Joint Optimization of Energy Efficiency and Data Compression in TDMA-Based Medium Access Control for the IoT - Extended Version
Chiara Pielli, Alessandro Biason, Andrea Zanella, Michele Zorzi
JJoint Optimization of Energy Efficiencyand Data Compression in TDMA-BasedMedium Access Control for the IoT
Chiara Pielli, Alessandro Biason, Andrea Zanella and Michele Zorzi{piellich,biasonal,zanella,zorzi}@dei.unipd.itDepartment of Information Engineering, University of Padova - via Gradenigo 6b, 35131 Padova, Italy
Abstract —Energy efficiency is a key requirement for theInternet of Things, as many sensors are expected to be com-pletely stand-alone and able to run for years without batteryreplacement. Data compression aims at saving some energy byreducing the volume of data sent over the network, but alsoaffects the quality of the received information. In this work, weformulate an optimization problem to jointly design the sourcecoding and transmission strategies for time-varying channelsand sources, with the twofold goal of extending the networklifetime and granting low distortion levels. We propose a scalableoffline optimal policy that allocates both energy and transmissionparameters (i.e., times and powers) in a network with a dynamicTime Division Multiple Access (TDMA)-based access scheme.
I. I
NTRODUCTION
In the Internet of Things (IoT) a large number of heteroge-neous devices are expected to exchange data gathered from thesurrounding environment. Although many IoT devices may beconnected to the energy grid all the time (e.g., in smart houseapplications), most of them will have to rely on their ownlimited energy supply and will likely be deployed in remote orharsh places [1]. The burden of replacing sensors or rechargingtheir batteries every few weeks may outweigh all the benefitsof collecting data, and nodes failure due to power depletionmay even lead to the breakdown of the whole architecture [2].Thus, it is crucial to conserve as much energy as possible. Onthe other hand, periodic sampling of environmental signalsleads to enormous amounts of raw data, the transmissionof which would rapidly deplete the sensor energy. One ofthe key strategies to solve this problem is data compression ,which allows to reduce the amount of transmitted data whilemaintaining high levels of Quality of Service (QoS). The goalof this work is to investigate the trade-offs between energyconsumption and data compression at the Medium AccessControl (MAC) layer.The energy efficiency problem has gained much interestin the last years and several protocols have been proposedwith the target of extending the network lifetime as much aspossible. A lot of effort has been put into the design of theMAC layer [3], since the usage of the RF chain may havea major impact on the energy consumption. Many works inthe literature study both offline and online policies for theenergy allocation problem [4], often in the presence of EnergyHarvesting (EH), but they usually focus on a single transmitter-receiver pair [5]–[7]. Also the idea of QoS provisioning at the MAC layer is notnew, but the QoS metrics considered are typically throughput,latency, and delivery ratio [8], [9] and much fewer workstake into account the effects that signal processing has onthe transmitted information. Indeed, although compressionallows for some energy savings, due to the reduced numberof symbols to be sent, it affects the received informationby introducing a certain degree of distortion. In [10], theauthors study energy allocation policies to minimize the signaldistortion when several sensors measure the same process ofinterest and exploit data fusion techniques, whereas in [11] anonline joint source coding and data transmission optimizationstrategy is investigated for sensors with EH capabilities thatgenerate correlated information.Often, in the literature, uncoordinated access schemes arechosen because of their flexibility and lower synchronizationcosts. Nevertheless, coordinated access schemes completelyavoid collisions and interference since there is no channel con-tention and, by adopting appropriate duty cycling mechanisms,also the energy wastage due to idle listening is prevented.Recently, the Internet Engineering Task Force introducedthe Time-Slotted Channel Hopping (TSCH) [12] mode asan amendment to the MAC portion of the IEEE802.15.4estandard. It is a TDMA-based scheme and adopts a channel-hopping mechanism to improve reliability in the presence ofnarrowband interference and multi-path fading. The standard-ization of 6TiSCH (IPv6 over TSCH) enabled a wide-spreaduse of the TSCH mode in industrial networks, leading to arevival of TDMA-based schemes.In this paper we propose a synchronized MAC protocol foran IoT network with an arbitrary number of nodes. We developa TDMA-like scheme based on an optimization framework,which adopts convex and alternate programming to minimizethe data distortion and extend the network lifetime simulta-neously, under QoS constraints. Realistic energy consumptionmodels that consider both the compression and transmissioncosts are taken into account.The paper is organized as follows. In Section II we describeour system model and introduce the optimization problem,which is divided in two parts, namely the Frame-OrientedProblem and the Energy-Allocation Problem, which are solvedin Sections III and IV, respectively. Section V shows thenumerical evaluation. Finally, Section VI concludes the paper.
Notation:
Boldface letters are used for matrices and vectors; a r X i v : . [ c s . I T ] S e p i refers to the i -th row of matrix E , E ( k ) to the k -thcolumn, and E ( k ) i is the ( i, k ) element. With “ ∀ i ” and “ ∀ k ”,we summarize i = 1 , . . . , N and k = 1 , . . . , n , respectively,where N and n are defined in the next section.II. S YSTEM MODEL
We consider a network of N users which access the uplinkchannel to send data packets to a central Base Station (BS)using TDMA. Time is divided into frames and frame k corresponds to the time interval [ t k , t k +1 ) .We consider the channel gains to be constant during eachframe, and we approximate the average physical rate of user i ∈ N def = { , . . . , N } as: r ( k ) i = W log (cid:16) γ ( k ) i (cid:17) = W log (cid:16) h ( k ) i P ( k )tx ,i (cid:17) , (1)where W is the bandwidth, γ ( k ) i the average Signal-to-NoiseRatio (SNR) of user i in frame k , P tx ,i the transmissionpower used by the node, and h ( k ) i the average channel gainnormalized with respect to noise.As in [13], we adopt an information-theoretic approach inwhich full Channel State Information (CSI) is available apriori. This allows us to derive the optimal policy to minimizethe average distortion over time and to find useful bounds onthe actual performance that can be obtained in practice. A. Data Generation and Compression
Nodes may generate data by collecting measurements fromthe environment or by serving as relays to the central BSfor farther nodes. Before transmission and according to thetype of signal generated, nodes can perform compressionin order to limit the physical amount of data to be sentover the network. The distortion degree D ( k ) i is a functionof the compression ratio, η ( k ) C,i = L ( k ) i /L ( k )0 ,i , where L ( k ) i is the size of the compressed packet, and L ( k )0 ,i is the sizeof the original data. We define the following mathematicalexpression, which approximates the rate-distortion curve of aGaussian source [14]: D ( k ) i = b i (cid:32) η ( k ) C,i ) α i − (cid:33) , (2)where α i , b i > . Notice that the distortion is null when thepacket is not compressed, i.e., η ( k ) C,i = 1 .We also introduce a QoS requirement on the quality of thereceived data D ( k ) i ≤ D ( k )th ,i , where D ( k )th ,i is a threshold distor-tion level: if the reconstruction error exceeds this threshold,the signal generated by the source node is no longer usefulfor the final destination. B. Energy Consumption Model
Devices are battery-equipped and the battery level of node i in frame k is B ( k ) i . Since no harvesting sources are considered,the initial battery levels B (0) will strongly impact the systemperformance. In every frame, an energy E ( k ) i ∈ [0 , B ( k ) i ] isused according to the following sources of energy consump-tion. Data processing the energy spent to process the data gath-ered from the environment or other nodes can be characterizedby exploiting the results of [15]: E ( k ) P,i = ˆ E ,i · L ( k )0 ,i · N C,i ( η ( k ) C,i ) , (3)where ˆ E ,i is the energy consumption per CPU cycle whichthus depends on the processor of the node, and N C,i ( η ( k ) C,i ) is the number of clock cycles per bit needed to compressthe input signal and is a function of the compression ratio. N C,i ( η ( k ) C,i ) depends on the compression algorithm (see [15]for further details), and, in the case of Lightweight TemporalCompression (LTC) and Fourier-based Low Pass Filter (DCT-LPF), it is linear: N C,i ( η ( k ) C,i ) = ˆ α i η ( k ) C,i + ˆ β i . We assume thedevices use one of these algorithms, so the energy consump-tion due to processing becomes: E ( k ) P,i = ˆ E ,i L ( k )0 ,i (cid:32) ˆ α i L ( k ) i L ( k )0 ,i + ˆ β i (cid:33) = E ,i L ( k ) i + β ( k ) P,i , (4)where we defined E ,i (cid:44) ˆ E ,i ˆ α i and β ( k ) P,i (cid:44) ˆ E ,i ˆ β i L ( k )0 ,i . Data transmission the energy spent for the data transmis-sion task can be expressed as: E ( k ) T X,i = P ( k )tx ,i · τ ( k ) i , (5)where τ ( k ) i is the transmission duration and P ( k )tx ,i is thetransmission power, which is assumed to be constant duringthe whole transmission. Data sensing and circuitry costs we also consider thecontributions to the energy consumption of both sensingoperations and energy losses due to circuitry, which include,e.g., the energy spent for node switches from sleep mode toactive mode and viceversa, the synchronization costs, and theadditional energy lost during the transmission. We can expressthese quantities in the following way: E ( k ) C,i = β ( k )sens ,i + β ( k ) C,i + E C,i · τ ( k ) i , (6)where β ( k )sens ,i and β ( k ) C,i represent the constant sensing andcircuitry contributions, respectively, and E C,i is the rate ofcircuitry energy consumption during data transmission. Notethat the energy consumption due to collisions and overhearingis avoided because of the exclusive use of the communicationchannel guaranteed by our TDMA approach.By combining Eqs. (4)-(6), the total energy consumption ofa node in a single frame k is: E ( k )used ,i = E ( k ) P,i + E ( k ) T X,i + E ( k ) C,i = β ( k ) i + E ,i L ( k ) i + ( P ( k )tx ,i + E ( k ) C,i ) τ ( k ) i , (7)where β ( k ) i (cid:44) β ( k ) P,i + β ( k )sens ,i + β ( k ) C,i . C. Optimization Problem
The goal of the system is to simultaneously satisfy the QoSrequirements and extend the network lifetime. To handle thesetwo conflicting objectives, we set up the following weightedoptimization problem: min { E (0) , E (1) , . . . } σ n n (cid:88) k =1 f ( k ) F OP ( E ( k ) ) − (1 − σ ) n, (8)here σ is the weight in [0 , and n is the effective lifetimeof the system, which we defined as the first frame in whichat least one node dies, i.e., it does not have enough energyin its battery to transmit any more data while satisfyingthe distortion constraint. Since the network lifetime is anoutcome of the energy assignment, we consider k ∈ N forthe optimization variables, i.e., the number of optimizationvariables is not known a priori. The second term of (8) is adecreasing function of n , and is used to express the trade-offbetween distortion and lifetime.Given the energy consumption vector E ( k ) , ∀ k , the lifetime n is uniquely determined, whereas the distortion dependson the parameters τ , L , and P tx . With the final goal ofminimizing the normalized distortion (the normalization isdone with respect to the QoS threshold), we define function f ( k ) F OP ( E ( k ) ) as f ( k ) F OP ( E ( k ) ) = min τ ( k ) , L ( k ) , P ( k )tx max i ∈N D ( k ) i D ( k )th ,i , (9)which will be presented in its extended form in (10).We have structured the problem in a modular fashion that in-troduces a level of independence between the building blocks.These can be slightly adapted to meet different requirementsin a separate way while keeping the overall framework. Thetwo blocks have the following objectives.1) Energy Allocation Problem (EAP): it is the main prob-lem, with the goal expressed in Eq. (8). EAP defines theenergy allocation over time { E (0) , E (1) , . . . } ;2) Frame-Oriented Problem (FOP): the focus is on singleframes and the goal of this sub-problem is to determinethe transmission durations and powers that minimizeEq. (9) given the energy consumed in that slot.In practice, the two problems are tightly coupled: EAPdefines the energy allocation to use in every slot, which isused by FOP to determine (9); on the other hand, the outputof FOP affects the choice of EAP (see (8)). In the next twosections, we discuss these two problems and how they areinterrelated. III. F
RAME -O RIENTED P ROBLEM
In this section, we present FOP, which defines the trans-mission durations and powers in a specific frame when theenergy consumption is given. According to (9), FOP envisagesa conservative approach and aims at minimizing the maximumnormalized distortion among all users.
A. Optimization Problem
The amount of energy consumed in a single frame, namely E ( k ) i , is given for all users and its optimal allocation isdetermined by EAP, which we will discuss in Section IV.Since FOP addresses a single frame, for ease of notation wewill omit the dependence on the time index k throughout thissection. Accordingly, boldface letters refer to column vectorsthat span over the N users for the considered frame. Theoptimization problem we set up and solve is the following:FOP: min τ , L , P tx max i ∈N D i D th ,i , (10a) subject to D i = b i (cid:18)(cid:18) L ,i L i (cid:19) α i − (cid:19) ≤ D th ,i , ∀ i, (10b) L i ≤ τ i r i , ∀ i, (10c) E ,i L i + β i + ( P tx ,i + E C,i ) τ i ≤ E i , ∀ i, (10d) P min ,i ≤ P tx ,i ≤ P max ,i , ∀ i, (10e) ≤ L i ≤ L ,i , ∀ i, (10f) N (cid:88) i =1 τ i ≤ T. (10g)The objective function represents a min max problem inorder to guarantee fairness among the users in the network.Constraint (10b) represents the distortion as defined in (2).Inequality (10c) is a capacity constraint derived from theShannon-Hartley theorem (see (1)). However, since larger datarates lead to smaller transmission times for a given datasize, the previous constraint can be taken with equality. Bydoing so, L can be removed from the optimization variablesand expressed as a function of P tx and τ . Inequality (10d)represents the relation between the given energy E i andthe consumed one (see (7)). Without loss of generality, weset Constraint (10d) with equality, as otherwise a positiveamount of energy would be wasted. We also set the real-istic bounds P min ,i and P max ,i on the transmission powerto reflect the physical transmission capabilities of a device. Constraint (10f) imposes that the number of transmitted bits L i is not larger than the number of generated bits L ,i . Finally,the last constraint combines together all the users in a TDMAfashion. Note that, without (10g), FOP could be decomposedinto N separate problems. B. Solution of
FOPWe now describe how to solve FOP. First, note that theobjective function can be formulated in an equivalent way byintroducing an auxiliary optimization variable Γ as follows:FOP Γ : minΓ , τ , P tx Γ , (11a)subject to D i D th ,i ≤ Γ , ∀ i, (11b)Constraints (10b) − (10g) . (11c)The optimal solution of FOP Γ , namely Γ (cid:63) , depends onthe energy allocated to each user by EAP in the consideredframe since the optimization variables are interrelated throughEq. (10d), and thus we express it as Γ (cid:63) = f FOP ( E ) . Noticethat we are only interested in solutions Γ (cid:63) ≤ (i.e., everydistortion is below threshold), as will be explained in Sec-tion III-C. We now propose an efficient technique to extractone optimal solution. Lemma 1.
In at least one optimal solution, Constraint (11b) is satisfied with equality for every i . Formally, we should also consider the case P tx ,i = 0 , but this would leadto an arbitrarily large distortion, which does not represent a relevant case forour study. i ( P tx ,i ) No solution P E i tx ,i ≡ P max ,i > x min P E i tx ,i ≡ P max ,i < x min Figure 1: Function g i ( P tx ,i ) . The dash-line represents different values of W/L i ( E i − E ,i L i + β i ) . The empty circle and triangle markers represent x min and P max ,i , respectively. The black square markers represent P E i tx ,i . Proof.
See Appendix A. (cid:4)
Throughout this subsection, consider a fixed Γ . First of all,notice that, the higher P tx ,i , the shorter τ i , according to (10c).Therefore, with higher transmission powers, it is more likelyto satisfy the constraint on the frame duration (10g). Based onthis consideration, we choose the highest P tx ,i that satisfiesboth (10d) and (10e). Hence, if we combine (10c) taken withequality and (10d), we obtain: g i ( P tx ,i ) ≤ WL i ( E i − E ,i L i + β i ) , (12)where we defined g i ( x ) (cid:44) ( x + E C,i ) / log (1 + h i x ) (seeFigure 1). Note that all the terms on the right-hand side arefixed since E i is given, L i is derived from Γ through Lemma 1and the remaining are system parameters. It can be shownthat g i ( x ) is a decreasing-increasing function of x and thatit admits only one minimum. We now make the followingtechnical assumption. Assumption 1. x min def = arg min x { g i ( x ) } < P max ,i , ∀ i . When the previous assumption is not satisfied, we choose P max ,i as transmission power. A method to find the optimal Γ (cid:63) is then based on the following reasoning. Let P E i tx ,i be themaximum power at which (12) is satisfied with equality. Then,the optimal P tx ,i will be P (cid:63) tx ,i = min { P max ,i , P E i tx ,i } . (13)If P E i tx ,i does not exist or P E i tx ,i < P min ,(cid:96) , then no solutionexists and the problem is infeasible for the given Γ .We now present a key result that will be used to solve FOP. Theorem 1. If FOP Γ is feasible for a fixed Γ (cid:48) ≤ , then FOP Γ is feasible for all Γ (cid:48)(cid:48) such that Γ (cid:48) ≤ Γ (cid:48)(cid:48) ≤ .Proof. See Appendix B. (cid:4)
Theorem 1 implies the following corollary.
Corollary 1. Γ (cid:63) can be found with a bisection search overthe interval [0 , . Thus few operations are required to find Γ (cid:63) with a very highprecision. When Γ (cid:63) has been found, all the other parameterscan be found using Lemma 1 and Eq. (13). We recall that choosing higher P tx ,i leads to lower τ i C. Infeasibility of the Problem
We consider FOP to be infeasible if either of these twoconditions occurs: i1) at least one constraint is not satisfied,i.e., there exists no allocation of τ and P tx that allows all usersto transmit their packets within the frame duration and withthe assigned energy levels, i2) the constraints are satisfied but Γ (cid:63) > , i.e., ∃ i ∈ N : D i > D th ,i , which can be interpretedas a violation of the QoS constraint.Since we consider hard constraints, if FOP were infeasible,the only strategies available would be: 1) allocating a largeramount of energy, 2) choosing a longer frame duration, or 3)removing some users. In this paper, we study technique 1)in Section IV, where we solve the energy allocation problem,and leave 2) and 3) as part of our future work. D. Notes on Convexity
Solving the energy allocation problem in the next sectionwould be much easier if f ( k )FOP ( E ( k ) ) were convex in E ( k ) . Inthis subsection we prove the convexity for the particular caseof low-SNR regime. While from our numerical evaluation thisproperty seems to hold in general, a formal proof of this factis left for future work. In any case, for the numerical results,we approximated f ( k )FOP ( E ( k ) ) with a convex function in orderto correctly solve EAP. Theorem 2.
In the low-SNR regime, FOP is convex in theinput energy E ( k ) .Proof. See Appendix C. (cid:4)
IV. E
NERGY -A LLOCATION P ROBLEM (EAP)Network lifetime and average maximum distortion of thenetwork are conflicting objectives (see (8)). EAP aims atfinding the optimal energy allocation over time that balancesthese two quantities, according to the weight σ . A. Optimization Problem
Assume the network lifetime n is fixed. In this case, theoptimization problem of Eq. (8) becomesEAP: min E n n (cid:88) k =1 f ( k )FOP ( E ( k ) ) , (14a)subject to E ( k ) i ≤ B (0) i − k − (cid:88) j =1 E ( j ) i , ∀ i, ∀ k, (14b) f ( k )FOP ( E ( k ) ) is feasible , ∀ k. (14c)The objective function (14a) represents the average over n frames of the maximum distortion achievable in every frame.It uses the function f ( k )FOP ( · ) as defined in Subsection III-Band is convex according to Subsection III-D. The size of theoptimization variable E is known, as n is fixed (see Eq. (8)).Eq. (14b) is the energy causality constraint, which should besatisfied for every frame and for all users, and can be rewrittenin equivalent but simpler form as n (cid:88) k =1 E ( k ) i ≤ B (0) i , ∀ i. (15)inally, according to Subsection III-C, the last constraintimposes that FOP is feasible for every frame and for all users(see Section III-C). Note that the constraints induce a convexfeasibility set because f ( k )FOP ( E ( k ) ) is convex in all the entriesof E ( k ) .In summary, EAP is a convex optimization problem andbased on this observation we now propose a technique to solveit. Consider matrix E , and focus on the following problem, inwhich we optimize the sequence E (1) (cid:96) , . . . , E ( n ) (cid:96) and keep allthe other variables fixed:EAP (cid:96) : min E (cid:96) n (cid:88) k =1 f ( k )FOP ( E ( k ) ) , (16a)subject to n (cid:88) k =1 E ( k ) (cid:96) ≤ B (0) (cid:96) , (16b) E ( k ) (cid:96) ≤ E ( k ) (cid:96) ≤ E ( k ) (cid:96) , ∀ k. (16c) E ( k ) (cid:96) is defined as the minimum amount of energy that node (cid:96) should use in frame k to obtain a feasible solution. Indeed,if E ( k ) (cid:96) were too low, it would not be possible to satisfythe distortion constraint (10b) or the time constraint (10g)of FOP. Similarly, E ( k ) (cid:96) is the energy value such that, forany E ( k ) (cid:96) ≥ E ( k ) (cid:96) , the objective function does not de-crease further (i.e., after level E ( k ) (cid:96) , using more energy isuseless). The values of E ( k ) (cid:96) and E ( k ) (cid:96) strictly depend on { E ( k )1 , . . . , E ( k ) (cid:96) − , E ( k ) (cid:96) +1 , . . . , E ( k ) N } . As discussed in Subsec-tion III-D, the solution of FOP for frame k is convex in E ( k ) (cid:96) and, in particular, it is strictly convex in ( E ( k ) (cid:96) , E ( k ) (cid:96) ) .Thus, EAP (cid:96) can be solved in the dual domain by using theLagrangian: max λ, E (cid:96) n (cid:88) k =1 f ( k )FOP ( E ( k ) ) − λ (cid:16) n (cid:88) k =1 E ( k ) (cid:96) − B (0) (cid:96) (cid:17) , (17a)subject to λ ≥ , (17b) E ( k ) (cid:96) ≤ E ( k ) (cid:96) ≤ E ( k ) (cid:96) , ∀ k. (17c)The Karush–Kuhn–Tucker conditions lead to E ( k ) (cid:96) = max { E ( k ) (cid:96) , min { E ( k ) (cid:96) , θ − ( λ ) }} , (18) θ ( E ( k ) (cid:96) ) (cid:44) ∂f ( k )FOP ( E ( k ) ) ∂E ( k ) (cid:96) , (19)where λ is such that (cid:80) nk =1 E ( k ) (cid:96) = B (0) (cid:96) .Since EAP (cid:96) focuses on the optimization of one user at atime, we propose Algorithm 1 to solve the general problem.Lines 4-11 perform the alternate optimization. In Line 5we use matrix E to solve (17) and update its (cid:96) -th row.Lines 6-11 distribute in a random fashion the residual energy B (0) (cid:96) − (cid:80) nk =1 E ( k ) (cid:96) in all the slots where E ( k ) (cid:96) is equal to ¯ E ( k ) ( χ {·} is the indicator function). Note that this operationdoes not change the distortion level obtained by solvingEAP (cid:96) , but simply provides a new E (cid:96) that allows the alternateoptimization to converge. Algorithm 1
Random Alternate Optimization Initialize a feasible E D ← ∞ while D has not converged do for (cid:96) = 1 , . . . , N do E (cid:96) ← solve EAP (cid:96) ( E ) v ← prob. vector of size (cid:80) k χ { E ( k ) (cid:96) = E ( k ) } S ← (cid:80) nk =1 E ( k ) (cid:96) v ind ← for k = 1 , . . . , n such that E ( k ) (cid:96) = E ( k ) do E ( k ) (cid:96) ← v ( v ind ) · ( B (0) (cid:96) − S ) v ind ← v ind + 1 D ← /n (cid:80) nk =1 f ( k )FOP ( E ( k ) ) We have the following result.
Theorem 3.
The alternate optimization approach of Algo-rithm 1, in which only user (cid:96) is considered in a single step,leads to the optimal solution.Proof.
See Appendix D. (cid:4)
In summary, EAP solves the energy allocation problem overtime using an alternate optimization procedure. At every stepof the algorithm (Line 5), EAP (cid:96) is solved, and FOP is invokedmultiple times to evaluate the derivative in (19), which relatesthe allocated energy to the corresponding distortion metric.This is iterated over all possible values of n and, by tuning σ , the network designer can choose a point in the trade-offbetween lifetime and QoS.V. N UMERICAL E VALUATION
In this section we show how the system parameters influencethe distortion of the system. We consider five groups of nodeswith different distortion curves placed at a fixed distance d from the BS.If not otherwise stated, we use the following parameters.The frame duration T is . ms. The channel gains arecomputed using the standard path loss model with a path-lossexponent equal to . (e.g., as in an urban scenario) and acentral frequency MHz. The bandwidth is W = 125 kHz,and the overall noise power is − dBm. The parameters ofthe distortion curves are α i ∈ [0 . , . , . , . , . and b i = [19 . , . , . , . , . , which have been derivedthrough empirical fittings of the realistic rate-distortion curvesof [15]. The energy consumption model assumes ε C ,i =5 · − W, β = 10 − J for all frames, and an initial batterylevel B ,i = 5 mJ. The minimum and maximum powerconsumptions are P min ,i = 0 W and P max ,i = 25 mW. Finally,we impose a distortion threshold D th = 8% for packets withfixed size L ,i = 500 bits. These values are not associatedto any specific protocol, but are reasonable and suitable forWSNs.In Figure 2 we plot the distortion and the lifetime obtainedas solutions of the optimization problem (8) for differentvalues of the number of nodes N , the curves have beenobtained by changing the weight factor σ . The distortiontends to increase with the lifetime, as expected, since smaller igure 2: Optimal normalized distortion as a function of the lifetime n fordifferent number of nodes when d = 250 m.Figure 3: Optimal normalized distortion as a function of the lifetime n fordifferent distances when N = 10 . amounts of energy can be allocated in each frame and thusnodes must compress more to transmit their data. For smallvalues of n , the graphs are constant because not all the energyin the batteries is used or because a zero distortion is achieved.Clearly, it is always better to choose the right extremes of theconstant regions rather than the other points since they providethe same QoS with longer lifetimes. The maximum lifetime isreached when the problem becomes infeasible (see conditioni2) of Section III-C), i.e., the curves in Figure 2 reach value and no energy allocation can satisfy all constraints of FOP withan acceptable distortion for all the frames of the consideredlifetime. We also remark that the lifetime strongly depends onthe initial battery level. However, even if larger batteries wereconsidered, the trend of the distortion curves would remainthe same. Note that, because of the symmetry of our setup,all curves coincide after a certain value of n . This happensbecause the only way to reach high values of n is to assignlow energy to every frame, which in turn corresponds to short τ ( k ) i and thus Constraint (10g) is always satisfied. Future workincludes the study of more complex scenarios using differentnode locations and Montecarlo simulations.In Figure 3, we show how the distance d (equal for allnodes) influences the network performance. As d increases,the performance of the system clearly decays and, when d is extremely high, the network cannot operate at all. It isinteresting to note that, because of the strong path loss, thehigher d , the higher the gap between two adjacent curves.Finally, Figure 4 shows the comparison between the optimal Figure 4: Normalized distortion evaluated optimally and with fixed transmis-sion durations as a function of the lifetime n when N = 10 and d = 250 m. approach of Eq. (8) which evaluates the transmission durationsaccording to FOP, and a sub-optimal approach which usesfixed durations (i.e., τ ( k ) i = T /N, ∀ i, k ). Note that all nodesare located at the same distance, thus there is no near-far effect(which however would further support our approach). Evenin this case, because of the different rate-distortion curves,the optimal solution may provide much better performancethan the naive scheme, which confirms the importance ofadapting the protocol parameters to the characteristics of thedata sources in a real deployment. In all cases we analyzed,the distortion levels stay below the tolerable threshold, i.e.,the sink can recover all data with pre-defined accuracy. Whenthe normalized distortion is 0, the information gathered fromthe sensors is sent over the channel with the highest possiblequality (i.e., no loss), whereas a normalized distortion closeto 1 corresponds to a very rough quantization of the signalsbeing sent. The correspondence between the actual values ofthe normalized distortion and the quality perceived by the userwill depend on the specific application, and a detailed studyof this relationship will be left as future work.VI. C ONCLUSIONS
In this paper, we studied a TDMA-based strategy forbattery-powered devices with QoS requirements. We definedan optimal scheduling policy that jointly considers sourcecoding operations and energy constraints, by determining thedata compression ratio and the energy allocation for each nodein each time frame, respectively. In particular, using convexand alternate programming, we presented an efficient methodto minimize the average of the maximum distortions for agiven lifetime. Numerical results show the importance of usingoptimized protocols to compensate the near-far effect.Future work includes more detailed numerical results, thestudy of latency on the data transmission, and the investigationof online schemes to derive the optimal performance.A
CKNOWLEDGMENTS
This work was partially supported by Intel’s CorporateResearch Council. A
PPENDIX AP ROOF OF L EMMA (cid:96) ∈ N suchthat a distortion D (cid:48) (cid:96) < Γ (cid:63) D th ,(cid:96) leads to an optimal solution,ut D (cid:48)(cid:48) (cid:96) = Γ (cid:63) D th ,(cid:96) does not. According to (2), this impliesthat longer (higher quality) packets are used in the firstcase, i.e., L (cid:48) (cid:96) > L (cid:48)(cid:48) (cid:96) . Then, if we chose P (cid:48)(cid:48) tx ,(cid:96) = P (cid:48) tx ,(cid:96) and τ (cid:48)(cid:48) (cid:96) = τ (cid:48) (cid:96) L (cid:48)(cid:48) (cid:96) /L (cid:48) (cid:96) , all constraints (10b)-(10g) would still besatisfied because τ (cid:48)(cid:48) (cid:96) < τ (cid:48) (cid:96) . Thus we would find a feasibleoptimal solution in which (11b) is satisfied with equality, i.e.,the initial assumption must be wrong.A PPENDIX BP ROOF OF T HEOREM (cid:96) . According to (13) and thanks toAssumption 1, the point P (cid:63) tx ,(cid:96) always falls in the increasingright branch of g (cid:96) ( x ) (otherwise it is equal to P max ,(cid:96) ).By assumption we have Γ (cid:48) ≤ Γ (cid:48)(cid:48) , which, using Defini-tion (2), implies L (cid:48) (cid:96) ≥ L (cid:48)(cid:48) (cid:96) . The right-hand side of (12) isa decreasing function of L (cid:96) , thus, by naming P (cid:48) and P (cid:48)(cid:48) the points P E i tx ,i corresponding to Γ (cid:48) and Γ (cid:48)(cid:48) , respectively,we obtain P (cid:48) ≤ P (cid:48)(cid:48) . Also, since P (cid:48) ≥ P min ,(cid:96) exists byassumption, also P (cid:48)(cid:48) exists.Finally, since the same energy E (cid:96) is considered in the twocases, P (cid:48) leads to a transmission duration τ (cid:48) (cid:96) longer thanwhat can be found with P (cid:48)(cid:48) . Thus, since with P (cid:48) all theconstraints of Problem (10) were satisfied, also using P (cid:48)(cid:48) theyremain satisfied (in particular, Constraint (10g) is still satisfiedbecause τ (cid:48)(cid:48) (cid:96) ≤ τ (cid:48) (cid:96) ). A PPENDIX CP ROOF OF T HEOREM L i (cid:39) τ i W h i P tx ,i log e , ∀ i. (20)By combining it with constraint (10d), we obtain that thesize L i of the transmitted packet is linear in the given energy E ( k ) i for each user. Then, according to (10b), and taking intoaccount that α i > ∀ i , D i is convex in E ( k ) i , ∀ i . Finally, sincethe maximum operation preserves convexity, f ( k )FOP ( E ( k ) ) isconvex. A PPENDIX DP ROOF OF T HEOREM Ψ (cid:63) , exists but,potentially, with multiple minimum points. We now show thatAlgorithm 1 produces a non-increasing sequence of distortionvalues ( Ψ ≥ Ψ ≥ . . . ), and, unless the minimum is achieved,there exists a non-null probability that Ψ m > Ψ m +1 , for some m . Since the minimum is unique, the sequence converges to Ψ (cid:63) thanks to [16, Proposition 2.7.1].Consider the generic step m of the algorithm, in whichthe optimization revolves around node (cid:96) , i.e., we focus thesequence E (cid:96) . As described in Section III-D, the objective Γ ( k ) (cid:96) is a convex function of E ( k ) (cid:96) . In particular, Γ ( k ) (cid:96) is strictlyconvex in [ E ( k ) (cid:96) , E ( k ) (cid:96) ] , and constant for E ( k ) (cid:96) ≥ E ( k ) (cid:96) . Thus,we can simplify the problem by restricting our attention tothe strictly decreasing region, and handle the constant regionsubsequently.Since the sum of convex functions is convex, when weaim at optimizing (cid:80) nk =1 Γ ( k ) (cid:96) in [ E (cid:96) , E (cid:96) ] , this is a convex optimization problem and can be solved using (17). Let E (cid:63)(cid:96) be the solution. Two cases should be considered:1) All the elements of E (cid:63)(cid:96) fall inside the strictly convexregion [ E (cid:96) , E (cid:96) ) ( E (cid:96) is excluded). This may happenbecause of the battery constraints. In this case, thereis only one optimal solution, and no other actions arerequired for the current step ( S = 0 in Line 7 of thealgorithm);2) Some of the elements fall at the beginning of theconstant region, i.e., E ( k ) (cid:96) = E ( k ) (cid:96) for some k ∈ K .In this case, also other solutions may be optimal forthe current iteration, since using all the feasible energycombinations with E ( k ) (cid:96) (cid:23) E ( k ) (cid:96) for k ∈ K lead to thesame solution. However, although all these sequencesprovide the same Ψ m in the current step, they mayinfluence the future values Ψ m +1 , Ψ m +2 , etc. Then, twosubcases should be distinguished: in the following N − steps of the algorithm (i.e., the next time that node (cid:96) is examined), the sequence of Ψ has either strictlydecreased, or remained constant. In the former case, thealgorithm proceeds toward the optimal solution. In thelatter, the algorithm cyclically returns to point m , thusthe sequence of Ψ has not improved. In this case, wechoose other points E ( k ) (cid:96) > E ( k ) (cid:96) , k ∈ K (e.g., with arandom approach as described in Lines 6-11) and repeatthe procedure. In an infinite horizon, all the possibleenergy combinations have been tested with a non-nullprobability, thus the algorithm has proceeded toward anoptimal solution.In the worst case scenario, Algorithm 1 may degeneratein almost an exhaustive search; however, in practical cases,very few iterations are required, and the algorithm rapidlyconverges. R EFERENCES[1] G. Lazarou, J. Li, and J. Picone, “A cluster-based power-efficientMAC scheme for event-driven sensing applications,”
Elsevier Ad HocNetworks , vol. 5, no. 7, pp. 1017–1030, Sept. 2007.[2] Y. Liang, “Efficient temporal compression in wireless sensor networks,”
Proc. IEEE 36th Conference on Local Computer Networks (LCN) , pp.466–474, Oct. 2011.[3] A. Ha´c,
Wireless Sensor Network Designs . John wiley & Sons, 2003.[4] C. G. Cassandras, T. Wang, and S. Pourazarm, “Optimal routing andenergy allocation for lifetime maximization of wireless sensor networkswith nonideal batteries,”
IEEE Transactions on Control of NetworkSystems , vol. 1, no. 1, pp. 86–98, Mar. 2014.[5] C. Ho and R. Zhang, “Optimal energy allocation for wireless communi-cations with energy harvesting constraints,”
IEEE Transactions on SignalProcessing , vol. 60, no. 9, pp. 4808–4818, Sept. 2012.[6] A. Fu, E. Modiano, and J. Tsitsiklis, “Optimal energy allocation fordelay-constrained data transmission over a time-varying channel,” in
Proc. IEEE 22nd Int. Conference on Computer Communications (IN-FOCOM) , vol. 2, Mar. 2003, pp. 1095–1105.[7] A. Biason and M. Zorzi, “Joint transmission and energy transfer policiesfor energy harvesting devices with finite batteries,”
IEEE J. Sel. Areasin Commun. , vol. 33, no. 12, pp. 2626–2640, Dec. 2015.[8] M. Natkaniec, K. Kosek-Szott, S. Szott, and G. Bianchi, “A survey ofmedium access mechanisms for providing QoS in ad-hoc networks,”
IEEE Communications Surveys and Tutorials , vol. 15, no. 2, pp. 592–620, Second Quarter 2013.[9] M. A. Yigitel, O. Durmaz Incel, and C. Ersoy, “QoS-aware MACprotocols for wireless sensor networks: A survey,”
Computer Networks ,vol. 55, no. 8, pp. 1982–2004, June 2011.10] S. Knorn, S. Dey, A. Ahlen, and D. E. Quevedo, “Distortion mini-mization in multi-sensor estimation using energy harvesting and energysharing,”
IEEE Transactions on Signal Processing , vol. 63, no. 11, pp.2848–2863, June 2015.[11] C. Tapparello, O. Simeone, and M. Rossi, “Dynamic compression-transmission for energy-harvesting multihop networks with correlatedsources,”
IEEE/ACM Transactions on Networking , vol. 22, no. 6, pp.1729–1741, Dec. 2014.[12] D. Dujovne, T. Watteyne, X. Vilajosana, and P. Thubert, “6TiSCH:deterministic IP-enabled industrial internet (of things),”
IEEE Commu-nications Magazine , vol. 52, no. 12, pp. 36–41, Dec. 2014.[13] J. Yang and S. Ulukus, “Optimal packet scheduling in an energy harvest-ing communication system,”
IEEE Transactions on Communications ,vol. 60, no. 1, pp. 220–230, Jan. 2012.[14] T. Berger,
Rate-Distortion Theory . John Wiley & Sons, Inc., 2003.[15] D. Zordan, B. Martinez, V. Ignasi, and M. Rossi, “On the performance oflossy compression schemes for energy constrained sensor networking,”
ACM Transactions on Sensor Networks , vol. 11, no. 1, pp. 15:1–15:34,Nov. 2014.[16] D. Bertsekas,