Robust Bayesian Learning for Wireless RF Energy Harvesting Networks

Transcription

1 t International Symposium on Modeling and Optimization in Mobile, Ad Hoc, and Wireless Networks (WiOpt) Robust Bayesian Learning for Wireless RF Energy Harvesting Networks Nof Abuzainab 1, Walid Saad 1, and Berouz Maam 2 1 Wireless@VT, Department of Electrical and Computer Engineering, Virginia Tec, Blacksburg, VA, USA s:{nof, walids}@vt.edu 2 Scool of Engineering, Nazarbayev University, Astana, Kazakstan, berouz.maam@nu.edu.kz Abstract In tis paper, te problem of adversarial learning is studied for a wireless powered communication network (WPCN) in wic a ybrid access point (HAP) seeks to learn te transmission power consumption profile of an associated wireless transmitter. Te objective of te HAP is to use te learned estimate in order to determine te transmission power of te energy signal to be supplied to its associated device. However, suc a learning sceme is subject to attacks by an adversary wo tries to alter te HAP s learned estimate of te transmission power distribution in order to minimize te HAP s supplied energy. To build a robust estimate against suc attacks, an unsupervised Bayesian learning metod is proposed allowing te HAP to perform its estimation based only on te advertised transmisson power computed in eac time slot. Te proposed robust learning metod relies on te assumption tat te device s true transmission power is greater tan or equal to advertised value. Ten, based on te robust estimate, te problem of power selection of te energy signal by te HAP is formulated. Te HAP optimal power selection problem is sown to be a discrete convex optimization problem, and a closed-form solution of te HAP s optimal transmission power is obtained. Te results sow tat te proposed robust Bayesian learning sceme yields significant performance gains, by reducing te percentage of dropped transmitter s packets of about 85% compared to a conventional Bayesian learning approac. Te results also sow tat tese performance gains are acieved witout jeopardizing te energy consumption of te HAP. I. INTRODUCTION Radio frequency (RF) energy arvesting is one of te most promising tecnologies to operate massive self-powered networks, suc as te Internet of Tings (IoT) [1]. Te reliance on RF signals for energy supply makes RF energy arvesting favourable since it will be easy to be implemented and integrated into current wireless systems. Moreover, RF energy arvesting offers a reliable metod to supply energy, as opposed to traditional energy arvesting tecniques tat rely on ambient sources suc as solar or wind in wic te amount of energy arvested strongly depends on environmental factors. Te RF energy source can be a wireless access point, known as a ybrid access point (HAP), tat can be configured to provide simultaneous communication and energy supply. Anoter example of suc RF energy sources could be a power beacon tat operates independently from te HAP. Tus, te reliable energy supply provided by tese dedicated RF energy sources allows te network to better serve Tis researc was supported by te US National Science Foundation under Grant CNS devices wit stringent quality-of-service (QoS) requirements. However, RF energy supply incurs extra energy expenditure by te HAP since te HAP must send dedicated signals for energy transfer to its associated devices. Hence, one of te main tecnical callenges in a wireless powered communication network (WPCN) is to find energy efficient resource allocation mecanisms tat determine te optimal energy tat sould be supplied by te HAP to its associated devices in order to meet teir QoS requirements, as pointed out in [2]. Tere as been considerable interest in designing energy efficient wireless resource allocation scemes suitable for WPCNs [3] [5]. In [3] and [4], a WPCN composed of a HAP serving multiple mobile users in a time division manner is considered. A centralized approac for maximizing te total network trougput in a WPCN is adopted in [3] by finding te optimal time fraction allocated to eac user. Te autors in [4] also propose a centralized approac for maximizing te proportional fairness sum of users rates by finding te optimal transmission power and te arvesting duration for eac user. A distributed noncooperative game teoretic approac is proposed in [5] wic considers a WPCN composed of several source destination pairs operating in te same frequency band. Tus, eac source finds te minimum transmit power tat meets te QoS and arvesting constraints of its associated destination. However, some of tese works assume tat te HAP transmits an energy signal wit fixed power, and tat te device consumes all of te arvested energy for transmission in te same slot. Moreover, te existing literature typically assumes tat te HAP knows apriori te QoS and arvesting requirements of all its associated devices. Tese assumptions are not very realistic especially in emerging IoT systems in wic devices ave very diverse caracteristics and requirements. Furter, te transmission power of eac wireless device depends on its adopted power control policy wic is often a function of te device s QoS requirements and its traffic caracteristics. One promising approac is to use macine learning tecniques [6], [7] in order to form a more realistic estimate of te distribution of te transmission power consumed by te wireless device. Tis enables te HAP to predict te transmission energy consumed by eac associated wireless device and determine te required energy to be supplied for te device. Tere is still little prior work tat considers learning for / IFIP

2 t International Symposium on Modeling and Optimization in Mobile, Ad Hoc, and Wireless Networks (WiOpt) RF energy arvesting [8] [10], and most of tis prior art as considered learning at te device s end. In [8], two algoritms based on supervised macine learning tecniques: te linear regression (LR) and te decision trees (DT) are proposed in order to predict te RF energy tat can be arvested in a certain frequency band and a given time slot. In [9], te problem of energy efficient RF energy arvesting for a wireless device is considered. To tis end, an unsupervised Bayesian learning approac is proposed tat allows te energy arvesting wireless device to predict te ambient RF energy availability in eac time slot. Ten, based on te predicted RF energy, te optimal sleep and arvesting policy are determined to minimize te consumed energy. An online convex optimization metod is proposed in [10] to allow an energy arvesting wireless transmitter to predict te energy available in a current time slot based on measurements from previous time slots. Despite its benefits, learning can be vulnerable to a man-inte-middle (MITM) attack by a malicious user tat can alter te data used by te learning algoritm and, consequently, degrade te system performance. MITM attacks constitute serious treat in te emerging IoT systems [11] due to te fact tat many IoT macine type devices ave limited computational capabilities and can not implement strong security mecanisms. Tus, teir security can be easily compromised. In WPCNs, an adversary, troug an MITM attack, can modify te transmission power consumption profile of te wireless device. Tus, te HAP will be misled into supplying less energy to te associated wireless device, wic will eventually exaust te device s battery. Hence, learning algoritms for WPCNs must be designed to be robust against suc attacks. Existing works tat study security for RF energy arvesting considered eiter jamming attacks [12], [13] or eavesdropping [14], [15]. To te best of our knowledge, tere is still no work tat considers attacks on learning witin RF energy arvesting networks. Te main contribution of tis paper is to introduce a novel learning sceme for RF energy arvesting tat allows te HAP to form a reliable estimate of te power consumption profile of eac associated wireless device. Te proposed learning sceme is based on unsupervised Bayesian learning, and it relies only on te received power from te wireless device in eac time slot and on cannel state information (CSI). Tus, it does not result in extra comunications and energy costs. However, te dependence of te proposed learning sceme on te device s received power makes it subject to attacks by an adversary tat is interested in depleting te battery of te wireless device. Te adversary can acieve tis end by altering te formed estimate of te power consumption profile troug performing MITM attack. To counter suc attacks, te estimate is built by te HAP based on te assumption tat te true value of te transmission power of te wireless transmitter is censored by a potential malicious user, and tat true transmission power value is greater tan or equal to te advertised value. Ten, based on tis robust estimate, te HAP determines te transmission power of te energy signal to be delivered to its associated device suc tat te HAP s payoff is maximized. We formulate te problem of optimal power selection by te HAP as discrete convex optimization problem, and we obtain a closed-form expression for te optimal transmission power. Te results sow tat te proposed robust Bayesian learning sceme yields significant performance gains, by reducing te percentage of dropped transmitter s packets of about 85% compared to te conventional Bayesian learning approaces. Te results also sow tat tese performance gains are acieved witout jeopardizing te energy consumption of te HAP. Te paper is organized as follows. Section I presents te system model. Section II presents te attacker model. Section III presents te defensive learning strategy of te HAP and te power selection mecanism of te energy signal. Section IV presents te simulation results. Finally, conclusions are drawn in section V. II. SYSTEM MODEL Consider a WPCN composed of a HAP [2] serving a set of wireless devices over ortogonal frequency cannels. For eac device i, te HAP can act as bot an energy supplying device tat performs wireless power transfer to te device and as an access point tat collects te information from te device. Te HAP is connected to a constant power supply suc as a smart grid wereas te device is not connected to any additional energy supply, and, ence, it relies on te energy arvested from te HAP. Simultaneous uplink and downlink transmissions are assumed [3] were te HAP transmits te energy signal and receives te uplink transmission from te device over two separate frequency bands. In eac time slot t of duration T seconds, te HAP transmits an energy signal wit power P at from a discrete set P a to device i. In te set P a, te power values are multiples of were 0 1. During te uplink, device i uses te arvested energy from te previous time slots to transmit its data to te HAP wit power P it wic takes value from a discrete set P i. In te uplink pase, device i determines te value of P it based on its QoS requirements, its traffic caracteristics as well as te energy available in its battery. Te uplink and downlink cannels between device i and te HAP are modeled as block Rayleig fading cannels wit coefficients D,it and U,it for downlink and uplink, respectively. Tese cannel gains do not cange witin time slot t. Tus, te amount of energy arvested by device i at time slot t is E it = η D,it 2 P at T were 0 < η < 1 is te energy arvesting efficiency. In our model, te devices served by te HAP ave eterogeneous traffic caracteristics and QoS requirements. Hence, it is not practical to assume tat te HAP as prior knowledge of te traffic caracteristic and te QoS requirement of eac device and, consequently, its power consumption profile. Instead, te HAP uses unsupervised Bayesian learning in order to estimate te power/energy consumption distribution of eac device. Te HAP relies only on te received signal power P r,it in te uplink in order to update its estimate in eac time period t. By using suc a metod, tere is no need for te device

3 t International Symposium on Modeling and Optimization in Mobile, Ad Hoc, and Wireless Networks (WiOpt) to explicitly send energy requests to te HAP. Suc energy requests would waste te energy stored in te device. Tus, assuming tat te HAP as full cannel state information, it computes te transmission power consumed by te device in Pr,it U,it. 2 time slot t as: P it = In nonparametric Bayesian learning, te Diriclet distribution [16] is often used to model a parameter wit unknown distribution since it is a conjugate prior of te multinomial distributon. Given tat N 1, N 2,..., N K independent observations of events E 1, E 2,...,E K are made, and under te assumption tat te prior distribution of te probability vector p = (p 1, p 2,..., p K ) of te events E 1, E 2,..., E K is Diriclet distributed wit parameter α = (α 1, α 2,..., α K ) (α 1, α 2,...,α K > 0), te posterior probability p given te observations N = (N 1, N 2,..., N K ) will follow a Diriclet distribution of order K wit parameter α + N as follows f(p N) = K α p i+n i 1 1 i B(α i + N i ), (1) i=1 were Γ(.) is te gamma function and B(.) is a normalizing factor given by K i=1 B(x) = Γ(x i) Γ( K i=1 x i). (2) Te posterior expected probability E[p i N] of observing event E i given te observations will ten be E[p i N] = α i + N i K j=1 α j + N j. (3) In our system, during eac slot t, te HAP seeks to estimate te probability distribution of te power consumption profile of device i based on te observations in te previous time slots to determine te suitable energy signal transmission power P at. Te observation made in eac time period t is te transmission power value P it tat is computed from te received uplink signal power P r,it of te device. Te HAP only considers positive transmission power values to form its estimate. Tis is because many wireless devices, especially macine type devices, ave very bursty traffic caracteristic, and tus, te number of time slots wen te device transmits will be negligble compared to wen te device does not transmit. In te considered model, we assume tat a malicious adversary seeks to launc a MITM attack on te HAP s learning mecanism so as to deplete te battery of te device by altering te learned power consumption distribution by te HAP. In te context of learning, tis is commonly known as data poisoning. Next, we explain ow te malicious user will interact wit te studied HAP s learning sceme. III. ATTACKER MODEL Te malicious user attempts to alter te power consumption distribution learned by te HAP by performing a wireless MITM attack [17]. In a wireless MITM, a malicious user is assumed to be capable of impersonating te HAP to lure te device to connect to it. After te device connects to te malicious user, te adversary adverstises a different uplink frequency band in order for te HAP not to receive te transmissions directly from te device. Ten, te advsersary obtains te device s information (suc as device ID and security parameters) and subsequently impersonates te device. In tis attack, te adversary intercepts te uplink packet transmitted by te device in time slot t and retransmits te packet to te HAP wit te minimum possible power value P mt {P j O t s.t. P j P it } tat maintains a low risk of being detected by te HAP. Te set O t P i is te set of device s transmission power values oberved by te malicious user up to time slot t. Full cannel state information is assumed to be available at te malicious user, and, tus, te malicious user can perfectly recover te transmission power value P it of te device from te received power value P r,it. In our model, te adversary as no knowledge of te HAP s exact attack detection metod or its defensive strategy. Tus, in order to limit te risk of being detected by te HAP, te malicious user cooses te minimum transmission power P mt at eac time slot t suc tat te Kullback Leibler (KL) distance between te estimates tat are based on te real and modified power values respectively do not exceed a predefined value r. Let R t and M t te estimates of te probability distribution based on te real and modified power values respectively at time slot t. Te malicious user uses te conventional Bayesian learning metod based on Diriclet distribution described in Section I to determine te estimates R t and M t. For te attacker, te value of r captures te risk of being detected by te HAP. Here, te iger te value of r, te iger te probability tat te attacker will be detected by te HAP. Tus, te attacker selects te transmission power according to te following optimization problem min P mt P mt s.t. D KL (R t M t ) r, 0 P mt P it, P mt O t. (4) In te studied system, te attacker as no prior information on te power consumption distribution, and, ence, te prior distribution of te probabilities of transmissions wit powers in O t is assumed to be uniform, i.e., Diriclet wit parameter 1. For a set of observed transmission power values O t, let φ i,t be te number of occurrences of transmission power value P i O t up to time slot t and ω i,t be te number of times te malicious user transmits wit power value P i up to time slot t. Define te vectors φ t = (φ i,t ) Pi O t and ω t = (ω i,t ) Pi O t. Tus, te posterior distributions R t and M t follow te Diriclet distribution wit parameters 1 + φ t and 1 + ω t respectively. Tus, te expected probabilities p i,t and q i,t of observing power value P i based on te estimates R t and M t are, respectively, p i,t = ω i,t+1 j O ω t j,t+ O t is ten given by D KL(R t M t) = φ i,t+1 j O φ t j,t+ O t = φi,t+1 t+ O t and q i,t = = ωi,t+1 t+ O t. Te KL distance of R t p i,t log pi,t = φ i,t + 1 q i,t t + O i O t i O t t and M t log φi,t + 1 ω i,t + 1. (5)

4 t International Symposium on Modeling and Optimization in Mobile, Ad Hoc, and Wireless Networks (WiOpt) Tus, te KL distance D KL (R t M t ) depends on te power value P mt cosen by te malicious user at time slot t. Te following proposition provides a simplified version of te constraint on te KL distance in (4) in order to avoid computing te KL distance for eac power value P mt to find te miminum transmission power P mt. Proposition 1. Let P l be te observed device power value at time slot t. Te attacker selects te minimum transmission power P mt = P k < P l at time slot t suc tat (t+ Ot ) r (t 1+ Ot 1 ) r ω k,t κl,t 1 ω k,t e φ k,t 1 +1, (6) were κ l,t 1 = (φ l,t 1 + 1) log( φ l,t 1+1 ω l,t 1 +1 ) (φ l,t 1 + 2) log( φ l,t 1+2 ω l,t 1 +1 ). Oterwise, Te attacker cooses Pmt = P l. Proof. First, let ω k,t 1 = ω k,t 1 +1 and φ k,t 1 = φ k,t Te KL distance at time slot t is given by D KL (R t M t ) = p i,t log p i,t q i,t i O t = (φ l,t 1 + 1) t + O t + i O t,i l,k log (φ l,t ω l,t 1 φ i,t 1 t + O t log φ i,t 1 ω i,t 1 ) φ k,t 1 + t + O t log( φ k,t 1 ω k,t 1 + 1). (7) Let F (R t M t ) = (t + O t ) D KL (R t M t ). Ten, F (R t M t ) F (R t 1 M t 1 ) = (φ l,t 1 + 1) log( φ l,t ω l,t 1 ) φ l,t 1 log( φ l,t 1 ω l,t 1 ) +φ k,t 1 log( ω k,t 1 ω k,t 1 + 1). (8) Given tat te value of te divergence at time slot t 1 is D KL (R t 1 M t 1 ) = r r. Te constraint on te divergence at time slot t translates to F (R t M t ) F (R t 1 M t 1 ) (t+ O t )r (t 1+ O t 1 )r. From (8), we get te constraint log( ω k,t 1 ω k,t 1 +1) (t+ Ot ) r (t 1+ Ot 1) r +κ l,t φ k,t 1 were κ l,t = φ l,t 1 log( φ l,t 1 ω l,t 1 Tus, we get ω (t+ Ot ) r (t 1+ Ot 1 ) r +κl,t k,t 1 ω k,t 1 +1 e φ k,t 1, and (t+1+ Ot ) r (t 1+ Ot 1 ) r ω k,t +1 ω k,t +2 e ) (φ l,t 1 +1) log( φ l,t 1 +1 ω ). l,t +κl,t 1 φ k,t Proposition 1 transforms te constraint on te KL distance given by (5) to a constraint on ω k,t 1 te number of times te malicious user transmits wit a power value P k up to time slot t 1. Tus, finding te optimal power value for te optimization problem does not require computing te KL distance to ceck te constraint for eac power value P k. It suffices to ceck te constraint on ω k,t 1 given by (6). Tus, by transmitting wit a power value less tan P it, te attacker misleads te HAP into believing tat te device is consuming a lower transmission power. To twart suc attacks, te HAP, on te oter and, utilizes a defensive/robust learning mecanism. Te details of te learning mecanism are explained in te following section. IV. HAP DEFENSIVE STRATEGY A. HAP Information Censoring Based Learning Mecanism In order to reduce te effect of a potential MITM on te updated estimate of probability distribution at eac time slot t, te HAP assumes tat te true transmission power of te device is iger tan te transmission power computed from te received signal at time period t i.e. te true transmission power belongs to te set {P j Ω t s.t. P j P mt } were Ω t is te set of power values observed by te HAP up to time slot t. Tus, te HAP constructs an estimate of te power consumption distribution based on tis belief. In tis case, te observation of te true transmission power of te device is considered to be censored. Te general definition of a censored observation [18] is given next. Definition 1. An observation is said to be censored wen it is not fully observable but rater it is reported tat it belongs to a subset C of te set of events {E 1, E 2,..., E K }. Tus, in te case of censored observations, te estimate of te probability distribution of te events {E 1, E 2,..., E K } will depend on te received reports about te censored observations [18]. In our problem, te report at time slot t is tat te true transmission power of te device belongs to te set C t = {P j Ω t s.t. P j P mt }. In tis case, te joint distribution of te probabilities p of te transmission powers in Ω t depends on λ C k, te conditional probability of getting a report C given tat te actual transmission power is P k. Denote by Λ te matrix of λ C k C, k and C t te set of reports up to time slot t. Ten, te likeliood of te reports given p and Λ will be f({c k } t k=1 p, Λ) = p i λ C i ) N C,t. (9) ( C C t i s.t.p i C were N Ct = (N C,t ) C Ct is te vector of counts of observed reports up to time slot t and N C,t is te number of times te set C is reported up to time slot t. As seen in (9), te likeliood f({c k } t k=1 p, Λ) depends on µ C,i = p i λ C i te joint probability of receiving report C wen te true transmission power is P i. Let µ be te matrix of µ C,i C, i. Te joint outcomes (P it, C t ) at time slot t are ten distributed wit parameter µ. Hence, as sown in [18], we can assume tat te prior distribution of µ follows a Diriclet distribution wit parameter a were eac entry a C,i is te parameter corresponding to µ C,i. Consequently, te prior distribution of te probability vector p at time slot t follows a Diriclet distribution wit parameter β t = (β i,t ) K i=1 were β i,t = C C i,t a C,i and C i,t is te set of all reported sets tat include P i up to time slot t.

5 t International Symposium on Modeling and Optimization in Mobile, Ad Hoc, and Wireless Networks (WiOpt) Under tese assumptions, te posterior distribution of te probability vector p of transmission powers in Ω t at eac time slot t is sown [18], [19] to belong to a class of generalized Diriclet distributions and is tus given by f(p N Ct, Λ) = D(β t, Λ, N Ct ). In general, te distribution D(b, Z, d) as a probability mass function g(p, b, Z, d) = f(p, b) k ( i z kip k ) di, (10) R(b, Z, d) were f(p, b) is te pdf of a Diriclet distribution wit parameter b and R(b, Z, d) is a Carlson s bidimensional ypergeometric function wic can be expressed as R(b, Z, d) = B(Z b+d) B(Z b) were B(.) is te normalizing factor of te Diriclet distribution given by (2). Te posterior mean of te probability p i of transmitting wit power value P i given te reports counts N Ct is [18] a E[p i N Ct ] = a + t E[p i] + t a + t ( N{i},t t + C C i,t\{i} N C,t t a ) C,i i C a, (11) C,i were a is te sum of elements of te Diriclet yperparameter a, and E[p i ] is te expectation of te prior distribution. Since te prior distribution is Diriclet wit paramter β t, te expectation is E[p i ] = j βi,t. βj,t Let I t be te updated estimate probability distribution by te HAP by te end of time slot t. Based on te formed estimate I t at te end of time slot t, te HAP selects in te subsequent time slot t + 1, te transmission power of te energy signal P a,t+1 tat maximizes its utility wile ensuring tat te battery of te device is not depleted, as explained next. B. Energy Signal Power Selection During te downlink at slot t, te HAP uses te last updated estimate I t 1 of te power consumption distribution to decide on te power value P at of te energy signal. Since in te first time slot te HAP as not received any observations, it transmits wit te maximum power P a,max. In te subsequent time slots, te objective of te HAP is to find te optimal transmission power value Pat tat maximizes its utility wile not depleting te device s battery. To acieve tis end, te HAP selects te transmission power suc tat te energy supplied is greater tan te expected transmission energy consumed by te device. In time slot t, I t 1 is te most updated estimate of te power consumption probability distribution at te HAP. Hence, te HAP assumes tat eac P ik is distributed according to I t 1. Tus, te constaint is given by ηt ( t 1 ) D,ik 2 Pak + D,it 2 P at k=1 t E[P ik ] T, (12) k=1 were Pak is te cosen transmission power value of te energy signal transmitted by te HAP at time slot k (1 k t 1) and te expectation is wit respect to te distribution I t 1. Tus, te expected transmission power value of te device E[P ik ](1 k t 1) is given by E[P ik ] = i Ω t p i,t P i were p i,t is te posterior expectation E[p i N Ct, Λ] given by (11). Since te transmission power values are positive, te constraint (12) becomes [ t k=1 P at E[P ik] η t 1 k=1 D,ik 2 P ak η D,it 2 ] +. (13) Furter, since P at P a, te lower bound on P at is redefined as [ t k=1 E[P ik] η t 1 k=1 D,ik 2 P ak ] + P at,lb = η D,it 2. (14) Te payoff of te HAP is expressed in terms ot its utility wic is te energy arvested by te device minus te cost C(P at ) of transmitting te energy signal. Te cost C(P at ) is typically defined as [20] C(P at ) = ap 2 at + bp at were te values of a and b (a, b > 0) depends on te caracteristics of te HAP. Hence, te payoff of te HAP is U at (P at ) = η D,it 2 P at C(P at ). (15) Let ξ t = η D,it 2, te payoff becomes U at (P at ) = (ξ t b)p at ap 2 at. (16) Hence to find te optimal power value P at, te HAP solves te following optimization problem max P at U at (P at ) s.t. P at P at,lb, P at P a. (17) Te optimal solution P at is found by first sowing tat te payoff function U at (P at ) is discrete concave in P at. Ten, te relaxed continuous version of te optimization problem in (17) is considered and its closed form solution P c at is obtained. Based on te solution of te continous version of te problem, te optimal solution of te original problem is obtained. Proposition 2. Te payoff U at is discrete concave in P at. Proof. A univariate discrete function f : Z R is discrete concave if f(x 1) + f(x + 1) 2f(x). Tus, te standard definition of discrete convexity/concavity assumes tat a discrete function f is defined over te set Z wile te set P a is not necessarily Z but it is assumed tat in P a, te power values are multiples of were 0 1. In order to sow tat U a is discrete concave, te variable P a is transformed into a variable P a defined in a subset in Z by defining P a = Pa. By substituting te P a in terms of P a in terms of te utility function U at, we get U at (P a) = (ξ t b)p a a 2 P a 2. Te payoff U at (P a) is discrete concave in P a since U at (P a 1) + U at (P a + 1) = 2(ξ t b)p a 2 (P a 2 + 1) 2(ξ t b)p a 2 P a 2 = 2U at (P a). A consequence of tis proposition is tat any local maximum is a global maximum of te optimization problem in (17). In order to caracterize te optimal solution, we consider te relaxed continuous version of te problem in (17). Remark 1. Te optimal solution for te relaxed optimization problem is

6 t International Symposium on Modeling and Optimization in Mobile, Ad Hoc, and Wireless Networks (WiOpt) ξ t b 2a, if P at,lb ξt b 2a P a,max, Pat c = P at,lb, if ξt b 2a < P (18) at,lb, P a,max, oterwise, were P a,max is te maximum power value in P a. Proof. It can be easily sown tat te utility function is continuous strictly concave in P a since te second order partial derivative is 2 a. Also, te value of P a at wic te derivative of te utility function is zero is P a = ξt b. Also, te only constraints of te optimizations are bound constraints on P a. Te results ten follows from te concavity of U at and te bound constraints. Proposition 3. Te optimal power Pat of te HAP is max( (ξ t b) Pat = P at,lb,, (ξ t b) ),if P at,lb (ξ t b) Pa,max, if (ξ t b) < P at,lb, P a,max, oterwise. (19) Proof. Wen P at,lb (ξt b) Pa,max, we ave P at,lb (ξt b) (ξt b), Pa,max since P at,lb and Pa,max are integers. Since U at () is strictly concave for continuous values of P a, te payoff for any integer power value d will be less tan te payoff of using te power value m = max( (ξt b), (ξt b) ) i.e. U a(m) U a (d). Hence in tis case, te optimal power value P a is max( (ξt b) (ξt b), ) and te corresponding optimal value in P a is Pa =, (ξt b) (ξt b) ). For te case wen is less max( (ξt b) tan te P at,lb, te optimal power value is P a = P at,lb since te payoff of any oter power value greater tan P at,lb is less tan U at (P at,lb ) due to te discrete concavity of U at and te corresponding optimal value in P a is Pa = P at,lb. Using te same concavity argument for te last case i.e. wen Pa max, te optimal value is P a = Pa,max and te corresponding power value in P a is P a = P a,max. Proposition 3 sows tat wen ξ t, te product of te device battery efficiency and te cannel gain, is considerably greater tan te energy cost parameters a and b, te utility of te HAP becomes iger tan te cost and tus te HAP transmits wit maximum power. Also, if ξ t is considerable smaller tan te energy cost parameters a and b, te HAP s cost becomes iger tan its utility and te HAP transmits wit te lowest feasible power. Oterwise, te HAP transmits wit te optimal power tat maximizes its payoff. (ξt b) V. SIMULATION RESULTS For our simulations, we set W = 10 khz, T = 2 msec, N 0 = 137 dbm, η = 0.8, P a,max = 2W, = 0.1, and P i = {0.1, 0.2, 0.3, 0.4} W. Te wireless transmitter is considered to be a video surveillance device [21] wic generates UDP Percentage of dropped packets (%) Energy consumed (J) Conventional (a,b)=(1,1) Conventional (a,b)=(1.5,1.5) Robust (a,b)=(1,1) Robust (a,b)=(1.5,1.5) Risk value Fig. 1: Percentage of packets lost vs. risk value Conventional (a,b)=(1,1) Conventonal (a,b)=(1.5,1.5) Robust (a,b)=(1,1) Robust (a,b)=(1.5,1.5) Risk value Fig. 2: Energy consumed vs. risk value packets of size M = 1000 bits. Te packets are generated according to a Poisson distribution of rate 30 packets/sec. Te video surveillance device cooses its transmission power in eac time slot suc tat te received SNR is greater tan or equal to te required tresold γ R to decode te packet at te HAP. Assuming tat te acieved rate and SNR are related by Sannon s capacity formula, te tresold is tus cosen M suc tat T = W log(1 + γ R). Te values considered for te energy cost parameters (a, b) of te HAP are (1, 1) and (1.5, 1.5) respectively [20]. In eac time slot, te device drops te packet if it does not ave enoug energy to transmit it. Eac simulation run simulates te network for time slots, i.e., 100 seconds. In eac run, te percentage of packets dropped by te surveillance device and te energy consumed by te HAP are computed wen te HAP uses te robust and conventional learning strategies respectively for te considered values of te energy cost parameters. Ten, te average percentage of packets dropped and te average energy consumed by te HAP is computed from 1000 simulation runs. Te simulation is performed for two scenarios. Te first is wen te malicious user s risk value takes values 0, 10 4, 10 3, 10 2, 10 1, 1 respectively wile te fading variance of te cannel between te HAP and te device for bot uplink and downlink is set to 0.3. Te second scenario is wen te fading variance is varied between 0.3 and 0.9 in steps of 0.1 wile te risk value is set to be r = Fig. 1 sows te percentage of dropped packets for bot te conventional and robust learning approaces versus te risk

7 t International Symposium on Modeling and Optimization in Mobile, Ad Hoc, and Wireless Networks (WiOpt) Percentage of dropped packets (%) Conventional (a,b)=(1,1) Conventional (a,b)=(1.5,1.5) Robust (a,b)=(1,1) Robust (a,b)=(1.5,1.5) Cannel variance Fig. 3: Percentage of packets lost vs. cannel variance Energy consumed (J) Conventional (a,b)=(1,1) Conventional (a,b)=(1.5,1.5) Robust (a,b)=(1,1) Robust (a,b)=(1.5,1.5) Cannel variance Fig. 4: Energy consumed vs. cannel variance value wen te values of te HAP s energy cost paramters (a, b) are (1, 1) and (1.5, 1.5) respectively. First, for te conventional learning approac, and wen te energy cost parameters values (a, b) are (1, 1), te percentage of dropped packets wit no attack (r = 0) is 33%. Tis percentage increases wit te risk and reaces 72% for risk values greater tan or equal to Wen te cost parameters (a, b) increases to (1.5, 1.5), te percentage of dropped packets increases for all considered risk values reacing up to 97% wen te risk value greater tan or equal to In contrast, for te proposed approac, wen (a, b) are set to (1, 1), te percentage of dropped packets 10.25% wen no attack occurs, and it remains around 10% wen te attacker s risk increases to 0.1. A more pronounced increase occurs wen te risk value is 1 as te percentage of dropped packets attains 37%. For suc a ig risk value, te optimal strategy of te attacker is to transmit wit minimum power, wic affects te effectiveness of te robust approac. However, in practice, te attacker will only coose low risk values in order not to be detected, and ence, te percentage of dropped packets wen r = 1 will not be attained. Wen te energy cost parameters increase to (1.5, 1.5), te percentage of dropped packets is around 15% for a risk value less tan or equal to 0.1 and increases to 44% wen te risk value is one. Tus, Fig. 1 sows tat te proposed robust learning strategy constitutes a better learning approac tan te conventional learning approac even wen no attack appens. Furter, te proposed robust learning approac is more robust to canges in te risk values unlike te conventional learning approac tat is sensitive to sligt variations in te risk value. From Fig. 1, we can also see tat, te proposed aproac can acieve a performance gain, in terms of te percentage of dropped packets, wic can reac up to 85% at r = 0.1 compared to te conventional learning approac. Fig. 2 sows te energy consumed for te conventional and robust learning approaces versus te risk value for different energy cost parameters. As sown in Fig. 2, te conventional learning approac maintains low energy consumption. Wen te energy cost paramteres are (1, 1), te energy consumed is 2.59 J wen no attack appens and drops to J for a risk value greater tan or equal to Wen (a, b) increases to (1.5, 1.5), te energy consumed decreases to 1.9 J wen no attack appens and drop to 0.03 for a risk value iger tan 0.3. On te oter and, te proposed robust strategy exibits iger energy consumption. Tis is because te robust approac overestimates te transmission power consumed by device, wic results in increasing te energy delivered to te device in eac time slot. Wen te value of te energy cost paramteres (a, b) is (1, 1), te energy consumed is around 62 J for a risk value lower tan or equal to 0.1 and drops to 10 J for a risk value equal to one. Wen (a, b) increases to (1.5, 1.5), te consumed energy decreases to 45 J for a risk value less tan 0.1 and drops to 8 J for a risk value equal to one. Tus, te results in Fig. 2 sow te tradeoff between maintaining a good performance in terms of te percentage of dropped packets and te energy consumed. Yet, te energy consumed by te robust learning is lower tan te energy consumed wen te HAP transmits wit fixed maximum power in eac time time slot. For te considered simulation values of te system parameters, te energy consumed by te fixed power policy is 200 J. Hence, te robust learning strategy can reac a gain in terms of energy efficiency up to 77% wile maintaining a low percentage of dropped packets. Fig. 3 sows te percentage of dropped packets for bot te conventional and robust learning approaces versus te cannel variance value for te considered values of te HAP s energy cost paramters. First, for te conventional learning appoac and wen te values of te energy cost parameters (a, b) are (1, 1), te percentage of dropped packets decreases significantly from 72% to 1.35% wen te value of te cannel variance increases from 0.3 to 0.4. Ten, te percentage of dropped packets tend to zero as te variance increases furter. Tis is due to te fact tat, wen te cannel quality improves, te received energy by te device increases, and te transmit power required by te device to deliver te packet successfully decreases wic allows for more successfull transmissions. Moreover, for energy cost parameters (1, 1), te probability tat te HAP s energy cost becomes lower tan its utility increases. Tus in tis case, te HAP is more likely to transmit wit maximum power P a,max. Next wen te HAP s energy cost parameters (a, b) are increased from (1, 1) to (1.5, 1.5), te percentage of dropped packets using te conventional learning approac increases considerably wen te cannel variance value is less tan or equal to 0.5. Tis is due to te fact tat, for iger values of te cost parameters, te energy cost increases, and te HAP is more likely to

8 t International Symposium on Modeling and Optimization in Mobile, Ad Hoc, and Wireless Networks (WiOpt) supply less energy to te device. Tus, te HAP s conservative strategy combined wit te altered estimate by te attacker will yield a significant increase in packet loss. On te oter and, te robust learning strategy maintains a low to negligible percentage of packet loss for bot considered values of energy cost parameters. Te increase in te percentage of packet loss due to te increase in te cost parameters is sligtly obervable wen te value of te cannel variance is 0.3 were te increase is from 10% to 15%. However, te percentage of dropped packets is negligible for iger values of te cannel variance. Clearly, from Fig. 3, we can see tat te proposed approac is more robust to more conservative energy policies by te HAP under different cannel conditions. Fig. 4 sows te energy consumed for bot te conventional and robust learning approaces as a function of te cannel variance. For te conventional learning approac and for cost parameters (1, 1), te energy consumed is only 0.46 J wen te value of cannel variance is 0.3 due to te ig packet loss as sown in Fig. 3. Ten, te energy consumed increases wit te cannel variance. Tis is because te percentage of packets lost decreases wit te cannel variance, as sown in Fig. 3, wic implies tat te device is transmitting successfully more packets and requires te HAP to transmit more energy. Wen te cost parameters increases to (1.5, 1.5), te energy consumed by te HAP decreases since te HAP adopts a more conservative energy policy. For te robust learning approac, for cost parameters (1, 1), te energy consumed decreases first wit te cannel variance wen te value of te cannel variance is less tan or equal to 0.6. Tis is because wen te cannel quality is low, te cannel gain takes low values wit ig probability. Tus, te HAP must spend more energy to meet eac device s energy requirements. However, for values of cannel variance iger or equal to 0.6, te energy consumed starts to increase due to te increase in te number of successfully transmitted packets by te device. Also, te energy consumed using te robust strategy becomes almost equal to te energy consumed using te conventional learning approac. Tis is because, wen te cannel quality becomes ig, te HAP can meet te energy requirements of te device wit minimal transmission power, wic makes te attacks by te malicious user ineffective. Te energy consumed by te robust learning strategy exibits a similar pattern wit te cannel variance value wen te values of te cost paramters are (1.5, 1.5) yet it is lower tan te energy consumed wen te value of te costs parameters are (1, 1). VI. CONCLUSION In tis paper, we ave introduced a robust Bayesian learning sceme for RF energy arvesting wic allows te HAP to form an estimate of te transmission power consumption profile of eac associated device based on te device s received power at eac time slot. Te proposed sceme takes into account potential man-in-te-middle-attacks by a malicious user tat tries to alter te learned estimate of te HAP in order to deplete te battery of te device. Based on te learned estimate, we ave considered te problem of optimal power selection by te HAP in eac time slot tat maximizes te HAP s payoff wile meeting device s energy requirements are met. Furter, we ave sown tat te payoff function is discrete concave and obtained a closed-form expression of te optimal power of te supplied energy signal. 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