be improved by increasingthe number of gateways, if they are sufficiently spaced fromeach other [20]. In this paper, our scenarios are restricted to thesingle gateway case, though our models could address multi-gateway scenarios. We assume that there is an uplink flowfrom each MR to the gateway and a downlink flow from thegateway to each MR. These flows require several resources tobe transmitted and are routed through multi-hop paths to becomputed (see Fig. 1).
B. Network Model and Notations
A wireless mesh network is a fixed infrastructure that con-sists of a set V of nodes composed of a set of mesh routers,denoted VMR, and a gateway Gw. This section is dedicated tothe modeling of the WMN.
1) Node Model: Each mesh router is characterized byits identity u ∈ VMR, its geographic position, and a weightdUL(u) (resp. dDL(u)) that reflects its uplink (resp. downlink)throughput requirement. The uplink throughput requirement isneeded to forward the uplink traffic generated by mobile clientsto the gateway.
During each time slot, a node can be either idle, receiving,or transmitting. When transmitting, the transmit power of thenode u is denoted Pt(u) and is bounded by a maximum valuePmax. The nodes have a continuous power control capability toreduce the interferences and to use the appropriate transmissionrate, as is explained in the following section. The energy con-sumption of a node, which depends on its activity, as detailedin Section III-C2, is denoted J(u).
In the following, we present the modeling of the links byintroducing an aggregated notion of logical links and a moredetailed notion of physical links. The former completes, withV , a graph representation of the network, which is convenientfor computing optimal routings. The latter describes all theparameters of a transmission needed for computing capacityand energy efficient resource allocations.
2) Links and SINR Interference Model: When a commu-nication occurs between two nodes, traffic is sent over the
link at a rate r which belongs to a set of transmission rateR = {rj}, Nr = |R|, 0 < r1 < r2 < . . . < rNr
. Note that eachtransmission rate rj is the result of the use of a modulation andcoding scheme MCSj . We introduce two notions of (directed)links. Let us denote a logical linke = (u, v) identified onlyby an origin-destination pair. E is the set of feasible logicallinks and G = (V,E) is the graph representation of the WMN.Such a representation is convenient for efficiently handlingrouting. However, to assess the achievability of a logical linkand hence to define E and cope with interference and energyissues, a more detailed notion of a link is required. Let us denotea physical link by l, identified by the following parameters(e, Pt, r).
• e = (o(l), d(l)) ∈ E the logical link between the origin-destination pair (o(l), d(l)).
• Pt ∈ [0, Pmax]: the transmit power of the node o(l) duringthis communication.
• r ∈ R: the transmission rate, in bits per second, usedduring this communication.
Each rate r has a corresponding SINR requirement β(r) forcommunication to be established with some given parameters,such as a maximum bit error rate (β(ri) > β(ri−1)). Thismeans that a physical link l = (e, Pt, r) is established if andonly if the power received from o(l) in d(l) is enough to reachthe SINR requirement of rate r. The power received at d(l) isproportional to Pt and to the channel gain function, denotedG(l), which takes into account a given radio propagationmodel (path loss, fading, and shadowing). Altogether, the SINRcondition at receiver d(l), in the presence of a set s of othersimultaneously active transmissions, is expressed as follows:
SINRd(l)=Pt ∗G (o(l), d(l))
μ+∑
l′=(e′,P ′t ,r
′) �=l,l′∈sP ′t∗G (o(l′), d(l))
≥β(r),
(1)
where μ ∈ R+ represents the thermal noise at the receiver.
The set of feasible physical links is denoted L and a logicallink e exists if and only if there exists Pt ∈ [0, Pmax] suchthat l = (e, Pt, r1) ∈ L. The set of logical links can thereforebe defined as E = {e = (u, v), ∃Pt < Pmax, (e, Pt, r1) ∈ L}.Note that L is infinite, while E is finite and tractable for routingissues.
3) Conflict Free Scheduling: A set I of physical links(l1, l2, . . . , ln) is said to be an independent set (ISet) if andonly if Eq. (1) holds at all receivers and ∀ li, lj ∈ s, i �= j,o(li) �= o(lj), d(li) �= d(lj), and o(li) �= d(lj). All links in thisset can be scheduled at the same time without creating anydecoding conflict. The set of all possible ISets is denoted I.
Note that, because we consider continuous power control,the set of physical links is infinite. However I can be reducedto a finite set of “minimal ISets” with respect to transmissionpowers: we only consider ISets in which transmission powercannot be reduced without modifying the transmission rate oflinks. This does not provide a tractable and easily generatedset of ISets; however, column generation allows for generatingonly a subset of useful ISets (this will be discussed in details inSection IV-B).
1134 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 14, NO. 2, FEBRUARY 2015
By scheduling only ISets, we ensure that the schedule isconflict free. Let w(I) be the time allocated to the ISet I, wehave
∑I∈I w(I) = T . Our optimization problems will compute
the w(I)’s to maximize the objective function.4) Routing Model: The activation of an ISet I provides to
each logical link, e ∈ E, a rate re(I) equal to r(l) ∈ R if itexists l = (e, Pt(l), r(l)) ∈ I, and to 0, otherwise. Hence, eachlogical link e sees a total rate equal to
∑I∈I re(I)w(I). These
rates are used to route the traffic between the mesh routers andthe gateway. We define a routing path as a set of logical linksthrough intermediate nodes from source to destination. For eachmesh router u ∈ VMR, let Pu
UL (resp. PuDL) denote the set of
uplink (resp. downlink) paths between u and the gateway, andlet PUL = ∪uPu
UL (resp. PDL = ∪uPuDL) denote the set of
uplink (resp. downlink) paths in the network. The uplink trafficis modeled by the flow function fUL : PUL → R
+. The trafficsent by u is hence
∑P∈Pu
ULfUL(P) (as it is for the downlink
traffic flow). The flow over a logical link e is the sum of theuplink and downlink traffic on the paths going through e. Thisflow has to be below the total rate of e. The problem of routingis to calculate the flow function that maximizes the throughputor minimizes the energy consumption.
C. Network Capacity and Energy Consumption Model
1) Network Capacity: We assume that the throughput re-quirements of the mesh routers are heterogeneous. This canbe explained by the number of clients connected to each meshrouter. To model this, each mesh router is allocated a weightthat reflects its greedy throughput requirement with respectto a common base λ. We consider a fair notion of networkcapacity in which every router receives at least its weightedshare of the global throughput. The resources are thereforeassigned so that each node u ∈ V receives an end-to-end uplinkthroughputλUL(u) (resp. downlinkλDL(u)), so that:λUL(u) ≥dUL(u) ∗ λ, where dUL(u) (resp. dDL(u)) is the uplink (resp.downlink) weight of node u and λ is the common base through-put (in bps) to be optimized. Hence, the network capacity isat least
∑u∈VMR
(λDL(u) + λUL(u)) ≥∑
u∈VMRdu ∗ λ, where
du = dUL(u) + dDL(u). Maximizing λ achieves a fair maxi-mization of the network capacity.
The idea behind throughput-optimal scheduling is to sched-ule as many links as possible in each time slot, that is, tomaximize the spatial reuse of system resources. This objectivehas to be mitigated with interferences and energy consumptionconstraints.
2) Energy Consumption Model: We propose a generic en-ergy consumption model that is based on node activity (idle,transmission, reception).1 A node can be operational or non-operational. When the radio part of the node is not in operation,some components are always on (due to signal processing,battery backup, as well as site cooling) and those componentsconsume a given quantity of power, denoted Cc. This state iscalled an Idle State. When the radio part is operational, the nodeu can either be in Transmission State (u = o(l)) or in Reception
1Our model is based on the model proposed in the EARTH project [13].
Fig. 2. Illustration of the power consumption model.
TABLE INETWORK MODEL PARAMETERS AND NOTATIONS
State(u = d(l)) and it consumes, respectively, (Cc+ a(u) ∗Pt(o(l))) and (Cc+ Pr(u)). The coefficient a(u) accounts forthe power consumption that scales with the average radiatedpower (due to the high power consumption of the amplifier).Here, we assume that Pr(u) is fixed for all nodes. The relationbetween transmission power and node energy consumption isnearly linear [13]; see Fig. 2. The fixed cost Cc is consumedregardless of the state of the nodes. Note that our optimizationproblem, detailed in Section IV, does not depend on thisparameter since it is static and consumed independently of nodestate. To reduce energy consumption, it is possible to turn off anode (Sleep State) when it is not in operation. This approach isstudied in several papers and is not investigated in this work.Each ISet I has power consumption (Watts), J(I), which iscalculated as follows:
J(I) = |V | ∗ Cc+∑l∈I
a (o(l)) ∗ Pt (o(l)) +∑l∈I
Pr (d(l))
(2)
The total energy consumption of the network, during theframe length T , is
∑I∈I w(I)J(I) when the scheduling is done
using the w(I)’s.Tables I and II summarize all the network model parameters
and notations.In the next section we formulate two different linear pro-
gramming problems: the first maximizes the network capacitysubject to a constraint on the total energy consumption, whilethe second minimizes the total energy consumption subject to acapacity constraint. We also present the column generation al-gorithm that we use to cope with the combinatorial complexityof the paths and the set of ISets.
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TABLE IILP MODEL NOTATIONS
IV. LINEAR MODELS FOR CAPACITY AND ENERGY
CONSUMPTION OPTIMIZATIONS
A. Master Formulation
The joint routing and scheduling problem can be expressedin two linear programs (LP) depending on the objective. Thefirst one maximizes the capacity with an energy budget con-straint. The Master Problem to Maximize Capacity (MPMC) isformulated as follows:
maxλ,(w(I))I∈I ,fUL(u)u∈V ,fDL(u)u∈V
λ
subject to ∀u∈VMR
∑P∈Pu
UL
fUL(P)≥dUL(u)∗λ (3)
∀u∈VMR
∑P∈Pu
DL
fDL(P)≥dDL(u)∗λ (4)
∀ e ∈ E T ∗
⎛⎝ ∑
P∈PDL,P�efDL(P)+
∑P∈PUL,P�e
fUL(P)
⎞⎠
≤∑I∈I
re(I)w(I) (5)
∑I∈I
w(I) ≤ T (6)
∑I∈I
w(I)J(I) ≤ EM (7)
λ>0, (w(I))I∈I ≥0, fUL(u)u∈V ≥0, fDL(u)u∈V ≥0
(8)
The objective function imposes the maximization of the end-to-end base throughput λ. Equations (3)–(5) express the routingpart as flows between the MRs and the gateway. Constraints(5) impose that the total traffic on the logical link e shouldnot exceed the capacity of the link itself while constraints (3)(resp. (4)) ensure that each MR achieves a maximum uplink(resp. downlink) throughput, taking into account the nodesweights. Equation (7) constrains the total energy expenditureof the network to a budget EM .
The second LP formulation minimizes the total energy ex-penditure under a capacity guarantee and is called the MasterProblem to Minimize Energy consumption (MPME):
minλ,(w(I))I∈I ,fUL(u)u∈V ,fDL(u)u∈V
∑I∈I
w(I)J(I)
subject to Equations (3)−(6) and λ ≥ λmin (9)
Fig. 3. The front Pareto description.
The flow equations of MPME remain the same as Eqs. (3)–(5)while the upper bound on the energy consumption Eq. (7) isreplaced by a lower bound on the network capacity Eq. (9).Finally, the objective is to minimize the energy expenditure ofthe network.
The physical link parameters (such as the transmission powerand the link rate) are explicitly taken into account by each ISetI ∈ I: recall that an ISet is a set of physical links and willbe calculated by a mixed integer linear program, detailed asfollows.
The MPMC and MPME formulations allow us to calculatethe Pareto front between the network capacity and the energyconsumption. Fig. 3 explains how we calculate this Paretofront. The first step is to calculate the two extremal points,P0 = (Emin, λmin) and P1 = (Emax, λmax), which presentthe minimum energy consumption, Emin, and the maximumbase throughput λmax. Recall that the network capacity is equalto
∑v dv ∗ λ. P0 and P1 are calculated as follows:
P0
⎧⎨⎩
Emin=min∑I∈I
w(I)J(I) | λ>0 (using MPMC)
λmin = maxλ |∑I∈I
w(I)J(I) ≤ Jmin (MPME)
P1
⎧⎨⎩
λmax = maxλ |∑I∈I
w(I)J(I) ≤ ∞ (MPMC)
Emax=min∑I∈I
w(I)J(I) | λ ≥ λmax (MPME)
Once the two extremal points have been determined, weuse one of the two linear programs to plot the rest of thecurve. For example, if we use the MPMC linear program,then we vary EM between Emin and Emax. This curve hasseveral properties. In particular, (i) the throughput is a strictlyincreasing function for E ∈ [Emin, Emax], (ii) for each EM ∈[Emin, Emax] there exists a saturation throughput λm such thatλ(EM ) = λm and λ(E) < λm for E < EM [18].
Because the number of variables (paths and ISets) are ex-ponential with the size of the network, these formulations arenot scalable as such. However, most of them will not be usedin an optimal solution. Therefore, it is not practical to generateall the variables. Column generation [4], [5] is a prominent andefficient technique to cope with this situation. Based on linear
1136 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 14, NO. 2, FEBRUARY 2015
programming duality results, it avoids the complete enumera-tion of the variable sets.
B. Column Generation
Column generation is an algorithmic technique for solvinglinear programs with an exponential set of variables, whichtakes its roots in duality theory [21]. Each linear program,denoted master in this context, has an associated and uniquedual program. For each constraint of the master, there is adual variable that is defined. Similarly, for each variable ofthe master, there is a constraint in the dual, which binds thedual variables related to the master constraints in which theconcerned master variable appears. This is done in such a waythat the duality association is reflexive (the dual of the dualof a LP is the original LP). The dual formulations of MPMCare detailed in the following section. Each instantiation of themaster variables is similarly associated to an instantiation ofthe dual variables, such that the master values represent a sub-optimal feasible solution if and only if the dual values are anon feasible solution, i.e., that at least one constraint of thedual is violated. Both sets of master and dual values representa feasible solution if and only if they are both optimal (with theproperty that the master and dual optimal objectives values arethe same).
Exploiting this property, the column generation principleinvolves first solving the master on a restricted set of variables(also called columns, hence the column generation), consid-ering that the non considered variables are zero. In our case,the variables are the flow over the paths and the weights of theISets. We then consider a restricted set of paths P0 and ISets I0which have to be carefully chosen to ensure the existence of aninitial feasible solution. Generally, P0 contains a shortest pathbetween each mesh router and the gateway (uplink/downlinkpaths), and I0 = {{l = (e, Pt, r1)}, e ∈ E, Pt =
β(r1)∗μG(l) }.
Thus, the solving of the master on this restricted set of vari-ables is fast and, if there exists a feasible solution, it is relatedto a set of dual values. If the master solution is suboptimal,the aforementioned property of the duality claims that what thedual values describe is a non feasible solution of the dual. Thereis, then, at least one constraint of the dual that is violated andwhich is in bijection with a variable of the master, which is herea path or an ISet. The separation theorem claims that solvingthe master problem on the set of variables, including this newvariable, will improve the solution [21]. The process loops untilno such variable exists, as depicted in Fig. 4. When this state hasbeen reached, the dual variables represent a feasible solution.Since the master also does, the theory of duality claims thatboth the master and the dual are optimal. Finding the newvariables in the column generation process consists of solvingthe auxiliary programs described in Section IV-B2.
1) Dual Formulation: Below, we present the dual formula-tion of MPMC. Note that the one for MPME is very similar.Recall that in this LP, there is a constraint for each variable ofthe master, be it the flow on a path or the weighting of an ISet.We denote θUL(.), θDL(.), γ(.), Ω, and σ, respectively, to be thedual variables associated to constraints (3)–(7). o(P) denotes the
Fig. 4. The column generation process.
source node of path P. J(u) is the power consumption (Watts)of node u:
min(θUL(u))u∈V ,(θDL(u))u∈V ,σ,Ω,(γ(e))e∈E
T ∗ Ω+ EM ∗ σ
subject to : ∀P∈PUL θUL(o(P))≤T ∗∑e∈P
γ(e) (10)
∀P ∈ PDL θDL (o(P)) ≤ T ∗∑e∈P
γ(e) (11)
I ∈ I∑e∈E
re(I)γ(e)− σJ(I)− Ω ≤ 0 (12)
∑u∈VMR
(θUL(u)d(u) + θDL(u)d(u)) ≥ 1 (13)
2) Auxiliary Programs: We now describe the two auxiliaryprograms which determine if there are uplink/downlink pathsor ISets that violate the constraints of the dual program. Thefirst program, associated to constraints Eqs. (10), (11), finds,for each source node, a weighted path with a weight lower thanthe dual variable associated to the source node. If the minimumweighted path fits the constraint, then so do all other paths.This problem is similar to the shortest path problem; hence, itcan be easily solved using linear programming (LP). This LPminimizes the weighted path with γ(e) under a conservationflow constraint, which defines the relation between incomingtraffic and outgoing traffic for each node [22].
The second auxiliary problem is associated with constraint(12). It is necessary to decide if there exists an ISet I suchthat
∑e∈E reγ(e)− σJ(I)− Ω > 0. Again, if the maximum
weight communication set respects (12), then so do all otherISets. Our auxiliary program considers two scenarios:
a) Generation of ISets with continuous power control andmulti-rate: In this case, each node continuously controls its
OUNI et al.: ENERGY AND THROUGHPUT OPTIMIZATION OF WIRELESS MESH NETWORKS 1137
transmission power and chooses the best MCS (or rate r ∈ R),depending on the SINR achieved at the receiver. Given a setof dual variables (γ(.), σ) obtained from the master problem(MPME or MPMC), we generate a new ISet by solving thefollowing Mixed Integer Linear Program:
maxΨ,Pt,J
∑e∈E
(reγ(e))− σ∑u∈V
J(u)− Ω (14)
∀u∈V J(u)≥a(u)∗Pt(u)+∑v∈V
∑1≤i≤Nr
Pr(u)Ψi(v,u)+Cc
(15)
∀ (u, v) ∈ E, i ∈ [1, Nr] Pt(u) ∗G(u, v) ≥ β(ri)
∗
⎛⎝ ∑
u′ �=u,v
Pt(u′)∗G(u′,v)+μ
⎞⎠−
(1−Ψi
(u,v)
)n∗Pmax
(16)
∀u ∈ V∑v∈V
∑1≤i≤Nr
Ψi(u,v) +
∑w∈V
∑1≤i≤Nr
Ψi(w,u) ≤ 1
(17)
∀ e = (u, v) ∈ E re =∑
1≤i≤Nr
riΨi(u,v) (18)
∀u ∈ V Pt(u) ≤ Pmax (19)
The decision variables of this linear program are Pt(u), J(u)and Ψi
(u,v) where (u, v) ∈ E and i ∈ [1, Nr]. The binary vari-
able Ψi(u,v) is equal to 1 if the communication between u and
v is active in the new ISet, with a transmission rate of at leastri, and to 0 otherwise. The goal is to find a new ISet I where(∑
e∈E reγ(e)− σ∑
u∈V J(u)) is maximum Eq. (14). If thisISet violates Eq. (12), it may improve the solution of the masterprogram. If not, no other ISet can, either, do and the solution ofthe master is optimal. The constraints of this ILP define the ISetstructure as follows. The energy consumption model, detailed inSection III-C2, is presented by constraints (15). The constraint(16) ensures that the SINR condition is satisfied for all activelinks in the ISet, taking into account the transmission rate usedby each one. Note that (1−Ψi
(u,v))n ∗ Pmax equals 0 when thelink (u, v) is active, hence the constraint (16) reverts back to theclassical interference constraint (1). Otherwise (Ψi
(u,v) = 0),and n ∗ Pmax ensures that Pt(u) can be equal to 0 (constraint(16) is always respected), where n is the number of nodes.Finally, constraint (17) implies that each node is active in atmost one link with one transmission rate in each time-slot. Thisconstraint also ensures the half-duplex property where a nodecannot transmit and receive simultaneously.
This auxiliary program builds a new ISet I which containsthe following physical links: for all e = (u, v) ∈ E such thatΨi
(u,v) = 1, l = (e, Pt(u), ri) ∈ I .b) Generation of ISets with single-rate: In this case, we
assume that only a single rate, r ∈ R, is available and thateach node can continuously control its transmission power. Westudy this case using the previous auxiliary program by set-ting Nr = 1.
TABLE IIIMODULATION AND CODING SCHEMES: MCS [19]
We have presented our linear programs, to optimize net-work capacity and energy consumption, and have presentedthe column generation to solve them. Next, we will discussthe energy-capacity tradeoff. We calculate an optimal systemsetting of the network to minimize the energy consumption(resp. to maximize the capacity) under the requirements of highnetwork capacity (resp. low energy consumption).
V. SINR BASED MODEL: CONTINUOUS POWER
CONTROL AND SINGLE-RATE
In this section, we assume that each node operates at afixed transmission rate (fixed MCS) and can tune its transmis-sion power at each transmission. We calculate optimal routesfor data, transmission power, resources allocation, and linkschedules.
A. Scenarios and Model Parameters
Both the capacity-oriented and energy-oriented formulationsand the column generation algorithm are implemented andtested using AMPL/CPLEX [23], [24]. In all of our numericalresults, we consider a multi-hop WMN with regular and ran-dom topologies. The regular network topology has its nodespositioned on a grid. The random topologies are generated witha Poisson process in the Euclidean plane. In all of our scenarios,we consider 24 MRs deployed in an area of 500 m ∗ 500 m anda gateway located in the center. Except when otherwise stated,all MRs have the same throughput requirement (the impactof a non-uniform throughput requirement is investigated inSection VI-B). The path-loss attenuation is equal to (d(u,v)d0
)−α
where α = 3.6 is the path loss exponent and d0 = 1 m isthe near-field crossover distance. The noise power density is−174 dBm/Hz. We consider five MCSs, presented in Table III,available to each node. The numerical values of the energyconsumption model are adapted from the models of the EARTHproject for small cells [2]. Combining (2) and (6), the energycost obtained is Cc ∗ |V | plus the variable part of the energycost which does not depend on Cc. Indeed, the fixed cost ofcircuit consumption has no impact on the optimization of thetransmit power assignment and therefore can be considered asnull in the following, up to a constant shift of the numericalresults. Table IV summarizes all physical parameters.
B. Capacity and Energy Tradeoff in the Case of 1 MCS
1) Insensitivity of the Mix of UL/DL Traffic to the Energy-Capacity Tradeoff: the Pareto front of the capacity/energytradeoff is depicted in Fig. 5 for a grid and a random networkusing only MCS4 with continuous power control. In this study,
1138 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 14, NO. 2, FEBRUARY 2015
TABLE IVPHYSICAL LAYER PARAMETERS
Fig. 5. Capacity and energy tradeoff, using MCS4 and Pmax = 15 dBm, inthe case of uplink-only, downlink-only and mixed traffic (25% uplink + 75%downlink). (a) Multi-hop grid network. (b) multi-hop random network.
we consider three scenarios: uplink-only, downlink-only andmixed traffic with 25% uplink and 75% downlink. In each case,a minimal energy budget for the network is required to route alltraffic between the MRs and the gateway. We observe that thereis no significant impact from the mix of uplink and downlinkflows on the energy-capacity tradeoffs. In fact, the capacity isconstrained by the activity inside a bottleneck zone around thegateway [20], [25]. In this area, there is no spatial reuse as onlyone link can be activated at each time, either in uplink or indownlink. Hence, the network capacity cannot be improved bycombining the uplink and downlink flows. Note that the uplinkand downlink flows paths are not necessarily the same, as theISets are different due to the asymmetric interferences.
Fig. 6. Impact of maximum power transmission on energy-capacity tradeoff:random network with MCS4.
2) Impact of Maximum Power Transmission on Energy-Capacity Tradeoff: Fig. 6 depicts the energy-capacity Paretofronts on a multi-hop random topology, when the maximumpower transmission takes one of three values (10 dBm, 15 dBm,and 21 dBm). It shows that increasing the maximum trans-mission power increases the magnitude of the energy-capacitytradeoff and the maximum network capacity. It also shows that alarger network capacity, with the same energy expenditure, canbe achieved. Indeed, higher transmission power induces, first,a higher connectivity in the network, in particular around thegateway. Intuitively, in the bottleneck area, going directly to thegateway saves time, which significantly increases the capacity.Second, new ISets can be generated with better spatial reuse andwith the same energy budget constraint. However, increasingtransmission power is a major contributor to increasing energyconsumption, which explains increasing numbers of the energy-capacity tradeoff solutions.
3) Benifit Due to Power Control: The benifit of enablingcontinuous power control is illustrated in Fig. 7(a) and (b)which present, respectively, network capacity and energy con-sumption in the cases of power control and of fixed power.2
Each result is averaged on 15 random instances. Let P1hop bethe transmission power which allows all MRs to communicatedirectly with the gateway. Fig. 7(a) shows that when Pmax <P1hop, the use of continuous power control is very beneficialfor increasing network capacity and energy consumption. Thetransmit power is adjusted to reduce the interferences, whichincreases the spatial reuse and thus improves the through-put. When Pmax ≥ P1hop, continuous power control and fixedpower leads to the same network capacity. In the case of fixedpower, this capacity is obtained with high transmission powerand, hence, with high energy consumption. Interestingly, powercontrol allows this capacity to be achieved with multi-hopcommunications and lower transmission power, which providesabout 70% of energy gain. Moreover, the average gain innetwork capacity reaches about 25%, and the energy gain is
2All nodes transmit at the maximum transmission power.
OUNI et al.: ENERGY AND THROUGHPUT OPTIMIZATION OF WIRELESS MESH NETWORKS 1139
Fig. 7. The impact of maximum transmission power and benifit due topower control: each result is averaged on 15 random instances using MCS4.(a) Network capacity. (b) Energy consumption.
between 25% and 70%. It is important to note that, in the caseof power control, the energy consumption increases only if thenetwork capacity increases.
VI. MULTI-RATE TRANSMISSION AND
OPTIMAL SYSTEM SETTING
Given an ISet I, each link l = (u, v, Pt, r) ∈ I is activatedduring w(I) with the transmission rate r(l) ∈ R. An optimalsystem setting consists of finding, for each communication,the best MCSj with a transmission power that minimizesthe overall energy consumption and maximizes the networkcapacity. The main question to be addressed is how MCSs andpower should be allocated to each transmission. In this section,we consider the five MCS presented in Table III. Note thatenergy consumption and capacity are linked to the MCS used.Intuitively, higher modulation means higher throughput andcapacity, but requires greater transmission power to meet theSINR threshold constraint. This increases the tradeoff betweencapacity and energy consumption.
TABLE VMCS VS ENERGY CONSUMPTION PER BIT/S (J/BIT/S)
Fig. 8. The energy and capacity tradeoff: fixed power vs continuous powercontrol (random network and multi-rate transmission).
To further illustrate this tradeoff, a simple scenario of a singlecommunication between a source and destination is shown.The energy consumption per bit per second (J/bit/s) for eachtransmission rate (or MCS) is depicted in Table V. We observethat MCS1 is the most energy efficient; however, it is the lowestin terms of throughput, while MCS5 leads to higher throughput.
In this scenario, with an isolated link, transmitting powerand throughput are bounded by the MCS characteristics whichresult in a tradeoff on the energy efficiency. As seen inSection V, in an example situation with several nodes andconcurrent communications, the interferences and the spatialreuse induce a tradeoff between the overall energy consumptionand capacity. In the following section, we study the tradeoff in anetwork when the nodes perform continuous power control anduse multi-rate transmission.
Next, we assume that the MCS presented in Table III areavailable for each node. For each network, an optimal solutionis calculated including: network capacity, energy consumption,routing, resource allocation, physical parameters of each node(transmit power and MCS used for each transmission), andactivation time of each communication.
A. Energy and Capacity Tradeoff
To reduce complexity and computing time, without lossof generality, we eliminate the MCS1 (which dramaticallyincreases the number of available links and leads to prohibitivecomputation times) and use only the four other MCSs. Thetradeoff between energy consumption and network capacityis depicted in Fig. 8, which presents the fixed power caseand the continuous power control case. This figure shows an
1140 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 14, NO. 2, FEBRUARY 2015
Fig. 9. Capacity and energy tradeoff with multi-rate transmission and contin-uous power control: Impact of topology.
important tradeoff between capacity and energy consumption.This tradeoff results from the use of different MCS and fromthe impact of spatial reuse. In the control power case, activatingonly one link on each time-slot with the lowest MCS is themost energy efficient solution. This is, of course, at the costof achieving the worst network capacity: increasing the numberof simultaneous communications and using high modulationsincreases the capacity but consumes more energy.
Comparing the energy-capacity tradeoff obtained with thetwo scenarios emphasizes that continuous power control in-creases the magnitude of the tradeoff (the capacity varies be-tween 140 and 450 Kb/s), and allows higher network capacityto be achieved with lower energy consumption.
B. Impact of Topology and Throughput Requirement
Most of the previous results were obtained with a singlerandom network and homogeneous throughput requirement.We investigate the impact of the throughput requirement dis-tribution (represented by the weight du) and the topology onthe energy-capacity tradeoff.
1) Impact of Topology: In Fig. 9, we illlustrate the energy-capacity tradeoff according to a selection of seven randomtopologies. Note that for all topologies, maximum transmissionpower is sufficient to reach maximum network capacity. Ourresults show that the topology has a significant impact oncapacity and energy consumption; however, all of the Pareto-Front curves show similar behavior. For example, topology“Random 4” provides the maximum capacity with an energyconsumption of 18% less than topology “Random 7.”
2) Impact of the Throughput Requirement: To examiningthe impact of the throughput requirement distribution on theenergy-capacity tradeoff, we compare the following distribu-tions with the same mean value.3
• Homogeneous distribution: all MRs have the same weight
3The optimization problem being linear and the results being reported asJ/b and Kn/s, the actual value of the mean requirement has no impact on thenumerical results. For our simulations, it was set to 2.
Fig. 10. Capacity and energy tradeoff with multi-rate transmission and con-tinuous power control: impact of weight distribution. (a) Random topology,continuous power control. (b) Grid topology, continuous power control.
• Uniform random distribution: weights are distributed uni-formly and independently
• Poisson random distribution: weights are distributed ac-cording to a Poisson distribution
The results are reported in Fig. 10, which illustrates theenergy-capacity tradeoff as a function of weight distribution,in the case of grid and random networks. The impact of thethroughput requirement on the energy-capacity tradeoff is verylow. Actually, the traffic load distribution is not very important;the bottleneck area around the gateway has the most impact oncapacity. In addition, the case of a high traffic load concentratedin an area was also studied. Fig. 11(a) showed that the impactof the weight distribution is significant when there is a trafficload concentrated in an area that creates another bottleneckarea. Fig. 11(b) shows the impact of the distance between thebottleneck and the gateway: the energy consumption decreaseswhen the bottleneck is near the gateway, while the networkcapacity is almost the same. Based on this observation, thebottleneck-gateway distance parameter should be taken intoconsideration in the network planning and design.
OUNI et al.: ENERGY AND THROUGHPUT OPTIMIZATION OF WIRELESS MESH NETWORKS 1141
Fig. 11. Impact of bottleneck on energy-capacity tradeoff with MCS3 andcontinuous power control. (a) Impact of the magnitude of the bottleneck on theenergy-capacity tradeoff. (b) Impact of the distance between the bottleneck andthe gateway on the energy-capacity tradeoff.
VII. DISCUSSION AND CONCLUSION
A. Discussion
In this section we discuss the main contributions of this paperwith respect to other results in the literature. This discussion isdivided into two Sections: the first is about our optimizationframework presented in Section IV. The second is about net-work design insights, which can be deduced from our results.
1) Optimization Framework: The main contribution of ouroptimization framework consists in using continuous powercontrol, which allows fine-tuning of transmit power; however,this adds more complexity to the optimization problem. Ourframework allows the optimal capacity to be computed under arealistic physical layer, based on SINR interferences, continu-ous power control, and multi-rate transmissions. One key ideais to model a single logical link as multiple parallel physicallinks with different radio transmission parameters. This allowsus to use a tractable scheduling and routing formulation. Bycomputing a restricted set of decision paths and ISets, column
generation enables this problem to be solved within a reason-able time.
Enhancing scalability, to handle a growing amount of nodes,remains a significant challenge in the literature. Currently, wecan study networks with 30 nodes using continuous powercontrol and multi-rate transmissions in a reasonable amount oftime.
2) Network Design Guidelines: Our optimization frame-work allows the offline computation of the optimal systemsettings of the network. This enables us to derive practicalengineering insights and effective benchmarking. This paperfocuses on the relationship between energy consumption andnetwork capacity. Our numerical results show that the energy-capacity tradeoff increases with continuous power control(Section V) and with multi-rate transmissions (Section VI).
The advantages of continuous power control are shown inSections V and VI. Network capacity and energy consumptionare optimized by reducing transmit power and interferences.This confirms the results of [5], [6], [8] which show that discretepower control (a set of power levels) improves the spatial reuseand hence improves the throughput. Ours results show thatthe use of multi-rate transmissions is beneficial for providinghighest capacity with low energy consumption. Moreover, weinvestigate several topologies and weight distributions. Wefound that the weight distribution has no impact on energy andcapacity: alone the congested area around the neighborhood ofthe gateway influences the energy-capacity tradeoff, which iscoherent with previous works on capacity [4], [5], [25].
These results can serve as a guide for the development ofprotocols which maximize the capacity with efficient energyconsumption. For example, a routing strategy and MCS distri-bution can be derived from our results. Indeed, we show that wecan significantly increase the network capacity by allowing thenodes communicate directly with the gateway in the congestionarea (around the gateway), using MCS with high throughput(MCS4 and MCS5). For the sake of energy consumption it ismore efficient to use multi-hop communications outside of thisregion, combined with spatial reuse. The implementation andtesting of a protocol based on this approach is one of our futuregoals.
B. Conclusion and Perspectives
In this paper, we have addressed the problem of network ca-pacity and energy consumption optimization in WMN. A set ofnovel linear programming models, using a column generationalgorithm, was presented. The later computes a linear relaxationof the routing and scheduling problem with a realistic SINRmodel and using continuous power control. Since the objec-tive of maximizing the network capacity is often in conflictwith the objective of energy minimization, we carried out athorough study of the tradeoff between them. We investigatedthe problem of joint resources, MCS, and transmission powerallocation, to compute an optimal offline configuration of thenetwork. This work assumed single-channel and single-radionodes. It is possible to extend our formulation to multi-channeland multi-radio; however, the price is obviously a dramaticincrease in complexity. Another challenge is to go beyond
1142 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 14, NO. 2, FEBRUARY 2015
the static and offline optimization approach presented in thispaper and investigate how to take into account the dynamics ofparameters such as throughput demand or channel state.
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Anis Ouni received the Engineering degreefrom Higher School of Communication of Tunis(Sup’Com), Tunis, Tunisia, in 2008, and theM.Sc. and the Ph.D. degrees from the NationalInstitute for the Applied Sciences (INSA-Lyon),Villeurbanne Cedex, France, in 2009 and 2013,respectively, all in the fields of computer networksand telecommunications. In 2013 and 2014, heheld a temporary Assistant Professor position atthe Industrial Engineering and Telecommunicationsdepartments at INSA Lyon. He is currently a
Post-doctoral Fellow with the Intitut of Telecom Paris-Tech, Paris, France.
Hervé Rivano received the Ph.D. degree from theUniversity of Nice-Sophia Antipolis in November2003, after graduating from the Ecole NormaleSupérieure de Lyon. He is an INRIA Researcher(Chargé de Recherche). He is the head of theUrbaNet team, which focuses on wireless networksfor digital cities. Prior to that, he was a CNRS Re-searcher in October 2004 and an INRIA Researchersince January 2011. His research interests includecombinatorial optimization and approximation algo-rithms applied to network design and provisioning.
Fabrice Valois received the Ph.D. degree incomputer science from University of Versailles,Versailles, France, in January 2000. He is a FullProfessor in the Telecommunication Departmentof INSA Lyon, Villeurbanne Cedex, France, since2008. He is also the head of the CITI-Inria lab-oratory, which is focused on smart and wirelesscommunicating objects. His research interests covernetworking issues for wireless sensor networks, andgenerally speaking for wireless multihop networks.More particularly, his research works are focused on
self-organization, routing and data aggregation, from theory to practise.
Catherine Rosenberg (F’11) received the Diplômed’Ingénieur from the Ecole Nationale Supérieure desTélécommunications de Bretagne, Brest, France, in1983, the M.S. degree in computer science from theUniversity of California, Los Angeles, CA, USA, in1984, and the Doctorat en Sciences in computer sci-ence from the Université de Paris XI, Orsay, France,in 1986. She is a Professor and a Canadian ResearchChair in the Department of Electrical and ComputerEngineering, University of Waterloo, Waterloo, ON,Canada. She was elected a Fellow of the Canadian
