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IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 34, NO. 5, MAY 2016 1477

Stability Analysis of Slotted Aloha WithOpportunistic RF Energy Harvesting

Abdelrahman M. Ibrahim, Ozgur Ercetin, and Tamer ElBatt, Senior Member, IEEE

Abstract—Energy harvesting (EH) is a promising technologyfor realizing energy-efficient wireless networks. In this paper,we utilize the ambient RF energy, particularly interference fromneighboring transmissions, to replenish the batteries of the EHenabled nodes. However, RF energy harvesting imposes new chal-lenges into the analysis of wireless networks. Our objective in thispaper is to investigate the performance of a slotted Aloha ran-dom access wireless network consisting of two types of nodes,namely Type I, which has unlimited energy supply and Type II,which is solely powered by an RF energy harvesting circuit. Thetransmissions of a Type I node are recycled by a Type II node toreplenish its battery. We characterize an inner bound on the sta-ble throughput region under half-duplex and full-duplex energyharvesting paradigms as well as for the finite capacity batterycase. Additionally, we analyze the case where RF energy har-vesting serves as a backup for an unlimited energy source. Wepresent numerical results that validate our analytical results, anddemonstrate their utility for the analysis of the exact system.

Index Terms—Wireless networks, slotted Aloha, opportunisticenergy harvesting, interacting queues.

I. INTRODUCTION

O NE of the prominent challenges in the field of commu-nication networks today is the design of energy efficient

systems. In traditional networks, wireless nodes are powered bylimited capacity batteries which should be regularly charged orreplaced. Energy harvesting has been recognized as a promisingsolution to replenish batteries without using any physical con-nections for charging. Nodes may harvest energy through solarcells, piezoelectric devices, RF signals, etc. In this paper, wefocus on RF energy harvesting. Recent studies present experi-mental measurements for the amount of RF energy that can be

Manuscript received March 29, 2015; revised October 21, 2015; acceptedDecember 4, 2015. Date of publication January 21, 2016; date of currentversion May 19, 2016. This work was made possible by grant numberNPRP 5-782-2-322 from the Qatar National Research Fund (a member ofQatar Foundation). The statements made herein are solely the responsibil-ity of the authors. The work of O. Ercetin was supported by the MarieCurie International Research Staff Exchange Scheme Fellowship PIRSES-GA-2010-269132 AGILENet within the 7th European Community FrameworkProgram.

A. M. Ibrahim was with the Wireless Intelligent Networks Center (WINC),Nile University, Giza, Egypt. He is now with the Department of ElectricalEngineering, Pennsylvania State University, University Park, PA 16802 USA(e-mail: [email protected]).

O. Ercetin is with the Faculty of Engineering and Natural Sciences, SabanciUniversity, Istanbul, Turkey (e-mail: [email protected]).

T. ElBatt is with the Department of Electronics and ElectricalCommunications Engineering, Faculty of Engineering, Cairo University,Giza, Egypt, and also with the Wireless Intelligent Networks Center (WINC),Nile University, Giza, Egypt (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/JSAC.2016.2520248

harvested from various RF energy sources. Two main factorsaffect the amount of RF energy that can be harvested, namely,the frequency of the RF signal and the distance between the“interferer” and the harvesting node, e.g., see Table I in [1].

Recently, an information-theoretic study of the capacity of anadditive white Gaussian noise (AWGN) channel with stochas-tic energy harvesting at the transmitter has shown that it isequal to the capacity of an AWGN channel under an averagepower constraint [2]. This, in turn, motivated the investiga-tion of optimal transmission policies [3] for single user [4]–[6]and multi-user [7]–[9] energy harvesting networks. The opti-mal policy that minimizes the transmission completion timewas studied in [4]. In [5], the authors studied the problem ofmaximizing the short-term throughput and have shown that itis closely related to the transmission completion time prob-lem [4]. The authors in [6] studied the optimal transmissionpolicies for energy harvesting networks under fading channels.Moreover, [7]–[9] extended the analysis to broadcast, multipleaccess, and interference channels, respectively. The authors in[10] introduced the concept of energy cooperation where a usercan transfer portion of its energy, over separate channels, toassist other users.

Significant research has also been conducted on RF energyharvesting. In [11], the author discusses the fundamental trade-offs between transmitting energy and transmitting informationover a single noisy link. The author derives the capacity-energyfunctions for several channels. The authors in [12], extendthe point-to-point results of [11] to multiple access and multi-hop channels. Recently, several techniques were proposed fordesigning RF energy harvesting networks (RF-EHNs), e.g.,[13]. The RF energy harvesting process can be classified asfollows: i) Wireless energy transfer, where the transmittedsignals by the RF source are dedicated to energy transfer,ii) Simultaneous wireless information and power transfer,where the transmitted RF signal is utilized for both informa-tion decoding and RF energy harvesting, and iii) Opportunisticenergy harvesting, where the ambient RF signals, consideredas interference for data transmission, are utilized for RF energyharvesting. The receiver architecture may also vary as follows[13], [14]: a) Co-located receiver architecture, where the radioreceiver and the harvesting circuit use the same antenna forboth decoding the data and energy harvesting, and b) Separatedreceiver architecture, where the radio receiver and the harvest-ing circuit are separated, and each is equipped with its ownantenna and RF front-end circuitry.

In [14], the authors discuss the practical limitations ofimplementing a simultaneous wireless information and powertransfer (SWIPT) system. A major issue is that energy

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1478 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 34, NO. 5, MAY 2016

harvesting circuits are not able to simultaneously decode infor-mation and harvest energy. Hence, the authors in [14] pro-posed and analyzed two modes of operation for the co-locatedreceiver architecture, that is, time switching and power split-ting. Furthermore, the RF energy harvesting transceivers mayalso be classified as: a) Half-duplex energy harvester, where aco-located RF energy harvesting transceiver can either transmitdata or harvest RF signals at a given instant of time, and b) Full-duplex energy harvester, where a node is equipped with twoindependent antennas and can transmit data and harvest RF sig-nals, simultaneously. In this paper, we investigate opportunisticRF energy harvesting under the half-duplex and full-duplexmodes of operation.

The cornerstone of random access medium access control(MAC) protocols is Aloha protocol [15], which is widely stud-ied because of its simplicity and decentralized nature. It offersseveral advantages over centralized scheduling, e.g., TDMA,where a central node has to collect state information about theusers queue lengths, and channels, in order to take the deci-sion. This not only involves high communication overhead,that scales with the number of nodes, but also computationalcomplexity. On the other hand, slotted Aloha does not requiregathering state information, hence stateless, which renders itzero-overhead and low-complexity. Furthermore, it is knownto achieve higher resource utilization and lower delay andhence favored, over plain TDMA, under light traffic condi-tions [16], [17].

The Aloha protocol has received remarkable attention fromthe community because of its relevance in the design and anal-ysis of modern communication systems [18]. One of the earlyapplications of slotted Aloha protocol is satellite networks [16],[17]. There are several other applications for Aloha-based pro-tocols, such as radio frequency identification (RFID) systems[19], underwater acoustic sensor networks (UW-ASN) [20],[21] and the emerging Machine-to Machine (M2M) communi-cations [22]. The valuable insights drawn from various modelsand analysis studies introduced for slotted Aloha has played amajor role in spurring research towards more efficient randomaccess protocols over the years.

Based on the Aloha protocol, nodes contend for the sharedwireless medium and cause interference to each other. Hence,the service rate of a node depends on the backlog of other nodes,i.e., the nodes queues become interacting as originally char-acterized in [23]. Tsybakov and Mikhailov [24] were the firstto analyze the stability of a slotted Aloha system with finitenumber of users. Rao and Ephremides [25] characterized suf-ficient and necessary conditions for queue stability of the twouser case, using the so-called stochastic dominance technique.In addition, they established conditions for the stability of thesymmetric multi-user case. Other works followed and studiedthe stability of slotted Aloha with more than two users [26]–[29]. The authors in [30] extended the stability analysis undercollision model to a symmetric multi-packet reception (MPR)model, which was later generalized to the asymmetric MPRmodel in [31].

Perhaps the closest to our work are [32], [33] which char-acterize the stability region of a slotted Aloha system withenergy harvesting capabilities, under the multi-packet reception

Fig. 1. System model.

model. The authors considered a system where the nodes har-vest energy from the environment at a fixed rate and, thus, theenergy harvesting process is modeled as a Bernoulli process.

In this work, we analyze the stability of a slotted Aloharandom access wireless network consisting of nodes with andwithout RF energy harvesting capability. Specifically, we con-sider a wireless network consisting of two nodes, namely a nodeof Type I which has unlimited energy supply and a node of TypeII which is powered by an RF energy harvesting circuit. The RFtransmissions of the Type I node are harvested by the Type IInode to replenish its battery. Our contribution in this paper ismulti-fold. First, we outline the difficulties in analyzing the sta-bility of the exact RF energy harvesting Aloha system SO andfor mathematical tractability we approximate it by the systemSG . Second, we generalize the stochastic dominance techniquefor analyzing RF EH-networks. Third, we characterize an innerbound on the stable throughput region of the system SG underthe half-duplex and full-duplex energy harvesting paradigms.We also derive the closure of the inner bound over all trans-mission probability vectors. Fourth, we investigate the impactof finite capacity batteries on the stable throughput region andinvestigate the case where RF energy harvesting serves as abackup for an unlimited energy source. Finally, we validateour analytical findings with simulations and conjecture that theinner bound of the system SG is also an inner bound for theexact system SO .

The rest of this paper is organized as follows. In Section II,we present the system model and the assumptions underlyingour analysis. In Section III, we describe the energy harvest-ing models for the systems SO and SG . Our main results arepresented in Section IV and proved in Section V. In SectionVI, we investigate the impact of finite capacity batteries, full-duplex energy harvesting and backup RF energy harvesting onthe stability region of our system. We corroborate our analyticalfindings by simulating the systems SO and SG in Section VII.Finally, we draw our conclusions and point out directions forfuture research in Section VIII.

II. SYSTEM MODEL

We consider a wireless network consisting of two sourcenodes and a common destination, as shown in Fig. 1. Weconsider a slotted Aloha multiple access channel [15], wheretime is slotted and the slot duration is equal to one packettransmission time. We use power and energy interchangeablythroughout the paper, since we assume unit time slots. We

IBRAHIM et al.: STABILITY ANALYSIS OF SLOTTED ALOHA 1479

assume two types of nodes in our system: Type I node has adata queue, Q1, and unlimited energy supply, while Type IInode has a data queue, Q2, and a battery queue, B, as shown inFig. 1. Moreover, packets arrive to the data queues, Q1 and Q2,according to independent Bernoulli processes with rates λ1 andλ2, respectively. A Type I node transmits with probability q1,whenever its data queue is nonempty and a Type II node trans-mits with probability q2, whenever its data and battery queuesare nonempty.

We assume packet erasure data channels, where the desti-nation successfully decodes a data packet from node i withprobability di and the packet is lost with probability 1 − di .Under Rayleigh fading, the probability di is given by

di = P

[Piκi r

−αi

N≥ Pth

]= exp

(− Pth Nrα

i

Pi

), (1)

where Pi is the transmission power of node i , κi ∼ Exp(1)

is the Rayleigh fading power gain, ri is the distance betweennode i and the destination, α is the pathloss exponent, N isthe noise power, and Pth is the decoding threshold power. Weconsider a collision channel model, where the destination suc-cessfully decodes a data packet, if only one node transmits. Iftwo nodes transmit simultaneously, a collision occurs and bothpackets are lost and have to be retransmitted in future slots.At the end of each time slot, the destination sends an immedi-ate acknowledgment (ACK) via an error-free feedback channel.Data packets of Type I and II nodes are stored in queues Q1 andQ2, respectively. The evolution of the queue lengths is givenby [29]

Qt+1i = max{Qt

i − Y ti , 0} + Xt

i , i = 1, 2. (2)

where Xti ∈ {0, 1} is the arrival process for data packets and

Y ti ∈ {0, 1} is the departure process independent of Qi status,

i.e., Y ti = 1 even if the data queue is empty [31]. Xt

i is aBernoulli process with rate λi and E[Y t

i ] = μi .We assume that Type II node operates under a half-duplex

energy harvesting mode, i.e., it either harvests or transmits butnot both simultaneously1. Hence, the harvesting opportunitiesare in those slots when a Type II node is idle while a Type Inode is transmitting. The channel between the two source nodesis a block fading channel, where the fading coefficient remainsconstant within a single time slot and changes independentlyfrom a slot to another. For Rayleigh fading, the instantaneouschannel power gain ht at time slot t is exponentially distributed,i.e., ht ∼ Exp(1). We consider a non-singular2 pathloss model(1 + lα)−1, where α is the pathloss exponent and l is the dis-tance between the two source nodes. In the next section, wedevelop a discrete-time stochastic process to model RF energyharvesting.

1We extend the analysis to the case of full-duplex RF EH in Section VI.2In order to harvest significant amount of RF energy, l is typically small.

Hence, we use a non-singular (bounded) pathloss model instead of a singular(unbounded) pathloss l−α model, because the singular pathloss model is notcorrect for small values of l due to singularity at 0, [34].

Fig. 2. A unit of energy is harvested whenever the accumulated energy in thebattery increases by γ .

III. ENERGY HARVESTING MODELS

Prior work largely models the energy harvesting processas an independent and identically distributed (iid) Bernoulliprocess with a constant rate [33]3. In the following, we firststudy the RF energy harvesting process of the exact systemSO and show that the inter-arrival time of the energy arrivalshas a general distribution. Second, we propose an approximatesystem SG where the inter-arrival time of energy arrivals isgeometrically distributed with the same mean as SO .

A. RF Energy Harvesting Model of the Exact System SO

The received power at a Type II node from the transmissionsof Type I node at time slot t is PR(t) = ηP1ht (1 + lα)−1, whereη is the RF harvesting efficiency [13]. Recent studies demon-strated that η typically ranges from 0.5 to 0.7, where its valuedepends on the efficiency of the harvesting antenna, impedancematching circuit and the voltage multiplier [35]. In order todevelop the analytical model underlying this paper, we assumethat the continuous energy arrival is accumulated in an inter-mediate energy storage device to form quantas of size γ joules,see Fig. 2(a). Conceptually, accumulating γ joules is equivalentto having a single energy packet arrival to the battery, see Fig.2(b). Let Z be the number of time slots needed to harvest oneenergy unit.

Lemma 1: For a persistent (q1 =1) and saturated Type Inode, the probability mass function (PMF) of Z , given the chan-nel between Type I and II nodes is modeled as a Rayleigh fadingchannel, is

P[Z = k] = e−θ θk−1

(k − 1)!, k = 1, 2, · · · , θ = γ (1 + lα)

ηP1(3)

3Typically, the energy harvesting process is not iid, because to harvest oneenergy unit, energy is accumulated over multiple slots.


Proof: We need k slots to harvest one energy unit if theaccumulated received energy over k slots is greater than orequal to γ , while the accumulated received energy up to slotk−1 is less than γ , i.e.,

P[Z = k] = P

[k∑

t=1

PR(t) ≥ γ,

k−1∑t=1

PR(t) < γ

](4)

(a)= P

[k−1∑t=1

ht < θ

]− P

[k∑

t=1

ht < θ

](5)

(b)= e−θ θk−1

(k − 1)!, k = 1, 2, · · · , (6)

(a) follows from applying the law of total probability, i.e.,

P

[k−1∑t=1

ht < θ

]= P

[k−1∑t=1

ht < θ,

k∑t=1

ht ≥ θ

]

+ P

[k−1∑t=1

ht < θ,

k∑t=1

ht < θ

], (7)

= P

[k−1∑t=1

ht < θ,

k∑t=1

ht ≥ θ

]+ P

[k∑

t=1

ht < θ

]. (8)

The distribution of sum of independent exponential ran-dom variables is an Erlang Distribution. Hence,

∑kt=1 ht ∼

Erlang(k, 1), where k is the shape parameter and ht ∼ Exp(1).From, the cumulative distribution function (CDF) of the Erlangdistribution, we know that

P

[k∑

t=1

ht < θ

]= 1 −

k−1∑j=0

e−θ θ j

j!, (9)

Hence, (b) is obtained by substituting (9) in (5). �From (6), we notice that the distribution of the inter-arrival

times of the energy harvesting process is a shifted Poissondistribution, i.e., Z = V + 1, where V is a Poisson randomvariable with mean θ . The expected inter-arrival time of the har-vesting process is given by E[Z ] = 1 + θ . In the general casewhere Type I node is unsaturated and transmits with probabil-ity q1, characterizing the PMF of Z is challenging because thequeue evolution process is not an iid process. For instance, ifQt

1 = 2, we know for sure that Qt+11 ≥ 1.

B. RF Energy Harvesting Model of the System SG

For mathematical tractability of the results derived in latersections, we consider a system SG , where the RF energy har-vesting process is an iid Bernoulli process with rate ph|{1},where ph|{1} can be interpreted as the probability of successin harvesting one energy unit given that Type I node is trans-mitting. Based on the fact that the inter-arrival time of aBernoulli process is geometrically distributed, the mean inter-arrival time is 1/ph|{1}. Hence, the relationship between theexact harvesting process in SO and the Bernoulli process in SG

is given by

ph|{1} = 1

1 + θ. (10)

Fig. 3. Markov chain model of the battery queue given that the data queuesQ1 and Q2 are backlogged. Note that δ′ =1−q1 ph − q2 and δ=1−q1 ph(1 −q2)− q2.

In general, we assume an iid Bernoulli process with rateph|M, where ph|M is the probability of harvesting one energyunit given a set of nodes M are transmitting. Under half-duplexenergy harvesting, Type II node only harvests from transmis-sions of Type I node, when Type II node is idle, i.e., theprobability of harvesting one energy unit given Type II nodeis transmitting ph|{2} = 0, and the probability of harvesting oneenergy unit given both nodes are transmitting ph|{1,2} = 0. Forconvenience, we denote ph|{1} by ph .

C. Analyzing the Battery Queue in SG

We assume that transmitting a single data packet costs TypeII node one energy unit. Hence, we model the battery of a TypeII node as a queue with unit energy packet arrivals from the har-vesting process, and unit energy packet departures when TypeII node transmits. Let Ht denotes the energy harvesting processmodeled as a Bernoulli process. Assuming half-duplex harvest-ing,4 the average harvesting rate is the difference between thefraction of time slots in which the Type I node is transmit-ting and the fraction of time slots in which both nodes aretransmitting, i.e.,

E[Ht ] = q1 ph (P[Q1 >0] − q2P[Q1 >0, B >0, Q2 >0]) .

(11)

The battery queue evolves as [33]

Bt+1 = Bt − μtB + Ht , (12)

where μtB ∈ {0, 1} represents the energy consumed in the trans-

mission of a data packet at time t . Under backlogged dataqueues Q1 and Q2, the average rate of harvesting becomes

E[Ht |Q1>0, Q2>0

] = q1 ph (1−q2P [B>0|Q1 >0, Q2 >0]).(13)

Now, the energy harvesting rate is only dependent on the bat-tery queue status. Thus, if the battery queue is empty, the energyharvesting rate is q1 ph , otherwise it is reduced to q1 ph(1 − q2)

because of the half-duplex operation. Hence, the battery queueforms a decoupled discrete-time Markov chain, as shown inFig. 3. By analyzing the Markov chain we find the probabilitythat the battery is non-empty, as given by the following lemma.

Lemma 2: For a half-duplex RF energy harvesting node withinfinite capacity battery, the probability that the battery is non-empty, given that the data queues Q1 and Q2 are backlogged,is given by

P [B > 0|Q1>0, Q2>0] = min

(q1 ph

q2 (1+q1 ph), 1

). (14)

4In the sequel, we will extend the model to full-duplex as well.


Fig. 4. The energy limited and interference limited sub-regions of the stablethroughput region of SD are characterized by the triangles BCO and BCD,respectively.

Proof: Let π = [π0, π1, ...] be the steady-state distribu-tion of the Markov chain shown in Fig. 3. Applying the detailedbalance equations, we obtain

πk =(

q1 ph

q2

)k

(1 − q2)k−1 π0, k = 1, 2, · · · .

Therefore, by substituting in the normalization condition∑i πi = 1, we get the utilization factor

ρ = 1 − π0 = q1 ph

q2(1 + q1 ph).

Hence, P[B > 0|Q1 > 0, Q2 > 0] = min {ρ, 1} . �

IV. MAIN RESULTS

In this section, we present our main results pertaining tothe stable throughput region of the opportunistic RF energyharvesting slotted Aloha network SG . We adopt the notion ofstability proposed in [29], where the stability of a queue isdetermined by the existence of a proper limiting distribution.A queue is said to be stable if

limt→∞P[Qt < x] = F(x) and lim

x→∞ F(x) = 1. (15)

The stability of a queue is equivalent to the recurrence ofthe Markov chain modeling the queue length. Loynes’ Theorem[36] states that if the arrival and service processes of a queueare strictly jointly stationary and the average arrival rate is lessthan the average service rate, then the queue is stable. If theaverage arrival rate is greater than the average service rate, thenthe queue is unstable and the queue size Qt approaches infin-ity almost surely. The stable throughput region of a system isdefined as the set of arrival rate vectors, (λ1, λ2) for our sys-tem, for which all data queues in the system are stable.

Next, we establish sufficient conditions on the stability ofthe opportunistic RF energy harvesting Aloha SG . Assuminghalf-duplex energy harvesting and unlimited battery capacity,the stability region is characterized by the following theorem.

Theorem 1: An inner bound on the stable throughput regionof the opportunistic RF energy harvesting slotted Aloha SG

is the triangle OBD, shown in Fig. 4. Assuming half-duplexenergy harvesting and unlimited battery size, the region ischaracterized by

RinnerG =

⎧⎨⎩(λ1, λ2) | λ1 ≤ d1q1

(1 − λ2

d2(1 − q1)

),

λ2 ≤d2(1 − q1) min

{q1 ph

(1+q1 ph), q2

}λ1

d1q1

(1 − min

{q1 ph

(1+q1 ph), q2

})⎫⎬⎭ . (16)

Proof: The proof is established in Sections V-A to V-E.�

Theorem 2: The closure of the inner bound RinnerG over all

transmission probability vectors q = (q1, q2) is given by

C(RinnerG ) =

⋃q

So(q) =⎧⎨⎩(λ1, λ2) | λ2 ≤ d2 phλ1

2d1

⎛⎝1− λ1

d1+

λ2

d2+√(

1+ λ1

d1− λ2

d2

)2

− 4λ1

d1

⎞⎠⎫⎬⎭ . (17)

Proof: The proof is established in Section V-F. �V. STABILITY ANALYSIS

For the majority of prior work on stability analysis of inter-acting queues, the service rate of a typical node decreaseswith respect to the transmissions of other nodes in the sys-tem. Perhaps, the most basic example is the conventional slottedAloha system [25], where increasing the service rate of an arbi-trary node comes at the expense of decreasing the service rateof other nodes. For our purposes, we refer to such systems as“interference-limited” systems.

On the other hand, in our RF energy harvesting system,transmissions from interfering nodes give rise to two oppos-ing effects on Type II (RF energy harvesting) nodes. Similarto classic interference-limited systems, the interfering nodescreate collisions and, thus, decrease the service rate of RFenergy harvesting nodes. Meanwhile, transmissions from inter-fering nodes are exploited by RF energy harvesting nodes toopportunistically replenish their batteries. Therefore, from theperspective of an RF energy harvesting node, a fundamentaltrade-off prevails between the increased number of energy har-vesting opportunities and the increased collision rate, whichare both caused by interference. As will be shown formally,this fundamental trade-off splits the stable throughput regionfor RF energy harvesting slotted Aloha networks into two sub-regions, a sub-region where interference is advantageous forthe RF energy harvesting node and another sub-region whereit is not. These two sub-regions map directly to two modes ofoperation for our system and are characterized as follows:

1. Energy-limited mode: is the sub-region of the stablethroughput region in which the transmissions of inter-fering nodes enhance the throughput of the RF energyharvesting node, i.e, the throughput enhancement dueto the increased harvesting opportunities outweights thedegradation due to collisions created by interfering nodes.


2. Interference-limited mode: is the sub-region of the stablethroughput region in which the transmissions of interfer-ing nodes do not increase the throughput of the RF energyharvesting node, i.e, the throughput degradation due tocollisions equals or outweighs the throughput increasedue to the increased energy harvesting opportunities.

The two sub-regions are shown in Fig. 4, where the energy-limited region is enclosed by the triangle BCO and theinterference-limited region is enclosed by the triangle BCD.

In order to characterize the inner bound on the stabilityregion of SG in Theorem 1, we introduce a deprived systemSD where Type II node transmits only in the time slots in whichQ1 is non-empty. We derive the stability region of SD , by goingthrough the following three steps discussed next. First, we char-acterize the average service rates of the two interacting dataqueues. Second, we generalize the stochastic dominance tech-nique proposed in [25] to capture our system dynamics andtwo-mode operation. Third, we derive the stability conditionsof SD using the generalized stochastic dominance approach.Finally, we prove in Section V-E that the stability region of SD

is an inner bound on the stability region of SG .

A. Service Rates of the Interacting Queues in SG

The average service rates of the data queues, Q1 and Q2, inSG are given by

μ1 = d1q1 (1 − q2 P[B >0, Q2 >0|Q1 >0]) , (18)

μ2 = d2 (q2 P[B >0, Q1 =0|Q2 >0]

+q2 (1 − q1) P[B >0, Q1 >0|Q2 >0]) , (19)

where the service rate of Type I node, μ1, is the fraction of timein which Type I node decides to transmit, excluding the fractionof time in which Type II node is also transmitting. A Type IInode transmits, if it is active, i.e., B >0 and Q2 >0, and decidesto transmit. Similarly, the service rate of Type II node, μ2, is thefraction of time in which Type II node has a non-empty batteryand decides to transmit, excluding the fraction of time in whichboth nodes are transmitting.

In our system, we have three interacting queues, namely Q1,Q2 and B. The analysis of three interacting queues is pro-hibitive and, hence, calculating the probability P[B >0, Q1 =0|Q2 >0]. Therefore, we consider a system SD where P[B >

0, Q1 =0|Q2 >0]=0, i.e., we consider a lower service ratefor the Type II node, since it transmits only in the time slotsin which Q1 is non-empty. The relationship between the twosystems is discussed in Section V-E.

B. Service Rates of the Interacting Queues in SD

In order to analyze the interaction between Q1 and Q2 in SD ,we decouple the battery queue, B, by substituting the probabil-ity of the battery queue being non-empty with the conditionalprobability given by Lemma 2. Hence, the average service ratesof the data queues Q1 and Q2 are given as

μ1 = d1q1 (1−q2P[B >0|Q1 >0, Q2 >0]P[Q2 >0|Q1 >0])

= d1q1

(1−min

{q1 ph

(1 + q1 ph), q2

}P[Q2 >0|Q1 >0]

).

(20)

μ2 = d2q2 (1 − q1)P[B >0|Q1 >0, Q2 >0] P[Q1 >0|Q2 >0]

= d2(1 − q1) min

{q1 ph

(1 + q1 ph), q2

}P[Q1 >0|Q2 >0].

(21)

From (20), we note that μ1 decreases with increasing P[Q2 >

0|Q1 >0]. Recall that for the interference-limited region,increasing the service rate of one node comes at the expenseof decreasing the service rate of other nodes. The system isinterference-limited from the perspective of Type I node, sinceincreasing λ2 comes at the expense of decreasing μ1. Moreover,from (21), we observe that μ2 is increasing in P[Q1 >0|Q2 >

0]. Hence, increasing λ1 increases μ2 until both data queuesare saturated. Thus, the system is energy-limited from the per-spective of the Type II node until the data queues becomesaturated.

In Fig. 4, we depict the stable throughput region RD of thesystem SD . The boundary between the energy-limited and inter-ference limited parts, from the perspective of Type II node,is λ1 = λ1, where λ1 is the arrival rate for Q1 at which bothdata queues, Q1 and Q2, become saturated. Accordingly, theenergy-limited sub-region is characterized by

RD|λ1≤λ1=

{(λ1, λ2) ∈ RD| λ1 ≤ λ1

}, (22)

and the interference limited sub-region is characterized by

RD|λ1>λ1=

{(λ1, λ2) ∈ RD| λ1 > λ1

}. (23)

C. The Generalized Stochastic Dominance Approach

In this section, we rely on stochastic dominance arguments[25], which are instrumental in establishing the stable through-put region of SD . However, the conventional stochastic domi-nance approach should be modified for our system, since thetransmission of dummy packets by Type I node increases theharvesting opportunities for Type II node. Thus, in order toconstruct a hypothetical “dominant” system in which the queuelengths are never smaller than their counterparts in the systemSD , the hypothetical system proposed in [25] is modified to cap-ture the two-mode operation inherent to our RF EH system.Recall, from (21), that the service rate of Q2 increases with λ1.Thus, it is straightforward to show that, using classic stochas-tic dominance arguments, saturating Q1 increases the servicerate of Q2. Hence, the queue length in this hypothetical system(particularly Q2) no longer dominates (i.e. could be smaller) itscounterpart in the system SD and, thus, the classic stochasticdominance argument fails. For example, if we consider the casewhere λ1 = 0 and λ2 > 0, we observe that λ2 ≤ q2 belongs tothe stable throughput region, which contradicts (21), where forλ1 = 0, we get λ2 = 0.

Hence, we define the hypothetical system for our RF EHsystem as follows:

• Arrivals at data queues Q1 and Q2 occur at the sameinstants as in the system SD .

• Transmission decisions, determined by independent cointosses, are identical to those in the system SD .


• In the energy-limited region, Type I node does not trans-mit dummy packets. Thus, the energy arrivals to thebattery queue of the harvesting node occurs exactly at thesame instants as in the system SD .

• In the interference-limited region, if the data queue isempty and the node decides to transmit, a dummy packetis transmitted, given the node has sufficient energy totransmit.

From the construction of the new hypothetical system pro-posed above, we observe that it behaves like the system SD

in the energy-limited region and dominates the system SD inthe interference-limited region. Thus, this hypothetical systemdominates our system SD , because the transmissions of dummypackets collide with the transmission of the other node and con-sume energy without contributing to the throughput of Type IInode.

D. Establishing the Stability Conditions of SD

In order to derive the stability conditions of the system SD ,we construct two dominant systems, where in the first dominantsystem Type II node is backlogged and in the second dominantsystem, Type I node is backlogged only in the interference-limited region. Note that the probability that Qi is non-empty,given that Q j is backlogged, is given by P[Qi > 0|Q j > 0] =λiμs

i, i = 1, 2, i = j , where μs

i is the service rate of Qi given

both data queues are saturated.1) First Dominant System: In this hypothetical system, we

consider the case where the Type II node continues transmittingdummy packets whenever its data queue, Q2, is empty giventhat its battery is non-empty. Since the system is interference-limited from the perspective of Type I node, our dominantsystem is identical to the one proposed in [25]. Hence, thesaturated service rate of Q1 is given by

μs1 = d1q1

(1 − min

{q1 ph

(1 + q1 ph), q2

}). (24)

By substituting P[Q1 > 0|Q2 > 0] = λ1/μs1 in (21), the ser-

vice rate of Q2 becomes

μ2 =d2(1 − q1) min

{q1 ph

(1+q1 ph), q2

}λ1

d1q1

(1 − min

{q1 ph

(1+q1 ph), q2

}) . (25)

Therefore, the stable throughput region of the first dominantsystem S1 is given by

R1 =⎧⎨⎩(λ1, λ2) | λ1 ≤ d1q1

(1 − min

{q1 ph

(1 + q1 ph), q2

}),

λ2 ≤d2(1 − q1) min

{q1 ph

(1+q1 ph), q2

}λ1

d1q1

(1 − min

{q1 ph

(1+q1 ph), q2

})⎫⎬⎭ . (26)

Furthermore, the system becomes interference-limited fromthe perspective of node II, when the data queues, Q1 and Q2,are backlogged, thus λ1 is given by

λ1 = μs1 = d1q1

(1 − min

{q1 ph

(1 + q1 ph), q2

}). (27)

2) Second Dominant System: In this hypothetical system,Q1 is backlogged only in the interference-limited region fromthe perspective of Type II node. In the interference-limited partof R2, the saturated service rate of Q2 is given by

μs2 = d2(1 − q1) min

{q1 ph

(1 + q1 ph), q2

}. (28)

Similarly, by substituting P[Q2 > 0|Q1 > 0] = λ2/μs2 in (20),

we obtain

μ1 = d1q1

(1 − λ2

d2(1 − q1)

). (29)

Therefore, the stable throughput region of the interference-limited part of the second dominant system is given by

R2|λ1≥λ1=

{(λ1, λ2) | λ1 ≤ λ1 ≤ d1q1

(1 − λ2

d2(1 − q1)

),

λ2 ≤ d2(1 − q1) min

{q1 ph

(1 + q1 ph), q2

}}. (30)

The stable throughput regions of the dominant systems R1and R2|λ1≥λ1

are shown in Fig. 4.3) Stability Region of the System SD: In the following

lemma we derive the relationship between the stable through-put region of the dominant systems R1 and R2|λ1≥λ1

and theoriginal system RD .

Lemma 3: The stable throughput region RD of the systemSD is given by the union of the stable throughput region of thefirst dominant system and the interference-limited part of thesecond dominant system, i.e., RD = R1 ∪ R2|λ1≥λ1

.

Proof: Based on [25], the stable throughput region of theoriginal system is the union of the two dominant systems, i.e.,RD = R1 ∪ R2. From the construction of the second dominantsystem, we know that the energy-limited region is identical tothe original system, i.e., R2|λ1<λ1

= RD|λ1<λ1. Hence,

R2 = RD|λ1<λ1∪ R2|λ1≥λ1

,

RD = R1 ∪ RD|λ1<λ1∪ R2|λ1≥λ1

.

Now, assume that the rate pair (x1, x2) ∈ RD|λ1<λ1. Hence,

x1 < λ1 and x2 ≤ μ2. The maximum service rate for Type IInode is achieved by backlogging Q2, thus from (26) we obtain

x2 ≤d2(1 − q1) min

{q1 ph

(1+q1 ph), q2

}x1

d1q1

(1 − min

{q1 ph

(1+q1 ph), q2

}) . (31)

Therefore, the rate pair (x1, x2) ∈ R1 and RD|λ1<λ1⊆ R1. �

Proposition 1: The stability region of system SD is given by

RD =⎧⎨⎩(λ1, λ2) | λ1 ≤ d1q1

(1 − λ2

d2(1 − q1)

),

λ2 ≤d2(1 − q1) min

{q1 ph

(1+q1 ph), q2

}λ1

d1q1

(1 − min

{q1 ph

(1+q1 ph), q2

})⎫⎬⎭ . (32)


Proof: For the purpose of the proof, we also define aconventional Aloha system with nodes with unlimited energysupplies, i.e., SAloha. The outline of the proof is as follows

(i) The generalized stochastic dominance approach provesthe necessity of the stability conditions on Q1 and Q2in (32). Meanwhile, it only proves the sufficiency of thestability conditions for Q2.

(ii) The stability condition on Q1 in SAloha is sufficient forstability of Q1 in SD .

(iii) The stability condition on Q1 is the same in both systemsSAloha and SD .

(iv) From (ii) and (iii), we establish the sufficiency of thestability condition on Q1 for SD .

The detailed proof is as follows(i) Recall that for systems with unlimited energy such as

SAloha, the sufficient and necessary stability conditions aregiven by the union of the stability regions of the two hypo-thetical systems in [25]. On the other hand, for a systemwith batteries the transmission of dummy packets in thehypothetical system wastes the nodes energy which lim-its data transmissions in future slots. For instance, in ahypothetical system where the first node is backlogged,there may exist instants at which the first node is unableto transmit due to energy outage, while it is able to trans-mit in the original system. Hence, the second node inthe hypothetical system may have a higher success ratecompared to the original system, and the sample pathdominance is violated. Consequently, the union of thestability conditions of two hypothetical systems is onlynecessary for the stability of the original system [33].We arrived to (32) by applying the generalized stochasticdominance approach, in which the first dominant systemis constructed such that Type II node continues trans-mitting dummy packets whenever its data queue, Q2, isempty. The transmission of dummy packets from Type IInode wastes energy. Therefore, Type I node in this dom-inant system may have a better success rate compared tothat of the system SD . Therefore, the generalized stochas-tic dominance approach only proves the necessity of thestability condition on Q1 in (32). On the other hand, thecondition on Q2 is sufficient and necessary, since Type Inode has unlimited energy and previous argument onlyapplies to nodes with batteries.

(ii) In SAloha the two contending nodes have unlimited energy,while in SD the transmissions of a Type II node are con-strained by the energy in the battery. Hence, the secondcontending node in SAloha, transmits more frequently thana Type II node with the same transmission probability q2in SD . Consequently, there are more collisions in SAlohacompared with those in SD and the service rate of TypeI node in SD cannot be smaller than the service rate ofthe first contending node in SAloha. Therefore, the stabil-ity condition on Q1 in SAloha is sufficient for stability ofQ1 in SD .

(iii) In the stability analysis of SD , we proved that the stabil-

ity condition on Q1 is λ1 ≤ d1q1

(1 − λ2

d2(1−q1)

), which is

identical to the stability condition on Q1 in SAloha [25].

(iv) The sufficiency of λ1 ≤ d1q1

(1 − λ2

d2(1−q1)

)for the sta-

bility of Q1 in SD follows from its sufficiency for Q1 inSAloha. Therefore, the stability conditions in (32) are nec-essary and sufficient conditions and the region RD is theexact stability region of the system SD . �

E. The Relationship Between Stability Regions of SG and SD

Lemma 4: The stability region of the system SD is an innerbound on the stability region of the system SG , i.e., RG ⊇ RD .

Proof: According to the assumption P[B >0, Q1 =0|Q2 >0]=0 in the system SD , a Type II node has a lower ser-vice rate compared to a Type II node in the system SG . Hence,the length of Q2 in the system SD is never smaller than itscounterpart in the system SG , i.e., the system SD dominatesSG from the perspective of Type II node. Consequently, thestability of Q2 in SD , is sufficient for the stability of Q2 inSG . Additionally, the assumption P[B >0, Q1 =0|Q2 >0]=0,implies that Type II node is not transmitting in the time slots atwhich Q1 is empty. Hence, the number of idle slots increases,since at those time slots Type I node has no data packets totransmit. Accordingly, the service rate of Type I node is notaffected, and the stability condition on Q1 is the same in SD

and SG . We conclude that the stability region of SD is an innerbound on the stability region of SG . �

Finally, from Proposition 1 and Lemma 4 we arrive toTheorem 1.

F. The Closure Over all Transmission Probabilities

In this subsection, we prove Theorem 2. The closure ofthe inner bound R

innerG , is defined by the union of all stability

regions for a given (q1, q2), i.e.,

C(RinnerG ) =

⋃(q1,q2)

RinnerG ((q1, q2)) .

In RinnerG , the service rate of Type II node is always lower than

the arrival rate of Type I node, i.e., μ2 < λ1. From (16), wenote that μ2 is increasing in min{ q1 ph

(1+q1 ph), q2}, while μ1 is

not affected by q2. Hence, in order to find the closure C(RinnerG ),

we need to find q2 that maximizes μ2. From (16), we observethat for maximizing μ2, the transmission probability of Type IInode q2 should be greater than or equal to q∗

2 = (q1 ph)/(1 +q1 ph), and increasing q2 beyond q∗

2 does not affect the servicerate μ2.

Interestingly, we can interpret q∗2 using renewal reward

theorem [37]. Assume that the data queues are backloggedand Type II node transmits whenever it receives an energypacket, i.e., q2 = 1. Let the expected reward R that Type IInode obtains, be the transmission of one data packet, i.e.,E[R] = 1. Moreover, the expected number of time slots, T ,needed for the transmission of one data packet is one slot fortransmission, and (q1 ph)−1 slots for harvesting one energypacket. Hence, the expected time needed for a transmis-sion E[T ] = 1 + (q1 ph)−1. Using renewal reward theorem, we


Fig. 5. The closure C(Rinner

G )= C1

OB ∪ C1BD, for d1 = d2 = 1 and ph = 0.2.

find that the effective transmission rate of Type II node isgiven by

q2,eff = E[R]

E[T ]= q1 ph

1 + q1 ph.

Therefore, the previous expression represents the maximumpossible transmission rate of Type II node, which is the min-imum transmission probability q2 that maximizes μ2. Now, theproblem of finding the closure C(So), reduces to finding theclosure of Rinner

G

((q1, q∗

2 ))

over all q1, i.e.,

C(RinnerG ) =

⋃q1∈[0,1]

RinnerG

((q1, q∗

2 )),

RinnerG

((q1, q∗

2 )) =

{(λ1, λ2) | λ1 ≤ d1q1

(1 − λ2

d2(1 − q1)

),

λ2 ≤ d2 ph

d1(1 − q1) λ1

}, (33)

which represents the triangle OBD in Fig. 6, where D=(d1q1, 0), and B=

(d1q1

1+q1 ph,

d2q1 ph(1−q1)1+q1 ph

). The region R

innerG

consist of two line segments. Thus, the closure C(RinnerG ) can

be found be taking the union of the closures of the line seg-ments OB, BD. First, we find the closure of the line segmentOB by solving x = d1q1

1+q1 ph, and y = d2q1 ph(1−q1)

1+q1 ph. The solu-

tion represents the trace of the point B for q1 ∈ [0, 1], which isy = d2 ph x

d1(1 − x

d1−ph x ). Hence, the closure of OB is given by

COB ={(λ1, λ2) | λ2 ≤ d2 phλ1

d1

(1 − λ1

d1 − phλ1

)}, (34)

which is represented in Fig. 5 by C1OB and C2

OB. Next, in orderto find the closure of BD, we solve

maxq1∈[0,1]

d1q1

(1 − λ2

d2(1 − q1)

).

The solution gives us the closure of the line segment extend-ing OB to the λ2-axis, represented by C1

BD and C2BD in Fig. 5.

However, since we want the closure of BD and not the exten-sion, CBD is limited from the left by the trace of point B. Thus,

closure of BD is the region bounded from the left by C2OB and

bounded by C1BD from the right. It is given by

CBD ={

(λ1, λ2) | λ1 >d1(1+2ph −√

1+4ph)

2p2h

,

√λ1

d1+

√λ2

d2≤ 1, λ2 >

d2 phλ1

d1

(1− λ1

d1− phλ1

)}. (35)

Note that the second condition is found by solving the two equa-tions of C2

OB and C1BD. Finally, the closure C(Rinner

G ) is the union

of COB and CBD, which is represented by C1OB and C1

BD in Fig. 5.After some algebraic manipulations, we obtain (17).

Remark 1: (17) can also be obtained by maximizing the ser-vice rate of Type II node μ2, subject to the stability conditionof Type I node λ1 ≤ μ1, i.e,

maxq1∈[0,1]

d2 phλ1

d1(1 − q1)

s.t λ1 ≤ d1q1

(1 − λ2

d2(1 − q1)

). (36)

Finding the closure of our system is equivalent to maximizingμ2, because μ2 is upper bounded by μ1. Thus, by maximizingμ2, we implicitly maximize μ1.

VI. MODEL EXTENSIONS

In this section, we discuss three extensions to the previousstability analysis. First, we investigate the impact of having afinite capacity battery on the stable throughput region. Second,we investigate the effect of having a full-duplex RF energy har-vesting node. Third, we consider the case where RF energyharvesting serves as a backup for an unlimited energy source.

A. Finite Capacity Battery

In this subsection, we investigate the impact of having a finitecapacity battery on the stable throughput region obtained inTheorem 1. Let M be the capacity of Type II node battery. Thus,the battery evolution equation becomes

Bt+1 = min{

Bt − μtB + Ht , M

}(37)

Similar to the unlimited battery capacity case, the probabilityof non-empty battery is given by the following lemma.

Lemma 5: For a half-duplex RF energy harvesting node withbattery of size M , the probability that the battery is non-empty,ζ , given the data queues Q1 and Q2 are backlogged, is

ζ =

⎧⎪⎪⎨⎪⎪⎩

ρ

(1−

(q1 ph (1−q2)

q2

)M

1−ρ(

q1 ph (1−q2)

q2

)M

), q2 = q1 ph

1+q1 ph,

1, q2 = q1 ph1+q1 ph

(38)

where ρ = q1 phq2(1+q1 ph)

is the probability that the battery is non-empty in the infinite capacity battery case.

Proof: Along the lines of Lemma 2. �


Fig. 6. The stable throughput region in the finite capacity battery is character-ized by the triangle OED.

Applying the same procedure used in proving the stabilityregion of the infinite battery capacity case, we get

Corollary 1: An inner bound on the stable throughput regionof the opportunistic RF energy harvesting slotted Aloha isthe triangle OED, shown in Fig. 6. Assuming half-duplexenergy harvesting and a battery of size M , the region isgiven by

RinnerG =

{(λ1, λ2)|λ1 ≤ d1q1

(1 − λ2

d2(1 − q1)

),

λ2 ≤ d2q2(1 − q1)ζλ1

d1q1 (1 − q2ζ )

}(39)

Proof: Along the lines of Theorem 1. �The reduction in the stability region due to finite capacity

battery is the triangle OEB shown in Fig. 6.

B. Full-Duplex Energy Harvesting

Now, we investigate the effect of having a full-duplex RFenergy harvesting Type II node, i.e., the harvesting circuit isseparated form the transmission circuit. Hence, a node cantransmit and harvest simultaneously. Full-duplex energy trans-mission is advantageous because harvesting self-interferencemay introduce high energy yield. In full-duplex RF energyharvesting paradigm, we have three types of harvesting oppor-tunities. First, harvesting transmissions of Type I node whileType II node is silent. Similar to the half-duplex case, we modelthis case by a Bernoulli process with mean ph|{1}. Second, har-vesting self-interference of Type II node while Type I nodeis silent, which is modeled by a Bernoulli process with meanph|{2}. Third, harvesting both transmissions of Type I nodeand self-interference of Type II node, which is modeled by aBernoulli process with mean ph|{1,2}.

In order to characterize the probabilities ph|{2} and ph|{1,2},we use a similar approach to the one used in characterizingph|{1} in Section III. We assume that the loopback interferencecoefficient c ∈ [0, 1] is known [38]. We also assume a Rayleighfading channel between the transmit antenna and the harvest-ing antenna, i.e., gt ∼ exp(1). In case only self-interference ispresent, the received power at time slot t is equal to PR(t) =

ηP2cgt . Hence, using the same approach as in the half-duplexcase, we obtain

ph|{2} = 1

1 + γηP2c

. (40)

In case both transmissions of Type I node and the self-interference are present, the received power at time slot t isequal to PR(t) = η(P1ht (1 + lα)−1 + P2 c gt ). The probabil-ity ph|{1,2} can be characterized in a similar fashion5.

For a full-duplex harvesting node, the average energy har-vesting process of the battery queue is given by

E[Ht ] = q1 ph|{1}P[Q1 > 0]

+ q2 ph|{2}P[Q1 = 0, B > 0, Q2 > 0]

+ (q2 ph|{2} + q1q2

(ph|{1,2} − ph|{2} − ph|{1}

))P[Q1 > 0, B > 0, Q2 > 0], (41)

where ph|M, is the harvesting probability given a set M ofnodes are transmitting. The battery queue forms a decoupledMarkov chain given that the data queues are backlogged. Byanalyzing the Markov chain we find the probability that thebattery is non-empty, which is given by the following lemma.

Lemma 6: For a full-duplex RF energy harvesting node withinfinite capacity battery, the probability that the battery is non-empty �, given the data queues Q1 and Q2 are backlogged, isgiven by

min

{q1 ph|{1}

q2(1−q1(ph|{1,2}− ph|{1})−(1−q1)ph|{2}

) , 1

}. (42)

Proof: Along the lines of Lemma 2. �Note that the probability of non-empty battery for the full-

duplex case is higher than that of the half-duplex case, i.e, � ≥min

{q1 ph|{1}

1+q1 ph|{1} , 1}

. The stable throughput region of the system

is given byCorollary 2: An inner bound on the stable throughput region

of the opportunistic RF energy harvesting slotted Aloha, underfull-duplex energy harvesting mode and infinite capacity bat-tery, is the region characterized by

RinnerG =

{(λ1, λ2) | λ1 ≤ d1q1

(1 − λ2

d2(1 − q1)

),

λ2 ≤ d2q2(1 − q1)�λ1

d1q1 (1 − q2�)

}(43)

Proof: Along the lines of Theorem 1. �

C. RF Energy Harvesting as a Backup

We consider a system where Type II node is powered bytwo sources of energy. An unlimited energy source that isavailable with probability pg and a half-duplex RF energy har-vesting circuit that serves as a backup for the unlimited energy

5In order to characterize ph|{1,2}, we need the distribution of the sum of inde-pendent gamma distributed random variables, all with integer shape parametersand different rate parameters, which is called the generalized integer gammadistribution (GIG) [39].


Fig. 7. Stability regions of RF EH-Aloha under half/full-duplex modes vs.Aloha with unlimited energy.

source, i.e., the RF energy harvesting circuit is active only whenthe unlimited energy source is unavailable. In this case, theexpected energy arrival rate to the battery queue is given by

E[H t ] = pg + (1 − pg)E[Ht ], (44)

where E[Ht ] is given by (11). Under the aforementionedassumptions, the probability that the battery queue is non-empty is given by

Lemma 7: For a Type II node equipped with an unlimitedenergy source and a backup half-duplex RF energy harvestingcircuit, the probability that the battery is non-empty, , giventhe data queues Q1 and Q2 are backlogged, is given by

= min

{pg + (1 − pg)q1 ph

q2(1 + (1 − pg)q1 ph

) , 1

}. (45)

Proof: Along the lines of Lemma 2. �

Fig. 8. The reduction in the stability region due to fading.

Finally, we arrive to the inner bound on the stability regionby replacing � with in (43).

VII. NUMERICAL ANALYSIS

A. RF EH-Aloha System SG vs. Slotted Aloha

In Fig. 7, we compare the stable throughput region ofthe conventional slotted Aloha with unlimited energy supplywith our RF EH-Aloha system SG . The stability regions areshown for q1 = 0.4, d1 = d2 = 1, ph|{1} = 0.2, ph|{2} = 0.2,and ph|{1,2} = 0.35. Also, we consider different values for q2to compare between full-duplex and half-duplex energy har-vesting. We observe that the stability region of slotted Alohais significantly reduced when RF energy harvesting is imple-mented, due to the energy limited sub-region. For small q2, i.e.,q2 < (q1 ph|{1})/(1 + q1 ph|{1}), we observe that the stabilityregions of half-duplex and full-duplex EH-Aloha are identical,because the service rate of Type II node is limited by the trans-mission probability q2. On the other hand, for large q2, i.e.,q2 > �, full-duplex RF energy harvesting expands the stabilityregion, which agrees with intuition. From Fig. 7(a), we noticethat increasing q2 beyond � enhances the throughput of TypeII node in the slotted Aloha system. However, the throughputof the Type II node in full-duplex EH-Aloha is limited by �.In Fig. 8, we show the effect of fading on the stable throughputregion, which is captured by the outage probabilities 1 − d1 and1 − d2.

B. Validating the Stability Conditions Using Simulations

In Figs. 9 and 10, we simulate the energy harvesting Alohasystem SD for q1 =0.4, q2 =0.4, d1 =d2 =1, ph =0.6 and thequeues are averaged over 105 time slots. In particular, Fig. 9shows the sum of the average queue lengths E[Qt

1]+E[Qt2]

versus the arrival rates (λ1, λ2). We observed that the systemexhibits unstable behavior, shown by the increase in the aver-age queue lengths, as we cross the boundaries of the stabilityregion. Fig. 10 shows the average service rate of Q2 versus thearrival rates (λ1, λ2). It shows that the maximum service rate(lower left corner points in the contour plot), for a given λ2, isachieved on the boundary of the stability region. These obser-vations support that RD in Proposition 1 is indeed the stabilityregion of SD .

For the aforementioned parameters, we simulate the systemSG in order to verify the inner bound in Theorem 1. Similarly,


Fig. 9. The stability regions RD illustrated with the simulated sum averagequeue lenghts.

Fig. 10. Simulation of the average service rate of Q2 in system SD .

Fig. 11. The stability region RinnerG illustrated with the simulated sum average

queue lenghts.

Fig. 12. Simulation of the average service rate of Q2 in system SG .

Fig. 13. The stability region RinnerG compared with the simulated sum average

queue lenghts of the exact system SO .

Fig. 14. Sample paths for the evolution of Q2 in the systems SD , SG and SO .

the sum of the average queue lengths and the average servicerate of Q2 are shown in Figs. 11 and 12, respectively. We noticefrom Fig. 11 that there exist rate pairs outside the left handside of the stability region for which the queues exhibit a sta-ble behavior. Additionally, Fig. 12 suggests that the maximumservice rate (lower left corner points in the contour plot), for agiven λ2, is achieved outside the stability region. These obser-vations indicates that Rinner

G is an inner bound on the stabilityregion of SG , as proposed in Theorem 1.

Finally, we show the utility of the stability region RinnerG ,

for the exact system SO which is described in Section III-A.We simulate the exact behavior of the system for η=0.7, γ =0.2335, P1 =1 and (1+lα)−1 =0.5. Hence, θ =0.667 and from(10) we get ph =0.6. The sum average queue lengths is shownin Fig. 13 versus the arrival rates (λ1, λ2). We observe that thebehavior of the average queue lengths in SO is very similar toSG . Henceforth, we conjecture that the stability region R

innerG is

also an inner bound for the stability region of the exact systemRO . To further support our analytical results, Figs. 14 and 15present sample paths for the evolution of the queues Q1 andQ2 in the systems SD , SG and SO . Note that the sample path ofthe evolution of an unstable queue should show an increasingtendency such that the queue size grow unboundedly as timeincreases. In Fig. 14, we show the evolution of Q2 for twoarrival rate pairs To and Ti . We observe that Q2 exhibits unsta-ble behavior at To in SD , while it exhibits stable behavior in SG


Fig. 15. Sample paths for the evolution of Q1 in the systems SD , SG and SO .

and SO . This supports our claim that the stability condition onQ2 in (16) is sufficient and necessary in SD , while it is only suf-ficient in SG and SO . While, Fig. 15 suggests that the stabilitycondition on Q1 in (16) is sufficient and necessary in SD , SG

and SO .

VIII. CONCLUSION

In this paper, we studied the effects of opportunistic RFenergy harvesting on the stability of a slotted Aloha systemconsisting of a Type I node, which has unlimited energy sup-ply, and a Type II node, which is solely powered by an RFenergy harvesting circuit. We illustrated the intricacy in ana-lyzing the exact behavior of such systems and proposed anapproximate system for which we were able to derive ana-lytical results. In particular, we characterized an inner boundon the stability region under the half-duplex and full-duplexenergy harvesting paradigms, by generalizing the stochasticdominance technique for RF EH-networks. We verified ouranalytical findings by simulating the exact and approximatesystems. The extension of our analysis to a random access net-work with multiple nodes can provide further insights to thedevelopment of efficient medium access protocols for networkswith RF energy harvesting capabilities, and presents itself asa promising future research direction. In general, no protocol

outperforms the others for all types of traffics, the suitabil-ity of a protocol depends on the traffic characteristics and theapplication. Therefore, comparing the performance of statefulcentralized algorithms with stateless random access protocols,specifically for EH-networks should be analyzed as a futureresearch direction.

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Abdelrahman M. Ibrahim received the B.Sc.degree in electrical engineering from AlexandriaUniversity, Alexandria, Egypt, in 2011, and theM.Sc. degree in wireless communications fromNile University, Giza, Egypt, in 2014. He iscurrently pursuing the Ph.D. degree with theDepartment of Electrical Engineering, PennsylvaniaState University, University Park, PA, USA. Hehas been a Graduate Research Assistant with theDepartment of Electrical Engineering, PennsylvaniaState University, since 2014. His research interests

include green communications, energy harvesting, and resource allocation forwireless networks.

Ozgur Ercetin received the B.S. degree in elec-trical and electronics engineering from the MiddleEast Technical University, Ankara, Turkey, in 1995,and the M.S. and Ph.D. degrees in electricaland computer engineering from the University ofMaryland, College Park, MD, USA, in 1998 and2002, respectively. Since 2002, he has been with theFaculty of Engineering and Natural Sciences, SabanciUniversity, Istanbul, Turkey, where he is currently aFull Professor. He was a Visiting Researcher at HRLLabs, Malibu, CA, USA, Docomo USA Labs, CA,

USA, and The Ohio State University, Columbus, OH, USA. His researchinterests include computer and communication networks with emphasis onfundamental mathematical models, architectures and protocols of wireless sys-tems, and stochastic optimization. He is currently an Associate Editor for theIEEE TRANSACTIONS ON MOBILE COMPUTING.

Tamer ElBatt (SM’06) received the B.S. and M.S.degrees in electronics and communications engineer-ing from Cairo University, Giza, Egypt, in 1993 and1996, respectively, and the Ph.D. degree in electri-cal and computer engineering from the University ofMaryland, College Park, MD, USA in 2000. From2000 to 2009, he was with major U.S. industryR&D labs, e.g., HRL Laboratories, LLC, Malibu,CA, USA, and Lockheed Martin ATC, Palo Alto,CA, at various positions. In July 2009, he joinedthe Electronics and Communications Department,

Faculty of Engineering, Cairo University, Giza, Egypt, as an AssistantProfessor, where he is currently an Associate Professor. He also has a jointappointment with Nile University, Giza, Egypt, since October 2009 and cur-rently serves as the Director of the Wireless Intelligent Networks Center(WINC) since October 2012. He was a Visiting Professor at the Departmentof Electronics, Politecnico di Torino, Turin, Italy, in August 2010, FENS,Sabanci University, Istanbul, Turkey, in August 2013, and the Departmentof Information Engineering, University of Padova, Padova, Italy, in August2015. His research has been supported by the U.S. DARPA, ITIDA, FP7,General Motors, Microsoft, and Google and is currently being supported byNTRA, QNRF, H2020, and Vodafone Egypt Foundation. He has authored morethan 90 papers in prestigious journals and international conferences. He holdsseven issued U.S. patents. His research interests include performance analy-sis, design, and optimization of wireless and mobile networks. He has servedon the technical program committees of numerous IEEE and ACM confer-ences. He served as the Demos Co-Chair of ACM Mobicom 2013 and thePublications Co-Chair of the IEEE Globecom 2012 and EAI Mobiquitous 2014.He currently serves on the Editorial Board of the IEEE TRANSACTIONS ON

MOBILE COMPUTING and International Journal of Satellite Communicationsand Networking (Wiley). He was the recipient of the 2014 Egypt’s StateIncentive Award in Engineering Sciences and the 2012 Cairo UniversityIncentive Award in Engineering Sciences.

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