Network-Level Cooperation in Energy Harvesting Wireless Networks
Nikolaos Pappas, Marios Kountouris, Jeongho Jeon, Anthony Ephremides, Apostolos Traganitis
NNetwork-Level Cooperation in Energy Harvesting WirelessNetworks
Nikolaos Pappas * , Marios Kountouris * , Jeongho Jeon ‡ , Anthony Ephremides ‡ , Apostolos Traganitis † * Sup´elec, Department of Telecommunications, Gif-sur-Yvette, France ‡ Department of Electrical and Computer Engineering and Institute for Systems ResearchUniversity of Maryland, College Park, MD 20742 † Computer Science Department, University of Crete, GreeceInstitute of Computer Science, Foundation for Research and Technology - Hellas (FORTH)Email: { nikolaos.pappas, marios.kountouris } @supelec.fr, { jeongho, etony@umd } .edu, [email protected] Abstract —We consider a two-hop communication network consisted ofa source node, a relay and a destination node in which the source and therelay node have external traffic arrivals. The relay forwards a fractionof the source node’s traffic to the destination and the cooperation isperformed at the network level. In addition, both source and relay nodeshave energy harvesting capabilities and an unlimited battery to store theharvested energy. We study the impact of the energy constraints on thestability region. Specifically, we provide inner and outer bounds on thestability region of the two-hop network with energy harvesting sourceand relay.
I. I
NTRODUCTION
Taking advantage of renewable energy resources from the environ-ment, also known as energy harvesting, allows unattended operabilityof infrastructure-less wireless networks. There are various forms ofenergy that can be harvested, including thermal, solar, acoustic, wind,and even ambient radio power. However, the additional functionalityof harvesting energy calls for our assessment of the system long-term performance such as in terms of the throughput and stability.In [1], the slotted ALOHA protocol was considered for a network ofnodes having energy harvesting capability and the maximum stablethroughput region was obtained for bursty traffic.Cooperative communication helps overcome fading and attenuationin wireless networks. Most cooperative techniques studied so farhave been on physical layer cooperation that achieves non-trivialbenefits [2]. Nevertheless, there is evidence that the same gains canbe achieved with network layer cooperation, which is plain relayingwithout any physical layer considerations [3]–[5]. A key differencebetween physical layer and network layer cooperation is that the lattercan capture the bursty nature of traffic.In [6], the authors studied the stability region of a cognitivenetwork under energy constraints. They employed an opportunisticmultiple access protocol that observes the priorities among the usersto better utilize the limited energy resources. The impact of network-level cooperation in an energy harvesting network with a pure-relay(without its own traffic) under scheduled access was studied in [7].In this paper, we study the impact of energy constraints on anetwork with a source-user, a relay and a destination under randomaccess of the medium. Specifically, we provide necessary and suffi-cient conditions for the stability of a network consisting of a source,a relay and a destination node as shown in Fig. 1. We consider thecollision channel with erasures and random access of the medium.The source and the relay node have external arrivals; furthermore, the
This research has been partly supported by the ERC Starting Grant 305123MORE (Advanced Mathematical Tools for Complex Network Engineering). relay is forwarding part of the source node’s traffic to the destination,the cooperation is taking place at the network-level.The analysis is not trivial even for such a simple network becausethe service process of a node not only depends on the status of itsbattery but also on the idleness or not of the other node. Note that thereason why the exact region is known only for the two-node and thethree-node cases (with or without energy availability constraints) isthe interaction between the queues of the nodes [8]–[10]. First, weobtain an inner bound of the stability region, afterwards we applythe stochastic dominance technique [9] and Loynes’ theorem [11] toobtain an outer bound of the stability region.The rest of this paper is organized as follows. In section II, wedefine the stability region, describe the channel model, and explainthe packet arrival and energy harvesting models. In III, we presentan inner and an outer bound of the stability region, the proofs of theresults are given in IV. Finally, we conclude our work in V.II. S
YSTEM M ODEL
We consider a time-slotted system in which the nodes randomlyaccess a common receiver as shown in Fig. 1, where both sourceand relay nodes are powered from randomly time-varying renewableenergy sources. Each node stores the harvested energy in a battery ofunlimited capacity. We denote with S , R , and D the source, the relayand the destination, respectively. Packet traffic originates from S and R . Because of the wireless broadcast nature, R may receive some ofthe packets transmitted from S , which in turn can be relayed to D .The packets from S that fail to be received by D but are successfullyreceived by R are relayed by R . A half-duplex constraint is imposedhere, i.e. R can overhear S only when it is idle.Each node has an infinite size buffer for storing incoming packets,and the transmission of each packet occupies one time slot. Node R has separate queues for the exogenous arrivals and the endogenousarrivals that are relayed through R . Nevertheless, we can let R havea single queue and merge all arrivals into a single queue as theachievable stable throughput region is not affected [12]. This is dueto the fact that the link quality between R and D is independent ofwhich packet is selected for transmission.The packet arrival and energy harvesting processes at S and R are assumed to be Bernoulli distributed with rates λ S , δ S and λ R , δ R , respectively, and are independent of each other. Q i and B i , i = S, R , denote the steady state number of packets and energyunits in the queue and the energy source at node i , respectively.Furthermore, a node i is called active if both its packet queue andits battery are nonempty at the same time, which is denoted by the a r X i v : . [ c s . I T ] S e p vent A i = { B i (cid:54) = 0 } ∩ { Q i (cid:54) = 0 } and idle otherwise (denotedby A i ). In each time slot, nodes S and R attempt to transmit withprobabilities q S and q R , respectively, if they are active. Decisionson transmission are made independently among the nodes and eachtransmission consumes one energy unit. We assume collision channelwith erasures in which if both S and R transmit at the same time slot,a collision occurs and both transmissions fail. The probability that apacket transmitted by node i is successfully decoded at node j ( (cid:54) = i ) is denoted by p ij , which is the probability that the signal-to-noiseratio (SNR) over the specified link exceeds a certain threshold forsuccessful decoding. These erasure probabilities capture the effect ofrandom fading at the physical layer. The probabilities p SD , p RD , and p SR denote the success probabilities over the link S − D , R − D ,and S − R , respectively. We also assume that node R has a betterchannel to D than S , i.e. p RD > p SD .The cooperation is performed at the protocol (network) level asfollows: when S transmits a packet, if D decodes it successfully,it sends an ACK and the packet exits the network; if D fails todecode the packet but R does, then R sends an ACK and takesover the responsibility of delivering the packet to D by placing itin its queue. If neither D nor R decode (or if R does not storethe packet), the packet remains in S ’s queue for retransmission. TheACKs are assumed to be error-free, instantaneous, and broadcastedto all relevant nodes.The average service rate for the source node is given by µ S = (cid:8) q S (1 − q R )Pr ( B S (cid:54) = 0 , A R ) + q S Pr( B S (cid:54) = 0 , A R ) (cid:9) × [ p SD + (1 − p SD ) p SR ] , (1)and for the relay is given by µ R = (cid:8) q R (1 − q S )Pr ( B R (cid:54) = 0 , A S ) + q R Pr( B R (cid:54) = 0 , A S ) (cid:9) p RD . (2)Denote by Q ti the length of queue i at the beginning of time slot t . Based on the definition in [10], the queue is said to be stable if lim t →∞ P r [ Q ti < x ] = F ( x ) and lim x →∞ F ( x ) = 1 Loynes’ theorem [11] states that if the arrival and service processesof a queue are strictly jointly stationary and the average arrival rateis less than the average service rate, then the queue is stable. If theaverage arrival rate is greater than the average service rate, then thequeue is unstable and the value of Q ti approaches infinity almostsurely. The stability region of the system is defined as the set ofarrival rate vectors λ = ( λ , λ ) for which the queues in the systemare stable. III. M AIN R ESULTS
This section presents the stability conditions of a network con-sisting of energy harvesting source and relay, and a destination, asdepicted in Fig. 1. The source and the relay are assumed to haveinfinite size queues to store the harvested energy.The next proposition presents an inner bound on the stability regionby providing sufficient conditions for stability.
Proposition III.1. If ( λ S , λ R ) ∈ R inner , where R inner is describedin (3), then the network in Fig. 1 is stable.Proof: The proof is given in Section IV-A.The following proposition describes an outer bound of the stabilityregion by obtaining necessary conditions for stability. Q S S Q R2 R D λ S λ R Q R1 B R δ R δ S B S S R Fig. 1: The wireless network topology with energy harvesting capa-bilities. R S S S R R RD q q p
S S RD q p min( , ) 1 min( , ) (1 )1 min( , ) min( , )(1 ) S S S S RD SD SD SRS S RD S S SD SR q q p p p pq p q p p min( , ) 1 min( , ) (1 )(1 )
R R S S RD SD SD SRSD SR q q p p p pp p ** min( , ) 1 min( , ) (1 )min( , ) 1 min( , ) min( , ) 1 min( , ) (1 ) S S S R R SD SD SRR R R S S RD S S R R SD SR q q p p pq q p q q p p R (a) R R S min( , ) R R RD q p S S R R RDS S R R SD SR q q pq q p p min( , ) 1 min( , ) (1 )
S S R R SD SD SR q q p p p R (b) R . Fig. 2: An outer bound of the stability region R = R (cid:83) R ,described in Proposition III.2. Proposition III.2.
If the network in Fig. 1 is stable then ( λ S , λ R ) ∈R , where R = R (cid:83) R . R and R are described by (4) and (5)respectively.Proof: The proof is given in Section IV-B.Fig. 2(a) and 2(b) illustrate the R and R described in Proposi-tion III.2. inner = { ( λ S , λ R ) : λ S < min ( δ S , q S ) [1 − min ( δ R , q R )] [ p SD + (1 − p SD ) p SR ] ,λ R + (1 − p SD ) p SR p SD + (1 − p SD ) p SR λ S < min ( δ R , q R ) [1 − min ( δ S , q S )] p RD (cid:27) . (3) R = (cid:26) ( λ S , λ R ) : (cid:20) δ S , q S )(1 − p SD ) p SR [1 − min( δ S , q S )] p RD (cid:21) λ S ++ min( δ S , q S ) [ p SD + (1 − p SD ) p SR ][1 − min( δ S , q S )] p RD λ R < min( δ S , q S ) [ p SD + (1 − p SD ) p SR ] ,λ R + (1 − p SD ) p SR p SD + (1 − p SD ) p SR λ S < min( δ R , q R ) [1 − min( δ S , q S )] p RD (cid:27) . (4) R = (cid:26) ( λ S , λ R ) : λ R + [1 − min( δ R , q R )] (1 − p SD ) p SR + min( δ R , q R ) p RD [1 − min( δ R , q R )] [ p SD + (1 − p SD ) p SR ] λ S < min( δ R , q R ) p RD ,λ S < min( δ S , q S ) [1 − min( δ R , q R )] [ p SD + (1 − p SD ) p SR ] } (5)IV. A NALYSIS
To derive the stability condition for the queue in the relay node,we need to calculate the total arrival rate. There are two independentarrival processes at the relay: the exogenous traffic with arrival rate λ R and the endogenous traffic from S . Denote by S A the event that S transmits a packet and the packet leaves the queue, then Pr( S A ) = [1 − q R Pr( A R )] [ p SD + (1 − p SD ) p SR ] . (6)Among the packets that depart from the queue of S , some will exitthe network because they are decoded by the destination directly,and some will be relayed by R . Denote by S B the event that thetransmitted packet from S will be relayed from R , then Pr( S B ) = [1 − q R Pr( A R )] (1 − p SD ) p SR . (7)The conditional probability that a transmitted packet from S isrelayed by R given that the transmitted packet exits node S ’s queueis given by Pr( S B | S A ) = (1 − p SD ) p SR p SD + (1 − p SD ) p SR . (8)The arrivals from the source to the relay are λ S → R = Pr( S B | S A ) λ S . (9)The total arrival rate at the relay node is given by λ R,total = λ R + (1 − p SD ) p SR p SD + (1 − p SD ) p SR λ S . (10) A. Sufficient Conditions
A queue is considered saturated if in each time slot there is alwaysa packet to transmit, i.e. the queue is never empty. Assuming saturatedqueues for the source and the relay node, the saturated throughputfor the source node is given by µ sS = { q S (1 − q R )Pr ( B S (cid:54) = 0 , B R (cid:54) = 0) + q S Pr ( B S (cid:54) = 0 , B R = 0) }× [ p SD + (1 − p SD ) p SR ] , (11) and for the relay is given by µ sR = { q R (1 − q S )Pr ( B S (cid:54) = 0 , B R (cid:54) = 0) ++ q R Pr( B S = 0 , B R (cid:54) = 0) } p RD . (12) Each node transmits with probability q i , i = S, R , whenever itsbattery is not empty and each transmission demands one energypacket. Each energy queue i is then decoupled and forms a discrete-time M/M/ queue with input rate δ i and service rate q i , thus theprobability the energy queue to be empty is given by Pr ( B i (cid:54) = 0) = min (cid:18) δ i q i , (cid:19) . (13)Then, after some calculations we obtain that the saturated through-put for the source is µ sS = min ( δ S , q S ) [1 − min ( δ R , q R )] [ p SD + (1 − p SD ) p SR ] , (14)and for the relay is µ sR = min ( δ R , q R ) [1 − min ( δ S , q S )] p RD . (15)The sufficient conditions ( R inner ) for the stability are obtainedby λ S < µ sS and λ R,total < µ sR and are given by (3), inProposition III.1. B. Necessary Conditions
The average service rates for the source and the relay are givenby (1) and (2), respectively. The average service rate of each queuedepends on the status of its own energy and also the queue size andthe energy statues of the other queues. The coupling between thequeues (both packet and energy) forms a four dimensional Markovchain which makes the analysis hard.The stochastic dominant technique [9] is essential in order todecouple the interaction between the queues, and thus to characterizethe stability region. That is we first construct parallel dominantsystems in which one of the nodes transmits dummy packets whenits packet queue is empty. Note that even in the dominant system anode cannot transmit if the energy source is empty (because even thedummy packet consumes one energy unit).We consider the first hypothetical system in which the source nodetransmits dummy packets when its queue is empty and all the otherassumptions remain intact. The average service rate for the relaygiven by (2) becomes µ R = { q R (1 − q S )Pr ( B S (cid:54) = 0 , B R (cid:54) = 0) + q R Pr( B S = 0 , B R (cid:54) = 0) } p RD . (16)he average service rate of the relay, µ R , in the first hypotheticalsystem is the same with the saturated throughput of the relay obtainedin (15). From Loyne’s criterion, the relay is stable if λ R,total < µ R . λ R + (1 − p SD ) p SR p SD + (1 − p SD ) p SR λ S < min ( δ R , q R ) [1 − min ( δ S , q S )] p RD . (17)The average number of packets per active slot for R is [1 − min ( δ S , q S )] q R p RD , thus the fraction of active slots is givenby Pr ( B R (cid:54) = 0 , Q R (cid:54) = 0) = λ R + (1 − p SD ) p SR p SD +(1 − p SD ) p SR λ S [1 − min ( δ S , q S )] q R p RD (18)After changing (13) and (18) into (1), the service rate for the sourcebecomes µ S = min ( δ S , q S ) (cid:34) − λ R + (1 − p SD ) p SR p SD +(1 − p SD ) p SR λ S [1 − min ( δ S , q S )] p RD (cid:35) × [ p SD + (1 − p SD ) p SR ] . (19)The queue in S is stable if λ S < µ S and after some manipulationswe obtain (cid:20) δ S , q S ) (1 − p SD ) p SR [1 − min ( δ S , q S )] p RD (cid:21) λ S ++ min ( δ S , q S ) [ p SD + (1 − p SD ) p SR ][1 − min ( δ S , q S )] p RD λ R < min ( δ S , q S ) [ p SD + (1 − p SD ) p SR ] . (20)The derived stability conditions from the first hypothetical systemare summarized in (4).In the second hypothetical system, the relay node transmits dummypackets and all the other assumptions remain intact. Thus, the averageservice rate for the source given by (1), becomes µ S = { q S (1 − q R )Pr ( B S (cid:54) = 0 , B R (cid:54) = 0) + q S Pr( B S (cid:54) = 0 , B R = 0) }× [ p SD + (1 − p SD ) p SR ] , (21)which is equal to saturated throughput of the source and is givenby µ S = min ( δ S , q S ) [1 − min ( δ R , q R )] [ p SD + (1 − p SD ) p SR ] . (22)From Loyne’s theorem, the queue in source is stable if λ S < µ S thus λ S < min ( δ S , q S ) [1 − min ( δ R , q R )] [ p SD + (1 − p SD ) p SR ] . (23)The average number of packets per active slot for S is q S [1 − min ( δ R , q R )] [ p SD + (1 − p SD ) p SR ] . The fraction of activeslots for the source S is Pr ( B S (cid:54) = 0 , Q S (cid:54) = 0) = λ S q S [1 − min ( δ R , q R )] [ p SD + (1 − p SD ) p SR ] . (24)After replacing from (13) and (24) into (2), the service rate for therelay is µ R = min ( δ R , q R ) (cid:20) − λ S [1 − min ( δ R , q R )] [ p SD + (1 − p SD ) p SR ] (cid:21) p RD . (25) The queue in the relay node R is stable if λ R,total < µ R and aftersome manipulations we obtain λ R + [1 − min ( δ R , q R )] (1 − p SD ) p SR + min ( δ R , q R ) p RD [1 − min ( δ R , q R )] [ p SD + (1 − p SD ) p SR ] λ S < min ( δ R , q R ) p RD . (26)The derived stability conditions from the second hypotheticalsystem are given by (5).An important observation made in [9] is that the stability conditionsobtained by using the stochastic dominance technique are not merelysufficient conditions for the stability of the original system but aresufficient and necessary conditions. However, the indistinguishability argument does not apply to our problem. In a system with batteries,the dummy packet transmissions affect the dynamics of the batteries.For example, there are instants when a node is no more able totransmit in the hypothetical system because of the lack of energy,while it is able to transmit in the original system, thus it may resultto a better chance of success for the other node.The obtained stability conditions are necessary conditions of theoriginal system and are summarized in Proposition III.2.V. C ONCLUSION
In this paper, we studied the effect of energy constraints on awireless network with energy harvesting source and relay and adestination. The source and the relay nodes have external arrivals andnetwork-level cooperation is employed in which the relay forwardsa fraction of the source’s traffic to the destination. We derivednecessary and sufficient stability conditions of the above cooperativecommunication scenario. A next step is to obtain the closure forthe inner and outer bounds presented here. Further extensions willinvestigate the effect of finite battery capacity and that of multi-packetreception. R
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