SelfSustainability of Energy Harvesting Systems: Concept, Analysis, and Design


 Ethelbert Shon Kennedy
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1 1 SelfSustainability of Energy arvesting Systems: Concet, Analysis, and Design Sudarshan Guruacharya and Ekram ossain Deartment of Electrical and Comuter Engineering, University of Manitoba, Canada s: arxiv: v2 [cs.it] 2 May 217 Abstract Ambient energy harvesting is touted as a low cost solution to rolong the life of lowowered devices, reduce the carbon footrint, and make the system selfsustainable. Most research to date have focused either on the hysical asects of energy conversion rocess or on otimal consumtion olicy of the harvested energy at the system level. A concet closely associated with the energy harvesting system is that of selfsustainability. In the context of energy harvesting systems, selfsustainability refers to the ability of the system to rovide the necessary energy for the consumer, without the need for external grid ower. owever, although intuitively understood, to the best of our knowledge, the idea of selfsustainability is yet to be made recise and studied as a erformance metric. In this aer, we rovide a mathematical definition of the concet of selfsustainability of an energy harvesting system, based on the comlementary idea of eventual energy outage. In articular, we analyse the harveststoreconsume system with infinite battery caacity, stochastic energy arrivals, and fixed energy consumtion rate. Using the random walk theory, two basic insights are obtained: 1) The system is selfsustainable when the rate of energy consumtion is strictly less the rate of energy harvest, and 2) assuming the first condition is satisfied, the robability of eventual energy outage is exonentially uer bounded. This uer bound guarantees that the eventual energy outage robability can be made arbitrarily small simly by increasing the initial battery energy. These gives us a valuable design insight. General formulas are given for the selfsustainability robability in the form of integral equations. For the secial case when the energy arrival follows a Poisson rocess, we are able to find the exact formulas for the eventual outage robability. We also show that the harveststoreconsume system is mathematically equivalent to a GI/G/1 queueing system, which allows us to easily find the outage robability, in case the necessary condition for selfsustainability is violated. MonteCarlo simulations verify our analysis. Index Terms Energy harvesting, selfsustainability, eventual energy outage, random walks, renewal theory, martingale I. INTRODUCTION In recent years, ambient energy harvesting and its alications have become a toic to great interest. Basically, a device is assumed to be able to harvest energy from random energy source in their surrounding environment for their future use. Such methods are useful when devices are low owered and energy constrained. They can hel in extending the life time of a device and lower its maintenance cost. Furthermore, they can hel lower carbon dioxide emissions and fight climate change. The readers are referred to [1] [5] for general surveys of this field. The work was suorted by a CRD grant from the Natural Sciences and Engineering Research Council of Canada (NSERC). Aart from traditional sources of ambient energy such as solar, wind, and wave, in the ast decade research has extended to energy scavenging techniques from diverse energy sources [6], [7] such as thermal [8], [9], ressure and vibrations [8], [1], ambient radiofrequency (RF) radiation [11], [12], bodily motions [13], magnetic field [14], ambient sound [15], and ambient light [16], [17]. Such energy harvesting techniques have been alied in the context of various technologies such as wireless sensor networks [18] [21], telecommunications [22] [24], cellular networks [25], cognitive radio network [26], [27], vehicular network [28], health care [29], [3], IoT (Internet of things) [31], [32], IoE (Internet of energy) [33], and smart grid [34] [36] technology. A. Nature of Energy Sources and General Architectures The ambient energy sources can be broadly classified as steady energy source or stochastic energy source. These diverse sources of energy are converted into electricity, which is then either directly consumed, or stored in a rechargeable battery for future use. As with harvesting, consumtion of energy can also be either steady or stochastic. Three general architectures of energy harvesting network are harvestconsume, harveststoreconsume, and harveststore/consume [5]. In the harvestconsume model, the harvested energy is immediately consumed by the consumer. This model is aroriate when a steady suly of harvested energy can be guaranteed and a batteryfree circuit is desired. The major roblem with this aroach is that, due to ossible random nature of the energy source, when the harvested energy is less than the minimum oerational energy required by consuming device, the device is disabled. We say that the consumer has exerienced an energy outage. In harveststoreconsume model, the harvested energy is first transferred to an energy storage facility or a rechargeable battery. The consumer then accesses the harvested energy from the battery. By this method the consumer can ensure a steady suly of energy, even though the harvested energy is randomly varying. This method is aroriate when the harvested energy is very small and is highly fluctuating, as in ambient RF energy harvesting. We can also have a hybrid model given by harveststore/consume. In this model the harvest energy is either stored for some future use or directly consumed. If the harvested energy is above minimum requirement of the consumer, then the excess energy can also be stored. This model is aroriate when the harvested energy does not fluctuate very raidly.
2 2 B. Motivation: Energy Outage, Eventual Energy Outage, and SelfSustainability A lot of research attention has been aid either on the hysical asects of energy harvesting mechanism [6] [17] or on the consumtion olicy of the harvested energy [18] [36]. Work dealing with former issue tend to focus on hysical modeling and otimization of energy harvesting devices, with the goal of imroving the efficiency of energy conversion rocess. Work dealing with the latter issue tend to focus on otimizing some erformance metric, under energy constrains of the energy harvesting device. Clearly, the work of the latter category is affected by the general architecture assumed. Also, the exact nature of the metric deends on what the harvested energy is being used for. For instance, in the field of energy harvesting communication, when the harvestconsume model is used, a ossible natural erformance metric is the joint energy and information outage robability [37], [38]. If instead the harveststoreconsume model is used, a ossible erformance metric is the data rate of the communication system [23]. While there have been lenty of work that have examined systems that have energy harvesting caability, very few work have focused on examining the concet of energy outage of the consumer. This is unfortunate because energy outage should be one of the key erformance metric of any energy harvesting system. The consumer is said to exerience energy outage if there is no energy available for its consumtion. For the harvestconsume model, the energy outage is determined by the randomness of the harvestertoconsumer energy transmission channel or the inherent randomness of the energy source. In [37], [38] the energy arrival rocess is assumed to be a stationary, ergodic rocess. As such, knowing the distribution of energy arrival allows us to calculate the energy outage robability. For the harveststoreconsume model, the harvested energy is modeled as discrete ackets of ossibly variable size [23]. The energy arrival rocess at the battery is generally modeled as a Poisson rocess, allowing the battery state to be modeled as a Markov chain [39]. owever, research on energy outage is sorely lacking for harveststoreconsume architecture. Another concet closely associated with energy harvesting systems is the concet of selfsustainability. Selfsustainability is commonly understood as the ability to suly one s own needs without external assistance. In the context of energy harvesting systems, it would mean the ability of the system to rovide the necessary energy for the consumer, without the need for external grid ower. To the best of our knowledge, this concet is yet to be mathematically defined. In this aer, we define this concet as follows: The selfsustainability of an energy harvesting system is the robability that the consumer will not eventually exerience an energy outage. ere we need to make a distinction between energy outage and eventual energy outage. We say that a consumer exeriences an eventual energy outage if the consumer undergoes an energy outage within finite time. If the consumer has to wait for infinite amount of time to exerience an energy outage, then we say that the energy harvesting system is selfsustainable. The eventual energy outage and the selfsustainability are comlementary in that the sum of their robabilities is unity. Symbolically, we will denote the selfsustainability robability by φ and eventual energy outage robability by ψ such that φ + ψ = 1. We will say that a system is selfsustainable if φ > (or equivalently ψ < 1) and unsustainable if φ = (or equivalently ψ = 1). Given these basic definitions, we can quickly oint out that for harvestconsume system, if both the energy harvest and consumtion rates are steady, then the system is selfsustainable if the rate of consumtion is less than the rate of harvest. Likewise, if the energy harvest is stochastic and the consumtion is steady (or vice versa), then the consumer will almost surely exerience an eventual outage. ence the harvestconsume system is not a selfsustaining architecture for stochastic energy harvest or consumtion. In general, we want the answers to the following questions: 1) Under what condition is selfsustainability ossible? 2) Can we come u with a formula or a bound for selfsustainability robability? 3) ow should we design a system, rovided a constraint on the eventual energy outage robability? C. Contributions The main contribution of this work is the establishment of the concet of selfsustainability for energy harvesting system. We secifically study the harveststoreconsume architecture, for the case where energy arrives imulsively, storage caacity is infinite and the rate of consumtion is constant. We analyze the system based on random walks, renewal theory, and martingales. In articular, we are able to adat many ideas from the ruin theory of actuarial science, making our work crossdiscilinary. This leads to the following findings: 1) The necessary condition for harveststoreconsume system to be selfsustaining is simly that the rate of consumtion be strictly less than the rate of harvest. That is, λ X >, where λ is the arrival rate, X is the average energy of an imulse, and is the fixed consumed ower. In this aer, we refer this condition as the selfsustainability condition. 2) We rovide general formulas for the selfsustainability robability of harveststoreconsume system. In articular, we demonstrate the relationshi between the selfsustainability robability and the maximum of the underlying random walk using three different formulas. Two of these formulas are renewal tye and Weinerof tye integral equations. 3) Since finding the analytical solution to the integral equations is difficult, using the concet of martingales, we rovide an exonential uer bound to eventual energy outage robability for harveststoreconsume system, rovided that the selfsustainability condition is satisfied. That is, ψ(u ) e r u, where u is the initial battery energy and r is some constant. We show that the eventual energy outage can be made arbitrarily small by simly increasing the initial battery energy. This leads
3 3 to simle design guidelines, given the eventual energy outage constraint. 4) Using the renewal tye integral equation for the selfsustainability robability, we rovide an asymtotic formula for the eventual energy outage robability based on key renewal theorem for defective distributions. That is, ψ(u ) Ce r u, for some constant C. 5) For the secial case when the arrival of energy ackets is modeled as a Poisson rocess, we give exact formulas for the eventual energy outage robability. That is, ψ(u ) = (1 r u λ )e r. 6) We rove that harveststoreconsume system is mathematically equivalent to a GI/G/1 queuing system and draw arallels between the two systems. This allows us to imort the results from queueing theory when the selfsustainability condition is not satisfied. While this is certainly not a new observation [23], we rovide a systematic roof of the equivalence and connect it to the idea of energy outage. From here on, without any ambiguity, we will simly refer to the energy outage as the outage and the eventual energy outage as the eventual outage. D. Organization The rest of the aer is organized as follows: Section II discusses the system model and assumtions, and also defines the robability of selfsustainability. Section III gives the random walk analysis of the energy surlus rocess and the selfsustainability robability. Section IV gives an exonential uer bound on eventual outage robability (which is the comlement of selfsustainability robability). Section V gives an asymtotic aroximation of the eventual outage robability and discusses the comutation of the adjustment coefficient, while Section VI studies the secial case when the energy arrivals is a Poisson rocess. Section VII examines the battery energy rocess, while Section VIII gives a numerical verification of the obtained formulas. Lastly, Section IX concludes the aer. II. SYSTEM MODEL, ASSUMPTIONS, AND TE CONCEPT OF SELFSUSTAINABILITY A. Definitions and Assumtions We will consider a harveststoreconsume (SC) system. That is, all the harvested energy is first collected, before being consumed. Thus, the consumer obtains the harvested energy indirectly from the storage facility (or the rechargeable battery). The harvested energy can arrive into the storage in continuous fashion (as in solar or wind or grid ower source) or in imulsive fashion. We will restrict our analysis to the case of imulsive energy arrivals. Thus, our main hysical assumtion is as follows: Assumtion 1. arvested energy arrives as imulses into the storage system. In other words, the harvested energy arrival is a countable rocess. The harvested energy arrives in the form of ackets into the storage system, and the size of each energy ackets may vary randomly. As such, we can model the energy surlus rocess of the system at any time t as U(t) = u t N(t) (u, t)dt + X i h(t t i ). (1) ere u > is the initial battery energy and (u, t) > is the ower consumtion from the battery, i.e. du dt = (u, t). The X i R + is the amount of energy in an ith energy acket that arrives at time t i, while N(t) = max{i : t i t} is the total number of energy ackets that have arrived at the storage by time t. Lastly, h(t) is the transient resonse of battery charging rocess, such that h(t) = for t < and h( ) = 1. Clearly, if U(t) >, then the system is roducing more energy than it is consuming. Likewise, if U(t) <, then the consumer takes in energy from the grid to comensate for energy deficit. We will now define a few concets that will be used in the aer: i=1 Definition 1 (Defective and roer distributions). A random variable with distribution F is said to be defective if lim x F (x) < 1, the amount of defect being 1 F ( ). The random variable is said to be roer if lim x F (x) = 1. Definition 2 (Renewal rocess). A sequence {S n } is a renewal rocess if S n = X 1 + +X n and S =, where {X i } are mutually indeendent, nonnegative random variables with common distribution F X such that F X () =. When F X is a roer distribution, the renewal rocess is said to be a ersistent renewal rocess. If F X is defective then the renewal rocess is said to be a transient or terminating renewal rocess. The variables X i in Definition 2 is often interreted as interarrival time. owever, X i need not always be time, as we will see later in the aer. We will now make further mathematical assumtions required to simlify our analysis. We will be working with the resulting simlified model for the rest of the aer. Assumtion 2. The storage caacity of the battery is infinite. Assumtion 3. The rate of energy consumtion is constant, i.e. (u, t) =. Assumtion 4. The transient resonse of battery charging rocess, h(t), is a unit ste function. Assumtion 5. The energy ackets {E i } arrive at corresonding times {t i }, for i =, 1, 2..., such that = t < t 1 < t 2 <, where the energy acket E arrives at time. The arrival times {t i } is (i) a renewal rocess. That is, the interarrival times {A i }, where A i = t i+1 t i, are such that A, A 1, A 2,... are indeendent and identically distributed. (ii) To avoid more than one energy arrivals at a time, we assume F A () =. (iii) Lastly, the exected interarrival time is finite, E[A i ] <. Assumtion 6. The amount of energy {X i } corresonding to the energy ackets {E i } for i =, 1, 2,..., is (i) a nonnegative, continuous random variable; (ii) {X i } are indeendent and identically distributed; (iii) to avoid energy ackets of
4 ρ = 1.2 Thus, the same results for our basic model can be shown to be valid for the dual model as well by a trivial variation in the arguments. For the rest of the work, we will focus on the first model given in (1). Energy Surlus Outage Time Fig. 1. Energy surlus, U(t), versus time, t. zero size, we assume F X () = ; and (iv) lastly, the exected energy in a acket is finitie, E[X i ] <. Assumtion 7. The interarrival times {A i } and the energy acket sizes {X i } are indeendent of each other. ere Assumtions 2 4 essentially simlify the hysical setu of our roblem, whereas Assumtions 5 7 are essentially technical in nature to facilitate the mathematical analysis. Note that in Assumtion 5, while not necessary, we have secifically assume that there is an arrival of an energy acket at time zero. Of all these Assumtions 2 7, Assumtion 2 regarding the infiniteness of the battery caacity is erhas be the most objectionable. owever, for the time being, we shall kee it for reliminary analysis of the system. Given these assumtions, our initial mathematical model of the energy surlus (1) simlifies to N(t) U(t) = u t + X i. (2) i= This is an instance of a random walk on real line through continuous time with downward drift. As shown in Fig. 1, the grah of U(t) versus t will look like a sawtooth wave with descending rams and with random jum discontinuities. The system exeriences an outage just before t = 12. B. Dual Model It is also ossible to have a dual model of the model given in (1). If instead of Assumtion 1, the harvested energy arrives steadily and the energy consumtion is imulsive, then we have the dual model given by N(t) U(t) = u X i h(t t i ) + i=1 t (u, t)dt. (3) Notice that in this dual model there is only a change in the signs and an interchange in the interretation of the symbols. C. Concet of SelfSustainability Let W (t) be the battery energy at time t. We say that the system undergoes outage when W (t) =. In other words, the case W (t) = reresents the situation when the battery is emty, and as such, the consumer needs to fetch its required energy from the grid to sustain its consumtion. We define the outage robability of the system as P out = P (W (t) = ), (4) that is, robability of finding the battery emty at any time t. Now consider the first time that the battery is emty, τ = inf{t > : W (t) =, W () = u } = inf{t > : U(t), U() = u }. We will refer to τ as the first time to outage. If τ =, then the system become selfsustaining and the system will not face an eventual outage. The occurrence of eventual outage is equivalent to the event {τ < }. owever, since τ itself is a random variable, we can only describe it robabilistically. Thus, we are interested in knowing the robability φ(u,, T ) = P su t T t N(t) i= X i u, that the energy surlus U(t) will not fall below zero through time t [, T ] where T <, and N(t) φ(u,, ) = P su t X i u, t i= of avoiding an eventual outage. Note that this is equivalent to the robability φ(u,, T ) = 1 P (τ < T u, ) = 1 ψ(u,, T ), where ψ(u,, T ) is the robability of an outage occurring within a finite time T. Definition 3. We define the selfsustainability robability of an energy harvesting system as the robability that the first time to outage τ is at infinity. Likewise, the eventual outage robability is the robability that the first time to outage τ is finite. That is, φ(u,, ) = 1 P (τ < u, ) = 1 ψ(u,, ), (5) where φ(u,, ) denotes the selfsustainability robability and ψ(u,, ) denotes the eventual outage robability. Since is held constant and T =, from here on, we will dro these two arameters in the argument, and simly refer to the selfsustainability robability and eventual outage robability as a function of u as given by φ(u ) and ψ(u ).
5 5 III. ENERGY SURPLUS PROCESS AND EVALUATION OF SELFSUSTAINABILITY PROBABILITY Let the exected interarrival time be E[A i ] and the arrival rate of energy ackets be defined as λ = 1 E[A i]. Also, let the average energy acket size be E[X i ] = X. Intuitively, if we want our system to be selfsustainable, then we would want the exected surlus energy to be ositive E[U(t)] >. For this to be true, it is sufficient that the consumtion rate,, be less than the energy arrival rate, λ X. That is, λ X >. We can state this condition rigorously as follows: Proosition 1. If λ X >, then lim t E[U(t)] > almost surely. Proof: The exected value of U(t) for any t is N(t) E[U(t)] = u t + E (a) i= X i = u t + E[N(t)]E[X i ] ( ) E[N(t)] = u + X t. t ere, the equality (a) is due to Wald s identity. From elementary renewal theorem, E[N(t)] t λ for large t almost surely [4, Th 5.8.4]. Thus, we have for large t, E[U(t)] = u + (λ X )t, almost surely. Given the assumtion λ X >, lim t E[U(t)] +, roving the statement. It should be noted that when the condition λ X > is satisfied, the roosition does not tell us that an outage will never occur. Rather, it tells us that there is a chance of such nonoccurrence of outage. A. Random Walk Analysis In general, it is difficult to determine the outage event. We therefore have to condition on the arrival rocess. We can tell that an outage has occurred if a new arrival finds the battery emty. The mathematical trick here is to reduce the continuous time rocess into a discrete time rocess by counting over the arrivals. In the analysis of random walks and ascending ladder rocess, we will basically follow the aroach laid out by Feller in [42]. A modern treatment of the subject can be found in [41]. Proosition 2. Let the sequences {Z i ; i } and {S i ; i 1} be defined as Z i = A i X i and S n = n 1 i= Z i where S =. The surlus energy {U n }, observed immediately before the arrival of nth energy acket, forms a discrete time random walk over real line, such that U n = u S n. (6) Proof: Let us denote t n = t n ɛ where ɛ >, as time immediately before the arrival of nth energy acket at t n. Since t n t n as ɛ, we can decomose the arrival time t n as t n = n 1 i= A i. If we follow the value of U(t) immediately before each arrival at t n, we have N(t n ) U n = u t n + i= X i n 1 = u (A i X i ). i= Let Z i = A i X i for n =, 1, 2,.... ere, Z i is no longer a nonnegative random variable; rather it can take any real value, Z i R. Since both {X i } and {A i } are indeendent and identical to each other, {Z i } are indeendent and identical to each other too. Our exression now becomes n 1 U n = u Z i, which is a discrete time random walk over real line. Defining {S n ; n 1} such that S n = n 1 i= Z i with initialization S =, gives us U n = u S n, which is our desired result. Remark: ere, the {U i } records the troughs of the sawtooth wave U(t) and can be interreted as the energy surlus that the nth arrival finds the system in. The existence of selfsustainable system can be directly roved with the hel of the following theorem from the theory of random walk on real line: i= Theorem 1. [41, Ch 8, Th 2.4] For any random walk with F Z not degenerate at, one of the following ossibilities occur: 1) (Oscillating Case) If E[Z i ] =, then P (lim su n S n = + ) = 1, P (lim inf n S n = ) = 1; 2) (Drift to + ) If E[Z i ] >, then P (lim n S n = + ) = 1; 3) (Drift to ) If E[Z i ] <, then P (lim n S n = ) = 1. The following roositions easily follow from the above theorem. Proosition 3. The SC system is selfsustaining only if it satisfies the selfsustainability condition given by λ X >. Proof: Consider the contraositive of the statement: If λ X, then the SC system will exerience eventual outage. Since E[Z i ] = E[A i X i ] = E[A] X, we have that λ X is equivalent to E[Z i ]. If λ X <, then this is equivalent to the condition E[Z i ] >. ence, from Case 2 of Theorem 1, S n + almost surely. Since U n = u S n, by the definition of limit, P ( lim n S n = + ) = P ( c, n : n > n, S n > c) = P ( c, n : n > n, u U n > c) = P ( c, n : n > n, U n < u c) Taking c = u, we have P ( n : n > n, U n < ) = 1. That is, U n < almost surely. Thus, the SC system will exerience eventual outage. Similarly, if λ X =, this is equivalent to the condition E[Z i ] =. Thus, according to the Case 1 of Theorem 1
6 6 lim su n S n = + and lim inf n S n = almost surely. It suffices to consider the latter case. From the definition of limit suremum, we have P (lim su S n = + ) n = P ( c, n : n > n, su S m > c) m n = P ( c, n : n > n, su (u U m ) > c) m n = P ( c, n : n > n, u inf U m > c) m n = P ( c, n : n > n, inf U m < u c). m n Taking c = u, we have that P ( n : n > n, inf m n U m < ) = 1. ere the inf m n U m < imlies the existence of a subsequence U mk which is strictly less than zero. Thus, combining the cases for λ X < and λ X =, we conclude that the SC system will exerience eventual outage. Remark: Since λ X > is a necessary, but not sufficient, condition for selfsustainability, its satisfaction does not guarantee that outage will not occur. owever, it rovides an easilytocheck condition under which selfsustainability is ossible. We will refer to this condition λ X > (or equivalently E[Z i ] < ) as the selfsustainability condition. Condition 1 (Selfsustainability). An SC system is said to be selfsustainable when λ X >, or equivalently, E[Z i ] <. In the following sections, we will investigate the two mathematical cases that arises when this condition is or is not satisfied. Below we give an immediate corollary of the Proosition 3. Corollary 1. The SC system will exerience an eventual outage almost surely if the selfsustainability condition is not satisfied. If the selfsustainability condition is satisfied, then the robability of eventual outage will be less than unity. That is, { = 1, if λ X ψ(u ) < 1, if λ X >. B. Ascending Ladder Process Now that we have succeeded in converting a continuous time rocess into discrete time rocess, consider the random walk {S n ; n 1} as S n = Z + + Z n 1, or recursively as S n+1 = S n + Z n, with initial value S =. Thus, we have U n as U n = u S n. Consider the ascending ladder rocess defined by M n = max i n S i, which is the artial maximum of artial sums. Since M = S =, the {M i ; i 1} is a ositive nondecreasing sequence, hence the name ascending ladder rocess. We can also relate the values of M n by the recursion M n = max(m n 1, S n ). Also, let M = su i S i be the maximum value attained by S i through the entire duration of its run. We have the key observation that the eventual outage is equivalent to {τ < } {M > u }. ence, φ(u ) = F M (u ) and ψ(u ) = 1 F M (u ). (7) Given the sequence {S i ; i 1}, the first strict ascending ladder oint (σ 1, 1 ) is the first term in this sequence for which S i >. That is, σ 1 = inf{n 1 : S n > } and 1 = S σ1. In other words, the eoch of the first entry into the strictly ositive halfaxis is defined by {σ 1 = n} = {S 1,..., S n 1, S n > }. (8) The σ 1 is called the first ladder eoch while 1 is called the first ladder height. Let the joint distribution of (σ 1, 1 ) be denoted by P (σ 1 = n, 1 x) = F,n (x). (9) The marginal distributions are given by P (σ 1 = n) = F,n ( ) (1) P ( 1 x) = F,n (x) = F (x). (11) n=1 The two variables have the same defect 1 F ( ). We can iteratively define the ladder eochs {σ n } and ladder heights { n } as σ n+1 = inf{k 1 : S k+σn > S σn }, (12) n+1 = S σn+1 S σn. (13) The airs (σ i, i ) are mutually indeendent and have the same common distribution given in (9). These ladder heights are related to record maximum at time n by M n = π n i=1 i, (14) where π n is the number of ladder oints u until time n, i.e. π n = min{k : σ σ k n}. An imortant observation related to the ladder oints is that the sums of {σ i } and { i }, σ σ n and n, form (ossibly terminating) renewal rocesses with interrenewal interval σ i and i. Clearly, we can have M < if and only if the ascending ladder rocess is terminating. As er Definition 2, this renewal rocess is terminating if the underlying distribution F is defective. Accordingly, let be a defective random variable with F () = and F ( ) = θ < 1, then the amount of defect given by 1 θ reresents the robability of termination. In other words, θ reresent the robability of another renewal, while 1 θ reresents the robability that the interrenewal interval is infinite. Thus the termination eoch is a Bernoulli random variable with failure being interreted as termination with robability 1 θ. The sum n has a defective distribution given by the nfold convolution F (n), whose total mass equals F (n) ( ) = F n ( ) = θ n. (15) This is easily seen by reinterreting the n ladder eochs as n successes of a Bernoulli random variable. The defect 1 θ n is thus the robability of termination before the nth ladder eoch.
7 7 Let us define the renewal function for the ladder heights by the sum on nfold convolutions of F as ζ(x) = F (n) (x), (16) n= where F (n) is the nfold convolution of F defined recursively as F (i) (x) = F (i 1) (x t)df (t) where i = 1, 2,..., n; and F () (x) is a unit ste function at the origin. Remark: ere the Lalace transform of F (n) can be () obtained recursively as: F (r) = 1/r, being a unit ste function. Similarly, F (1) () (r) = L[F f () (x)] = F (r) f (r) = (2) (1) f (r)/r. And, F (r) = L[F f (1) (x)] = F (r) f (r) = [ f (r)] 2 (n) /r. And so on, until F (r) = [ f (r)] n /r. Thus, the Lalace transform of the renewal function is ζ(r) = 1 r n= [ f (r)] n 1 = r(1 f, since for the geometric sum (r)) f (r) < f () = 1. The renewal function ζ(x) is equivalent to the exected number of ladder oints in the stri [, x], where the origin counts as a renewal eoch. Thus, ζ() = 1. For terminating renewal rocess, the exected number of eochs ever occurring is finite, as from (15), we have ζ( ) = n= F (n) ( ) = n= θ n = 1 1 θ. If the ascending ladder rocess terminates after nth eoch, then n = M, the all time maximum attained by the random walk S n. The robability that the nth ladder eoch is the last and that {M x} is given by P (M x, terminate after n) = (1 θ)f (n) (x). (17) Using (15), the marginalization of (17) over M shows us that the robability of the ladder rocess terminating after the nth eoch follows a geometric distribution, P (terminate after n) = (1 θ)f (n) ( ) = (1 θ)θn. Similarly, marginalizing (17) over n, we have P (M x) = (1 θ) F (n) (x) = (1 θ)ζ(x). (18) n= We now need a criteria to determine whether the ascending ladder rocess terminates or not, as well as a method to find the value of θ. The following roosition also immediately follows from Theorem 1 and our discussion about ascending ladder rocess: Proosition 4. Given the selfsustainability condition, λ X >, the ascending ladder height rocess { i } of an SC system is terminating almost surely. The robability of selfsustainability given in terms of the renewal function ζ is φ(u ) = (1 θ)ζ(u ). (19) Proof: From Case 3 of Theorem 1, since the random walk S n drifts to when E[Z i ] <, the maximum M < almost surely, and thus { i } terminates. Equation (19) is obtained from (7) and (18). We now relate the selfsustainability robability with two convolution formulas. Proosition 5. Given the selfsustainability condition, the selfsustainability robability satisfies the following equivalent integral equations: φ(u ) = (1 θ) + φ(u ) = u φ(u x)f (x)dx, (2) φ(x)f Z (u x)dx. (21) Proof: We begin with the fact that φ(u ) = P (M u ). The roofs of the two equations follow from the standard renewal tye argument: (1) The event {M u } occurs if the ascending ladder rocess terminates with M or else if 1 assumes some ositive value x u and the residual rocess attains the age u x. So, φ(u ) = P (M = ) + = (1 θ) + u u P ( 1 u 1 = x)f (x)dx P (age u x)f (x)dx. Since the ascending ladder rocess renews at 1, the robability of the age of the residual rocess is P (age u x) = φ(u x). Thus φ(u ) = P (M u ) satisfies the renewal equation φ(u ) = (1 θ) + u φ(u x)f (x)dx. (2) The event {M u } occurs if and only if max(z, Z + Z 1, Z + Z 1 + Z 2,...) u. Conditioning on Z = y, this is equivalent to Z = y u and max(, Z 1, Z 1 + Z 2,...) u y. ere, P (max(, Z 1, Z 1 +Z 2,...) u y) = φ(u y). Deconditioning over all ossible y, we get φ(u ) = u φ(u y)f Z (y)dy, which by change of variable x = u y becomes φ(u ) = φ(x)f Z (u x)dx. Equations (2) and (21) relates φ with (a defective random variable) and Z (a roer random variable) resectively. Equation (2) can be recognized as a renewal equation while (21) can be recognized as a Wienerof integral. Corollary 2. Given the selfsustainability condition, φ(u ) is a roer distribution with (i) φ(u ) = for u <, (ii) φ() = 1 θ, and (iii) φ( ) = 1. Proof: (i) Follows from the fact that u only takes nonnegative values. (ii) By utting u = in (2). (iii) Putting u = in (2), we have φ( ) = (1 θ) + φ( ) f (x)dx. Recalling that is defective, with F ( ) = θ, we have φ( ) = 1. Remark: Corollary 2 can also be roved as a consequence of Proosition 4, since ζ() = 1 and ζ( ) = 1/(1 θ). In the above corollary, it is remarkable that the system can be selfsustaining even when there is no initial battery energy. In other words, this is robability that the random walk U n starting from the origin will always be ositive. This also
8 8 allows us to interret θ as the eventual outage robability when there is no initial battery energy, i.e. ψ() = θ. The corollary also guarantees that as u becomes large, the eventual outage becomes zero. Thus, a ossible strategy in reducing the eventual outage is to simly increase the initial battery energy. We will later show that the rate at which the eventual outage decreases with u is exonential. Remark: Equations (19) and (2) can also be exressed in terms of Lalace transform. Taking the Lalace transform of (2), we have φ(r) = 1 θ + r φ(r) f (r), φ(r) 1 θ = r(1 f (r)). (22) The exact relationshi between and Z is given by the well known Weinerof factorization identity [42, Ch XII.3] [41, Ch VIII.3] in terms of their moment generating functions (MGFs) as: 1 M Z = (1 M )(1 M ), (23) where is the descending ladder height, defined in a manner similar to the ascending ladder height rocess. The is defined over (, ]. The above identity is also written in terms of convolution as F Z = F + F F F. (24) Likewise, the distributions of Z,, and are related to the renewal function ζ by F (x) = ζ F Z (x), x >, (25a) F (x) = ζ F Z (x), x, (25b) where ζ is the renewal function defined by F in a manner similar to ζ given in (16). When E[Z] <, while F is defective, F is roer. Since is defined over (, ], this gives us the condition that F () = 1. Lastly, since is a roer distribution, the descending ladder rocess is a roer renewal rocess. The factorization (23) is in itself difficult to erform exlicitly. As such, we will focus on obtaining bounds and asymtotic aroximations of φ (or equivalently, ψ). IV. BOUND ON EVENTUAL OUTAGE PROBABILITY Thus far we have described the energy surlus rocess and related the various associated concets to the eventual outage / selfsustainability robability. These robabilities can be evaluated by solving the formulas given in Proositions 4 and 5. owever, doing so is not trivial. As such, we wish for a simle bound to estimate the eventual outage robability. ere we will establish a tight exonential bound for the eventual outage robability using the concet of martingales. Definition 4 (Adjustment Coefficient). The value r is said to the adjustment coefficient of X if E[ex(r X)] = e r x df X = 1 Definition 5 (Martingale). A rocess {X i } is said to be a martingale if E[X n+1 X n,..., X ] = X n. Proosition 6. Let {Z i }, {S i } be as before. Suose there exists an adjustment coefficient r > such that E[ex(r Z i )] = 1, then ex(r S n ) for n =, 1, 2,... is a martingale. Proof: We have E[ex(rS n+1 ) S n,..., S 1 ] = E[ex(r(S n + Z n+1 )) S n,..., S 1 ] = E[ex(rZ n+1 )] E[ex(rS n ) S n,..., S 1 ] = E[ex(rZ n+1 )] ex(rs n ). Since there exists a constant r > such that E[ex(r Z n+1 )] = 1, then E[ex(r S n+1 ) S n,..., S 1 ] = ex(r S n ), satisfying the definition of a martingale. Remark: Note that since Z is a roer random variable, it is trivially true that E[ex(r Z i )] = 1 if r =. In the following roosition, we show that there exists a nontrivial value of r > for which this roerty holds true. We will now give the conditions under which the adjustment coefficient will exist. Proosition 7. Suose that E[Z i ] <, which is the selfsustainability condition. Also, assume that there is r 1 > such that the moment generating function (MGF) M Z (r) = E[e rzi ] < for all r 1 < r < r 1, and that lim r r1 M Z (r) =. Then, there is a unique adjustment coefficient r >. Proof: Since the MGF exists in a neighborhood of zero, derivative of every order exists in ( r 1, r 1 ). By definition, M Z () = 1. Since E[Z i ] < by assumtion, we have M Z () = E[Z i] <, which means that M Z (r) is decreasing in the neighborhood of. Also, since M Z (h) = E[Z2 i erzi ] > (due to the fact that the exectation of ositive random variable is ositive), it follows that M Z is convex on ( r 1, r 1 ). Again, by assumtion, we have lim r r1 M Z (r) =. It now follows that there exists a unique s (, r 1 ) such that M Z (s) < 1 and M Z (s) = ; and that on the interval (s, r 1) the function M Z ( ) is strictly increasing to +. As such, there exists a unique r (, r 1 ) such that M Z (r ) = 1. Since the MGF does not exist on [r 1, ), it follows that r is unique on (, ). Proosition 8. Assume that the selfsustainability condition E[Z i ] < holds, and that the adjustment coefficient r > exists. Then the eventual outage robability is bounded by ψ(u ) = P (τ(u ) < ) ex( r u ), u >. (26) Proof: Put τ(u ) = inf(n : S n > u ). The τ(u ) is the the first assage time that the random walk {S n } exceeds u > in the ositive direction. The event that {τ(u ) k} is equivalent to the union of events k i= {S i > u }. Similarly, for a >, let σ(a) = inf{n : S n < a}. ere σ(a) denotes the first assage time that the random walk {S n } exceeds a < in the negative direction. Since E[Z i ] <, by Case 3 of Theorem 1, P (σ(a) < ) = 1. ence, (τ(u ) σ(a)) min(τ(u ), σ(a)) is a stoing time with P ((τ(u ) σ(a)) < ) = 1. Since ex(r S n )
9 9 for n =, 1, 2,... is a martingale, using otional samling theorem, we now get 1 = E[e r S ] = E[ex(r S (τ(u) σ(a)))] = E[ex(r S τ(u)) τ(u ) < σ(a)] + E[ex(r S σ(a) ) σ(a) < τ(u )] E[ex(r S τ(u)) τ(u ) < σ(a)] e r u P (τ(u ) < σ(a)) as S τ(u) > u. Since P (lim a σ(a) = ) = 1, thus letting a we get P (τ(u ) < ) = lim a P (τ(u ) < σ(a)) e r u, as required. Remark: This remarkable roosition guarantees that the eventual outage robability can be made arbitrarily small simly by increasing the initial battery energy. Thus, during the design of a system, where we are willing to tolerate an arbitrarily small eventual outage robability, our task is to determine the initial battery energy. We can use the above bound to roughly calculate the required initial battery energy. The following corollaries immediately follows. Corollary 3. Assuming that the selfsustainability condition holds and r > exists, then lim u φ(u ) =. Corollary 4. Assuming that the selfsustainability condition holds and the adjustment coefficient r > exists, for a given tolerance ɛ >, if the eventual outage robability is constrained at ψ(u ) = ɛ, then the maximum initial battery energy required is u = 1 r log( 1 ɛ ). V. ASYMPTOTIC APPROXIMATION OF EVENTUAL OUTAGE PROBABILITY While the exonential bound given in the revious section is simle to use, we can sharen our estimates using the key renewal theorem for defective distribution. The basic idea behind this aroach is to transform a defective distribution into a roer distribution using the adjustment coefficient, and then aly the key renewal theorem for roer distribution. A. Asymtotic Aroximation First we define the renewal equation and then give the related theorem. Definition 6. The renewal equation is the convolution equation for the form Z = z + F Z, where Z is an unknown function on [, ), z is a known funciton on [, ) and F is a known nonnegative measure on [, ). Often F is assumed to be a robability distribution. If F ( ) = 1, then the renewal equation is roer. If F ( ) < 1, then the renewal equation is defective. Theorem 2. [41, Ch. V, Pro 7.6,. 164] [42, Ch. IX.6, Theo. 2,. 376] Suose that for defective distribution F, there exists an adjustment coefficient r > such that µ = te r t df < exists. If in the defective renewal equation Z = z + F Z, z( ) = lim t z(t) exists and e r t (z(t) z( )) is directly Riemann integrable, then the solution of the renewal equation satisfies µe r t [Z( ) Z(t)] z( ) r + e r s [z( ) z(s)]ds. (27) Proosition 9. Given the selfsustainability condition, the eventual outage robability is asymtotically given by ψ(u ) 1 θ r e r u, (28) µ where r is the adjustment coefficient of and µ = xe r x f (x)dx <. Proof: Recall that (2) is in the form of a renewal equation with the defective distribution of. As such, in Theorem 2, we have z(t) 1 θ and Z φ(u ). Thus, the integral in (27) vanishes and µ e r u [φ( ) φ(u )] 1 θ r. From Corollary 2, we know that φ is roer. Thus φ( ) φ(u ) = 1 φ(u ) = ψ(u ). ence, we have the desired result. Note that the adjustment coefficient r for is the same as that for Z. This is easily demonstrated, since M Z (r ) = E[e r z ] = 1 where r >, we have from the Weinerof factorization (23), (1 M (r ))(1 M (r )) = 1 M Z (r ) =. ere since is defined over (, ], we have M (r ) = (a) < (b) = 1, e r x f ( x)dx, f ( x)dx, since in (a) r > and f is ositive, while in (b) is a roer distribution. Thus, 1 M (r ) >, and this imlies that M (r ) = 1, making r the adjustment factor of as well. The above result is asymtotic in the sense that higher values of u gives more accurate results. owever, without further modeling assumtions, this is as far as we can roceed, since exlicit evaluation of F is difficult. In Section VI, we will exlore the case when the energy acket arrival is modeled as Poisson rocess. B. Comuting the Adjustment Coefficient The comutation of the adjustment coefficient r is not trivial. As given by Definition 4, the adjustment coefficient must satisfy the condition E[e r Z ] = M Z (r ) = 1. For the comutational urose, it is more convenient to use the cumulant generating function (CGF) of Z rather than its MGF. The CGF of Z is defined as K Z (r) = log M Z (r). In terms of CGF, the adjustment coefficient satisfies the condition K Z (r ) = log M Z (r ) = log 1 =. ence, we see that the
10 1 adjustment coefficient r is a real ositive root of CGF K Z (r), and can be found by solving the equation K Z (r ) =. (29) Since Z i = A i X i, due to the linearity of CGFs, K Z (r) = K A (r) + K X ( r). For the imortant case when the energy acket arrival is a Poisson rocess, the interarrival time A i Ex(λ). As such, M A (r) = λ/(λ r), and K A (r) = log(1 r λ ). Thus for) the Poisson arrivals, the equation (29) becomes log (1 r λ + K X ( r ) =, which after exonentiation can be exressed in fixed oint form as r = λ (1 M X( r )). (3) Given the Poisson arrival, some ossible distributions for the energy ackets sizes and their corresonding solutions are as follows: 1) If we further assume that size of the energy ackets is exonentially distributed, X i Ex(1/ X), then we have M X (r) = 1/(1 r X). So, the solution to (3) is r = 1 X ( ) λ X 1. (31) 2) If instead we assume that the size of the energy ackets is deterministic, X i = c, then we have M X (r) = e cr. So the solution to (3) is obtained by solving the equation e cr + r λ 1 =. (32) 3) If instead we assume that the size of the energy ackets is uniformly distributed, X i U(, 2 X), then we have M X (r) = e2 Xr 1 2 Xr. So the solution to (3) is obtained by solving the equation e 2 Xr 2 X λ r2 + 2 Xr + 1 =. (33) A simle aroximation of r can be obtained by making a formal ower exansion of M X ( r) in terms of the moments X2 of X i u to second order term as M X ( r) 1 Xr+ 2 r2. Using this exression in (3) and solving for r >, we obtain r 2 ( ) λ X λx 2 1. (34) More generally, since K Z (r) can be exanded in terms of the mean and variance of Z as K Z (r) µ Z r+ σ2 Z 2 r 2, we have the aroximate solution r for (29) as r 2µ Z σz 2. (35) Since µ Z = E[Z] <, the above aroximation will correctly give r >. This value can be used as an initial oint for a root finding algorithm. Better aroximations can be found by including higher order terms in the exansion and reverting the series using Lagrange inversion. VI. SPECIAL CASE: EVALUATION OF EVENTUAL OUTAGE PROBABILITY FOR POISSON ARRIVALS In general, the density of Z i is given in terms of the densities of A i and X i as f Z (z) = 1 ( ) z + x f A f X (x)dx. (36) max(, z) When the energy acket arrival is assumed to be a Poisson rocess, the interarrival time A i is exonentially distributed, A i Ex(λ). As such, the density of Z i = A i X i is f Z (z) = λ e λz max(, z) When z, the density of Z has the form f Z (z) = λ e λz e λx fx (x)dx. (37) e λx fx (x)dx, z. Since the above integral is indeendent of z, the right tail of the density is exonential. That is, f Z (z) = Ce λz for z, where C is some constant given by C = λ e λx fx (x)dx. Now, from (25a), we have f (x) = ζ f Z (x) = C e λ(x s) ζ (s)ds. Again we see that regardless of the exression for ζ, f takes an exonential form given by f (x) = Ce λx e s ζ (s)ds, where the integral on the right hand is some constant. Since we know that is defective, we can rewrite f in a manner similar to roer exonential distribution as f (x) = ( λ r ) e λx, such that θ = F ( ) = 1 r λ. ence, the amount of defect is 1 θ = r λ. When f (x) is multilied by e r x we have the roer distribution e r x f (x) = ( ) λ r e ( λ r )x, and the mean of this roer exonential distribution is µ = ( λ ) 1. r Thus, we have from (28), ( ψ(u ) 1 r λ ) e r u. (38) When X is also exonentially distributed, we have the exact value of r from (31). ence, we have ψ(u ) u e r. (39) λ X In fact, the (38) and (39) are not just asymtotic aroximations, but also exact formulas (see [42, Ch XII.5, Ex 5(b)]). From these arguments, we have the following roosition: Proosition 1. Assume that the selfsustainability condition holds and the adjustment coefficient r > exists. If the energy
11 11 ackets arrive into an SC system as a Poisson rocess, then the eventual outage robability is given by ( ) ψ(u ) = 1 r e r u. (4) λ Furthermore, if the energy acket size is also exonentially distributed, then ψ(u ) = { ex 1 X ( ) } λ X λ X 1 u. (41) VII. BATTERY ENERGY PROCESS So far we have directed our attention to the energy surlus U(t) and the case when the selfsustainability condition is satisfied. For the sake of comleteness, let us now consider the battery energy W (t) at time t and the case when the selfsustainability condition is not satisfied. A. Equivalence with Queueing Systems We will first rove that the battery energy rocess is a Lindley rocess. Making this identification will then allow us to comare the SC system to a GI/G/1 queue, which in turn will allow us to exloit the results from queueing theory, for which the Lindley rocess was first studied. When the selfsustainability condition is not satisfied, the battery energy rocess W (t) is stationary and ergodic. Thus, it makes better sense to talk about the outage robability P (W (t) = ) of the system rather than the eventual outage robability which is always unity, i.e. ψ(u ) = 1. Definition 7. [41, Ch 3.6] A discretetime stochastic rocess {Y i } is a Lindley rocess if and only if {Y i } satisfies the recurrence relation Y n+1 = max(, Y n + X n ), n =, 1,... (42) where Y = y and {X i } are indeendent and identically distributed. This recursive equation is called Lindley recursion. 1 Proosition 11. The battery energy rocess W (t) observed just before the energy acket arrivals is a Lindley rocess and satisfies the recursion W n+1 = max(, W n + Z n), for n =, 1,..., where W = u and Z n = Z n. Proof: As with the energy surlus in the revious section, let W n be the amount of battery energy immediately before the arrival of nth energy acket. The initial battery energy immediately before the arrival of the first energy acket E is W = u. The amount of battery energy just before the arrival of next (n+1)th energy acket, W n+1, is then the sum of W n and X n, the amount of energy contributed by the nth acket into the battery, minus the amount consumed during the interarrival eriod of the (n + 1)th acket, A n. Thus we have W n+1 = W n + X n A n if W n + X n A n. Likewise, the battery will be emty, W n+1 =, if W n +X n A n. Putting both of them together, we have { Wn + Z W n+1 = n, if W n + Z n, if W n + Z n (43). 1 The Lindley rocess is referred to as queueing rocess in [42] TABLE I QUEUEING ANALOGUE Parameter Energy arvesting Queueing  Consumer Server  Battery Buffer E n nth energy acket nth customer λ Packet arrival rate Customer arrival rate W n Battery energy Virtual waiting time X n Packet size Service time A n Energy consumed Interarrival time (scaled) P (W (t) = ) System outage System idle  Selfsustainability System always busy ρ = λ X Utilization factor Utilization factor (traffic intensity) where Z n = X n A n. Since {X n } and {A n } are IID, the {Z n} is clearly IID. We can write (43) in comact form as W n+1 = max(, W n + Z n), which is a Lindley recursion as given in (42). Since {W i } satisfies the Lindley recursion, {W i } is a Lindley rocess. The fact that {W i } is a Lindley rocess gives rise to a number of imortant consequences, as stated in the following roositions. Proosition 12. The SC system is equivalent to a GI/G/1 queueing system. Proof: The roosition follows from the fact that the virtual waiting time (i.e. the amount of time the server will have to work until the system is emty, rovided that no new customers arrive, or equivalently the waiting time of a customer in a firstinfirstout queueing disciline) of an nth customer arriving into a GI/G/1 queueing system is a Lindley rocess [41, Ch 3.6, Ex 6.1]. Since SC Consume system is also a Lindley rocess by Proosition 11, the equivalence is established. Proosition 13. Suose < E[ Z i ] <. The SC system is ergodic and stationary if and only if E[Z i ] >, i.e. λ X <, when the selfsustainability condition is not satisfied. Under this condition, there will exist a unique stationary distribution for W n, indeendent of the initial condition W, which is given by the Lindley s integral equation: F W (w) = F Z (w x)df W (x), w (44) Proof: This is a standard result from queueing theory. See [41, Coro 6.6]. Remark: We see that when the selfsustainability condition is not satisfied, the battery energy rocess of the system is ergodic and stationary. This allows us to easily comute the outage robability, P (W (t) = ), for such case by invoking the ergodicity and stationarity of W (t). The equation (44) is again a Weinerof integral; and excet for some secial cases, its general solution is difficult to obtain. B. Discussion 1) By comaring the Lindley recursion for the energy harvesting system with that of a queueing system, we can translate the terms and concets of one system into those
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