Coded Status Updates in an Energy Harvesting Erasure Channel
We consider an energy harvesting transmitter sending status updates to a receiver over an erasure channel, where each status update is of length $k$ symbols. The energy arrivals and the channel erasures are independent and identically distributed (i.…
Authors: Abdulrahman Baknina, Sennur Ulukus
Coded Status Updates in an Ener gy Harv esting Erasure Channel Abdulrahman Baknina Sennur Ulukus Departmen t of Electrical and Computer Engineer in g University of Maryland , College Park, MD 20742 abakn ina@umd.edu ulukus@umd. edu Abstract —W e consider an energy harvesting transmitter se nd - ing statu s upd ates to a receiver over an erasure channel, where each status update is of length k symbols. The energ y arrivals and the channel erasures a re independent and identically distributed (i.i.d.) and Bernoulli di stributed in each slot. In order to co mbat the effects of th e erasur es in the channel and the u ncertainty in the energy arri vals, we use channel coding to encode the status upd ate symbols. W e consider two types of chann el coding: maximum distan ce separable (MDS) codes and rateless erasure codes. F or each o f these models, we study two achieva ble schemes: best-effort and sav e-and-transmit. In the best-effo rt scheme, th e transmitter starts transmission right away , and send s a symbol if it has energy . In t he sa ve-and-transmit scheme, the transmitter remains silent in the b eginning in order to sav e some energy t o minimize ener gy outages in future slots. W e analyze th e a verage age of information (AoI) un der each of th ese p olicies. W e sh ow through numerical results that as th e aver age rechar ge rate decreases, MDS codin g with save -and-t ransmit outp erf orms all best-effort schemes. W e show t hat rateless co din g with sa ve-a nd- transmit outp erf orms all the other schemes. I . I N T RO D U C T I O N W e consider an ene rgy h arvesting single-u ser system, where the com munication channel between the tran smitter and the receiver is an erasure channel. The tr ansmitter collects mea- surements of a certain ph enomen on and sends updates on this pheno m enon to the receiver; these up d ates are r e ferred to as status u pdates . T he pur pose of sending status u pdates is to minimize the age of infor mation (AoI) at the receiver . Energy harvesting commun ications with the ob jectiv e of maximizing the throughpu t has bee n extensi vely studied, for example, see [1]–[2 5]. The single-u ser channel is studied in [1]–[4], extende d to multi-user settings in [5]– [7], multi-ho p channels in [8]–[10], and two-way channels in [ 11], [12]. Effects of imperfect circuitry , receiver side processing , and temperatur e increases are considered in [13]–[25]. In this paper , we c o nsider an energy har vesting comm u- nication system with the o bjective of m inimizing the av erag e AoI at th e receiver . Status up dates and AoI metric is studied in many dif fere nt settings, for example, see [26]–[40]. References [26]–[30] study m inimizing the AoI with a queuing theo retic approa c h ; penalty functions and non-lin e ar costs are studied in [31], [32]; the o ptimality of last-co me-first-serve for multi-ho p settings is sho wn in [33]; and erasure channels are considered in [ 34], [35]. The energy h arvesting case an d when the energy This w ork wa s supporte d by NSF Grants CNS 13-14733, CCF 14-22111, CNS 15-26608, and CC F 17-13 977. E i Tx measurements Rx erasure channel channel coding channel decoding Fig. 1. An energy harv esting transmitte r with an infinite batt ery . The transmitt er colle cts measurements and sends updates to the rec eiv er over an erasure channel. arriv als are known only causally is studied in [36]–[38]. The optimality of th reshold policies for th e case of un it batteries is sho wn in [38]. Energy harvesting single-user and multi-hop settings with n on-causal energy arri val knowledge are studied in [ 39], [40]. This p aper is closely relate d to [35], in which coded status updates are proposed in o rder to overcome channel errors. W e c o nsider a single- user cha nnel shown in Fig. 1, where the transmitter is energy harvesting and further transmission errors may occur due to energy outag es. W e co n sider two different types of channel cod es to e ncode the statu s u pdates. First, we consider maximum distanc e separ able (MDS) codes. W ith M DS cod ing, the tra n smitter enco des the k status update symbols into n symbols. The receiv er re ceiv es th e update successfully if it receives any k of these n encoded sym bols. Next, we con sider rateless co des, for examp le, fountain codes. In this case, the tr ansmitter encodes the k up date symb ols into as many symbols as nee d ed until k of these symbo ls are received successfu lly . For each o f these models, we c o nsider two d ifferent policies: best-effort and sa ve-and-tran smit. Best- effort and sa ve-and -transmit schemes were o riginally consid- ered in [4 1], in the co n text of achieving the capacity of the energy h a r vesting A WGN chan nel. I n the b est-effort scheme, in eac h slot, the transmitted symbo l may suf fer from two errors: ch annel er asure and energy o utage. In the sa ve-and - transmit scheme, the transmitter remains silent at th e beginning to save energy and to r educe the error s du e to energy outage. For all these cases, we derive the average Ao I . Throug h numerical results, we sho w that as the av erage rech arge rate decreases, MDS codes with save-and-transmit outperforms all the best-effort schemes. The gain beco mes significant for low values of av erage energy arr ivals. W e observe that rateless coding with sav e-an d-transmit outp erform s all other policies. I I . S Y S T E M M O D E L W e consider a single-u ser ch a nnel with a transmitter which has an infinite- sized battery , see Fig. 1. The energy arri vals are Bernoulli and i.i.d.: in slot i , a unit energy arriv es with probab ility p or n o energy arrives with probab ility 1 − p , i.e., P [ E i = 1] = 1 − P [ E i = 0] = p . T he tran smitter ob tains the measuremen ts (status updates), which are packets o f length k , which should be sent to the recei ver in a way to minim ize the av erag e AoI at the receiver . The total AoI up to time T is, ∆ T = Z T 0 ( t − u ( t )) dt (1) where u ( t ) is the time stamp o f the latest recei ved status update packet an d ∆( t ) = t − u ( t ) is the instantan e o us AoI. An example ev olution o f the AoI is shown in Fig. 2. The av erag e long- ter m AoI in this case is calculated as, ∆ = lim T →∞ ∆ T T = lim i →∞ P i j =1 Q j P i j =1 T j (2) In all the subsequen t analysis we will assume ren ew al po licies, i.e., where Q j and T j are i.i.d. The AoI then redu c es to, ∆ = lim i →∞ 1 i P i j =1 Q j 1 i P i j =1 T j = E [ Q ] E [ T ] (3) where we drop ped the subscript j as Q j and T j are i.i.d. The chan nel between the tra nsmitter a nd the rece iver is an i.i.d. er a sure chan nel. The p robability of sym bol era su re (loss) in each slot is δ . In or der to combat the ch annel erasures and the energy outag es, the transmitter encodes the status up dates before sending them throu gh the channel. W e consider tw o types of channel codes: MD S and rateless codes. W e first con sid e r MDS channel codes. For th is case we have an ( n, k ) channel codin g scheme, where k is the length of an unco ded status u pdate an d n is th e len g th of an en coded codeword which is sent th r ough the chann el with n ≥ k . When the transmitter is d one with sending the n sym b ols, it generate s a ne w update and begins sending it. This is irrespectiv e o f the success o f the transmission o f the se n symbols. The o ptimal value of n d epends on k , δ , and p . For MDS chann el coding, we study two achievable schemes. W e first study a sav e-an d- transmit scheme in which the tran smitter sa ves energy from the incoming energy arr i vals un til it has at least n un its o f energy in its battery . This in effect makes sure that errors which can occur during the codeword transmission are only du e to the erasures in the chann el. T o ensure tha t the syn chroniza tio n is maintained b etween the transmitter and th e rec e i ver, the transmitter remains in the saving ph ase fo r a nu mber o f slots which is multiple o f n . W e then study a best-ef for t sch e me, in which the transmitter attempts transmission in each slot. In this case, the erro r in each symbol can b e either du e to an energy outag e or a chan nel erasur e or bo th. W e n ext study the case of rateless coding in which the trans- mitter keeps send ing the up date until k symbols are suc cess- t ∆( t ) Q 3 Q 4 Q 1 T 1 T 2 T 3 T 4 Q 2 Fig. 2. An example for the evo lution of the age of informa tion. fully r e cei ved. For this case, we also study two schemes: best- effort and sa ve-and-tran sm it. In the best-effort schem e, once the update is successfully received, the tr ansmitter generates a new update and begins tr ansmitting it immediately . In the sa ve- and-tran smit schem e, once the u pdate is successfully rec ei ved, the tr ansmitter waits some time in or d er to sav e some energy in the b attery to prevent future ene rgy outages. The transmitter sa ves for m slots, where the o ptimal m should b e obtained as a function of the system parameter s δ , k , an d p . I I I . A O I U N D E R M D S C H A N N E L C O D I N G A. Save-an d-T r an smit P o licy In the save-and-transmit po licy , befo re the transmitter at- tempts to transmit th e cod ed update, the transmitter rem ains silent f or an integer multiple of n slots u ntil the batter y has energy at least eq ual to n . T he duration the transmitter remains silent for th e j th time while tr ansmitting the i th update is a random variable denoted by Z ij ∈ { n, 2 n, 3 n, . . . } which depends on the energy arri val distribution. The random variable Z ij can be expressed as: Z ij = W i n n (4) where W i is th e random variable which d enotes the numb er of slots needed to save n units o f en ergy and ⌈ x ⌉ denotes the smallest integer greater than o r eq ual to x . Since th e energy arriv als f ollow an i.i. d. Bernou lli d istribution, W i will follow a negativ e bino mial distribution as follows: P W i ( w ) = w − 1 n − 1 p n (1 − p ) w − n , w = n, n + 1 , . . . (5) The distribution of Z ij can be o btained u sing (5) as follows: P Z ij ( z ) = z X w = z − n +1 P W i ( w ) , z = n, 2 n, . . . (6) After the saving ph ase, the transmission resumes f or n slots. After the transmitter is done transmitting the n co ded symbo ls, the tr ansmitter aga in goes to the sa ving phase until it recharges its battery to at least n . T h e transmitter alterna tes be twe en sa vin g an d tr a nsmission p hases. The upd ate is successful if at least k symbols are rec ei ved without be in g erased; th e re will be no energy outa g e due to t ∆( t ) n Z i 1 Z i 2 ˜ X i T i Q i ˜ X i Fig. 3. An example for the ev olution of the age of information under the sav e-and-transmit sc heme for the MDS channel coding case. the sa ving phase. Hence , th e prob a b ility of having a succ e ss in a n slot of dur a tio n is, ǫ k,n ( δ ) = n X x = k x − 1 k − 1 (1 − δ ) k δ x − k (7) Thus, in the n consecutive slots the transmission is successfu l with pro bability ǫ k,n ( δ ) . Now , the update will be suc c e ssful in the V th transmission, wh ere V is a geo m etrically distributed random variable with a the following pm f, P V ( n ) ( v ) = ǫ k,n ( δ )(1 − ǫ k,n ( δ )) v − 1 , v = 1 , 2 , . . . (8) Hence, we may need to re peat the sa ve-and-tr a nsmit phases for V times befor e we have a successful status update. W e now characte r ize the random variable which identifies the instant at which the u pdate will be successful w ith in the n co nsecutive slots. W e den ote this ra n dom variable by ˜ X i which has a cond itional pmf P X i | X i ≤ n ( x ) where P X i ( x ) = x − 1 k − 1 (1 − δ ) k δ x − k , x = k , k + 1 , . . . (9) Hence, ˜ X i is distributed as: P ˜ X i ( x ) = x − 1 k − 1 (1 − δ ) k δ x − k ǫ k,n ( δ ) , x = k , k + 1 , . . . , n (10) An example which illustrates the AoI evolution is shown in Fig. 3. In this figur e , the transmitter at first waits 3 n slots in or d er to recharge the battery to at least the le vel n . It then attemp ts to transmit. The tra n smission in this case is not successful d ue to the channel erasures so the transmitter again waits for n slots in orde r to ch arge the battery . Th e tra n smission then proceeds again in the next slot. The transmission is then successful and the r e cei ver received the upda te after ˜ X i transmissions, wh ere k ≤ ˜ X i ≤ n . W e now con sider a renewal policy which serves as an up per bound for the sa ve-and-tra n smit policy d escribed above. W e assume that at the end of the up date period, the transmitter depletes all its batter y . Th u s, the tr ansmitter r enews its state at the en d of each succ e ssfu l u pdate a nd always begins with a depleted battery . In this case, the AoI can be wr itten as: ∆ M DS − S T = E [ Q i ] E [ T i ] (11) Next, we evaluate E [ Q i ] and E [ T i ] . W e first obtain Q i as, Q i = n n ( V i − 1) + ˜ X i + V i X j =1 Z ij + 1 2 n ( V i − 1) + ˜ X i + V i X j =1 Z ij 2 + n 2 2 − ˜ X 2 i 2 (12) = n 2 V i 2 2 + nV i ˜ X i + n V i X j =1 Z ij + h n ( V i − 1) + ˜ X i i V i X j =1 Z ij + 1 2 V i X j =1 Z ij 2 (13) W e th en o btain T i as, T i = nV i + V i X j =1 Z ij (14) Now , it rem a ins to calculate the exp e ctation of Q i and T i . W e first calculate the first and seco n d moments of P V i j =1 Z ij , using [42, Theorem 6.13] , as follows: E V i X j =1 Z ij = E [ Z ] ǫ k,n ( δ ) (15) Similarly , we have: E V i X j =1 Z ij 2 = E Z 2 ǫ k,n ( δ ) + 2 − 2 ǫ k,n ( δ ) ǫ 2 k,n ( δ ) E [ Z ] 2 (16) W e th en co mbine a ll these to obtain: E [ T i ] = n ǫ k,n ( δ ) + E [ Z ] ǫ k,n ( δ ) (17) and E [ Q i ] = n 2 (2 − ǫ k,n ( δ )) 2 ǫ 2 k,n ( δ ) + nµ ˜ X ǫ k,n ( δ ) + n (2 − ǫ k,n ( δ )) E [ Z ] ǫ 2 k,n ( δ ) + µ ˜ X E [ Z ] ǫ k,n ( δ ) + 1 2 E Z 2 ǫ k,n ( δ ) + (1 − ǫ k,n ( δ )) E [ Z ] 2 ǫ 2 k,n ( δ ) (18) where E [ Z ] and E Z 2 can be calculated using (6) and µ ˜ X can b e calcu lated using (10). Hence, the average AoI ∆ M DS − S T in (1 1) can be found b y substituting with the expressions in (1 7) and (1 8). B. Best-Effort P olicy W e n ow consider the case when the transmitter does not wait at the be gin ning in o rder to save energy , in stead it begins transmission im mediately . The e rror events in this case can b e either an erasure in the commu nication ch annel or a n energy outage at the tran sm itter . These two ev ents may occu r f or each transmitted symbol. Hence, fo r th e symbol to be received without an error, there should be no energy o utage and no channel erasure ; th is form s a Bernoulli rand o m variable with t ∆( t ) Q i m Z i |{z} Y i + Z i Y i − 1 + Z i − 1 update generated Y i |{z} no energy outage best-effort Fig. 4. An example for the ev olution of the age of information under the sav e-and-transmit sc heme for the rate less channel coding case. probab ility of success equal to q , p (1 − δ ) . The ev olution of AoI is similar to Fig. 3 but in this case, Z ij is eq ual to ze r o as the transmitter does not wait to sa ve energy . Using analy sis similar to the pr evious scheme, b ut with having the pro bability o f success equal to q , the a verage AoI in this case can be written as: ∆ M DS − B E = n ǫ n − n 2 + k ǫ k +1 ,n +1 ( q ) q ǫ k,n ( q ) (19) This can also be obtained using the same analysis as in [35], but with pro b ability of success equ al to q , I V . A O I U N D E R R A T E L E S S C H A N N E L C O D I N G A. Best-Effort P olicy W e consider her e the case wh en the transmitter begins to transmit im mediately . In each slot, the tr ansmitter su ffers two possible error e vents. The first is ch annel erasure and the second is energy o utage. Hence, a sym bol will be re- ceiv ed successfully if neither er ror occur s, which h appens with probab ility equ al to q . Th e channel is now equ iv alent to an erasure channel, similar to the on e co nsidered in [35], but with probab ility of success equal to q . Following analysis similar to th e on e in [35], but with pro bability of su c cess eq ual to q , the average AoI in this case is equal to: ∆ RC − B E = k q 3 2 + 1 − q k (20) B. Save-an d-T r an smit P olicy In this po licy , we consider the case wh en the transmitter does not generate a new update immediately o n ce the tran s- mission of the previous upd ate is succ essful, but it waits for a deterministic tim e of m slots. Here, m is a deterministic number which b oth the tra n smitter and the re c ei ver know in advance; this m should then b e optimized to m in imize the av erag e AoI and will be a fun ction of δ , p and k . The transmission in this policy proceeds as follows: o nce the previous up date is successfu l, the tran smitter b egins a saving phase of d u ration m slots. Then, the transmitter generates a new update an d begins transmittin g it to the recei ver . While Y i m E 1 E 2 E 1 num ber of time duration energy arriv als E 2 E 3 Fig. 5. An example to i llustrate the rand om v ariable Y i . transmitting the u pdate, th e tran sm itter may receive more en- ergy arriv als; howev er, the amount of energy in th e battery will always be non-in creasing as the transmitter tran sm its a symbol in e a ch slot while the energy may not arr i ve at e very slot. The transmitter keeps tran smitting the update un til its batter y state hits zer o; this dec lares the end of th e n o-outage phase. W e denote the n umber of symbols sent su ccessfully in this phase by k i . If k i ≥ k , then n o more tr ansmission is r e quired and the update is successful. Otherwise, the tran smitter transmits the r e m aining k − k i using the best-effort scheme d escribed in Su bsection IV -A. W e denote the duration the tr ansmitter transmits with n o outage by Y i and we denote the duratio n we transmit using the best-ef fort scheme by Z i . An example for the ev olution of the A o I in this c ase is sh own in Fig. 4. The a verage AoI can be calculate as follows, ∆ RC − S T = E [ Q i ] m + E [ Y i + Z i ] (21) = E h ( m + Y i + Z i ) 2 + 2 ( m + Y i + Z i ) ( Y i − 1 + Z i − 1 ) i 2 m + 2 E [ Y i + Z i ] (22) This AoI can be ca lc u lated explicitly once E [ Y i ] , E [ Y 2 i ] , E [ Z i ] , E [ Z 2 i ] and E [ Y i Z i ] are calculated . W e note that Y i and Z i are depend ent on each other wh ile Y i and Y i − 1 are indep endent due to u sing a renewal policy . W e now define the rando m variables { E i } ∞ i =1 ; th e random variable E 1 represents th e amount of energy harvested in the first m slo ts. For i ≥ 2 , th e ra n dom variable E i represents the amount of energy harvested durin g th e p revious E i − 1 slots. Hence, we have E i ≤ E i − 1 . W e n ow characterize the rando m variable Y i , Y i = ∞ X i =1 E i (23) where E 1 is Bin ( m, p ) , and for i ≥ 2 , E i giv en E i − 1 = e i − 1 is Bin ( e i − 1 , p ) ; Bin ( . ) den otes binomial distrib ution. An example for the ev olution of Y i is shown in Fig. 5. W e can obtain the marginal pm f for th e random variables E i , i ≥ 2 , by applying [42, Th eorem 6 .12] and using [42, T able 6.1]. E ach E i consists o f a sum of i.i.d. Bern oulli random variables an d the number of these ran dom variables is distributed according to a binomial d istribution of E i − 1 which is indep endent o f the Bern oulli rand om variables. Hence, th e marginal pm f of the ran d om variable E i is Bin( m , p i ). W e can now calculate E [ Y i ] as, E [ Y i ] = ∞ X i =1 E [ E i ] = mp 1 − p (24) Next, we want to ca lculate E [ Y 2 i ] which we calculate as E [ Y 2 i ] = var ( Y i ) + E [ Y i ] 2 . The term var ( Y i ) can be ca lcu lated as follows var ( Y i ) = ∞ X i =1 var ( E i ) + 2 ∞ X i j . T o calculate the covariance, we fir st calculate the condition al probab ility P ( E j +1 | E i ) . For j > i , we have that P ( E j | E i ) is distributed as Bin( E i , p j − i ). This again f o llows b y applying [42, Theor em 6.12] and using [42, T able 6.1]. W e now calculate for j > i cov ( E j , E i ) as follows: cov ( E j , E i ) = E [ E j E i ] − E [ E j ] E [ E i ] = mp j (1 − p i ) ( 2 7) Next, we calcu late P i
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