On the Delay Performance of In-network Aggregation in Lossy Wireless Sensor Networks

Transcription

1 O the Delay Performace of I-etwork Aggregatio i Lossy Wireless Sesor Networks Chaghee Joo, Member, IEEE, ad Ness B. Shroff, Fellow, IEEE Abstract I this paper, we study the implicatio of wireless broadcast for data aggregatio i lossy wireless sesor etworks. Each sesor ode geerates iformatio by sesig its physical eviromet ad trasmits the data to a special ode called the sik, via multi-hop commuicatios. The goal of the etwork system is to compute a fuctio at the sik from the iformatio gathered by spatially distributed sesor odes. I the course of collectig iformatio, i-etwork computatio at itermediate forwardig odes ca substatially icrease etwork efficiecy by reducig the umber of trasmissios. O the other had, it also icreases the amout of the iformatio cotaied i a sigle packet ad makes the system vulerable to packet loss. Istead of retrasmittig lost packets, which icurs additioal delay, we develop a wireless system architecture that exploits the diversity of the wireless medium for reliable operatios. To elaborate, we show that for a class of aggregatio fuctios, wireless broadcastig is a effective strategy to improve delay performace while satisfyig reliability costrait. We provide scalig law results o the performace improvemet of our solutio over uicast architecture with retrasmissios. Iterestigly, the improvemet depeds o the trasmissio rage as well as the reliability costrait. Idex Terms Data aggregatio, lossy wireless etworks, delay performace. I. INTRODUCTION Wireless sesor etworks cosist of a large umber of sesor odes with limited resources of eergy, trasmissio power, etwork badwidth, ad computatio power. Each sesor ode moitors the physical eviromet i its eighborhood, collects data, ad processes iformatio. I may applicatios, the goal of wireless sesor etworks is to compute a global fuctio of the iformatio gathered by spatially distributed sesors at a special ode called the sik. Multi-hop commuicatio is ofte used to relay the iformatio from the source ode to the siks. Distributed i-etwork computatio (or aggregatio [] ca improve the commuicatio efficiecy of the system. It allows for a itermediate ode to participate i the computatio of the global fuctio: a sesor ode ca collect iformatio from a subset of sesors ad aggregate it by performig computatios with partial iformatio. Compared with previous edto-ed iformatio delivery paradigms, i which itermediate Mauscript received XXXX; revised XXX. This work was supported i part by NSF Awards CNS-0700 ad CNS , ad by the Army Research Office MURI awards W9NF (SA08-03, ad W9NF , ad i part by the Basic Sciece Research Program through the Natioal Research Foudatio of Korea (NRF, fuded by the Miistry of Educatio, Sciece, ad Techology (No Earlier versio of this work has bee preseted at the 46-th Aual Allerto Coferece o Commuicatios, Cotrol, ad Computig, Moticello, IL, USA, 008. Chaghee Joo is with the School of Electrical ad Computer Egieerig, UNIST, Ulsa , Korea ( cjoo@uist.ac.kr Ness B. Shroff is with the Depts. of ECE ad CSE, The Ohio State Uiversity, Columbus, OH 430 USA ( shroff@ece.osu.edu odes simply relay the received iformatio without chage, distributed i-etwork computatio ca result i sigificat performace improvemets i eergy cosumptio, memory usage, badwidth, ad delay. I this paper, we focus o the delay performace of ietwork aggregatio i lossy wireless etworks. Uder a oisy wireless chael, maitaiig the overall reliability of the fuctio computatio while performig distributed computatios at itermediate odes is a major challege [] [4]. Sice the iformatio cotaied i a sigle packet is highly itesified after several i-etwork computatios, a packet loss ca sigificatly impact the computatio result, ad thus a higher level of protectio is required for each packet trasmissio. A packet ca be protected by Error Correctig Code (EEC [5] or ca be restored by retrasmittig the lost packet. I either case, additioal delay is uavoidable. I may applicatios, it is importat to compute the global fuctio i a timely ad reliable maer, ad thus limitig the amout of additioal delay is importat. To this ed, we develop a ew etwork architecture for i-etwork computatio for a class of geeralized maximum fuctios. We focus o the delay performace of the fuctio computatio subject to reliability costrait i lossy wireless eviromets. We show that aggregatio with wireless broadcast ca substatially reduce the delay while satisfyig the reliability costrait. Our scalig law result clarifies the relatioship betwee delay performace, reliability, ad trasmissio rage. We also provide distributed algorithms ad evaluate their performace through simulatios. I-etwork aggregatio has also bee studied i may other aspects []. The maximum achievable computatio rate for a class of fuctios has bee ivestigated i [6] [8]. Eergy efficiecy i lossy eviromets has bee cosidered i [3], [4], [9]. Time ad eergy complexity of distributed computatio has bee provided i [0], []. Our work ca be differetiated from the previous work i that i we focus o the delay performace of i-etwork computatio, ii we cosider reliability costraits i lossy wireless etworks, ad iii we ivestigate the effect of wireless broadcast o the delay performace. The paper is orgaized as follows. We first describe the system model i Sectio II. We provide scalig law results of asymptotic delay bouds uder differet reliability costraits ad trasmissio rages i Sectio III. I Sectio IV, we develop distributed algorithms to implemet i-etwork aggregatio that exploits wireless broadcastig i the presece of iterferece. I Sectio V, we evaluate our schemes through simulatios. Fially, we coclude our paper i Sectio VI.

2 II. SYSTEM MODEL We cosider a sesor etwork G(V, E havig a set V of sesor odes ad a set E of liks, i which sesor odes are deployed. The goal is to compute a global fuctio with iformatio obtaied from each sesor ode. We assume that the fuctio should be calculated at a special ode, called the sik. Each sesor ode ot oly geerates its ow data but also relays others data to the sik via multi-hop wireless commuicatios. The wireless chael is assumed to be lossy. A packet loss ca be restored by retrasmittig the lost packet, which, however, results i additioal delay. Sice may applicatios have both reliability ad delay costraits, we focus o the relatioship betwee the reliability ad the delay performace ad show how they ca improve whe ietwork aggregatio is appropriately employed i the wireless system. We are iterested i a class of fuctios that satisfy all the followig three properties. Symmetric: A fuctio f is symmetric if f( x, y = f( y, x. Decomposable: A fuctio f is decomposable if f( x, y = f(f( x,f( y. Compoetwise Trasitive: A fuctio f is compoetwise trasitive if [f( x, y] i = [f( x] i ad [f( y, z] i = [f( y] i imply that [f( x, z] i = [f( x] i, where [ ] i deote the i-th elemet of the vector. We deote this class of fuctios by Geeralized Maximum (GM fuctios, sice the fial result correspods to a elemet (could be a vector elemet of the sesed data. Some examples iclude max or mi, ragig (i.e., [mi, max], ad -largest (or smallest values. May sesor etwork services ca be realized through this class of fuctios: itrusio detectio by collectig biary iformatio, object trackig by collectig -closest locatios to the object ad their distaces, ad multi-modal evirometal moitorig (e.g., fidig highest temperature with humidity exceedig a certai threshold [], [3]. Also, i geeral wireless etworks, this type of fuctios might eed to be calculated frequetly to update system parameters such as the largest ode degree, the logest queue legth, the worst lik quality, etc [4], [5]. The properties of the GM fuctios promote i-etwork aggregatio. Specifically, a itermediate ode ca collect iformatio from other sesor odes, ad istead of directly relayig the received packets, it processes ad aggregates them ito a uit of iformatio, i.e., a packet. It the forwards the computed value to the sik or to the ext hop. Appropriate use of i-etwork aggregatio ca sigificatly reduce the amout of traffic geerated over the etwork [3], [4], [6]. Aother importat feature of the GM fuctios is that they allow duplicate data, i.e., isertig aother copy of data does ot affect the fuctio results. We actively exploit this feature to battle agaist lossy wireless chaels. Our model is based o the followig assumptios. Assumptio. The iformatio geerated at each sesor ode is exact without error. Assumptio. The message passig computatio model [] is assumed, i.e., all the iformatio has to be explicitly trasmitted ad silece periods (icludig listeig to others activities caot be used to covey iformatio. Hece, if a sesor ode does ot trasmit a packet, its iformatio caot cotribute to the global fuctio computatio. Assumptio 3. Time is slotted (each slot is equal to a samplig period ad all sesor odes are assumed to be sychroized. Schedulig is perfect i TDMA etwork systems. Assumptio 4. Routig is fixed. We first cosider a tree topology rooted at the sik, which is a popular structure i wireless sesor etworks because iformatio heads for a fusio ceter (sik. We also modify the topology later to icorporate wireless broadcast. Assumptio 5. The wireless chael betwee each pair of trasmittig ad receivig odes is assumed to be idepedet across liks ad times, ad modeled as a biary chael with o-zero packet loss probability p. To avoid trivialities, We assume that p is bouded as 0 < p p p <. At time slot t, each sesor ode ν geerates iformatio β ν by sesig its physical eviromets. Our objective is to calculate a GM fuctio value f(β,β,...,β at the sik, that coveys the aggregated iformatio from the sesor odes i a timely ad reliable maer. Let β deote the correct fuctio value that has to be reported, i.e., β := f(β,...,β. The iformatio value of β j is said to be critical if the fuctio result without β j is differet from β, i.e, f(β,...,β j,β j+,...,β β. For istace, let f( = mi{ } ad β = 5, β =, ad β 3 = 9. The β has the critical iformatio value because f(β,β 3 = 5 = f(β,β,β 3. Note that if the iformatio is represeted by a vector with m compoets, there ca be at most m critical iformatio values, sice the compoet-wise trasitive property implies that a sigle critical iformatio determies a compoet of the fuctio result. I the sequel, for easy of expositio, we assume that m =, ad f( is a max fuctio with a sigle elemet. However, sice the three properties allow the critical iformatio to be duplicated ad to be aggregated i ay order ad i ay itermediate ode, our aalysis ca be easily exteded to ay GM fuctio with m >, ad as log as m is a costat, our asymptotical results do ot chage. Let ˆβ deote the iformatio of the critical value. Sice β = ˆβ for m =, we use β ad ˆβ iterchageably i the remaider of the paper. III. ASYMPTOTIC ANALYSIS OF THE DELAY BOUND Let P s deote the miimum probability that the sik computes the fuctio correctly. We study the asymptotic delay performace of the sesor system for the followig reliability costrait: ( P s = O c(, ( i.e., there exists 0,c 0 > 0 such that for all > 0, P s c 0 c(, wherec( is a icreasig fuctio ofwithc( as, idicatig the speed of covergece rate at which reliability is achieved as a fuctio of the umber of odes. A. Aggregatio with uicast We first cosider a poit-to-poit commuicatio system with a tree topology [], where a ode has a paret ad

3 multiple childre (except the root ode ad leaf odes. Each ode obtais iformatio i two ways: from its ow sesor ad from its childre. Oce a ode collects iformatio from all its childre, it aggregates the iformatio icludig its ow ito a sigle packet usig the GM fuctio ad trasmits the packet to its paret over a poit-to-poit commuicatio lik (uicast. The procedure repeats from leaf odes to the root. We call this etwork architecture as aggregatio with uicast ad deote it by U. Sice routes follow the tree structure rooted at the sik, each ode ν has a uique paret µ(ν. Let p deote the probability of loss for trasmissio over lik (ν, µ(ν. Let r u ( 0 deote the maximum umber of retrasmissios allowed at each lik, ad let P s (ν deote the probability of success over lik (ν, µ(ν whe success occurs by takig the maximum umber of allowed retrasmissios. We ca obtai P s (ν as P s (ν = Pr {all trasmissios fail} = p +ru(. We defie the depth d(ν as the umber of hops betwee ode ν ad the sik. Let d ( deote the maximum depth over all sesor odes, i.e., d ( := max ν V d(ν. The, P s, the worst-case probability of success, is give by the success probability that the iformatio of the critical value arrives at the sik through the logest path. Lettig ˆν deote the ode that geerates the iformatio ˆβ of the critical value, we have d(ˆν P s = mi P s (ν k = ˆν=ν;ν V k= d ( k= ( P s (ν k, (3 where ν := ˆν ad ν k+ := µ(ν k for all k >. The last equality holds because i the worst-case, ˆν has the largest depth d (. By substitutig ( ito (3, we ca obtai the followig iequality: c d ( p +ru( P s c d ( p +ru(, (4 where c ad c are some costat. From the left side of (4 ad from the reliability costrait of (, we have +r u ( c 3 log(d ( c(+c 4, for some costat c 3 ad c 4. Also, from the right side of (4, the iequality with some costat is sufficiet for the reliability costrait. Hece, a scheme that satisfies the reliability costrait ( should have r u ( Θ(log(d ( c(, (5 ad the boud is tight i the sese that some scheme with the equality ca satisfy the reliability requiremet. We ow cosider the delay caused by retrasmissios to achieve the give reliability costrait. Estimatig the delay by the umber of trasmissios, the worst-case delay D u ca be preseted asd u = mi r u({d ( (+r u (}. From (5, we havemi(+r u ( = Θ(log(d ( c(, which implies that a packet should be trasmitted at leastθ(log(d ( c( times to satisfy the reliability costrait. Hece, we obtai the worst-case delay uder aggregatio with uicast as D u = Θ(d ( log(d ( c(. (6 B. Aggregatio with wireless broadcast I this sectio, we propose a ew etwork architecture with wireless broadcast to improve the delay performace while achievig the same level of reliability. We explicitly exploit diversity from wireless broadcast. We first describe the system architecture ad the aalyze its delay performace. We modify the tree structure i Sectio III-A by allowig odes to broadcast a packet to multiple parets. Assumptio 4.. Each ode (at depth d has at least x( parets (at depth d, ad trasmits a packet through the wireless broadcast chael to all parets (+r b ( times. At the root, we assume that the sik has x( ateas ad it ca process sigals from multiple ateas. I this architecture, we say that a ode successfully trasmits a packet if the broadcasted packet is successfully received by oe of x( parets. Note that each packet cotais aggregated iformatio abstractig all the iformatio successfully collected by the trasmitter. Due to the properties of the GM fuctios, it is sufficiet that each ode successfully trasmits the aggregated iformatio to oe of its parets i order to esure that the iformatio ˆβ of the critical value is successfully delivered to the sik. We call this architecture as aggregatio with broadcast ad deote it by B. The ituitio ca be better described usig Fig.. Assumig that liks are bidirectioal, the dotted lies i the figure is a lik betwee two odes, ad arrows idicate a trasmissio from a child to a paret. A failed trasmissio is marked by a cross. Fig. (a illustrates that two trasmissios from ode fail uder U. O the other had, Fig. (b shows that a sigle broadcast ca trasmit the packet to ode 6 successfully. Hece, U requires four trasmissios to deliver iformatio B to the sik, whereas B eeds two trasmissios. Note that the aggregatio with broadcast B appears to be a little like floodig, but there are sigificat differeces. While floodig is very ieffective because of broadcastig multiple duplicate packets, B reduces this iefficiecy by i-etwork aggregatio. Moreover, it takes advatage of the diversity of wireless broadcast, which is ot exploited i floodig. We ow estimate the worst-case probabilityp s of successful fuctio computatio uder B. Assumig idepedet packet losses over liks (Assumptio 5, a packet trasmissio from ode ν is successful with probability P s (ν p x( (+r b(, (7 where r b ( is the maximum umber of retrasmissios. Agai, sice the iformatio ˆβ of the critical value has to be delivered via at most d ( hops to reach the sik, the guarateed probability P s of a successful iformatio delivery ca be represeted by d(ˆν P s mi P s (ν k = ˆν=ν;ν V k= d ( k= P s (ν k, (8 This implies that there are at least x( disjoit paths from a ode to the sik. Sice there are x( differet first-hop odes from the ode s parets ad each of these first-hop odes has x( parets, we ca fid at least x( disjoit two-hop paths. The by iductio, we ca show that there are at least x( disjoit paths to the sik. 3

4 BCD (a Aggregatio with uicast (U (b Aggregatio with wireless broadcast (B Fig.. Trasmissios over lossy wireless liks. Each trasmissio is deoted by a arrow, ad a failed trasmissio is deoted by a cross at the ed of the arrow. Uder aggregatio with uicast, it eeds four trasmissios for iformatio B to be successfully delivered to the sik, while it eeds two trasmissios uder aggregatio with broadcast. where ν := ˆν ad ν k+ is oe of parets of ν k. Note that ulike U, the first equality i (3 is chaged to a iequality i (8 because the iformatio ˆβ ca take multiple (at least x( paths to the sik. From (7, we ca obtai P s d ( p x((+r b(. (9 The forc 5 := logp ad a costatc 6, the followig iequality suffices to satisfy the reliability costrait (: x( (+r b ( c 5 log(d ( c(+c 6. (0 Note that if each ode broadcasts its packet to c 5 log(d ( c( parets, the reliability costrait ( ca be satisfied with r b ( = 0. Sice the delay boud Db ca be represeted as Db = mi r b ({d ( (+r b (}, we have ( Db Θ d ( max{, log(d ( c( x( }. ( C. Performace i geometric etworks We ow cosider a popular sceario i which sesor odes are radomly deployed i a geometric space, ad evaluate the delay performace of aggregatio schemes with uicast U ad with broadcast B. We derive the gai of B over U for geometric etworks, where the reliability costrait ad trasmissio rage are a fuctio of the umber of odes. We show that i geeral a higher gai ca be achieved with a stroger reliability requiremet ad a larger trasmissio rage. We first start with the gai for the previous (o-geometric tree etwork. We defie the maximum delay gai of B over U as G := Du/D b. From (6 ad (, we have G := D u D b = Ω d ( log(d ( c( d ( max{, log(d ( c( ( x(log(d ( c( = Ω x(+log(d. ( c( x( } ( Suppose that the etwork has depth d ( = with the reliability costrait c( =. From (6 ad (, f( = Ω(g( meas that there exists costats, c > 0 such that for all, f( cg(. we have Du = Θ( log uder U, ad D b Θ( max{,log/x(} = Θ( uder B whe x( = Θ(log ad r b ( = O(. Hece, if each ode ca broadcast to Θ(log parets, B outperforms U by G = Ω(log. However, the achievability of x( = Θ(log log depeds o the topology of the uderlyig etwork. I geometric etworks, both d ( ad x( are related to the topological structure ad we eed to icorporate some topological otio ito our aalysis. To this ed, we study the delay performace of aggregatio schemes i radom etworks, where odes are uiformly placed, subject to reliability costrait. I our aalysis, we do ot take ito accout edge effects, assumig that all odes have the same order of paret odes 3. Note that i sesor etworks, most traffic heads for the sik. Hece, by carefully locatig the sik, there would be few trasmissios o the edge of the etwork. The assumptio ca be further supported by our developmet of a distributed algorithm i Sectio IV. Assumptio 4.. Give a etwork of sesor odes uiformly ad idepedetly distributed o a disk of radius, each ode has a idetical trasmissio rage ad has the same order of parets x( The sik is located at the ceter with x( ateas. Straight-lie routig has bee employed, thus achievig d ( =, ad all the paths from a ode to the sik have asymptotically the same legth. I the ext sectio, we show that this ca be achieved by a simple routig scheme. Uder aggregatio with uicast U, the delay boud directly comes from (6. By replacig d ( with, we have ( Du = Θ c( log. (3 O the other had, uder aggregatio with broadcast B, we have x( Θ( because each ode has eighborig odes i its trasmissio rage. We ca achieve the equality by settig the parets of each ode to the set of 3 If the odes are uiformly distributed i space, they asymptotically have the same order of eighbors [6]. The, as show i Sectio IV, it is ot hard to develop a scheme, uder which each ode asymptotically have the same order of parets. 4

5 odes withi a sector of its trasmissio rage (to the directio of the sik. The, from ( ad d ( =, we ca obtai the delay boud as { } Db (max Θ, c( log. (4 3 From (3 ad (4, we ca preset the gai G geo of B over U i geometric etworks as log c( G geo := D u D b = Ω +log c(. (5 As a example, we cosider a radom geometric etwork with miimal coectivity. It has bee show i [7] that should be at least Θ( for the etwork to be asymptotically coected with high probability. Usig = ( Θ(, we have d ( = Θ. Suppose that c( = log, i.e., the reliability requiremet eforces that P s = O(. I this case, the delay boud of U ca be writte as Du = Θ( from ( (3. ( For B, it suffices to satisfy x(( + r b ( = Θ log c( to achieve the same level of reliability. Sice each ode ca have x( = Θ( = Θ( parets, the coditio ca be satisfied with r b ( = O( whe c( =. Further, if x( c 5, there is o eed of retrasmissio( uder B. Hece, we ca obtai the delay boud Db = O ad the gai G geo = Ω(. Note that = is the umber of odes i the trasmissio area of a ode. This implies that B ca potetially achieve a gai i delay as large as the diversity gai of wireless broadcast. I geeral, from (5, the gai depeds o both ad c(. We tabulate the gais for various etwork eviromets i Table I. The first colum shows that the gai is domiated by the broadcastig areas i multi-hop etworks with miimal coectivity. The last colum shows that the gai is domiated by the reliability costrait i sigle-hop etworks. The results also show that we ca improve the delay performace by exploitig multicast trasmissios, with a smaller trasmissio rage whe a low level of reliability is required (i.e., whe c(, ad with a larger trasmissio rage whe a high level of reliability is required (i.e., whe c(. TABLE I GAINS (Du/D b OFB OVERU UNDER VARIOUS TRANSMISSION RANGES AND RELIABILITY CONSTRAINTS. = = = c( = Ω( Ω(log Ω(log c( = Ω(log Ω(log Ω(log c( = exp Ω( Ω( Ω( IV. DISTRIBUTED ALGORITHMS I this sectio, we develop a practical solutio for aggregatio with broadcast usig a tiered routig structure. Although the tiered structure has appeared i the literature for lightweight routig [8] ad efficiet sleep/wake schedulig [9], [0], the purpose of our desig is quite differet. Ulike [8] [0], we assume that wireless liks are lossy, ad that the etwork has a specific goal of computig a GM fuctio. By exploitig the diversity of the wireless medium, we ited to improve the delay performace while satisfyig reliability costrait. We first describe our solutio, ad show that the algorithm achieves the delay performace of (4. To this ed, we show that uder the algorithm, each sesor ode has at least Θ( parets ad the maximum hop distace to the sik is at most Θ(. We exted our schemes to resource costraied etworks, ad revisit performace aalysis i the presece of wireless iterferece. We close this sectio with developmet of a hybrid scheme that ca combie the uicast ad the broadcast architecture. A. Algorithm with tiered structure We assume that wireless sesor odes are uiformly deployed over a disk of radius. Our results ca be exteded to more geeral etworks of differet sizes ad topologies, which impact o our aalysis by a costat factor ad do ot affect our scalig results. Uder Assumptio 4., each ode has a idetical trasmissio rage of, ad we divide the etworks ito δ circular tiers as show i Fig., with 0 < δ <. Each tier has a idetical width of δ. Let T i deote the set of odes i the i-th tier, which is a area withi distace of (δ (i,δ i] from the sik. The sik is the oly ode i T 0. The etwork is a time-slotted TDMA system. At the begiig of each time slot, each sesor ode geerates a packet with the sesed iformatio. A time slot is further divided ito mii-slots ad i each mii-slot, a sigle packet ca be trasmitted. Let D b deote the delay performace of the algorithm, which is estimated i the umber of trasmissios, i.e., mii-slots for the sik to compute the fuctio. Routig is simplified usig the tiered structure; Every ode µ i T i is a paret of odeν it i+ if its distace is o greater tha. Trasmissios are scheduled from the outermost tier to the sik tier-by-tier oe at a time, so that odes i T i ca trasmit oly after all odes i T i+ fiish their trasmissios. We group odes i each tier ito mutually exclusive subsets such that all odes i a subset ca trasmit simultaeously. Let H(i,j deote the j-th subset i T i, ad let h i deote the total umber of subsets i each tier T i such that hi j= H(i,j = T i. Clearly, all odes i T i ca fiish a sigle trasmissio i h i mii-slots. If there is o iterferece betwee simultaeous trasmissios withi a tier, we will have a sigle group with h i = for all tier i. The reaso that we itroduce groupig will become clearer i the ext sectio whe we take ito accout wireless iterferece. We will have the total delay D b to compute the fuctio as D b = /δ i= h i (+r b (. (6 5

6 i Fig. 3. Parets (i T i of ode ν (i T i is located i the shaded area. Fig.. Network with tiered structure. Algorithm Distributed aggregatio with wireless broadcast. for i = δ to do for j = to h i do Each ode ν i H(i, j broadcasts its (aggregated iformatio (+r b ( times. if ode µ T i receives the packet the Node µ does aggregatio ad updates its iformatio. ed if ed for ed for The overall algorithm proceeds as follows: At the begiig of every time slot, each ode origiates a packet with sesed iformatio. Nodes i H(i, j broadcast their packets ( + r b ( times i decreasig order of i ad icreasig order of j, such as H( δ,,h( δ,,...,h( δ,h,h( δ δ,,h( δ,,...,h(,h. Note that with this orderig, odes i T i start their trasmissios after all odes i T i+ fiish trasmissios. The odes i T i who receive a packet from a ode i T i+ do aggregatio usig the GM fuctio, ad update their packet if ecessary. The detailed algorithm is as show i Algorithm. Now we show that uder Algorithm, each ode has at least Θ( parets ad the maximum hop distace from a ode to the sik is Θ(. The the miimum umber of parets x( ca be bouded as follows. Suppose that ode ν is located i T i as show i Fig. 3. The umber of parets of ode ν i T i is o smaller tha the umber of odes i the shaded area. For each ode ν V, there exists δ < δ ν < such that the distace betweeν ad the shaded area isδ ν. Let δ := max ν V δ ν. Sice odes are uiformly distributed with desity π, it ca be easily show that the umber of odes i the shaded area is bouded below by 4 ( cos δ x( δ δ π = Θ(. π (7 4 Although we implicitly assume that the shaded area is completely icluded i T i, the same order results ca be obtaied whe the shaded area stretches to ier tiers. Sice each ode has at least Θ( parets, Algorithm ca achieve the required reliability ( by satisfyig (0 with some r b ( = Θ(. Further, sice each tier has the width δ ad a packet is trasmitted tier-by-tier, there are at most tiers ad we δ have the maximum umber of hops to the sik as d ( = ( δ = Θ. (8 Hece, from (7 ad (8, Algorithm achieves the delay performace (4 ad the gai (5 with some r b ( = Θ(. However, this ca be achieved oly whe there is o iterferece betwee simultaeous trasmissios ad all the odes i each tier i belog to the same group with h i =. B. Performace i the presece of iterferece I Sectio III, we have aalyzed the performace (e.g., (5 ad Table I without cosiderig wireless iterferece. However, if the etwork is resource-costraied ad has limited frequecy chaels, the wireless iterferece will restrict the umber of simultaeous trasmissios, e.g.,h i of Algorithm, ad this has to be factored ito calculatio of the gais. Assumptio 6. We cosider a protocol model for the iterferece costraits [6], where two liks withi two times of trasmissio rage caot trasmit simultaeously. Multiple odes withi a tier ca trasmit simultaeously if the distace betwee ay two of them is greater tha. We show that h i = Θ(, ad obtai the delay performace of Algorithm i the presece of wireless iterferece. We first aalyze the delay performace of Algorithm by providig a algorithm that multiple odes i a tier ca be scheduled without iterferece. The we compare the solutio with a realizatio of U, which appears i [] with = i a lossless etwork, ad exteded accordigly. We evaluate their performace ad clarify the improvemet of B over U i differet etwork settigs. From (6, we eed to estimate h i to obtai D b, which is determied by the schedulig policy withi a tier. To this ed, we first estimate H(i,j, whereh(i,j is the subset of odes i T i that are scheduled simultaeously, ad deotes the cardiality of the set. We partitio T i ito subsets {Ci m} as show i Fig. 4. Note that sice each cell Ci m has a width more tha at the boudary of the ier tier, there are at 6

7 C i m- Fig. 4. Partitio {C m i } of T i. C i m- d C i m C i m+ most π(i δ cells, where a is the closest iteger o smaller tha a. Let H(i,j iclude a ode from every three cells, i.e.,h(i,j has a ode from cellsci m,cm+3 i,..., so o. Sice ay two odes i H(i, j are separated more tha, they do ot have ay commo paret ad their trasmissios do ot iterfere with each other. Moreover, sice all cells has the same umber of odes (possibly except oe cell, which may have a smaller umber of odes, the umber of odes i each H(i,j is idetical ad would be about a third of the umber of Ci m. Specifically, H(i,j = 3 π(i δ = 3 π(i δ, (9 for all i >, where a is the closest iteger o greater tha a. The umber of odes i T i ca be bouded by π(i δ δ π T i πiδ δ π, (0 for all i >. Sice odes are uiformly distributed, we ca obtai from (9 ad (0 the umber of mii-slots h i eeded for all odes i T i to make a sigle trasmissio as ( Ti h i = Θ = Θ(, H(i, j for all i >. For i =, it is clear h = Θ( because T = δ. Hece, we have h i = Θ( for all i. From (0 ad (6, we obtai D b = /δ i= h i (+r b ( ( = Θ δ (+r b ( ( ( = Θ + c( log. ( Remarks: Ituitively, each tier has width Θ( ad thus icludes Θ( odes. Sice we ca schedule a set of odes, where distace betwee ay two odes is o smaller tha, the umber of scheduled odes will be at most Θ(/. Hece, it takes at least Θ( mii-slots to fiish trasmissios i each tier. We ca obtai the above equatio by multiplyig the term, which explais the wireless iterferece, to (4. Now we cosider the realizatio of U preseted i []. The algorithm is desiged i lossless etworks with miimal trasmissio rage for coectivity, i.e., =, ad show to be optimal. We exted it ito lossy etworks with geeral trasmissio rage as follows: Amog sesor odes, there are Θ( odes who locally collect iformatio from its eighbors ad do aggregatio. They ca be placed such that they form a tree with depth Θ( ad odes of the same depth do ot iterfere with each other. At the begiig of each time slot, each ode trasmits its packet over a poit-to-poit commuicatio lik to the earest collectig ode. Due to retrasmissios for lost packets, it takes Θ( (+r u ( times. 3 After the above procedure, all iformatio is ow located i collectig odes. The, each collectig ode, startig from leaf ode, trasmits packet to its immediate paret up to ( + r u ( times. After receivig all packets from childre, each collectig ode does aggregatio ad trasmits the data to its parets. This procedure takes Θ( (+r u( times util all iformatio arrives at the sik. From the above ad (5, the algorithm has the delay performace (( D u = Θ = Θ + (( + (+r u ( log c(. ( Note that the wireless iterferece is icorporated i the first term. Ulike Algorithm, it is added to (3 istead of beig multiplied. This is because the iterferece matters oly whe odes trasmit packets to collectig odes. O the other had, i Algorithm, the iterferece remais through the procedure because it cotiuously exploits the wireless broadcast. From ( ad (, the gai ca be obtaied as G = Θ (+3 log c( +log c( Table II summarizes the gai of Algorithm over the istace of U i the presece of wireless iterferece for various etwork settigs. Algorithm outperforms the istace of U i most cases. However, i some cases, e.g., c( = log ad =, the istace of U has better delay performace tha Algorithm. Such a case occurs whe either of the followig two coditios holds: log c( <, if t 3, log c( <, if t 3. Note that poor delay performace could be caused either by a limited umber of simultaeous trasmissios due to wireless iterferece or by a large umber of retrasmissios required for reliability. Although Algorithm exploits user ad path diversity improvig delay performace by reducig the umber of retrasmissios, the improvemet may ot be sigificat due to wireless iterferece. Table II shows that whe the trasmissio rage is small, Algorithm does ot perform very well sice the delay from iterferece domiates.. 7

8 I cotrast, whe the trasmissio rage is large, Algorithm ca improve the delay performace substatially, while the gai depeds o the reliability costrait. The results imply that broadcastig is more useful whe a larger trasmissio rage is required, e.g., due to topological restrictio or delay deadlie of sesed data. TABLE II GAINS OF ALGORITHM OVER AN INSTANCE OFU UNDER VARIOUS WIRELESS STRUCTURES AND RELIABILITY CONSTRAINTS. = = = c( = Θ( Θ( log Θ(log c( = Θ( Θ( Θ( c( = exp Θ( Θ( ( 3 Θ( It is also worthwhile otig that whe there is o loss i liks ad =, Algorithm has the delay of Θ( ad the istace of U achieves Θ(. Hece, the istace of U is better i lossless etworks with miimal coectivity. C. A hybrid scheme Motivated by the cases that the istace of U outperforms the istace of B, we cosider a hybrid method that bleds U ad B. Uder the hybrid scheme, the iformatio obtaied by a idividual sesor ode is collected by some special odes called collectig odes, ad these collectig odes are resposible for data aggregatio ad iformatio delivery to the sik. The scheme seems similar to U, but there are importat differeces i that all packet trasmissios are doe by wireless broadcast ad that (uiformly distributed collectig odes ca iterfere with each other. We first describe the implemetable algorithm ad provide a sufficiet coditio to satisfy the reliability costrait. The, we aalyze the delay performace ad the gai of the hybrid method. We use the tiered structure of Algorithm. Assumptio 4.. I additio to Assumptio 4., we further assume that amog all sesor odes, there are collectig odes that are uiformly deployed over the etwork. Each ode has at least y( [, ] collectig odes withi its trasmissio area. The algorithm cosists of two phase: Phase : Each o-collectig ode broadcasts its packet (+r b ( times. All earby collectig odes receive the packet ad do aggregatio. Phase : From the outermost tier, each collectig ode i T i broadcasts its packet (+r b ( time. Collectig odes i T i receive the packet ad do aggregatio. This procedure repeats tier-by-tier as Algorithm. Remarks: The algorithm has some similarity with the solutio i [], which, however, operates with uicast, does ot take ito accout packet losses, ad requires specific placemet of collectig odes. Sufficiet coditio for the reliability costrait: Lettig P s ad P s deote the probability of successful packet trasmissio i phase ad at each tier i phase, respectively. The probability P s of successful delivery of critical iformatio value ca be writte as d ( P s P s k= P s ( p y( (+r b( ( p y( (+r b( d ( p y( (+r b( d ( p y( (+r b(. To simplify equatios, we drop ( i the sequel. Usig d = Θ( t, we obtai P s Θ ( c, if y (+r b Θ(logc ad y (+r b Θ(log c t. (3 Hece,y (+r b c 5 logc ady (+r b c 5 log c t with c 5 = logp (ad some r b = Θ(,r b = Θ( are sufficiet coditios to satisfy the reliability costrait (. Delay performace: Let D h deote the delay boud of the hybrid scheme. Lettig D h ad D h deote the delay icurred by phase, ad the delay icurred by phase, respectively, we have D h = D h +D h. Usig the techiques provided i the previous sectios so far ad from (3, the followig ca be easily show ( D h = Θ t (+ y logc, D h = Θ ( y t ( + y log c t, where i D h, the term t is the time for all o-collectig odes to broadcast a packet ad the followig ( + y logc is required for retrasmissios, ad i D h, the term y t is the time for collectig odes (y i a sigle tier to broadcast a packet multiplied by the umber of tiers ( t, ad the followig (+ y log c t is required for retrasmissios. The we obtai D h = Θ (t yt t + + y logc+ t log c. (4 t The gai G h of the hybrid scheme of the istace of U ca be preseted from ( as ( G h := D ( u t + t log c t = Θ D h t + y t + t y logc+ t log. (5 c t We deote G,N,D respectively, as G = N ( t D := + t log c t t + y t + t y logc+ t log c t Differetiatig both sides by y, we obtai dg dy = N ( t D y logc. t ( Note that the sig of dg dy is determied by t y logc t,. 8

9 Probability of failure (-P s broadcast =0 = = =3 =4 Delay (time uit h broadcast =0 = = =3 =4 Probability of failure (-P s broadcast =0 = = =3 = Probability of lik error (p Probability of lik error (p Probability of lik error (p (a Loss rate (b Delay (c Loss rate with delay costrait. Fig. 5. Loss rate ad delay of iformatio delivery. The loss rates of aggregatio with uicast improves with the umber of retrasmissios r u i Fig. 5(a. However, the icrease of r u leads to higher delay performace i Fig. 5(b. I cotrast, there is o retrasmissio uder the broadcast-based scheme (r b = 0 ad uder the uicast-based scheme with r u = 0. Thus their delays remai costat regardless of the loss rates. Whe accoutig for iformatio delivery withi delay boud (mi + uit times, i.e., ĥ, broadcastig without retrasmissios shows better performace tha uicast with retrasmissios. which is a mootoically ( decreasig fuctio of y [,t ]. dg t Hece, dy < 0 if y logc t y= < 0, which implies that G also mootoically decreases i [,t ], ad dg thus ca be maximized whe y =. Similarly, ( dy > 0 y=t t if y logc t > 0, ad G ca be maximized ( whe y = t. Otherwise, G will be maximized whe t y logc t = 0, which leads to the settig of y = t3 logc. Summarizig, we obtai the optimal settig for the hybrid scheme as Θ(, if logc < t, 3 y( = Θ(t, if t < logc, Θ( (6 t 3 logc, if t logc 3 t. Therefore, the optimal desity of collectig ode depeds o the reliability costrait ad the trasmissio rage (or the distace betwee a ode ad the sik. The exact gai is also determied by the choice of t, c, ad y from (5. Sice oe of the four terms of (4 will domiate the others, the gai ca preset as Θ((+ t log c 3 t, G h Θ( (t,c,y = y (+t3 log c t, Θ(y(+ t log c 3 t /logc, Θ(+t 3, where the cases deped o t,c,y. Sice c ad t [,], we have Θ(log c t > Θ( ad Θ(log c t /logc Θ(. The for all four cases, we achieve G h Θ( if y [,Θ(t ] is chose accordigly. This result is expected: Sice the istace of U is equivalet to the hybrid scheme with y =, the performace of the hybrid scheme with optimal parameter y must be o smaller tha that of the istace of U. Further, if > 3, a gai strictly greater tha Θ( will be achieved. V. SIMULATION RESULTS I this sectio, we simulate our solutios, ad evaluate their performace. We are iterested i reliability i terms of successful trasmissios as well as the delay. We first simulate scearios of TDMA etworks without iterferece, ad proceed to resource-costraied etworks with wireless iterferece. A. TDMA etworks without iterferece We compare the performace of uicast-based ad broadcast-based schemes i a wireless sesor etwork with 00 odes, which are radomly placed i a disk of radius. The trasmissio rage of each ode is set to 0.5. The tiered structure has the width 0.5 (δ =, ad a paret-child relatioship has bee established betwee every pair of odes if the two odes are located i eighborig tiers ad their distace is less tha 0.5. I this settig, there are four tiers. We locate the sesor ode that geerates the critical iformatio values at the boudary of the etwork, i.e., i the 4-th tier. Sice routig follows the tiered structure, both aggregatios with uicast ad broadcast takes at least four trasmissios for the packet geerated from the sesor ode to arrive at the sik. We assume that for all tiers i, it takes the same umber of mii-slots ĥ for all odes i T i to fiish a sigle trasmissio, ad cosider ĥ as a time uit for the delay performace. For aggregatio with uicast, we chage the umber of retrasmissios r u from 0 to 5, ad for aggregatio with broadcast we set r b = 0. All liks are assumed to fail trasmissio with the same probability p. Chagig p, we cout the umber of time uits (ĥ required for the sik to receive the critical iformatio value ad measure the rate of failure, i.e., loss of the iformatio. We ru each simulatio 000 times ad average the results. Fig. 5 illustrates the loss rate of the critical iformatio value ad the delay performace. Fig. 5(a shows that aggregatio with uicast ca improve the loss rate with more retrasmissios. However, it also icreases the delay as show i Fig. 5(b. I cotrast, there is o retrasmissio uder the broadcast-based scheme (r b = 0 ad uder the uicastbased scheme with r u = 0. Thus their delays remai costat regardless of the loss rates. If we have the delay boud of 6ĥ, which is the miimum achievable delay plus ĥ, the it is observed i Fig. 5(c that the retrasmissio strategy caot 9

10 B(, 500 B(, B(, Delay B(3, B(3, B(3,3 B(4, B(4, B(4,3 B(4, No. Fowarders Per-lik trasmissios Fig. 6. Coflict graph of blocks. Each blocks cotais 0 odes. If a ode i a block trasmits, o other ode i the coected blocks ca trasmit at the same time. Each vertex that represets a block is colored such that o two coected vertices have the same color. improve the loss rate beyod a certai threshold, ad that aggregatio with broadcast achieves better performace. B. Resource-costraied etworks with iterferece We evaluate our hybrid schemes of Sectio IV takig ito accout wireless iterferece. The difficulty i the simulatios lies i implemetig a optimal scheduler. Sice trasmissio time ĥ i a tier chages with the umber of collectig odes, we eed detailed implemetatio of schedulig fuctioality, which however ofte requires high computatioal complexity eve uder a very simple iterferece model. To facilitate implemetatio of the schedulig compoet, we cosider the followig block-based etwork, which captures essetial features of wireless iterferece i tiered etworks. Network topology: We group 0 earby odes as a block (like Ci m. Nodes i a block are withi commuicatio rage of each other, ad they caot trasmit simultaeously due to iterferece costraits. Two blocks are coected whe trasmissio of ay ode i a block ca be received by all odes i the other ode. Also, we assume that o two odes i the coected blocks ca trasmit simultaeously due to wireless iterferece. We assume that blocks have a tiered structure. There is oly oe block of odes that ca trasmit to the sik, ad this block cosists of the first tier T. I the secod tier T, there are two blocks, each of which is coected to the block i T, They are also coected with each other. Similarly, we assume that there are i blocks i each T i. Let B(i,j deote the j-th block i T i. For i-tier iterferece, we assume that B(i, j is coected with B(i, j ad B(i, j +, where the additio ad the substractio is modular-(i + operatio for the circular property of the tier, i.e.,b(i, is coected with B(i, i. For data forwardig ad iter-tier iterferece, we coect each B(i,j to two blocks i T i : to Fig. 7. Delay performace (time slots i presece of iterferece, with differet umber of forwarders k =,...,0, ad differet umber of perlik trasmissios r =,..., 0. Loss probability p does ot affect the delay performace. Delay icreases more quickly with icrease i r tha with icrease i k. B(i,j ad B(i,j, where the substractio is agai modular-i operatio. We further assume that odes i the block of the first tier are directly wired to the sik ad hece, trasmissios at the last hop, i.e., from odes i T to the sik, are either lost or iterfere with other trasmissios. We cosider a etwork with total 0 tiers ad 55 blocks. Iterferece: We ca draw a equivalet coflict graph by represetig a block as a vertex. A vertex B(i,j has a edge with vertices of B(i,j ad B(i,j + i tier i (siblig blocks, B(i,j ad B(i,j i tier i (paret blocks, ad correspodig blocks i tier i+ (child blocks. Assumig that there is o iterferece betwee o-coectig blocks, the iterferece relatioship ca be described i a simple form 5 as ay iteded trasmittig ode i a block should be the oly trasmitter withi the block ad its coected blocks. Fig. 6 illustrates the coflict relatioship amog blocks. Data trasmissio: We assume a time-slotted system, where each time slot is further divided ito mii-slots. Data is geerated at the begiig of each time slot, ad trasmitted to the sik durig the mii-slots i two steps: collectig ad forwardig. I each block, we choose k out of 0 odes as a collectig ode (also deoted by a forwarder. First, each o-collectig ode i a block broadcasts its data to all the collectig odes i the block. The the collectig odes aggregate the received data ad trasmit to collectig odes i the upper tier (i.e., to odes i the paret blocks. Note that i our etwork structure, each collectig ode i T i have total k paret odes i T i. It is a istace of broadcast model B if k = 0, ad it is close to a istace of uicast model U if k =. Schedulig: For collectig data withi a block, we schedule as follows. We first color blocks usig 6 colors such 5 The model takes ito accout wireless iterferece at the seder side, ad does ot cosider iterferece at the receiver side. 0

11 per-lik TX, r= per-lik TX, r= per-lik TX, r=4 per-lik TX, r= per-lik TX, r= per-lik TX, r= per-lik TX, r=4 per-lik TX, r= per-lik TX, r= per-lik TX, r= per-lik TX, r=4 per-lik TX, r= Lost data Lost data Lost data Lik error probability Lik error probability Lik error probability (a k = (b k = 3 (c k = 6 Fig. 8. Lost iformatio (umber of sesor odes i the presece of iterferece, with differet lik loss probabilities p = 0.,...,0.9, ad differet umbers of per-lik trasmissios r =,,4,8. Results with differet umbers of forwarders k =,3,6 show that a small icrease of forwarders ca sigificatly improve reliability. that o two coected blocks have the same color 6 as show i Fig. 6. We use 3 colors at each tier. Nodes i the blocks of the same color, oe ode per block, ca trasmit at the same time without iterferece. Hece, it takes 6 (0 k mii-slots for each o-collectig ode to trasmit oce. We assume that the odes retrasmit (i.e., re-broadcast r x ( 0 times for reliable collectig. After collectig the data withi blocks, the aggregated data is forwarded to the sik tier-by-tier. Note that blocks of the same color ca trasmit at the same time. Hece, all the collectig odes i a tier ca fiish a trasmissio for 3k mii-slots. Nodes retrasmit r y ( 0 times for forwardig. We simulate our schemes chagig the umber of forwarders k from to 0. Each lik betwee two odes has loss probability p, which chages i the rage of [0., 0.9]. The umber of (retrasmissios r per lik also chages from to 0, i.e., +r x = +r y = r [,0]. Fig. 7 illustrates the delay performace for the sik to get all the data uder differetk ad r i terms of mii-slots. Lik loss probability p does ot affect the delay. The results show sharp icreases i delay whe the umber of retrasmissios r per lik icreases tha whe the umber of forwarders k icreases. Fig. 8 shows the impact of forwarders o reliability. We measure the umber of lost iformatio with differet lik loss probabilities, umbers of retrasmissios, ad umbers of forwarders. The results show that a small umber of forwarders sigificatly improve the reliability, especially whe the lik loss probability is high, i.e., uder a harsh eviromet like uder-water scearios. The gais of wireless broadcast are more visible i Fig. 9, which presets the delay ad the data loss for the give umber of forwarders. For each k forwarders, the lowest-delay poit is a result whe there is o retrasmissio (i.e., r =, the the ext lowest-delay poit is a result whe there is per-lik retrasmissio, ad so o. As the umber of retrasmissios icreases, the reliability improves while the delay performace deteriorates. The curve for k forwarders ca be cosidered as a achievable performace boudary with differet umber 6 Ideed, it is sufficiet with 5 colors i our particular case. If the umber of blocks does ot icrease by oe per tier, we may eed 6 colors. Delay Forwarders Forwarders 3 Forwarders 4 Forwarders 5 Forwarders 6 Forwarders per-lik TX, r = Lost data Fig. 9. Delay performace ad the umber of lost iformatio i the presece of iterferece, with differet umbers of forwarders k =,..., 6, ad differet umbers of per-lik trasmissios r =,...,0. Loss probability p = 0.8. The performace boudary for each k forwarders improves approachig the origi, whe the umber of forwarders k icreases. retrasmissios. The results show that the boudary improves, i.e., gets closer to the origi, as the umber of forwarders icrease. VI. CONCLUSION I a wireless sesor etwork, i-etwork aggregatio ca sigificatly improve efficiecy whe the goal of the etwork is to compute a global fuctio. However, sice the loss of a aggregated packet is far more harmful tha a uaggregated packet, a higher level of protectio is required for reliable operatios i lossy wireless eviromets. I this paper, we use wireless broadcast as a meas of protectig the aggregate iformatio for a class of geeralized maximum fuctios. Exploitig the diversity of wireless medium, broadcastig spreads iformatio spatially, ad the properties of the fuctio eable distributed i-etwork computatio with the spread iformatio. We show that aggregatio with broadcast ca improve delay performace while satisfyig the same level of reliability. The gai ca be preseted as a fuctio of reliability costrait ad trasmissio rage. Usig a tiered etwork topological structure, we develop solutios for aggregatio that exploit wireless diversity, ad

12 are ameable to implemetatio i a distributed maer. We evaluate the schemes i a resource-costraied etwork with wireless iterferece. Further, we also develop a hybrid scheme that combie the uicast ad multicast architecture. Simulatio results show that aggregatio with broadcast outperforms aggregatio with uicast, especially, i severely lossy etwork eviromets. There are may iterestig ope questios to cosider. Aggregatio fuctios besides the geeralized maximum fuctios should be cosidered. A ope questio is whether the performace bouds i the presece of iterferece i Sectio IV-B are tight or ot. Although we focus o the delay performace, other performace metrics such as time complexity ad achievable samplig rate are also of importace. It would be iterestig to study the relatioship betwee these metrics with aggregatio fuctios ad etwork topologies. REFERENCES [] A. Giridhar ad P. R. 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Bahk, Tier Based Aycast to Achieve Maximum Lifetime by Duty Cycle Cotrol i Wireless Sesor Networks, i IWCMC, August 008. Chaghee Joo received his Ph.D degree from Seoul Natioal Uiversity, Korea, 005. He is a assistat professor at Ulsa Natioal Istitute of Sciece ad Techology (UNIST, Korea. Before joiig UNIST, he worked at Purdue Uiversity ad the Ohio State Uiversity, USA, ad at Korea Uiversity of Techology ad Educatio, Korea. His research iterests spa a wide area of etworkig techologies icludig aalysis, modellig, cotrols, ad optimizatio. He has served o the techical committees of several primary cofereces, icludig IEEE INFOCOM, ACM MobiHoc, ad IEEE SECON. He is a member of IEEE, ad a recipiet of the IEEE INFOCOM 008 best paper award. Ness B. Shroff received his Ph.D. degree i Electrical Egieerig from Columbia Uiversity i 994. He joied Purdue uiversity immediately thereafter as a Assistat Professor i the school of ECE. At Purdue, he became Full Professor of ECE i 003 ad director of CWSA i 004, a uiversity-wide ceter o wireless systems ad applicatios. I July 007, he joied The Ohio State Uiversity, where he holds the Ohio Emiet Scholar edowed chair professorship i Networkig ad Commuicatios, i the departmets of ECE ad CSE. From 009-0, he served as a Guest Chaired professor of Wireless Commuicatios at Tsighua Uiversity, Beijig, Chia, ad curretly holds a hoorary Guest professor at Shaghai Jiaotog Uiversity i Chia. His research iterests spa the areas of commuicatio, social, ad cyberphysical etworks. He is especially iterested i fudametal problems i the desig, cotrol, performace, pricig, ad security of these etworks. Dr. Shroff is a past editor for IEEE/ACM Tras. o Networkig ad the IEEE Commuicatio Letters. He curretly serves o the editorial board of the Computer Networks Joural, IEEE Network Magazie, ad the Networkig Sciece joural. He has chaired various cofereces ad workshops, ad co-orgaized workshops for the NSF to chart the future of commuicatio etworks. Dr. Shroff is a Fellow of the IEEE ad a NSF CAREER awardee. He has received umerous best paper awards for his research, e.g., at IEEE INFOCOM 008, IEEE INFOCOM 006, Joural of Commuicatio ad Networkig 005, Computer Networks 003 (also oe of his papers was a ruer-up at IEEE INFOCOM 005, ad also studet best paper awards (from all papers whose first author is a studet at IEEE WiOPT 0 ad IEEE IWQoS 006.