Medium Access Control Protocols for Wireless Sensor Networks with Energy Harvesting
aa r X i v : . [ c s . I T ] D ec Medium Access Control Protocols for WirelessSensor Networks with Energy Harvesting
F. Iannello, O. Simeone and U. Spagnolini
Abstract
The design of Medium Access Control (MAC) protocols for wireless sensor networks (WSNs) hasbeen conventionally tackled by assuming battery-powered devices and by adopting the network lifetimeas the main performance criterion. While WSNs operated by energy-harvesting (EH) devices are notlimited by network lifetime, they pose new design challenges due to the uncertain amount of harvestableenergy. Novel design criteria are thus required to capture the trade-offs between the potentially infinitenetwork lifetime and the uncertain energy availability.This paper addresses the analysis and design of WSNs with EH devices by focusing on conventionalMAC protocols, namely TDMA, Framed-ALOHA (FA) and Dynamic-FA (DFA), and by accountingfor the performance trade-offs and design issues arising due to EH. A novel metric, referred to as delivery probability , is introduced to measure the capability of a MAC protocol to deliver the measureof any sensor in the network to the intended destination (or fusion center , FC). The interplay betweendelivery efficiency and time efficiency (i.e., the data collection rate at the FC), is investigated analyticallyusing Markov models. Numerical results validate the analysis and emphasize the critical importance ofaccounting for both delivery probability and time efficiency in the design of EH-WSNs.
Index Terms
Wireless sensor networks, multiaccess communication, energy harvesting, dynamic framed ALOHA.
F. Iannello is with both Politecnico di Milano, Milan, 20133, Italy and the Center for Wireless Communications and SignalProcessing Research (CWCSPR), New Jersey Institute of Technology (NJIT), Newark, New Jersey 07102-1982 USA (e-mail:[email protected]). U. Spagnolini is with Politecnico di Milano. O. Simeone is with the CWCSPR, NJIT.
I. I
NTRODUCTION
Recent advances in low-power electronics and energy-harvesting (EH) technologies enablethe design of self-sustained devices that collect part, or all, of the needed energy from thesurrounding environment. Several systems can take advantage of EH technologies, ranging fromportable devices to wireless sensor networks (WSNs) [1]. However, EH devices open new designissues that are different from conventional battery-powered (BP) systems [2], where the mainconcern is the network lifetime [3]. In fact, EH potentially allows for perpetual operation of thenetwork, but it might not guarantee short-term activities due to temporary energy shortages [2].This calls for the development of energy management techniques tailored to the EH dynamics.While such techniques have been mostly studied at a single-device level [4], in wireless scenarioswhere multiple EH devices interact with each other, the design of EH-aware solutions needs toaccount for a system-level approach [5][6]. This is the motivation of this work.In this paper, we focus on system-level design considerations for WSNs operated by EH-capable devices. In particular, we address the analysis and design of medium access control(MAC) protocols for single-hop WSNs (see Fig. 1) where a fusion center (FC) collects datafrom sensors in its surrounding. Specifically, we investigate how performance and design of MACprotocols routinely used in WSNs, such as TDMA [7], Framed-ALOHA (FA) and Dynamic-FA(DFA) [8], are influenced by the discontinuous energy availability in EH-powered devices.
A. State of the Art
In recent years, WSNs with EH-capable nodes have been attracting a lot of attention, alsoat commercial level. To provide some examples, the Enocean Alliance proposes to use a MACprotocol for EH devices based on pure ALOHA strategies [11], while an enhanced self-poweredRFID tag created by Intel, referred to as WISP [12], has been conceived to work with the EPCGen 2 standard [13] that adopts a FA-like MAC protocol.However, while performance analysis of MAC protocols in BP-WSNs have been investigatedin depth (see e.g., [7][8][14]), analyses of MAC protocols with EH are hardly available. A notable h ( n ) h m ( n ) h M ( n ) S E ( n ) E H,1 ( n ) S M E M ( n ) E H,M ( n ) FC ••• S m E m ( n ) E H,m ( n ) EnergyHarvestingUnit (EHU)Energy StorageDevice (ESD)
SensorCPU E m ( n ) E H,m ( n ) S m Radio
Figure 1. WSN with a single Fusion Center (FC) gathering data from M sensors, which are equipped with an energy storagedevice (ESD) and an energy-harvesting unit (EHU). exception is [6], where data queue stability has been studied for TDMA and carrier sense multipleaccess (CSMA) protocols in EH networks. We remark that routing for EH networks has insteadreceived more attention, see e.g., [2][15]. B. Contributions
In this paper we consider the design and analysis of TDMA, FA and DFA MAC protocolsin the light of the novel challenges introduced by EH. In Sec. III we propose to measure thesystem performance in terms of the trade-off between the delivery probability , which accountsfor the number of sensors’ measurements successfully reported to the FC, and the time efficiency ,which measures the rate of data collection at the FC (formal definitions are in Sec. III). We thenintroduce an analytical framework in Sec. IV and Sec. V to assess the performance of EH-WSNsin terms of the mentioned trade-off for TDMA, FA and DFA MAC protocols. In Sec. VI wetackle the critical issue in ALOHA-based MAC protocols of estimating the number of EH sensorsinvolved in transmission, referred to as backlog , by proposing a practical reduced-complexityalgorithm. Finally, we present extensive numerical simulations in Sec. VII to get insights intothe MAC protocol design trade-offs, and to validate the analytical derivations.
II. S
YSTEM M ODEL
In this paper, we consider a single-hop WSN with a FC surrounded by M wireless sensorslabeled as S , S , ..., S M (see Fig. 1). Each sensor (or user ) is equipped with an EH unit (EHU)and an energy storage device (ESD), where the latter is used to store the energy harvested by theEHU. The FC retrieves measurements from sensors via periodic inventory rounds (IRs), onceevery T int seconds [ s ] . Each IR is started by the FC by transmitting an initial query command (Q),which provides both synchronization and instructions to sensors on how to access the channel.Time is slotted, with each slot lasting T s [s] . The effective duration of the n th IR, during whichcommunication between the FC and the sensors takes place, is denoted by T IR ( n ) . We assumethat T IR ( n ) ≪ T int for all IR n , and also that the query duration is negligible, so that the ratio T IR ( n ) /T s indicates the total number of slots allocated by the FC during the n th IR.In every IR, each sensor has a new measure to transmit with probability α , independentlyof other sensors and previous IRs. If a new measure is available, the sensor will mandatoryattempt to report it successfully to the FC as long as enough energy is stored in its ESD (seeSec. II-B for details). Each measure is the payload of a packet, whose transmission fits withinthe slot duration T s . Sensors’ transmissions within each IR are organized into frames , each ofwhich is composed of a number of slots that is selected by the FC. Depending on the adoptedMAC protocol, any user that needs to (and can) transmit in a frame either chooses or is assigneda single slot within the frame for transmission as it will be detailed below. Moreover, after auser has successfully transmitted its packet to the FC, it first receives an acknowledge (ACK)of negligible duration by the FC and then it becomes inactive for the remaining of the IR. Weemphasize that the FC knows neither the number of sensors with a new measure to transmit,nor the state of sensors’ ESDs. A. Interference Model
We consider interference-limited communication scenarios where the downlink packets trans-mitted by the FC are always correctly received (error-free) by the sensors, while uplink packetstransmitted by the sensors to the FC are subject to communication errors due to possibleinterference arising from collisions with other transmitting sensors. The uplink channel powergain for the m th sensor during the n th IR is h m ( n ) . Channel gain h m ( n ) is assumed to beconstant over the entire IR but subject to random independent and identically distributed (i.i.d.)fading across IRs and sensors, with pdf f h ( · ) and normalized such that E [ h m ( n )] = 1 , for all n, m . In the presence of simultaneous transmissions within the same slot during the k th frameof the n th IR, a sensor, say S m , is correctly received by the FC if and only if its instantaneoussignal-to-interference ratio (SIR) γ m,k ( n ) is larger than a given threshold γ th , i.e., if γ m,k ( n ) = h m ( n ) P l ∈I m,k ( n ) h l ( n ) ≥ γ th , (1)where I m,k ( n ) denotes the set of sensors that transmit in the same slot selected by S m in frame k and IR n . We assume γ th > dB so that, in case a slot is selected by more than one sensor,at most one of the colliding sensor can be successfully decoded in the slot.According to the interference model (1), any slot can be: empty when it is not selected byany sensor; collided when it is chosen by more than one sensors but none of them transmitssuccessfully; successful when one sensor transmits successfully possibly in the presence of other(interfering) users. Successful transmission in the presence of interfering users within the sameslot is often referred to as capture effect [14]. Remark 1 : Errors in the decoding of downlink query packets can be accounted for throughthe parameter α as well. In fact, let α Q be the probability that a user correctly decodes thedownlink packet sent by the FC at the beginning of an IR. Moreover, assume that downlinkdecoding errors are i.i.d. across sensors and IRs, and let α N be the probability that a user has anew measure to transmit in any IR. Then, the probability that any user S m has a new measureand correctly decodes the FC’s query is given by the product α = α Q α N . B. ESD and Energy Consumption Models
We consider a discrete ESD with N + 1 energy levels in the set E = { , δ, δ, ..., N δ } , where δ is referred to as energy unit . Let E m ( n ) ∈ E be the energy stored in the ESD of the m thuser at the beginning of the n th IR. Energy E m ( n ) is a random variable that is the result of theEH process and the energy consumption of the sensor across IRs; its probability mass function(pmf) is p E ( n ) ( · ) and the corresponding complementary cumulative distribution function (ccdf)is G E ( n ) ( x ) = Pr[ E m ( n ) ≥ x ] . Note that, the initial energy distribution p E (1) ( · ) is given, whilethe evolution of the pmf p E ( n ) ( · ) for n > depends on both the MAC protocol and EH process.We assume that each time a sensor transmits a packet it consumes an energy ε , which accountsfor the energy consumed in the: a ) reception of the FC’s query that starts the frame (see Fig.2); b ) transmission; c ) reception of FC’s ACK or not ACK (NACK) packet. At the beginning ofeach IR, a sensor with a new measure to transmit can participate to the current IR only if theenergy stored in its ESD is at least ε . Let ε δ = ε/δ be the number of energy units δ requiredfor transmission, where ε δ is assumed to be an integer value without loss of generality. Let F ε = N δ/ε = N/ε δ be the (normalized) capacity of the ESD, which is assumed to be an integerindicating the maximum number of (re)transmissions allowed by a fully charged ESD. T int Q FC query FC ACK T int QTDMA S S S D D D Q S S D E D T IR TD T IR TD T IR ( n+1 ) DFA
QDFA T IR ( n ) DFA S ,S S Q S S C E D D D Q S ,S Q S C E E DS m Sensor packet
E/D/C
Empty/ Successful/ Collided slotIR n IR n +1 = { S , S , S } = { S , S } = { S , S , S } = { S , S } = { S , S } = { S } Figure 2. Examples for TDMA and DFA MAC protocols for M = 3 . FA is not depicted since it is a special case of DFAwith only one frame. The backlog for each frame is indicated above each query. Some sensors might not be in the backlog dueto energy shortage and/or absence of a new measure to report. C. Energy Harvesting Model
During the time T int between the n th and ( n + 1) th IRs the m th sensor S m harvests an energy E H,m ( n ) , which is modeled as a discrete random variable, i.i.d. over IRs and sensors, with pmf q i = Pr[ E H,m ( n ) = iδ ] , with i ∈ { , , , ... } , and for all m and n . For technical reasons that wediscuss in Sec. V-B, we assume that the probability q and q of harvesting zero and one energyunit respectively, are both strictly positive, namely q > and q > .We assume that the EH dynamics is much slower than the IR duration T IR ( n ) , so that theamount of energy harvested within T IR ( n ) can be considered as negligible with respect to ε (recallalso that T IR ( n ) ≪ T int ). Hence, the only energy that a sensor can actually use throughout anIR is the energy initially available at the beginning of the IR itself (i.e., E m ( n ) ).III. P ERFORMANCE M ETRICS AND M EDIUM A CCESS C ONTROL P ROTOCOLS
We first introduce in Sec. III-A the considered performance metrics, namely delivery proba-bility and time efficiency, and then in Sec. III-B we review the considered MAC protocols.
A. MAC Performance Metrics1) Delivery Probability:
The delivery probability p d ( n ) measures the capability of the MACprotocol to successfully deliver the measure of any sensor, say S m , to the FC during the n th IR p d ( n ) = Pr [ S m transmits successfully in IR n | S m has a new measure in IR n ] . (2)The statistical equivalence of all sensors makes the probability (2) independent of the specificsensor. Notice that a sensor fails to report its measure during an IR if either it has an energyshortage before (re)transmitting the packet correctly, or the MAC protocol does not providethe sensor with sufficient retransmission opportunities. Given the potentially perpetual operationenabled by EH, it is relevant to evaluate the delivery probability when the system is in steady-state . The asymptotic delivery probability is thus obtained by taking the limit of p d ( n ) for largeIR index n , provided that it exists, as p ASd = lim n →∞ p d ( n ) . (3)
2) Time Efficiency:
The time efficiency p t ( n ) measures the probability that any slot allocatedby the MAC within the n th IR is successfully used (i.e., it is neither empty nor collided, seeSec. II-A) p t ( n ) = Pr [ The FC correctly retrieves a packet in any slot of the n th IR ] . (4)By taking the limit of (4) for n → ∞ , we obtain the asymptotic time efficiency p ASt = lim n →∞ p t ( n ) . (5) Remark 2:
Informally speaking, the time efficiency p t ( n ) measures the ratio between the totalnumber of packets successfully received by the FC and the total number of slots allocated by theMAC protocol (i.e., T IR ( n ) /T s , see Sec. II). As it will be shown in Sec. III-B, the IR duration T IR ( n ) is in general a random variable, and consequently, time efficiency p t ( n ) differs frommore conventional definitions of throughput (see e.g., [8]) which measure the number of packetsdelivered over the interval between two successive IRs T int , instead of T IR ( n ) . The rationale for this definition of time efficiency is that it actually captures more effectively the rate of datacollection at the FC. Whereas, the delivery probability accounts for the fraction of users, witha new measure to transmit at the beginning of the current IR, which are able to successfullyreport their payload to the FC within the IR, where delivery failures are due to collisions andenergy shortages.In contention based MACs (e.g., ALOHA), there is a trade-off between delivery probabilityand time efficiency. In fact, increasing the former generally requires the FC to allocate a largernumber of slots in an IR to reduce packet collisions, which in turn decreases the time efficiency. B. MAC Protocols
In this section, we review the standard MAC protocols that we focus on.
1) TDMA:
With the TDMA protocol, each user is pre-assigned an exclusive slot that itcan use in every IR, irrespective of whether it has a measure to deliver or enough energy totransmit. Recall that such information is indeed not available at the FC. Every n th IR is thuscomposed by one frame with M slots and has fixed duration T T DIR = M T s , as shown in Fig. 2.Since TDMA is free of communication errors in the considered interference-limited scenario, itsdelivery probability p d ( n ) is only limited by energy availability and it is thus an upper boundfor ALOHA-based MACs. However, TDMA might not be time efficient due to the many emptyslots when the probability of having a new measure α and/or the EH rate are small.
2) Framed-ALOHA (FA) and Dynamic-FA (DFA):
Hereafter we describe the DFA protocolonly, since FA follows as a special case of DFA with no retransmissions capabilities as discussedbelow. The n th IR, of duration T DF AIR ( n ) , is organized into a set of frames as shown in Fig.2. The backlog B k ( n ) for the k th frame is the set composed of all sensors that simultaneouslysatisfy the following three conditions: i ) have a new measure to transmit in the n th IR; ii ) havetransmitted unsuccessfully (because of collisions) in the previous k − frames (this conditiondoes not apply for frame k = 1 ); iii ) have enough energy left in the ESD to transmit in the k th frame. All the users in the set B k ( n ) , whose cardinality |B k ( n ) | = B k ( n ) is referred to as backlog size , thus attempt transmission during frame k . To make this possible, the FC allocatesa frame of L k ( n ) slots, where L k ( n ) is selected based on the estimate ˆ B k ( n ) of the backlog size B k ( n ) (estimation of B k ( n ) is discussed in Sec. VI) as L k ( n ) = l ρ ˆ B k ( n ) m , (6)where ⌈·⌉ is the upper nearest integer operator, and ρ is a design parameter. Note that, if thebacklog size is B , the probability β ( j, B, L ) that j ≤ B sensors transmit in the same slot in aframe of length L is binomial [16] β ( j, B, L ) = (cid:18) Bj (cid:19) (cid:18) L (cid:19) j (cid:18) − L (cid:19) B − j . (7)Finally, FA is a special case of DFA where only one single frame of size L ( n ) is announcedas retransmissions are not allowed within the same IR.IV. A NALYSIS OF THE
MAC P
ERFORMANCE M ETRICS
In this section we derive the performance metrics defined in Sec. III-A for TDMA, FA andDFA. The analysis is based on two simplifying assumptions: A . Known backlog : the FC knows the backlog size B k ( n ) = |B k ( n ) | before each k th frame; A . Large backlog : the backlog size B k ( n ) , in any IR n and any frame k of size L k ( n ) = ⌈ ρB k ( n ) ⌉ , is large enough to let the probability (7) be approximated by the Poisson distri-bution [16]: β ( j, B k ( n ) , L k ( n )) ≃ e − ρ ρ j j ! . (8)Assumption A . simplifies the analysis as in reality the backlog can only be estimated by theFC (see Sec. VI and Sec. VII for the impact of backlog estimation). Assumption A . is standardand analytically convenient, as it makes the probability β ( j, B k ( n ) , L k ( n )) dependent only onthe ratio ρ between the frame length L k ( n ) and the backlog size B k ( n ) . The assumptions aboveare validated numerically in Sec. VII. A. Delivery Probabilities
Here we derive the delivery probability (2) within any n th IR under the assumptions A . and A . for the considered MAC protocols. The IR index n is dropped to simplify the notation.
1) Delivery Probability for TDMA:
As the TDMA protocol is free of collisions, each sensor S m that has a new measure to report in the current IR cannot deliver its payload to the FC onlywhen it is in energy shortage, namely if E m < ε . Provided that user S m has a new measure totransmit, the delivery probability (2) reduces to p T Dd = Pr [ E m ≥ ε ] = G T DE ( ε ) , (9)which is independent of the sensor index m and dependent only on the ccdf G T DE ( · ) of theenergy stored in sensor ESD at the beginning of the considered IR. The ESD energy distributionfor any arbitrary n th IR is derived in Sec. V.
2) Delivery Probability for FA:
In the FA protocol, each sensor S m that has a new measureto report in the current IR is able to correctly deliver its payload to the FC only if: a) it transmitssuccessfully in the selected slot, possibly in the presence of interfering users provided that itsSIR is γ m, ≥ γ th ; and b) it has enough energy to transmit. From (1), the probability that sensor S m , with S m ∈ B , transmits successfully in the selected slot, given that |I m, | = j users selectthe same slot of S m (thus colliding), is given by p c ( j ) = Pr " h m ≥ γ th j X l =1 h l , (10)where, without loss of generality, we assumed that I m, = { S , ..., S j } , and S m / ∈ I m, , as usersare stochastically equivalent. Under the large backlog assumption A . , the probability that thereare j interfering users is Poisson-distributed (see (8)), and thus the unconditional probability p c that S m captures the selected slot can be approximated as p c ≃ e − ρ ∞ X j =0 ρ j j ! p c ( j ) . (11)Note that, in (11) we also extended the number of possible interfering users up to infinity as p c ( j ) rapidly vanishes for increasing j . Moreover, depending on the channel gain pdf f h ( · ) , probabilities (10) can be calculated either analytically (e.g., when f h ( · ) is exponential, see [17])or numerically.Finally, under assumption A . , the successful transmission event is independent of the ESDenergy levels (which in principle determine the actual backlog size in (7)), and thus the deliveryprobability (2) for the FA protocol can be calculated as the product between the probability G F AE ( ε ) = Pr [ E m ≥ ε ] that sensor S m has enough energy to transmit and the (approximated)capture probability (11) as p F Ad ≃ G F AE ( ε ) e − ρ ∞ X j =0 ρ j j ! p c ( j ) , (12)where the ESD energy ccdf G F AE ( ε ) for any arbitrary n th IR is derived in Sec. V.
3) Delivery Probability for DFA:
DFA is composed of several instances of FA, one for each k th frame of the current IR. As DFA allows retransmissions, we need to calculate the probability p c,k ( j ) that any sensor active during frame k , say S m ∈ B k , transmits successfully in the selectedslot given that there are |I m,k | = j users that transmit in the same slot, with I m,k ⊆ B k . Thecomputation of p c,k ( j ) , for k > , is more involved than (10). In fact, packets collisions introducecorrelation among the channel gains of collided users, as any sensor in the backlog B k , for k > ,might have collided with some other sensors in the set B k . We recall that, even though the channelgains are i.i.d. at the beginning of the IR, they remain fixed for the entire IR.While the exact computation of probabilities p c,k ( j ) is generally cumbersome, the large backlogassumption A . enables some simplifications. Specifically, correlation among channel gains canbe neglected, since for large backlogs it is unlikely that two users collide more than once withinthe same IR. By assuming independence among the channel gains at any frame, calculation of p c,k ( j ) requires only to evaluate the channel gain pdf f ( k ) h ( · ) at the k th frame for any user within B k , which is the same for all users by symmetry. The computation of pdf f ( k ) h ( · ) can be donerecursively, starting from frame k = 1 , so that at frame k we condition on the event that theSIR (1) was γ m,k − < γ th . Under assumption A . , this can be done numerically.Now, let ˜ h ( k ) m , for m ∈ { , ..., M } and k ∈ { , ..., F ε } , be random variables with pdf f ( k ) h ( · ) independent over m , where ˜ h (1) m = h m . The conditional capture probabilities p c,k ( j ) can then beapproximated as (compare to (10)) p c,k ( j ) ≃ Pr " ˜ h ( k ) m ≥ γ th j X l =1 ˜ h ( k ) l , (13)for any m / ∈ { , ..., j } as users are stochastically equivalent. By exploiting the Poisson ap-proximation similarly to (11), the unconditional probability that any user within the backlogsuccessfully transmits in the selected slot during the k th frame becomes p c,k ≃ e − ρ ∞ X j =0 ρ j j ! p c,k ( j ) . (14)Recalling that a user keeps retransmitting its message until it is successfully delivered to theFC, then the successful delivery of a message in a frame is a mutually exclusive event with respectto the delivery in previous frames. Therefore, the probability of transmitting successfully in the k th frame, given that enough energy is available, is p c,k Q k − i =1 (1 − p c,i ) . Finally, by accountingfor the probability G DF AE ( kε ) = Pr [ E m ≥ kε ] of having enough energy in each k th frame, theDFA delivery probability can be obtained, under assumption A . , as p DF Ad ≃ F ε X k =1 G DF AE ( kε ) p c,k k − Y i =1 (1 − p c,i ) , (15)where the ESD energy ccdf G DF AE ( kε ) for any arbitrary n th IR is derived in Sec. V. B. Time Efficiencies
In this section we derive the time efficiency (4) for the three considered protocols.
1) Time Efficiency for TDMA:
Let M m be the event indicating that user S m has a newmeasure to report in the current IR, with Pr[ M m ] = α , then the TDMA time efficiency (4) isgiven by the probability that the m th user has enough energy to transmit and a new measure toreport: p T Dt = Pr [ E m ≥ ε, M m ] = Pr [ E m ≥ ε ] Pr [ M m ] = αG T DE ( ε ) , (16)where we exploited independence between energy availability E m and M m . Note that in principle the backlogs B , B ... are correlated, and therefore the exact p DF Ad should be obtained by averagingover the joint distribution of the backlog sizes. However, the assumption A . removes the dependence on the backlog size.
2) Time Efficiency for FA:
Since we assumed γ th > dB , then when more than one usertransmits within the same slot, only one of them can be decoded successfully, that is, successfultransmissions of different users within the same slot are disjoint events. Therefore, the probabilitythat a slot, simultaneously selected by j users, is successfully used by any of them is given by jp c ( j − , where p c ( j − is (10) by recalling that any user have ( j − interfering users.Furthermore, under assumption A . , the probability that exactly j users select the same slot is e − ρ / ( ρ j j !) , and by summing up over the number of simultaneously transmitting users j we get p F At ≃ e − ρ ∞ X j =1 ρ j j ! jp c ( j −
1) = e − ρ ∞ X j =0 ρ ( j +1) j ! p c ( j ) (17)Note that, a consequence of assumption A . is to make the FA time efficiency (17) independentof the ESD energy distribution. Moreover we remark that, when ρ = 1 , p c ( j ) = 1 for j = 0 and p c ( j ) = 0 for j > , then we have p F At = e − , which is the throughput of slotted ALOHA [8].
3) Time Efficiency for DFA:
The derivation of the DFA time efficiency p DF At follows fromthe FA time efficiency by accounting for the presence of multiple frames within an IR similarlyto Sec. IV-A3. Since the time efficiency is defined over multiple frames, we first derive the timeefficiency in the k th frame, similarly to (17) but considering (13) instead of (10), as p DF At,k ≃ e − ρ ∞ X j =0 ρ ( j +1) j ! p c,k ( j ) . (18)We then calculate p DF At by summing (18) up, for all k ∈ { , ..., F ε } , weighted by the (ran-dom) length of the corresponding frame L k normalized to the total number of slots in the IR P F ε k =1 L k . Note that, under assumption A . the random frame length L k is well-represented byits (deterministic) average value L k ≃ E [ L k ] = ρE [ B k ] and thus the DFA time efficiency results p DF At ≃ P F ε k =1 p DF At,k E [ B k ] P F ε k =1 E [ B k ] , (19)where the average backlog size E [ B k ] in frame k , can be computed, under assumption A . , as E [ B k ] = M αG
DF AE ( kε ) Q k − i =1 (1 − p c,i ) . In fact, M α indicates the average number of users thathave a new measure to report in the current IR, G ( kε ) is the probability that kε energy units are stored in the ESD at the beginning of the IR, thus allowing k successful transmissions, and Q k − i =1 (1 − p c,i ) is the probability that a sensor collides in all of the first ( k − frames.V. ESD ENERGY EVOLUTION
In Sec. IV we have shown that the performance metrics for the n th IR depend on the energydistribution in the sensor ESD at the beginning of the IR itself. The goal of this section is toderive the ccdf G E ( n ) ( · ) , for any IR n , in order to obtain the asymptotic performance metrics(3) and (5) from Sec. IV-A and Sec. IV-B respectively.In general, the evolution of sensor ESDs across IRs in DFA are correlated with each other, dueto the possibility of retransmitting after collisions. However, under the large backlog assumption A . , similarly to the discussion in Sec. IV-A3, the evolution of sensor ESDs become decoupledand can thus be studied separately. Accordingly, we develop a stochastic model, based on adiscrete Markov chain (DMC) that focuses on a single sensor ESD as shown in Fig. 3. Inaddition, we concentrate on the DFA protocol as ESD evolutions for TDMA and FA follow asspecial cases. Note that, in TDMA (or FA), the evolution of sensor ESDs are actually independentwith each other as retransmissions are not required (or allowed). A. States of a Sensor
The state of a sensor is uniquely characterized by: i ) sensor activity or idleness (see below); ii ) the amount of energy stored in its ESD; iii ) the current frame index if the sensor is active.A sensor is active if it has a new measure still to be delivered to the FC in the current IR andenough energy in its ESD, while it is idle otherwise. States in which the sensor is active, referredto as active states , are denoted by A kj and they are characterized by: a ) the current frame index k ∈ { , ..., F ε } ; and b ) the number j ∈ { , ..., N } of energy units δ stored in the sensor ESD.States in which the sensor is idle, referred to as idle states , are instead denoted by I j and theyare uniquely characterized by the number j ∈ { , ..., N } of energy units stored in the sensor …… Energy Shortage a) I I I F r a m e A N …… … … …… … … … … …… … … … … b.1) S m has a new meas.: TX & coll. S m reTX & coll. I I S m has harvested ;IR n starts I S m in shortage; IR n ends for S m S m in shortage;IR n ends for S m IR ( n +1) ends for S m A A A S m has harvested 5 ;IR n starts b.2) S m has a new meas.: Successful TX S m has harvested 3IR ( n +1) starts a c p a c p a c p … d e q N q ,0 q N q , d e q q q NN q ,2 d e- c p Harvesting transitionDischarge transition (TX/RX) F r a m e F r a m e F r a m e k F r a m e F ε d N de d ( ) de d - d e- N I d e - N I d e I - d e I N I d e A d e A d e A k A d e ed e F A - d e q d e- N A d e- N A k kN A d e )1( -- dd d d e- N A Figure 3. a) Discrete Markov chain used to model the evolution of the energy stored in the discrete ESD of a sensor in termsof the energy unit δ . In b.1) and b.2) there are two outcomes of possible state transition chains for ε δ = 3 . Grey shaded statesindicate energy shortage condition. Some transitions are not depicted to simplify representation. ( ¯ α = 1 − α and ¯ p c,k = 1 − p c,k ). ESD. EH is then associated to idle states given the assumption that any energy arrival in thecurrent IR can only be used in the next IR (see Sec. II-C).
B. Discrete Markov Chain (DMC) Model
Operations of a sensor across IRs are as follows. When sensor S m is not involved in an IR, itis in an idle state, say I j , waiting for the next IR. When a new IR begins, the energy harvestedin the last interval T int is added, so that, if the ESD is not in energy shortage, the state makes atransition I j → A l toward an active state, with l ≥ ε δ ≥ j . Otherwise, if it is in energy shortage,it makes a transition I j → I l toward another idle state, with j ≤ l < ε δ . If sensor S m is not inenergy shortage, it remains in state A j at the beginning of the IR only if it has a new measure totransmit, which happens with probability α . Instead, with probability ¯ α = 1 − α the state makes atransition toward an idle state as A j → I j . If there is a new measure, the sensor keeps transmittingit in successive frames until either the packet is correctly delivered to the FC, or its ESD falls in energy shortage, or both. A collision in frame k happens with probability ¯ p c,k = 1 − p c,k (seeSec. IV-A3) and leads to a transition either A kj → A k +1 j − ε δ , for j ≥ ε δ (no shortage after collision)or A kj → I j − ε δ , for j < ε δ (shortage after collision). Successful transmissions in frame k , whichhappens with probability p c,k , instead leads to a transition A kj → I j − ε δ . Transition probabilitiesare summarized in Fig. 4, where we have defined q j,N = Pr[ E H,m ≥ ( N − j ) δ ] = 1 − P N − j − i =0 q i .Note that, the probability α of having a new measure is only accounted for in active states inthe first frame (i.e., in states A j , for j ∈ { , ..., N } , see Fig. 4-b)). In fact, being in any state A kj for k > already implies that a new measure was available at the beginning of the IR. Noticethat, according to the model above, state transitions in the DMC at hand are event-driven anddo not happen at fixed time intervals. A sketch of the considered DMC is shown in Fig. 3-a),while we show two outcomes of possible state transition chains in Fig. 3-b.1) and 3-b.2).From Fig. 3-a), it can be seen that, when q > , q > and p c,k > , for k ∈ { , ..., F ε } ,the DMC at hand is irreducible and aperiodic and thus, by definition, ergodic (see [18]). In fact,if q > , any state of the Markov model can be reached from any other state with non-zeroprobability, and therefore the Markov chain is irreducible. Moreover, the probability of having aself-transition from state I to itself is q > , and therefore state I is aperiodic. The presence ofan aperiodic state in a finite state irreducible Markov chain is enough to conclude that the chain isaperiodic [18, Ch. 4, Th. 1]. Since the DMC is ergodic it admits a unique steady-state probabilitydistribution φ = [ φ I , ..., φ I N , φ A εδ , ..., φ A FεN ] , regardless of the initial distribution, which can becalculated by resorting to conventional techniques [18]. This also guarantees the existence oflimits (3) and (5). Vector φ represents the steady-state distribution in any discrete time instantof the interrogation period (i.e., during either any frames of an IR or idle period). However,to calculate (3) and (5) we need the DMC steady-state distribution φ + conditioned on being atthe beginning of the IR. This can be calculated by recalling that between the end of the lastissued IR and the beginning of a new one, sensor S m can only be in any idle states I j , with j ∈ { , ..., N } , and thus its state conditional distribution φ − = [ φ − I , ..., φ − I N , φ − A εδ , ..., φ − A FεN ] , is given by φ − I j = φ I j / P Ni =0 φ I i , ∀ j ∈ { , ..., N } and φ − A kj = 0 , for all j, k . The desired distribution φ + of the state at the beginning of the next IR can be obtained as φ + = φ − P , where P isthe DMC probability transition matrix of the DMC in Fig. 3-a) that can be obtained throughFig. 4. Note that, according to the transition probabilities in Fig. 4, starting from any state I j ,with j ∈ { , ..., N } , only states I j , with j ∈ { , ..., ε δ − } and states A j , with j ∈ { ε δ , ..., N } can be reached. Therefore, the only possibly non-zero entries of distribution φ + are φ + I j for j ∈{ , ..., ε δ − } and φ + A j for j ∈ { ε δ , ..., N } .Once the DMC steady-state distribution φ + at the beginning of any (steady-state) IR isobtained, we can calculate the steady-state distribution p E ( n →∞ ) ( · ) of the energy stored in thesensor ESD at the beginning of any (steady-state) IR, denoted by π E = [ π E (0) , ..., π E ( N )] , bymapping the DMC states into the energy level set E as follows π E ( j ) = φ + I j for j ∈ { , ..., ε δ − } φ + A j for j ∈ { ε δ , ..., N } . (20)The ccdf G E ( n →∞ ) ( · ) is immediately derived from π E . Finally, we remark that analysis of FAand TDMA can be obtained by limiting the set of active states to A ε δ , A ε δ +1 , ..., A N (i.e., noretransmission), and recalling that sensor S m after transmission returns to idle states regardlessof the success of transmission. From / ToFrom / To }1,...,{ ; -˛ δ l ε jlI }1,...,{ ; -˛ N ε lA δ l NlA l = ; j I jl q - jl q - ∑ --= -= jNi iNj qq j I d e - j I =-˛ kjA kj dd ee aa c p a kc p , =˛ kNjA kj d e +- kj A d e kc p , c p a a)b) a >-˛ kjA kj dd ee >˛ kNjA kj d e Figure 4. State transition probabilities for the DMC model in Sec. V-B: a) transition probabilities due to energy harvesting; b)transition probabilities due to the bidirectional communication with the FC. The transition matrix P can be derived accordingto the probabilities in a) and b) for all the values of k ∈ { , ..., F ε } and j ∈ { , ..., N } . VI. B
ACKLOG E STIMATION
In this section we propose a backlog estimation algorithm for the DFA protocol (extension toFA is straightforward). Unlike previous work on the subject [16][19], here backlog estimation isdesigned by accounting for the interplay of EH, capture effect and multiple access. Computationalcomplexity of optimal estimators is generally intractable for a large number of sensors even forconventional systems (see e.g., [19]). We thus propose a low-complexity two-steps backlogestimation algorithm that, neglecting the IR index, operates in every IR as follows: i ) the FCestimates the initial backlog size B based on the ccdf G E ( ε ) of the ESD energy at the beginningof the current IR; ii ) the backlog estimates for the next frames are updated based on the channeloutcomes and the residual ESD energy.For the first frame, the backlog size estimate and the frame length are ˆ B = M αG E ( ε ) and L = l ρ ˆ B m , respectively. For subsequent frames, let us assume that the FC announced a frameof L k = l ρ ˆ B k m slots. The FC estimates the backlog size for frame k + 1 by counting the numberof slots that are successful ( N D,k ) and collided ( N C,k ) within the k th frame of length L k slots.Since the FC cannot discern exactly how many sensors transmitted in each successful slot, theestimate of the total number C D,k of sensors that collided in N D,k successful slots is ˆ C D,k = ( β D,k − N D,k , with β D,k being the conditional average number of sensors that transmit in a slotgiven that the slot is successful (with no capture β D,k = 1 ). Similarly, for the collided slots weobtain ˆ C C,k = β C,k N C,k , where β C,k is now conditioned on observing a collided slot. Derivationsof β D,k and β C,k are in Appendix A. Since the estimate of the total number of sensors thatunsuccessfully transmitted is ˆ C k = ˆ C C,k + ˆ C D,k , the backlog size estimate ˆ B k +1 for the ( k + 1) thframe is obtained by accounting for the fraction of sensors within ˆ C k that are not in energyshortage: ˆ B k +1 = ˆ C k G E (( k + 1) ε | kε ) , where G E (( k + 1) ε | kε ) = Pr [ E m ≥ ( k + 1) ε | E m ≥ kε ] .The proposed backlog estimation scheme thus works as follows: ˆ B k = M αG E ( ε ) if k = 1ˆ C k − G E ( kε | ( k − ε ) if k > . (21)Algorithm (21) can be applied to any n th IR by deriving the ESD distribution p E ( n ) ( · ) (or G E ( n ) ( · ) ) from any initial distribution p E (1) ( · ) , by exploiting the DMC model in Sec. V-B.VII. N UMERICAL R ESULTS
In this section, we present extensive numerical results to get insight into the MAC protocolsdesign. Moreover, to validate the analysis proposed in Sec. IV and Sec. V, we compare theanalytical results therein with a simulated system that does not rely on simplifying assumptions A . and A . . The performances of the backlog estimation algorithm proposed in Sec. VI arealso assessed through a comparison with the ideal case of perfectly known backlog at the FC. A. MAC Performance Metrics Trade-offs
The energy E H,m ( n ) harvested between two successive IRs is assumed as geometrically-distributed with q i = Pr[ E H,m ( n ) = iδ ] = ξ (1 − ξ ) i , where ξ = δ/ ( δ + µ H ) . The averageharvested energy normalized by ε , referred to as harvesting rate , is E [ E H,m ( n ) /ε ] = µ H .The asymptotic time efficiencies (5) for TDMA, FA and DFA protocols, are shown in Fig. 5versus design parameter ρ (recall (6)). System performance is evaluated by considering: µ H ∈{ . , . } , M = 400 , γ th = 3 dB , α = 0 . ; ε is normalized to unity, energy unit is δ = 1 / so that ε δ = 50 and F ε = 10 . We compare the analytical performance metrics derived in Sec. IV withsimulated scenarios for both known and estimated backlog. While the performance of TDMA isclearly independent of ρ , in FA and DFA there is a time efficiency-maximizing ρ , which is closeto one (in [8] the optimal value was ρ = 1 since the capture effect was not considered). Theeffect of decreasing (or increasing) the harvesting rate µ H on the TDMA time efficiency is dueto the larger (or smaller) number of sensors that are in energy shortage and whose slots are notused, while it is negligible for FA and DFA due to their ability to dynamically adjust the framesize according to backlog estimates ˆ B k . The tight match between analytical and simulated resultsalso validates assumptions A . and A . and the efficacy of the backlog estimation algorithm.The asymptotic delivery probability, for harvesting rate µ H ∈ { . , . , . } , versus designparameter ρ is shown in Fig. 6 with the same system parameters as for Fig. 5. Unlike for thetime efficiency, TDMA always outperforms FA and DFA in terms of delivery probability. In fact,sensors operating with TDMA and FA have the same energy consumption since they transmitat most once per IR, while possibly more than once in DFA. However, TDMA does not suffercollisions and thus it is able to eventually deliver more packets to the FC. The delivery probabilitystrongly depends on the harvesting rate µ H , which influences the ESD energy distribution andconsequently the energy shortage probability. Moreover, DFA outperforms FA thanks to theretransmission capability when the harvesting rate is relatively high (e.g., µ H = 0 . ), while forlow harvesting rate (e.g., µ H ∈ { . , . } ) DFA and FA perform similarly. In fact, for lowharvesting rates, most of the sensors are either in energy shortage or have very low energy intheir ESDs. Hence, most of the sensors that are not in energy shortage are likely to have only onechance to transmit, and thus retransmission opportunities provided by DFA are not leveraged.The trade-off between asymptotic delivery probability (3) and asymptotic time efficiency (5)is shown in Fig. 7 for different values of the harvesting rate µ H ∈ { . , . , . } . Systemparameters are the same as for Fig. 5. For TDMA, the trade-off consists of a single point on theplane, whereas FA and DFA allow for more flexibility via the selection of parameter ρ . When increasing ρ more sensors might eventually report their measures to the FC, thus increasing thedelivery probability to the cost of lowering time efficiency (see Fig. 5 and 6). For FA and DFA,the trade-off curves are obtained as max ρ (cid:8) p ASd (cid:9) , s.t. p ASt = λ for each achievable λ .The impact of the capture effect on the performance metrics trade-offs is shown in Fig. 8,where we vary the SIR threshold γ th ∈ { . , , } dB and keep the harvesting rate µ H = 0 . fixed (other parameters are as in Fig. 5). As expected, the lower the SIR threshold γ th the higherthe probability that the SIR of any of the colliding sensors is above γ th , and thus the higher theperformance obtained with ALOHA-based protocols. TDMA is insensitive to γ th . r A sy m p t o t i c t i m e e ff i c i en cy p t AS TDMA − AnalyticalFA − AnalyticalDFA − AnalyticalTDMAFA − Known B k DFA − Known B k FA − Estimated B k DFA − Estimated B k m H = 0.35 m H = 0.15 Figure 5. Asymptotic time efficiency (5) versus ρ , for different energy harvesting rates µ H ∈ { . , . } . Comparisons arebetween analytical derivations and simulated results with both known ( B k ) and estimated backlog ( ˆ B k , see (21)), ( M = 400 , γ th = 3 dB , α = 0 . , F ε = 10 , ε = 1 , δ = 1 / ). r A sy m p t o t i c de li v e r y p r obab ili t y p d AS TDMA − AnalyticalFA − AnalyticalDFA − AnalyticalTDMAFA − Known B k DFA − Known B k FA − Estimated B k DFA − Estimated B k m H = 0.35 m H = 0.05 m H = 0.15 Figure 6. Asymptotic delivery probability (3) versus ρ , for different energy harvesting rate µ H ∈ { . , . , . } .Comparisons are between analytical derivations and simulated results with both known ( B k ) and estimated backlog ( ˆ B k , see(21)), ( M = 400 , γ th = 3 dB , α = 0 . , F ε = 10 , ε = 1 , δ = 1 / ). t AS A sy m p t o t i c de li v e r y p r obab ili t y p d AS TDMA − AnalyticalFA − AnalyticalDFA − AnalyticalTDMAFA − Estimated B k DFA − Estimated B k m H = 0.35 m H = 0.15 m H = 0.05 Figure 7. Trade-off between asymptotic delivery probability (3) and asymptotic time efficiency (5) for different energy harvestingrate µ H ∈ { . , . , . } . Comparisons are between analytical derivations and simulated results with estimated backlog ( ˆ B k ,see (21)), ( M = 400 , γ th = 3 dB , α = 0 . , F ε = 10 , ε = 1 , δ = 1 / ). t AS A sy m p t o t i c de li v e r y p r obab ili t y p d AS TDMA − AnalyticalFA − AnalyticalDFA − AnalyticalTDMAFA − Estimated B k DFA − Estimated B k g th = 0.01dB g th = 3dB g th = 10dB m H = 0.15 Figure 8. Trade-off between asymptotic delivery probability (3) and asymptotic time efficiency (5) for different SIR threshold γ th ∈ { . , , } dB values and fixed energy harvesting rate µ H = 0 . . Comparisons are between analytical derivations andsimulated results with estimated backlog ( ˆ B k , see (21)), ( M = 400 , α = 0 . , F ε = 10 , ε = 1 , δ = 1 / ). VIII. C
ONCLUSIONS
The design of medium access control (MAC) protocols for single-hop wireless sensor networks(WSNs) with energy-harvesting (EH) devices offers new challenges as compared to the standardscenario with battery-powered (BP) nodes. New performance criteria are called for, along withnew design solutions. This paper addresses these issues by investigating the novel trade-offbetween the delivery probability, which measures the capability of a MAC protocol to deliverthe measure of any sensor in the network to the intended destination (i.e., fusion center, FC) andthe time efficiency , which measures the data collection rate at the FC. The analysis is focused onstandard MAC protocols, such as TDMA, Framed-ALOHA (FA) and Dynamic-FA (DFA). Noveldesign issues are also discussed, such as backlog estimation and frame length selection. Extensivenumerical results and discussions validate the proposed analytical framework and provide insightinto the design of EH-WSNs. A PPENDIX AA VERAGE N UMBER OF S ENSOR T RANSMISSIONS PER T IME - SLOT
The conditional averages β D,k and β C,k are calculated similarly to [8] by accounting for thecapture effect and an arbitrary ρ . Let Y be the number of simultaneous transmissions in thesame slot, and let U k and C k respectively be the event of successful and collided slot in frame k , the average number of sensors per successful and collided slot are respectively β D,k = ∞ X j =1 j Pr [ Y = j |U k ] ; β C,k = ∞ X j =2 j Pr [ Y = j |C k ] (22)To calculate β D,k consider A . and A . and allow the number of possible interfering users up toinfinity as in Sec. IV-A2. By exploiting the Bayes rule, we have Pr [ Y = j |U k ] = Pr [ U k | Y = j ] Pr[ Y = j ]Pr[ U k ] ,where Pr [ U k | Y = j ] = jp c,k ( j − , Pr [ Y = j ] = e − ρ / ( ρ j j !) and Pr [ U k ] = p DF At,k (see 18). Wecan similarly obtain β C,k given that
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