Ani-Jamming Schedules for Wireless Daa Broadcas Sysems Paolo Codenoi 1, Alexander Sprinson, and Jehoshua Bruck Absrac Modern sociey is heavily dependen on wireless neworks for providing voice and daa communicaions. Wireless daa broadcas has recenly emerged as an aracive way o disseminae dynamic daa o a large number of cliens. In daa broadcas sysems, he server proacively ransmis he informaion on a downlink channel; he cliens access he daa by lisening o he channel. Wireless daa broadcas sysems can serve a large number of heerogeneous cliens, minimizing power consumpion as well as proecing he privacy of he cliens locaions. The availabiliy and relaively low cos of anennas resuled in a number of poenial hreas o he inegriy of he wireless infrasrucure. In paricular, he daa broadcas sysems are vulnerable o jamming, i.e., he use of acive signals o preven daa broadcas. The goal of jammers is o cause disrupion, resuling in long waiing imes and excessive power consumpion. In his paper we invesigae efficien schedules for wireless daa broadcas ha perform well in he presence of a jammer. We show ha he waiing ime of clien can be reduced by adding redundancy o he schedule and esablish upper and lower bounds on he achievable minimum waiing ime under differen requiremens on he saleness of he ransmied daa. I. INTRODUCTION Modern sociey has become heavily dependen on wireless neworks o deliver informaion o diverse users. The emerging wireless infrasrucure provides opporuniies for building efficien daa disribuion sysems ha allow users o access he laes daa, such as sock quoes and raffic condiions, a any ime, wheher hey are a home, a heir office, or raveling. Wireless daa disribuion sysems also have a broad range of applicaions in miliary neworks, such as ransmiing up-odae bale informaion o acical commanders in he field. Due o heir open and ubiquious naure, wireless informaion sysems are exremely vulnerable o aack and misuse. Wireless sysems can be aacked in various ways, depending on he objecives and capabiliies of an adversary. Due o high availabiliy and relaively low cos of powerful anennas, jamming, i.e., he use of acive signals o preven daa disribuion, has emerged as an aracive way of aack. Jamming is a common mehod of aack in miliary neworks, where he adversary has he capabiliy and he moivaion o disrup communicaion links. One common characerisic of wireless infrasrucure is an asymmery beween he downlink and uplink channels. Typically, he downlink channel has higher bandwidh and more energy resources han he uplink channel. This inrinsic asymmery of wireless sysems impacs he way informaion is delivered o cliens. In paricular, he sandard clien-server paradigm, in which he daa ransfer is iniiaed by cliens, is no adequae for wireless sysems. Wireless daa broadcas [1] [] has recenly emerged as an aracive way o disseminae daa o a large number of cliens. In daa broadcas sysems, he server proacively ransmis he informaion on he downlink channel and he cliens access daa by lisening o he channel. This approach enables he sysem o serve a large number of heerogeneous cliens, minimizing cliens power consumpion as well as proecing he privacy of he cliens locaions. Figure 1 depics a ypical daa broadcas sysem. The sysem includes he following componens: he server (scheduler), he broadcas channel, he informaion source, and he wireless users. The server periodically accesses he informaion source, rerieves he mos recen daa, encapsulaes i ino packes and sends he packes over he broadcas channel. Each ransmied packe carries he mos recen updae on he sae of he informaion source. There are wo key performance characerisics of a wireless daa disribuion sysem. The firs characerisic is he waiing ime, i.e., he amoun of ime spen by he clien waiing for an updae. Waiing ime is an imporan parameer, as imely informaion delivery is essenial for many pracical applicaions. In addiion, i is closely relaed o he amoun of power spen by he clien o obain he informaion. The second characerisic is saleness, i.e., he amoun of ime ha passes from he momen he updae is generaed, unil i is delivered o he clien. The saleness of he schedule usually depends on he amoun of redundancy used by he sysem, as informaion becomes less and less relevan wih ime. Jamming Aacks: The goal of he jammer is o disrup he normal operaion of he broadcas sysem by sending acive signals over he channel ha inerfere wih he signal sen by he server (see Figure 1). Jamming signals may resul in long waiing ime and excessive power consumpion of Source Jammer Server 1 Deparmen of Compuer Science, Universiy of Chicago, Chicago, Illinois, USA. Email: paoloc@uchicago.edu Deparmen of Elecrical and Compuer Engineering, Texas A&M Universiy, Texas, USA. Email: spalex@ece.amu.edu Parallel and Disribued Compuing Group, California Insiue of Technology, Pasadena, California, USA. Email: bruck@paradise.calech.edu Fig. 1. A ypical daa broadcas sysem.
he cliens. The radiional defences agains jamming include spread specrum echniques such as direc sequence and frequency hopping [], [5]. While hese echniques consiue an imporan ool for combaing jamming, an addiional proecion is required a he packe level due o he following reasons. Firs, he pseudo-random noise code or frequency hopping sequence may be known o he adversary. Second, even if no informaion abou he spread-specrum proocol is available o he adversary, i can sill desroy a small porion of each ransmied packe by sending a srong jamming signal of shor duraion. If no oher proecion mechanism is used a he packe-level, he few desroyed bis will resul in dropping of he enire packe. In his paper we invesigae efficien ani-jamming schedules for daa broadcas. In our schedules, each updae is encoded by an error-correcing code, such as a Reed-Solomon code, which allows he schedule o minimize boh waiing ime and saleness. As power supply is he mos imporan consrain for pracical jammers, we focus on jammers ha have cerain resricions on he duraion of jamming pulses. To he bes of our knowledge, his is he firs sudy ha invesigaes anijamming schedules for wireless daa disribuion sysems. Relaed Work: The design of opimal broadcas schedules araced a large body of research (see [6] [9] and references herein). Daa broadcas over lossy communicaion channels was sudied in [10]. This work proposes efficien coding soluions ha reduce performance degradaion due o packe loss. Sudies [11], [1] focused on he design of universal schedules ha guaranee low waiing ime for any user, regardless of he access paern. These sudies show ha a good universal schedule has o combine boh encoding and randomizaion echniques. II. MODEL A. Schedules As menioned in he inroducion, he goal of he sysem is o enable each clien o access he mos updaed informaion is a imely manner. To ha end, he server periodically ransmis updaes, each updae capures he mos recen sae of he informaion source. We assume ha each updae includes exacly k informaion symbols and he ransmission of one updae wihou encoding requires one ime uni. Each updae is encoded ino a packe by using a Maximal Disance Separable (MDS) code, e.g., a Reed-Solomon code [1]. The encoding ensures ha any k symbols of he packe are sufficien in order o reconsruc he updae. We enumerae updaes according o he ime of heir creaion. Definiion 1 (Schedule S): A schedule is a sequence {r 1, r,... }, r i 1, such ha r i 1 is he number of ime unis allocaed for he ransmission of updae i. Noe ha according o he above definiion each packe includes r i k k symbols. A schedule S = {r 1, r,... } can also be defined by is ransmission sequence { 1,,... }, where i represens he saring ime of he ransmission of packe i, i.e., 1 = 0 and i = i 1 j=1 r j for i > 1. For example, consider he schedules depiced in Figures (a) and (b). In he firs schedule, each packe is allocaed r ime unis, hence he ransmission sequence of he schedule is 0, r, r,.... The second schedule ransmis packes of varying lengh. 0 Message Message Message 0 r r r 0 Fig.. r1 (a) Message Message Message r (b) h1 l1 h l h l h l (c) Message Message Message 0 6 9 1 (d) r r Examples of schedules and jamming packes A wireless clien begins o lisen o he wireless channel upon a reques for new informaion. In order o saisfy is reques, he clien mus receive a leas k symbols from he curren or subsequen packes. In paricular, if he clien fails o receive k symbols from he curren packe, i coninues o lisen o he channel, unil i receives a leas k symbols from one of he subsequen packes. The are wo key performance characerisics of he schedule: he expeced waiing ime and he maximum saleness of he received daa. Definiion (Waiing ime WT (S)): Le S be a broadcas schedule. Suppose ha he clien s reques was placed a ime. Le n be he number of he packe currenly ransmied over he channel, i.e., he packe for which i holds ha n < n+1. Le be he firs ime he clien receives a leas k symbols from a packe n, n n. Then, he waiing ime of he clien is defined as WT = 1. We subrac one from he waiing ime, because he clien will lisen a leas ha long for any updae. We assume ha he cliens requess are disribued uniformly over ime. Accordingly, he expeced waiing ime of he cliens is defined as follows: Definiion (Expeced Waiing Time EWT(S)): Le S be a broadcas schedule. Then, he expeced waiing ime is defined as follows: T 1 EWT(S) = lim WT (S)d (1) T T 0 The waiing ime is an exremely imporan parameer for many ime-sensiive applicaions. In addiion, i is closely relaed o he amoun of power spen by he clien o obain he informaion. The saleness of he daa is defined o be he amoun of ime ha passes from he momen an updae is generaed unil i is delivered o he clien. The saleness capures he qualiy of delivered informaion, because in dynamic seings he informaion becomes less and less relevan wih ime. Definiion (Saleness ST (S)): Le S be a broadcas schedule. Suppose ha he clien s reques was placed a ime. Le n be he number of he packe currenly ransmied over he channel, i.e., he packe for which i holds ha n < n+1. Le be he firs ime he clien receives a leas k symbols from an updae n, n n. Then, he saleness
of he daa is defined o be ST = n 1, where n is he ime he updae n was generaed. Again, we subrac one from he saleness, because he clien will lisen a leas ha long for any updae. For example, consider he schedule depiced in Figure (a). Le be ime he clien pus a reques for he new informaion and le n be he number of he packe currenly ransmied over he channel. Then, he number of symbols received by he clien from packe n is equal o n = ( r r )k. If n k, hen he clien will be able o decode his packe, hence is waiing ime is zero. Oherwise, he clien needs o wai for he nex packe, hence is waiing ime is equal o n k. I is easy o verify ha if he requess are disribued uniformly over ime, hen he expeced waiing ime is k n = 1 r. While redundan ransmission improves he expeced waiing ime of a schedule, i comes a a price in erms of he saleness of he received daa. Indeed, if n k, hen he updae received by he clien a ime, was generaed in ime r r, hence he saleness of he daa is r r. On he oher hand, if n < k, hen he clien will ge a new updae, hence he saleness is zero. A schedule ha ransmis all updaes over ime inervals of equal lengh is referred o as a regular schedule. Regular schedules provide firm guaranees on he saleness of he received daa. In paricular, schedule S r = {r, r,... } ensures ha he saleness of he received informaion is a mos r 1 ime unis. In addiion, a regular schedule uses he same encoding for each packe, which simplifies he design of he mobile device and reduces is cos. Jammer model We focus on pulse erasure jammers. Such jammers produce a sequence of pulses, each pulse resuls in an erasure in he channel. Definiion 5 (Jamming Sequence J ): A jamming sequence is a sequence {h 1, l 1, h, l,... }, such ha h 1 is he beginning ime of he firs pulse, l i is he lengh of pulse i, and h i, i is he lengh of ime inerval beween pulses i 1 and i. Figure (c) depics an example of a jamming sequence. I has been recognized [1] ha he power supply is he mos imporan limiaion for he majoriy of pracical jammers. A ypical jammer is powered by a baery, which can be recharged from an exernal source, such as a solar cell array. Accordingly, in our model, we limi he lengh of pulses in he jamming sequence by a consrain l max, i.e., l i l max for all i 1. Since afer each pulse he baery mus be recharged we also consrain he lengh of he inerval beween wo consecuive pulses o be a leas h min, i.e., h i h min for all i. We refer o a jamming sequence ha saisfies he energy limiaions as an admissible jamming sequence. We denoe by WT (S, J ) he waiing ime of schedule S in he presence of jammer J. Similarly, he expeced waiing ime of a schedule S in he presence of jammer J is denoed by EWT(S, J ), i.e., EWT(S, J ) = 1 T lim T T 0 WT (S, J )d. For example, suppose ha a jamming sequence J = {1, 1, 1,... } is applied o he schedule S = {,,... } (see Figure (d)). In his case, he expeced waiing ime is equal o r Lower Bound Upper Bound r < r < 1 + r 1 + r r = < r < 1 + 51 1 + 51 r < i r < i + 1, i =,... i + 1 r < i + i =,,... i + r < (i + 1) i =,,... 18 6r r r+ r + 10+r r r +11 + δ 5δ+10 TABLE I 18 + δ δ+10 + δ δ+10 + 10+r r + δ δ+10 + δ δ+10 UPPER AND LOWER BOUNDS ON THE MAXIMUM EXPECTED WAITING TIME MWT(S r) OF REGULAR SCHEDULES S r = {r, r,... }. HERE, Expeced Waiing Time 1.8 1.6 1. 1. 1 δ = r r 6 8 10 1 1 Fig.. The lower and upper bounds on MWT(S r). The lower and upper bounds are marked by solid and dashed lines, respecively. EWT(S, J ) = 11/1, which is by 5 1 more han he expeced waiing ime of he same schedule wihou jamming. For a given schedule S, we define is maximum waiing ime MWT(S) o be he maximum value of EWT(S, J ) over all admissible jamming sequences J. The maximum waiing ime characerizes he wors-case behavior of he schedule in he presence of an adversarial jammer. Resuls In his sudy we invesigaed he performance of broadcas schedules in he presence of a jammer. Firs, we esablished lower and upper bounds on he maximum expeced waiing ime MWT(S r ) of regular schedules S r = {r, r,... }. Then, we exended our resuls for general schedules and invesigaed he rade-off beween he maximum expeced waiing ime MWT(S r ) and he saleness consrain. Our resuls for regular schedules are summarized in Table I and in Figure. All of our lower bounds are consrucive as we presen jamming sequences ha yield he expecing waiing imes equal o he value of he lower bound. We also idenify opimal jamming sequences for many classes of regular schedules. All resuls are up o erms of order ε = 1 k. I is imporan o noe ha he size of he packe r in a regular schedule S r is closely relaed o he saleness of he delivered informaion. Indeed, he maximum saleness of he daa is always lower or equal o r 1, while is average saleness does no exceed r 1 r. Hence our resuls esablish a r
' M I 1 I I I B B 1 B B B 5 B 6 B 7 1 5 6 " M Fig.. Division of a inerval I M ino blocks and subinervals I 1,..., I. rade-off beween he expeced waiing ime of he cliens and he saleness. In paricular, we idenify he bes schedule for any given saleness consrain. We observe ha he schedule S has a clear advanage over oher schedules: i achieves low expeced waiing ime wih minimum penaly in erms of he saleness of he delivered daa. In addiion, we esablished upper and lower bounds on he wors case waiing ime MWT(S) for a general class of non-regular schedules. This class includes schedules in which he lengh of each packe is differen and he schedules ha employ randomizaion, i.e., he lengh of each packe is disribued according o some probabiliy disribuion. We assume ha in he case of random schedules he jammer knows he probabiliy disribuion bu has no access o he server s random bis. Theorem 6: Le S be a schedule and le r be he expeced lengh of he packes in S. Then he wors case expeced waiing ime MWT(S) of he schedule in he presence of an admissible jammer is bounded by + r MWT(S) + 11 () Due o he space consrains, mos of he proofs and echnical deails are omied from his paper and can be found in [15]. III. UPPER BOUNDS FOR REGULAR SCHEDULES In his secion we esablish upper bounds on he opimal jamming sequences for regular schedules S r for a cerain range of values of r. We begin by inroducing he noion of added waiing ime. Le I = [, ] be a ime inerval, S be a schedule, and J be a jamming sequence. Then, he added waiing ime for inerval I is defined o be AWT(I, J ) = (WT (S, J ) WT (S))d. Inuiively, AWT(I, J ) capures he oal amoun of addiional waiing ime experienced by cliens requess ha arrive during inerval I due o jamming. Le S r, r be a regular schedule, J be an admissible jamming sequence, and I M = [ M, M ] be a ime inerval allocaed for ransmission of a single updae. Our goal is o obain an upper bound on AWT(I, J ). Such a bound would immediaely ranslae in an upper bound on MWT(S r ). We denoe by 1,..., h he finishing imes of he jamming pulses ha belong o I M and divide I M ino h + 1 blocks B 1,..., B h+1, such ha B 1 = [ M, 1], B i = [ i 1, i ] for i h, and B h+1 = [ h, M ] (See Figure ). We also divide I M ino subinervals I 1,...,I such ha ha I 1 = B 1, I includes blocks B,...,B h 1, I = B h, and I = B h+1. We denoe by T i he lengh of he subinerval I i. In he following lemma we esablish an upper bound for each of he subinervals I 1,...,I. Lemma 7: T1 / if T 1 1 AWT(I 1, J ) T 1 1/ if 1 T 1 AWT(I, J ) AWT(I, J ) AWT(I, J ) { T+δ 5δ if T is odd, ( T ) oherwise, T (T )+T +T 1 if T < 1 T 1 if T 1 T if T < 1 T +T 1 if T 1 () where δ = T T. Proof: See [15]. We assume, wihou loss of generaliy, ha inervals I 1, I, or I do no conain an unjammed inerval whose lengh is longer han one ime uni. Indeed, if his is he case, such an inerval can be shorened a he expense of one of he unjammed inervals in I, wih no increase in he value of AWT(I M, J ). This implies ha T 1, T, T. The added waiing ime for he inerval I M equals o he sum of he added waiing imes for is subinervals I 1,...,I, i.e., AWT(I M, J ) = AWT(I 1, J )+AWT(I, J )+AWT(I, J )+AWT(I, J ). An upper bound on AWT(I M, J ) can be found by solving he following maximizaion program: maximize AWT(I M, J ) subjec o T 1 + T + T + T = r T i 0 i = 1,..., T i i = 1,, I can be shown, using he ools of he heory of consrained opimizaion ha for r, he opimal value of AWT(I M, J ) is achieved when he following condiions are saisfied: (a) T 1 = r T T T, (b) T = r (c) T = (d) T = 1. If r and r is an even number his implies he following upper bound on AWT(I M, J ): AWT(I M, J ) which, in urn, implies ha r + 8 + r r, MWT(S r ) r r + 10 +.. (5) If r and r is an odd number, hese condiion imply he following upper bound on AWT(I M, J ): r + 8 δ + δ AWT(I M, J ), where δ = r r. This, in urn, implies ha MWT(S r ) + δ δ + 8. (6) We conclude our discussion by he following heorem: Theorem 8: Le S r = {r, r, } be a regular schedule. Then if r and he ineger par of r is an odd number ()
Fig. 5. r=. 1-1 0.5 1.+ 1 1 Message 5 Message (a) 1-1. 1-1 1 Message 0. 6.6 (b) Message 7.5 10 Message.6+ 1 Message 9.9 1. Jamming sequences for regular schedules J r (a) r=.5 (b) hen MWT(S r ) + 10+δ δ, where δ = r r. If r and he ineger par r of r is an even number, hen MWT(S r ) + r r +10. IV. LOWER BOUNDS Theorem 9: Le S r be a regular schedule. Then, up o erms of order ε = 1 k, MWT(S r ) 1 + r if r < 6r if r < 1 + r r+ 51 51 r if 1 + r < + 10+r r if i r < i + 1, i =,,... r +11 if i + 1 r < i +, i =,,... + δ 5δ+10 if i + r < (i + 1), i =,,... where δ = r r. Proof: We prove he heorem by presening, for each schedule S r, r a jamming sequence J r such ha EWT(S r, J r ) is equal o he values of he lower bounds saed in he heorem. For r <, J r = {1 ε, 1, 1 + δ, 1, 1 + δ,... }, where δ = r and ε = 1 k. An example of his schedule for r =.5 is depiced on Figure 5(a). For r < 1 + 51, J r = {1+δ+ε, 1, 1, 1 ε, 1+δ, 1 ε, 1,1, +ε+δ, 1, 1, 1 ε, 1+ δ, 1 ε, 1, 1, + ε + δ,... }, where δ = r r and ε = 1 k. This schedule is depiced on Figure 5(b). For 1+ 51 r <, J r = {1 ε, 1, 1, δ, 1, 1, 1, δ,..., where ε = 1 k and δ = r. For i r < i + 1, i =,,..., J = {1 ε, 1, 1 + ε, 1 1, 1+δ ε, 1, 1+ε, 1,, 1, 1+δ ε, }, where ε = 1 k and δ = r r. For i+1 r < i+, i =,,..., J = {1 ε, 1, 1+δ + ε, 1, ε, 1, 1+ δ +ε,... }, where ε = 1 k and δ = r r. For i + r < (i + 1), i =,,..., J R = (1 ε, 1, 1, δ, 1,, 1, δ, ), J R = {1 ε, 1, 1, δ, 1,, 1, δ, }), where ε = 1 k and δ = r r. V. CONCLUSION We invesigaed he design of efficien ani-jamming schedules for wireless daa disribuion sysems. For such schedules, waiing ime and saleness are he key performance parameers. (7) The goal of he jammer is o induce large delays in daa ransmission and o increase he saleness of he daa by forcing he schedule o ransmi he daa wih high level of redundancy. Our conribuion can be summarized as follows. Firs, we idenify opimal and near opimum jamming sraegies for he imporan class of regular schedules. In such schedules, he same encoding is used for all packes, which simplifies he design of he mobile device and reduces is cos. Nex, we provided lower and upper bounds on he performance of more general class of non-regular schedules. Our resuls esablish a rade-off beween he expeced waiing ime of he clien and he saleness of he informaion in he presence of a jammer. As a fuure research, we inend o exend our resuls o he case in which he broadcas channel is shared by wo or more informaion sources. We also would like o invesigae he performance of random ani-jamming schedules for wireless daa broadcas. REFERENCES [1] T. Imielinski, S. Viswanahan, and B.R. Badrinah. Daa on Air: Organizaion and Access. IEEE Transacions on Knowledge and Daa Engineering, 9():5 7, 1997. [] D. Cherion. Disseminaion-Oriened Communicaion Sysems. Technical repor, Sanford Universiy, 199. [] S. Acharya, R. Alonso, M. 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