Age Minimization in Energy Harvesting Communications: Energy-Controlled Delays

Age Minimizat ion in Ener gy Harv esting Communications: Ener gy-Controlle d Delays Ahmed Arafa Sennur Ulukus Departmen t of Electrical and Computer Engineer in g University of Maryland , College Park, MD 20742 arafa@umd.edu ulukus@umd. edu Abstract — W e consider an energy harvesting source that is col- lecting measurements from a physical phenomenon and sendi n g updates to a destin ation wi t hin a communication session time. Updates incur transmission delays that are function of the energy used in their transmission. T he more transmission ener gy used per update, the faster it re aches th e destination. The goal is to transmit updates in a timely manner , namely , such that th e total ag e of information is min imized by the end of th e communication session, subject to ener gy causality constraints. W e consider two varia tions of this p roblem. In the first settin g, the source controls th e number of measurement updates, their transmission times, and the amounts of energy used in their transmission (which gov ern their delays, or service ti mes, incurred). In the second setting, measurement u pdates externally arri ve o ver time, and theref ore the number of updates beco mes fixed, at the expense of addi ng data causality constraints to the problem. W e characterize age-minimal p olicies i n the two settings, and discuss the r elationship o f the a ge of information metric to other metrics used in the en ergy harv esting literature. I . I N T RO D U C T I O N A source collects me a su rements from a physical phe- nomeno n and sends information u pdates to a destina tio n. The source relies solely on energy har vested from nature to commun icate, an d the goal is to send these updates in a timely manner during a giv en co mmunicatio n session time , n amely , such that the to tal age of informa tion is min imized by the end of the session time. The age of information is the time elapsed since the freshest update has reach e d the destination. Power scheduling in en ergy harvesting comm unication sys- tems has been e x tensiv ely studied in th e r e c ent liter ature. Earlier works [1]–[4] c onsider the single-user setting u n- der different battery capacity assumptions, with and witho ut fading. Referen ces [5] –[8] extend this to m u ltiuser setting s: broadc a st, multiple access, and inter f erence ch a nnels; an d [9 ]– [13] consider two-hop, relay , an d two-way channe ls. Minimizing th e age of informatio n metric has been stud- ied mostly in a q ueuing- theoretic framework; [1 4] studies a source-d e stination link und er rand om an d d eterministic service times. This is exten ded to multiple sources in [ 15]. References [16]– [18] co nsider v ariatio ns of the sin g le sour ce system, such as rand omly arriving u pdates, update m anagemen t and co n trol, and n onlinear ag e m etrics, while [19 ] shows that last-come- first-serve po licies are o ptimal in m u lti-hop networks. This work was support ed by NSF Grants CNS 13-14733, CCF 14-22111, CCF 14-22129, and CNS 15-26608 . Our work is most closely related to [20], [ 21], where age minimization in single-user energy har vesting systems is con- sidered; the d ifference of these works fr o m energy harvesting literature in [1]–[13 ] is that the objective is age of inf ormation as opp o sed to th rough put or transmission co mpletion time, and the difference of them from age minimization literature in [1 4]–[19 ], [22] is that sen ding updates in curs energy ex- penditur e where energy beco mes available interm ittently . [20 ] considers rand om serv ice time (time fo r the upd ate to take effect) and [2 1 ] co nsiders ze ro service time. Recently in [23 ], we considered a fixed no n-zero service time in two-hop and single hop settings. In our work h ere, we consid er an energy- controlled ( variable) service time in a single- u ser setting. W e consider a source-de stin a tio n pair where the source relies on e n ergy h arvested from nature to send in formation updates to the destination. Different from [2 0], [2 1], upda tes’ service tim e s de p end on the am o unts of energy u sed to send them; the higher the en ergy used to send an u pdate, the faster it re aches the destination . Hence, a tradeoff arises; given an amount of ene rgy available at the source, it can e ither send a few n umber of up dates with r elati vely small service times, o r it can send a larger n umber of u pdates with relati vely higher service times. In this paper, we investi gate th is tra deoff and characterize th e optimal solution in the offli ne setting . W e formu late the m o st gener al setting of this p roblem where the source decides on the n umber of updates to be sent, wh en to send them, and the amoun ts of energy co nsumed in their transmission (an d therefore the amounts of serv ice times or delays they incu r), such th a t the total age of in formatio n is minimized by the en d of the session time, subjec t to energy causality con stra in ts. W e presen t some structura l insigh ts of the optimal solution in th is g eneral settin g , and p ropose an iterativ e so lution. Our results show that th e optimal numb er of u p dates depend s on the parameters of the problem: the amounts and times of the harvested e n ergy , delay -energy consump tion relationship, and the session time. W e also consider the scenario wh ere upd ate arriv a l times at the source (measu rement time s) cannot be co ntrolled; they arrive during the communication session. Thus, two main changes occ ur to the previously mentioned mo del. First, the total number of up d ates gets fixed; and second , data causality constraints are enfor ced, since the sou rce cannot transmit an update before receiving it. W e f ormulate the problem in this setting and characterize its optimal solution . time age t 1 t 2 t 3 0 t 1 + d 1 T t 2 + d 2 x 2 x 4 t 3 + d 3 x 3 x 1 L Q 2 Q 1 Q 3 Fig. 1. Age ev olution versus time in a controlled measurement times s ystem, with N = 3 upda tes. I I . S Y S T E M M O D E L A N D P RO B L E M F O R M U L A T I O N A source n ode acquires m easurement u pdates f rom some physical ph enomen o n and sends them to a destination d uring a commun ication session of dura tio n T tim e un its. Updates need to be sent as timely as possible, i.e., such th at the total age of information is minimized by time T . The age of info r mation metric is defined as a ( t ) , t − U ( t ) , ∀ t (1) where U ( t ) is the time stamp of the latest received info rmation (measurem e nt) update, i.e., the time at which it was acquir ed at the source. W itho u t loss of generality , we assume a (0) = 0 . The objective is to minim ize the following qu antity A T , Z T 0 a ( t ) dt (2) The source powers itself using energy harvested fr o m nature, and is equ ipped with an infinite batter y to store its incoming energy . Ene rgy is harvested in p ackets of sizes E j at times s j , 1 ≤ j ≤ M . Without lo ss of gen erality , we assume s 1 = 0 . T he to tal energy harvested b y tim e t is E ( t ) = X j : s j ≤ t E j (3) W e denote by e i , the energy u sed in transmitting upd a te i , and denote by d i , its tran smission d e la y (servic e time) until it reaches the destination. These are related as follows e i = f ( d i ) (4) where f is a deceasing conv ex fun ction 1 . Let t i denote th e transmission time of upda te i . The following the n ho lds k X i =1 f ( d i ) ≤ k X i =1 E ( t i ) , ∀ k (5) which rep resent the energy causality con straints [ 1], which mean that energy canno t be used in transmission prio r to being 1 This relat ionship is v alid, for instance, if the channel is A WGN. W ith normaliz ed bandwidt h and noise varian ce, we have f ( d ) = d  2 2 B/d − 1  , with B denot ing the size of the update pac ket in bits [24]. time age t 2 t 3 0 T t 2 + d 2 t 3 + d 3 Q 2 L t 1 + d 1 a 1 a 3 a 2 t 1 Fig. 2. Age e volution versus time in a system where N = 3 update measurement s arrivi ng during communication. harvested. W e also h a ve the service time co nstraints t i + d i ≤ t i +1 , ∀ i (6) which ensu r e that th ere can be only o ne tra nsmission at a time. A. Contr olled Measur em ents In this setting, the sourc e controls when to take a new measuremen t upd ate, and the goal is to choose total numbe r of updates N , transmission times { t i } N i =1 , and d elays { d i } N i =1 , such th a t A T is min imized, subject to energy causality co n- straints in (5) and service time con straints in (6). W e no te that the sour ce sh ould start the tra nsmission of an upd ate measuremen t whenever it is acquired. Otherwise, its age can only increase. In Fig. 1, an examp le ru n o f th e age ev olution versus tim e is presented in a system with N = 3 up dates. The ar ea und er the age c u rve is given by the sum of the areas of the three tr a pezoids Q 1 , Q 2 , and Q 3 , plus the area of the triangle L . Th e area of Q 2 for instan ce is given by 1 2 ( t 2 + d 2 − t 1 ) 2 − 1 2 d 2 2 . Compu ting th e area for a general N updates, we formulate the problem as follows min N , t , d N X i =1 ( t i + d i − t i − 1 ) 2 − d 2 i + ( T − t N ) 2 s.t. t i + d i ≤ t i +1 , 1 ≤ i ≤ N k X i =1 f ( d i ) ≤ k X i =1 E ( t i ) , 1 ≤ k ≤ N (7) with t 0 , 0 an d t N +1 , T . B. Externa lly Arriving Measurements In this setting, m e asurement upd ates arrive du ring the com- munication session at times { a i } N i =1 , where N is n ow fixed. W e now have the following con straints t i ≥ a i , ∀ i (8) representin g the data causa lity constraints [1], which mean that updates canno t b e transmitted prior to being received at the sou rce. I n Fig. 2, we show an example of the age ev olu tion in a system with N = 3 arriving updates. Th e area of Q 2 in this case is given by 1 2 ( t 2 + d 2 − a 1 ) 2 − 1 2 ( t 2 + d 2 − a 2 ) 2 and the area of L is th e co n stant term 1 2 ( T − a 3 ) 2 . Computing the area for gener al N u p date arriv als, we wr ite th e objective function as P N i =1 ( t i + d i − a i − 1 ) 2 − ( t i + d i − a i ) 2 , with a 0 , 0 . This can be fur ther simplified after some algebr a to get the following prob lem for mulation 2 min t , d N X i =1 ( a i − a i − 1 ) ( t i + d i ) s.t. t i + d i ≤ t i +1 , 1 ≤ i ≤ N k X i =1 f ( d i ) ≤ k X i =1 E ( t i ) , 1 ≤ k ≤ N t i ≥ a i 1 ≤ i ≤ N (9) W e note that both problems ( 7) and (9) ar e no n-conve x. One main reaso n is that the total energy arriving up to time t , E ( t ) , is no t con c ave in t . Hencefo rth, in the next sections, we solve the two pr oblems when all the energy packets arriv e at the beginn ing o f comm unication, i.e., wh en M = 1 energy arriv a l. I n this case E ( t ) = E , ∀ t . The solu tions in the case of multiple energy ar riv als fo llow similar structures. I I I . C O N T R O L L E D M E A S U R E M E N T S In this section, we f o cus o n prob lem (7) with a sing le energy arriv a l. W e first have the fo llowing lemma. Lemma 1 In pr oblem (7), all energy is consume d by the end of commun ication. Proof: By direct fir st derivati ves, we o bserve that the objec ti ve function is increasing in { d i } . Thus, if not all energy is consumed , then o ne can simply use th e remaining amo unt to decrease the last service time and achieve lower age.  Next, we app ly the chang e of variables x 1 , t 1 + d 1 , x i , t i + d i − t i − 1 for 2 ≤ i ≤ N , an d x N +1 , T − t N . Then, we m ust h av e P N +1 i =1 x i = T + P N i =1 d i , which reflects the depend ence relationship between the variables. This can also be seen geometr ica lly in Fig. 1. Then, the prob lem beco m es min N , x , d N +1 X i =1 x 2 i − N X i =1 d 2 i s.t. x 1 ≥ d 1 x i ≥ d i + d i − 1 , 2 ≤ i ≤ N x N +1 ≥ d N N +1 X i =1 x i = T + N X i =1 d i N X i =1 f ( d i ) ≤ E (10) The variables { x i } N +1 i =1 control th e inter-upda te times, which are lower boun ded by the service times { d i } N i =1 , which are in turn con trolled by the a m ount of harvested energy , E . W e 2 An inheren t assumption in this model is that 0 < a 1 < a 2 < · · · < a N . Otherwise the areas of the trapezoids become 0 and the problem beco mes dege nerate . Also, the para meters of the problem are such tha t it is feasi ble. propo se an iterativ e alg o rithm to find the optima l inter-update times { x ∗ i } given the op tim al nu mber of updates N ∗ and the optimal service times { d ∗ i } . This is described as follows. Let { ¯ x i } N +1 i =1 denote th e output of th is algorith m, and let us define the stopp ing con dition to b e when P N ∗ +1 i =1 ¯ x i = T + P N ∗ i =1 d ∗ i . W e initialize by setting ¯ x 1 = d ∗ 1 ; ¯ x i = d ∗ i + d ∗ i − 1 , 2 ≤ i ≤ N ; and ¯ x N +1 = d ∗ N . W e then che ck the stopping condition . If it is n o t satisfied, we co mpute m 1 , arg min ¯ x i , and increase ¯ x m 1 until either the stopping con d ition is sat- isfied, o r ¯ x m 1 is equal to min i 6 = m 1 ¯ x i . I n th e latter case, we compute m 2 , arg min i 6 = m 1 ¯ x i , and increase both ¯ x m 1 and ¯ x m 2 simultaneou sly until either the stopp ing con dition is satisfied, or they are b oth eq ual to min i / ∈{ m 1 ,m 2 } ¯ x i . In the latter case, we compute m 3 and pro ceed similar ly as above until the stopp ing condition is satisfied. Note that if m k is not unique at some stag e k o f the algor ithm, we incr ease the whole set { ¯ x i , i ∈ m k } simultaneo u sly . The above algorith m has a water-filling flav or; it evens o ut the x i ’ s to the extent allo w e d by the service times d i ’ s and the session time T , while keeping them as low as possible. The next lemm a shows its optimality . Lemma 2 In pr oblem (10), given N ∗ and { d ∗ i } N i =1 , the opti- mal x ∗ i = ¯ x i , 1 ≤ i ≤ N + 1 . Proof: First, note th at the algorith m initializes x i ’ s by th eir least po ssible values. If this satisfies th e stopp ing (feasibility) condition , then it is optimal. Otherwise, since we ne e d to increase at least on e of the x i ’ s, the alg orithm ch ooses the least one; this giv es the least objecti ve f unction since y < z im plies ( y + ǫ ) 2 < ( z + ǫ ) 2 for y , z ≥ 0 and ǫ > 0 . Next, ob serve that while increasing one of th e x i ’ s, if the stop ping condition is satisfied, the n we have reached the minimal f e asible solution. Otherwise, if two x i ’ s beco me equal, th en by convexity of the sq uare function, it is optimal to increase both of them simultaneou sly [2 5]. This shows that each step o f the algorithm is o ptimal, and henc e it ach iev es the age-minim al solution.  W e n ote that the above algorith m is essentially a variation of the solution of the single-hop pro blem in [23]. There , all the inter-update delays are fixed, while h e re they can b e different. Next, we present an example to show ho w the choice of the number o f upd ates and inter-update delays affect the solution, in a sp e c ific scen ario. In par ticular , we f o cus on the case where the inter-update delays are fixed for all u pdate packets, i.e., d i = d, ∀ i . In this case, by Le m ma 1, for a g i ven N , the optimal inter-update delay is g i ven by d = f − 1 ( E / N ) . W e can then use the algorithm above to find the optimal x i ’ s, a s shown in Lemma 2 . For example, w e consider a system with energy E = 2 0 energy un its, with f ( d ) = d  2 2 /d − 1  , and T = 10 time units. W e plot the optimal age in this case versus N in Fig. 3. W e see that the o ptimal num ber of up dates is equal to 5 ; it is not op timal to send too fe w or too many update s (the max imum feasible is 7 in this example). This echoes the early results in [14], where the optimal rate of upd ating is n ot the maxim u m (th rough put-wise) or the min im um ( delay-wise), but rather lies in between . 1 2 3 4 5 6 7 Number of up dates 12 14 16 18 20 22 24 26 28 Age of Information Fig. 3. Age of info rmation versus number of updates. I V . E X T E R NA L L Y A R R I V I N G M E A S U R E M E N T S In th is section, we solve pro blem (9) with a sing le energy arriv a l. W e observe that the problem in this case is conve x and can b e so lved by standard techn iques [25 ]. W e first hav e the following lemma. Lemma 3 In pr oblem (9) , the optimal upd ate times satisfy t ∗ 1 = a 1 (11) t ∗ i = max    a i , a i − 1 + d ∗ i − 1 , . . . , a 1 + i − 1 X j =1 d ∗ j    , i ≥ 2 (12) Proof: This follows directly from the constraints of p roblem (9); the optimal update times should always be equ al to th e ir lower b ounds. Hence, we have t ∗ 1 = a 1 , t ∗ 2 = max { a 2 , t ∗ 1 + d ∗ 1 } = max { a 2 , a 1 + d ∗ 1 } , t ∗ 3 = max { a 3 , t ∗ 2 + d ∗ 2 } = max { a 3 , a 2 + d ∗ 2 , a 1 + d ∗ 1 + d ∗ 2 } , and so on.  By the previous lemm a, the problem n ow reduc es to finding the o ptimal inter-update de la y s { d ∗ i } . W e n ote that starting from t ∗ 1 = a 1 , we have two choices fo r t ∗ 2 ; either a 2 or a 1 + d ∗ 1 . Once t ∗ 2 is fixed, t ∗ 3 in tu r n h a s two choices; either a 3 or t ∗ 2 + d ∗ 2 . Now observe that once a choice pattern is fixed, the objective function o f prob lem (9 ) will b e given by P N i =1 c i d i where c i > 0 is a constant th a t depen ds o n the choice p attern. For instance, for N = 3 , choosing the pattern t ∗ 2 = a 1 + d ∗ 1 and t ∗ 3 = a 3 giv es c 1 = a 2 , c 2 = a 2 − a 1 , and c 3 = a 3 − a 2 . W e introdu c e the following Lagran gian for this pro blem [25 ] L = N X i =1 c i d i + λ N X i =1 f ( d i ) − E ! (13) where λ is a non-negative Lagrang e m ultiplier . Th e KKT condition s are c i = − λf ′ ( d i ) (14) Hence, the optimal λ ∗ is giv en by the un ique solu tion of N X i =1 h ( − c i /λ ∗ ) = E (15) where h , f ◦ g and g , ( f ′ ) − 1 . T o see this, n ote th at since f is conve x, it follows that g exists and is increasing. By (1 4), we the n h ave d ∗ i = g ( − c i /λ ∗ ) . Substituting in the en e rgy constraint, which has to b e satisfied with equality , gives (1 5). By monotonicity o f f and g , h is also monoto ne, and therefore (15) has a uniqu e solution in λ ∗ . Therefo re, we solve problem ( 9) b y first fixing a choice pattern fo r th e upd a te times, which giv es us a set of co nstants { c i } allowing u s to solve for λ ∗ using (15). W e go th r ough all po ssible choice patterns an d choose the one that is feasible and gives minimal age. W e fin ally n o te that the measurements’ arrival times can be so c lose to each oth er that th e o ptimal solutio n is su c h that t ∗ i > a i + l for som e i an d l ≥ 1 . T hat is, there would be l + 1 measuremen ts waiting in the data queue before t ∗ i . If th e total number of updates c a n be ch anged, then this solution can be made better by transmitting only the fresh e st, i.e. , the ( i + l ) th, measuremen t p acket at t ∗ i and ign oring all the rest. Th is strictly improves the a g e and saves some energy as well. The solution can b e fu rther op timized by re-solving the pro blem with ˜ N = N − l arriving measurem ents at times ˜ a 1 = a 1 , . . . , ˜ a i − 1 = a i − 1 , ˜ a i = a i + l , . . . , ˜ a ˜ N = a N . V . D I S C U S S I O N : R E L A T I O N S H I P T O O T H E R M E T R I C S In this section we discuss the relationship between the pro- posed pr oblems in this work and other well-known problem s in the energy har vesting literatu re: tra nsmission c o mpletion time minim ization, and delay min imization. Reference [1 ] introdu c ed the tran smission completion time minimization problem . In this pr oblem, g i ven some amounts of data arriving during the commun ication session, th e objective is to minimize the time by which all the d ata is deliv ered to th e destination, subject to energy and data causality co nstraints. Reference [2 6] studies this p roblem from a different pe r- spectiv e. Instead of minimizing the completion time of all the data, th e ob jecti ve is to min imize the delay experienced by each b it, which is equal to the difference between the time of its receptio n at th e receiv er and the time of its arrival at the transmitter . Delay-minimal policies are fund amentally different than those minimizing comp letion time. For in stance, in [1], due to the concave rate- power relationsh ip , tran smitting with constant powers in between e n ergy ha rvests is optimal. While in [26], th e o ptimal d elay-minima l powers are d ecreas- ing over time in b etween energy har vests, since earlier arriving bits con tribute mo re to th e cum ulativ e dela y and are th us given higher priorities (transmission powers and rates). W e n ote tha t min imizing the age o f inf ormation pro blem is similar to the delay m inimization problem f ormulated in [26]. In both p roblems, there is a time co unter that co unts time between d a ta tran sm ission s an d receptio ns. In the a g e of information pro blem, the time counter star ts in creasing time t 2 t 3 0 T t 2 + d 2 t 3 + d 3 t 1 + d 1 a 1 a 3 a 2 t 1 B 2 B 3 B dela y Fig. 4. Cumulati ve dat a pac ket s arri ving (blue) and de partin g (black) versus time with N = 3 data pack ets. The shaded area in yello w between the two curve s represents the total dela y of the system. from the beginn ing of the com munication session. While in the d elay problem , th e time co unter is bit- dependen t; it starts increasing on ly from the momen t a new bit enters the system and stops when it reaches the destination. The delay minimization problem w as previously form ulated in [27] fo r the case where th e delay is computed per packet, as opposed to p er bit in [26] ( n ote that the age is also comp uted per packet and not per bit) . The transmitter in [2 7] was e nergy constrained but not h arvesting energy over time, which m odels the case where all energy packets ar r i ve at the beginning of commun ication. For the sake o f com parison, we extend the delay minimization problem in [2 7] to the en ergy har vesting case as in [26] an d relate it to the ag e minimization prob lem considered in this work. Follo wing the model in Sectio n II-B, the i th a r riving d a ta packet waits f or t i − a i time in qu eue, a nd then gets served in d i time u nits. Follo wing [2 6 ], the total delay is define d as the ar e a in b etween the cumula tive departin g data c u rve, and the cumulative arri ving data cu rve. In Fig. 4, we show an example realization using the same transmission, arriv al, and service times used in Fig. 2. The solid blu e curve rep resents the cumulative received data packets over time; the dotted black curve represents cumulative departed (served) data packets over tim e; an d the shaded area in yellow rep resents the total delay D T . The delay of the fir st data p acket f o r instance is giv en by B ( t 1 − a 1 ) + 1 2 B d 1 , wher e B is the length of the data packet in bits. Computing the area for general N arriv als, the delay minimization problem is giv en by min t , d N X i =1 2 t i + d i s.t. problem (9) constrain ts (16) W e see that min im izing delay in problem (16) is almost th e same as min im izing age in p roblem (9). The main dif ference is that to minimize ag e, transmission and service times ar e weighted by arriv al times, while this is not the case when minimizing delay . 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