Energy Harvesting Networks with General Utility Functions: Near Optimal Online Policies
aa r X i v : . [ c s . I T ] M a y Energy Harvesting Networks with General UtilityFunctions: Near Optimal Online Policies
Ahmed Arafa Abdulrahman Baknina Sennur Ulukus
Department of Electrical and Computer EngineeringUniversity of Maryland, College Park, MD 20742 [email protected] [email protected] [email protected]
Abstract — We consider online scheduling policies for single-user energy harvesting communication systems, where the goalis to characterize online policies that maximize the long termaverage utility, for some general concave and monotonicallyincreasing utility function. In our setting, the transmitter relies onenergy harvested from nature to send its messages to the receiver,and is equipped with a finite-sized battery to store its energy.Energy packets are independent and identically distributed (i.i.d.)over time slots, and are revealed causally to the transmitter. Onlythe average arrival rate is known a priori. We first characterizethe optimal solution for the case of Bernoulli arrivals. Then, forgeneral i.i.d. arrivals, we first show that fixed fraction policies [1]are within a constant multiplicative gap from the optimal solutionfor all energy arrivals and battery sizes. We then derive a setof sufficient conditions on the utility function to guarantee thatfixed fraction policies are within a constant additive gap as wellfrom the optimal solution.
I. I
NTRODUCTION
A single-user communication channel is considered, wherethe transmitter relies on energy harvested from nature to sendits messages to the receiver. The transmitter has a batteryof finite size to save its incoming energy, and achieves areward for every transmitted message that is in the formof some general concave increasing utility function of thetransmission power. The goal is to characterize online powercontrol policies that maximize the long term average utilitysubject to energy causality constraints.
Offline power scheduling in energy harvesting communica-tion systems has been extensively studied in the recent liter-ature. Earlier works [2]–[5] consider the single-user setting.References [6]–[12] extend this to broadcast, multiple access,and interference settings; and [13]–[16] consider two-hopand relay channels. Energy cooperation and energy sharingconcepts are studied in [17], [18]. References [19]–[24] studyenergy harvesting receivers, where energy harvested at the re-ceiver is spent mainly for sampling and decoding. Other works[24]–[29] study the impact of processing costs, i.e., the powerspent for circuitry, on energy harvesting communications.Recently, [1] has introduced an online power control policyfor a single-user energy harvesting channel that maximizes thelong term average throughput under the AWGN capacity utilityfunction log(1 + x ) . The proposed policy is near optimal inthe sense that it performs within constant multiplicative and This work was supported by NSF Grants CNS 13-14733, CCF 14-22111,CCF 14-22129, and CNS 15-26608. additive gaps from the optimal solution that is independent ofenergy arrivals and battery sizes. This is extended to broadcastchannels in [30], multiple access channels in [31], [32], andsystems with processing costs in [33], [34] (for examples ofearlier online approaches see, e.g., [35]–[37]).In this paper, we generalize the approaches in [1] to work forgeneral concave monotonically increasing utility functions forsingle-user channels. That is, we consider the design of onlinepower control policies that maximize the long term averagegeneral utilities. One motivation for this setting is energyharvesting receivers. Since power consumed in decoding ismodelled as a convex increasing function of the incoming rate[19], [20], [23], the rate achieved at the receiver is then aconcave increasing function of the decoding power. In oursetting, energy is harvested in packets that follow an i.i.d.distribution with amounts known causally at the transmitter.The transmitter has a finite battery to store its harvested energy.We first study the special case of Bernoulli energy arrivals thatfully recharge the battery when harvested, and characterize theoptimal online solution. Then, for the general i.i.d. arrivals,we show that the policy introduced in [1] performs withina constant multiplicative gap from the optimal solution forany general concave increasing utility function, for all energyarrivals and battery sizes. We then provide sufficient conditionson the utility function to guarantee that such policy is withina constant additive gap from the optimal solution.II. S
YSTEM M ODEL AND P ROBLEM F ORMULATION
We consider a single-user channel where the transmitterrelies on energy harvested from nature to send its messages tothe receiver. Energy arrives (is harvested) in packets of amount E t at the beginning of time slot t . Energy packets follow ani.i.d. distribution with a given mean. Our setting is online: theamounts of energy are known causally in time, i.e., after beingharvested. Only the mean of the energy arrivals is known apriori. Energy is saved in a battery of finite size B .Let u be a differentiable, concave, and monotonically in-creasing function representing some general utility (reward)function, with u (0) = 0 and u ( x ) > for x > , and let g t denote the transmission power used in time slot t . By allocat-ing power g t in time slot t , the transmitter achieves u ( g t ) instantaneous reward. Denoting E t , { E , E , . . . , E t } , afeasible online policy g is a sequence of mappings { g t : E t → + } satisfying ≤ g t ≤ b t , min { b t − − g t − + E t , B } , ∀ t (1)with b , B without loss of generality (using similar argu-ments as in [1, Appendix B]). We denote the above feasibleset by F . Given a feasible policy g , we define the n -horizonaverage reward as U n ( g ) , n E " n X t =1 u ( g t ) (2)Our goal is to design online power scheduling policies thatmaximize the long term average reward subject to (online)energy causality constraints. That is, to characterize ρ ∗ , max g ∈F lim n →∞ U n ( g ) (3)III. M AIN R ESULTS
In this section, we present the main results of this paper. Wenote that problem (3) can be solved by dynamic programmingtechniques since the underlying system evolves as a Markovdecision process. However, the optimal solution using dynamicprogramming is usually computationally demanding with fewstructural insights. Therefore, in the sequel, we aim at findingrelatively simple online power control policies that are prov-ably within a constant additive and multiplicative gap fromthe optimal solution for all energy arrivals and battery sizes.We assume that E t ≤ B ∀ t a.s., since any excess energyabove the battery capacity cannot be saved or used. Let µ = E [ E t ] , where E [ · ] is the expectation operator, and define q , E [ E t ] B (4)Then, we have ≤ q ≤ since E t ≤ B a.s. We define thepower control policy as follows [1] ˜ g t = qb t (5)That is, in each time slot, the transmitter uses a fixed fractionof its available energy in the battery. Such policies were firstintroduced in [1], and coined fixed fraction policies (FFP).Clearly such policies are always feasible since q ≤ . Let ρ (˜ g ) be the long term average utility under the FFP { ˜ g t } . Wenow state the main results. Lemma 1
The optimal solution of problem (3) satisfies ρ ∗ ≤ u ( µ ) (6) Theorem 1
The achieved long term average utility under theFFP in (5) satisfies ≤ ρ (˜ g ) u ( µ ) ≤ (7)We note that the results in Lemma 1 and Theorem 1 indicatethat the FFP in (5) achieves a long term average utility that iswithin a constant multiplicative gap from the optimal solutionthat is equal to . This result is proved in [1] for u ( x ) = log(1 + x ) . Here, we are generalizing it to work for anyconcave increasing function u with u (0) = 0 .Next, we state the additive gap results. We first define h θ ( x ) , u ( θx ) − u ( x ) (8)for some ≤ θ ≤ , and define the following two classes ofutility functions. Definition 1 (Utility Classes)
A utility function u belongs toclass ( A ) if h θ ( x ) does not converge to 0 as x → ∞ , andbelongs to class ( B ) if lim x →∞ h θ ( x ) = 0 . Now let us define the following function for < θ < h ( θ ) , inf x h θ ( x ) (9)whenever the infimum exists. Note that the infimum exists forclass ( B ) utility functions since h θ ( x ) < for x > bymonotonicity of u , and h θ (0) = 0 . We state some propertiesof the function h in the next lemma. The proof follows bymonotonicity and concavity of u and is omitted for brevity. Lemma 2 h ( θ ) is non-positive, concave, and non-decreasingin θ . The next two theorems summarize the additive gap resultsfor utility functions in classes ( A ) and ( B ) in Definition 1. Theorem 2 If h ( θ ) exists, and if r , (1 − q ) lim t →∞ − lim x → ¯ x t +1 u (cid:0) (1 − q ) t +1 x (cid:1) /u ( x )1 − lim x → ¯ x t u ((1 − q ) t x ) /u ( x ) < (10) where ¯ x t ∈ arg inf x h (1 − q ) t ( x ) ; then the achieved long termaverage utility under the FFP in (5) satisfies u ( µ ) + α ≤ ρ (˜ g ) ≤ u ( µ ) (11) where α , P ∞ t =0 q (1 − q ) t h ((1 − q ) t ) is finite. Theorem 3
For class ( B ) utility functions, the achieved longterm average utility under the FFP in (5) satisfies lim µ →∞ ρ (˜ g ) = ρ ∗ (12)We note that the results in Lemma 1 and Theorem 2indicate that the FFP in (5) achieves a long term averageutility, under some sufficient conditions, that is within aconstant additive gap from the optimal solution that is equal to | P ∞ t =0 q (1 − q ) t h ((1 − q ) t ) | . One can further make this gapindependent of q by minimizing it over ≤ q ≤ . We discussexamples of the above results in Section VI, where we alsocomment on FFP performance under utility functions that donot satisfy the sufficient conditions in Theorem 2.IV. B ERNOULLI E NERGY A RRIVALS
In this section, we characterize the optimal solution of aspecial case of the energy arrival i.i.d. process: the Bernoullirocess. Let { ˆ E t } be a Bernoulli energy arrival process withmean µ as follows ˆ E t ∈ { , B } , with P [ ˆ E t = B ] = p, and pB = µ (13)where P [ A ] denotes the probability of A . Note that undersuch specific energy arrival setting, whenever an energy packetarrives, it completely fills the battery, and resets the system.This constitutes a renewal . Then, by [38, Theorem 3.6.1] (seealso [1]), the following holds for any power control policy g lim n →∞ ˆ U n ( g ) = lim n →∞ n E " n X t =1 u ( g t ) = 1 E [ L ] E " L X t =1 u ( g t ) a.s. (14)where ˆ U n ( g ) is the n -horizon average utility under Bernoulliarrivals, and L is a random variable denoting the inter-arrival time between energy arrivals, which is geometric withparameter p , and E [ L ] = 1 /p .Using the FFP defined in (5) in (14) gives a lower bound onthe long term average utility. Note that by (13), the fraction q in (4) is now equal to p . Also, the battery state decaysexponentially in between energy arrivals, and the FFP is ˜ g t = p (1 − p ) t − B = (1 − p ) t − µ (15)for all time slots t , where the second equality follows since pB = µ . Using (14), problem (3) in this case reduces to max g ∞ X t =1 p (1 − p ) t − u ( g t ) s.t. ∞ X t =1 g t ≤ B, g t ≥ , ∀ t (16)which is a convex optimization problem. The Lagrangian is, L = − ∞ X t =1 p (1 − p ) t − u ( g t ) + λ ∞ X t =1 g t − B ! − ∞ X t =1 η t g t (17)where λ and { η t } are Lagrange multipliers. Taking derivativewith respect to g t and equating to 0 we get u ′ ( g t ) = λ − η t p (1 − p ) t − (18)Since u is concave, then u ′ is monotonically decreasing and f , ( u ′ ) − exists, and is also monotonically decreasing. Bycomplementary slackness, we have η t = 0 for g t > , and theoptimal power in this case is given by g t = f (cid:18) λp (1 − p ) t − (cid:19) (19)and it now remains to find the optimal λ . We note bymonotonicity of f , { g t } is non-increasing, and it holds that g t = f (cid:18) λp (1 − p ) t − (cid:19) > ⇔ λ < p (1 − p ) t − u ′ (0) (20) Hence, if u ′ (0) is infinite, then (20) is satisfied ∀ t , and theoptimal power allocation sequence is an infinite sequence. Inthis case, we solve the following equation for the optimal λ ∞ X t =1 f (cid:18) λp (1 − p ) t − (cid:19) = B (21)which has a unique solution by monotonicity of f .On the other hand, for finite u ′ (0) , there exists a time slot N , after which the second inequality in (20) is violated since λ is a constant and p (1 − p ) t − is decreasing. In this casethe optimal power allocation sequence is only positive for afinite number of time slots ≤ t ≤ N . We note that N is thesmallest integer such that λ ≥ p (1 − p ) N u ′ (0) (22)Thus, to find the optimal N (and λ ), we first assume N isequal to some integer { , , , . . . } , and solve the followingequation for λ N X t =1 f (cid:18) λp (1 − p ) t − (cid:19) = B (23)We then check if (22) is satisfied for that choice of N and λ . If it is, we stop. If not, we increase the value of N andrepeat. This way, we reach a KKT point, which is sufficient foroptimality by convexity of the problem [39]. We note that for u ( x ) = log(1+ x ) whose u ′ (0) is finite, [1] called N , ˜ N . Wegeneralize their analysis for any concave increasing function u . This concludes the discussion of the optimal solution in thecase of Bernoulli energy arrivals.V. G ENERAL I . I . D . E NERGY A RRIVALS :P ROOFS OF M AIN R ESULTS
A. Proof of Lemma 1
In this section, we derive the upper bound in Lemma 1 thatworks for all i.i.d. energy arrivals. Following [1] and [33], wefirst remove the battery capacity constraint setting B = ∞ .This way, the feasible set F becomes n X t =1 g t ≤ n X t =1 E t , ∀ n (24)Then, we remove the expectation and consider the offlinesetting of problem (3), i.e., when energy arrivals are knowna priori. Since the energy arrivals are i.i.d., the strong lawof large numbers indicates that lim n →∞ n P nt =1 E t = µ a.s.,i.e., for every δ > , there exists n large enough such that n P nt =1 E t ≤ µ + δ a.s., which implies by (24) that the feasibleset, for such ( δ, n ) pair, is given by n n X t =1 g t ≤ µ + δ a.s. (25)Now fix such ( δ, n ) pair. The objective function is given by n n X t =1 u ( g t ) (26)ince u is concave, the optimal power allocation minimizingthe objective function is g t = µ + δ , ≤ t ≤ n [39] (see also[2]). Whence, the optimal offline solution is given by u ( µ + δ ) .We then have ρ ∗ ≤ u ( µ + δ ) . Since this is true ∀ δ > , wecan take δ down to 0 by taking n infinitely large. B. Proof of Theorem 1
We first derive a lower bound on the long term averageutility for Bernoulli energy arrivals under the FFP as follows lim n →∞ ˆ U n (˜ g ) ( a ) = p ∞ X i =1 p (1 − p ) i − i X t =1 u (˜ g t )= ∞ X t =1 p (1 − p ) t − u (cid:0) (1 − p ) t − µ (cid:1) (27) ( b ) ≥ ∞ X t =1 p (1 − p ) t − u ( µ )= 12 − p u ( µ ) ≥ u ( µ ) (28)where ( a ) follows by (14), ( b ) follows by concavity of u [39],and the last inequality follows since ≤ p ≤ . Next, weuse the above result for Bernoulli arrivals to bound the longterm average utility for general i.i.d. arrivals under the FFPin the following lemma; the proof follows by concavity andmonotonicity of u , along the same lines of [1, Section VII-C],and is omitted for brevity. Lemma 3
Let { ˆ E t } be a Bernoulli energy arrival process asin (13) with parameter q as in (4) and mean qB = µ . Then,the long term average utility under the FFP for any generali.i.d. energy arrivals, ρ (˜ g ) , satisfies ρ (˜ g ) ≥ lim n →∞ ˆ U n (˜ g ) (29)Using Lemma 1, (28), and Lemma 3, we have u ( µ ) ≤ ρ (˜ g ) ≤ ρ ∗ ≤ u ( µ ) (30) C. Proof of Theorem 2
By Lemma 1 and Lemma 3, it is sufficient to study the lowerbound in the case of Bernoulli arrivals. By (27) we have lim n →∞ ˆ U n (˜ g ) = ∞ X t =1 p (1 − p ) t − u (cid:0) (1 − p ) t − µ (cid:1) ( c ) ≥ ∞ X t =1 p (1 − p ) t − (cid:0) u ( µ ) + h (cid:0) (1 − p ) t − (cid:1)(cid:1) = u ( µ ) + ∞ X t =0 p (1 − p ) t h (cid:0) (1 − p ) t (cid:1) , u ( µ ) + α (31)where ( c ) follows since h ( θ ) exists, and is by definition nolarger than h θ ( x ) , ∀ x, θ . Now to check whether α is finite,we apply the ratio test to check the convergence of the series P ∞ t =0 (1 − p ) t h ((1 − p ) t ) . That is, we compute r , lim t →∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (1 − p ) t +1 h (cid:0) (1 − p ) t +1 (cid:1) (1 − p ) t h ((1 − p ) t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (1 − p ) lim t →∞ inf x − u (cid:0) (1 − p ) t +1 x (cid:1) /u ( x )inf x − u ((1 − p ) t x ) /u ( x ) (32)where the second equality follows by definition of h . Next,we replace inf x by lim x → ¯ x t since ¯ x t ∈ arg inf h (1 − p ) t ( x ) ,and take the limit inside (after the 1). Finally, if r < then α is finite; if r > then α = −∞ ; and if r = 1 then the testis inconclusive and one has to compute lim T →∞ P Tt =0 p (1 − p ) t h ((1 − p ) t ) to get the value of α . D. Proof of Theorem 3
For utility functions of class ( B ) , we have lim x →∞ u ( θx ) − u ( x ) = 0 . Thus, ∀ ǫ > there exists ¯ µ large enough such that u (cid:0) (1 − p ) t − µ (cid:1) > u ( µ ) − ǫ, ∀ µ ≥ ¯ µ (33)whence, for Bernoulli energy arrivals we have lim n →∞ ˆ U n (˜ g ) = ∞ X t =1 p (1 − p ) t − u (cid:0) (1 − p ) t − µ (cid:1) ≥ u ( µ ) − ǫ, ∀ µ ≥ ¯ µ (34)It then follows by Lemma 1 and Lemma 3 that ρ ∗ ≥ ρ (˜ g ) ≥ u ( µ ) − ǫ ≥ ρ ∗ − ǫ, ∀ µ ≥ ¯ µ (35)and we can take ǫ down to 0 by taking µ infinitely large.VI. E XAMPLES AND D ISCUSSION
In this section we present some examples to illustrate theresults of this work. We first show that the utility function u ( x ) = log(1 + x ) considered in [1] belongs to class ( A ) .Indeed we have h ′ θ ( x ) = θ − θx )(1+ x ) , which is negativefor all < θ < , and therefore h θ ( x ) is decreasing in x and does not converge to 0. We then show that the sufficientconditions of Theorem 2 are satisfied: h ( θ ) exists, and is equalto lim x →∞ log θx x = log( θ ) ; r = 1 − q and hence thegap α is finite. Furthermore, [1] showed that minimizing α over all q gives a constant additive gap, independent of q , thatis equal to . .Next, we note that all bounded utility functions belong toclass ( B ) . These are functions u where there exists someconstant M < ∞ such that u ( x ) ≤ M, ∀ x . Examplesfor these include: u ( x ) = 1 − e − βx for some β > , and u ( x ) = x/ (1 + x ) . To see that these functions belong toclass ( B ) , observe that lim x →∞ u ( x ) = M by monotonicityof u , and hence lim x →∞ u ( θx ) − u ( x ) = 0 . We also note thatclass ( B ) is not only inclusive of bounded utility functions.For example, the unbounded function u ( x ) = p log(1 + x ) satisfies lim x →∞ p log(1 + θx ) − p log(1 + x ) = 0 andtherefore belongs to class ( B ) . For such unbounded functionsin class ( B ) , the FFP is not only within a constant additivegap of the optimal solution, it is asymptotically optimal aswell, as indicated by Theorem 3.ote that one can find a (strict) lower bound on h ( θ ) forsome utility functions if it allows more plausible computationof α , or if h ( θ ) itself is not direct to compute. For instance, forany bounded utility function u , the following holds: h ( θ ) ≥ ( θ − M , where M is the upper bound on u . To see this,observe that by concavity of u and the fact that u (0) = 0 we have inf x u ( θx ) − u ( x ) ≥ ( θ −
1) sup x u ( x ) . This gives α ≥ P ∞ t =0 q (1 − q ) t ((1 − q ) t − M , which is no smallerthan − M if we further minimize over q . Another exampleis u ( x ) = log (1 + √ x ) , which belongs to class ( A ) . Weobserve that h ( θ ) in this case is lower bounded by log( θ ) .Hence, this function admits an additive gap no larger than . calculated in [1] for u ( x ) = log(1 + x ) .Finally, we note that the conditions of Theorem 2 are onlysufficient for the FFP defined in (5) to be within an additivegap from optimal. For instance, consider u ( x ) = √ x . Thisfunction belongs to class ( A ) as h θ ( x ) does not converge to0. In fact, h θ ( x ) is unbounded below and h ( θ ) does not exist.This means that any FFP of the form ˜ g t = θb t , for any choiceof < θ < , is not within a constant additive gap from theupper bound √ µ . However, there exists another FFP (with adifferent fraction than q in (4)) that is optimal in the case ofBernoulli arrivals. Since u ′ (0) = ∞ , we use (21) to find theoptimal λ , where f ( x ) = 1 / (4 x ) , and substitute in (19) toget that the optimal transmission scheme is fractional: g t =ˆ p (1 − ˆ p ) ( t − B, ∀ t , where the transmitted fraction ˆ p , − (1 − p ) . This shows that one can pursue near optimality resultsunder an FFP by further optimizing the fraction of power usedin each time slot, and comparing the performance directly tothe optimal solution instead of an upper bound. While in thiswork, we compared the lower bound achieved by the FFP to auniversal upper bound that works for all i.i.d. energy arrivals.R EFERENCES[1] D. Shaviv and A. Ozgur. Universally near optimal online power controlfor energy harvesting nodes.
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