Towards Mobility as a Network Control Primitive

Transcription

1 Towards Moblty as a Network Control Prmtve Davd K. Goldenberg Dept. Comp. Sc. Yale Unversty New Haven, CT 6517 davd.goldenberg@yale.edu Brad E. Rosen Dept. Comp. Sc. Yale Unversty New Haven, CT 6517 brad.rosen@yale.edu Je Ln Dept. Elec. Eng. Yale Unversty New Haven, CT 6517 je.ln@yale.edu Y. Rchard Yang Dept. Comp. Sc. Yale Unversty New Haven, CT 6517 yry@cs.yale.edu A. Stephen Morse Dept. Elec. Eng. Yale Unversty New Haven, CT 6517 as.morse@yale.edu ABSTRACT In the near future, the advent of large-scale networks of moble agents autonomously performng long-term sensng and communcaton tasks wll be upon us. However, usng controlled node moblty to mprove communcaton performance s a capablty that the moble networkng communty has not yet nvestgated. In ths paper, we study moblty as a network control prmtve. More specfcally, we present the frst moblty control scheme for mprovng communcaton performance n such networks. Our scheme s completely dstrbuted, requrng each node to possess only local nformaton. Our scheme s self-adaptve, beng able to transparently encompass several modes of operaton, each respectvely mprovng power effcency for one uncast flow, multple uncast flows, and many-to-one concast flows. We provde extensve evaluatons on the feasblty of moblty control, showng that controlled moblty can mprove network performance n many scenaros. Ths work consttutes a novel applcaton of dstrbuted control to networkng n whch underlyng network communcaton serves as nput to local control rules that gude the system toward a global objectve. Categores and Subject Descrptors C..1 [Computer-Communcaton Networks]: Network Archtecture and Desgn wreless communcatons General Terms Algorthms, Performance, Desgn Keywords Moblty Control, Routng, Self-Confguraton n Ad Hoc Networks 1. INTRODUCTION As technology rapdly progresses, dverse sensng and moblty capabltes wll become more readly avalable to devces. For ex- Permsson to make dgtal or hard copes of all or part of ths work for personal or classroom use s granted wthout fee provded that copes are not made or dstrbuted for proft or commercal advantage and that copes bear ths notce and the full ctaton on the frst page. To copy otherwse, to republsh, to post on servers or to redstrbute to lsts, requres pror specfc permsson and/or a fee. Mobhoc 4 May 4-6, 4, Roppong, Japan. Copyrght 4 ACM /4/5...$5.. ample, many modern moble robots are already equpped wth varous sensng capabltes. As another example, there are presently research actvtes on low-power robotc nsects that move n a varety of ways, ncludng flyng and skmmng across the surface of water (e.g. [1]). Once moblty becomes feasble, we envson that large systems of such moble autonomous agents performng varous mportant tasks wll be soon to come. Communcaton wll undoubtedly be one of the essental functonaltes of these moble networks. The objectve of ths paper s to explore the novel capablty of these networks to optmze ther communcatons usng controlled moblty. One can envson many settngs n whch moblty can potentally be used to mprove network communcatons. One such scenaro s a long term buggng deployment of self-organzng moble sensors whose purpose s to ntercept or record, and then report as much data as possble from a target such as an enemy communcaton tower or command center. If the sensor nodes are able to move nto postons that mnmze the energy cost of reportng ths stream of data out of the network, the amount of useful nformaton the network can transport would be maxmzed. Smlar arguments for moblty can apply to long-term concast data gatherng [6] or to aggregaton of large data events n a GHT []. One can also magne moble networks beng unformly deployed over space wth the ntenton that when a large, geographcally dspersed user such as a mltary dvson moves n and sets up a base, the network wll adapt ts confguraton n order to best serve the specfc communcaton demands of that user. Such adaptve wreless networks wth the capablty to autonomously algn themselves to ft user needs would be tremendously useful. In general, long-term deployments whch exhbt persstent or habtual communcaton patterns are prme canddates for the applcaton of moblty to mprove network performance. In such settngs, the traffc wll be regular enough and hgh enough n volume to warrant nodes expendng energy movng n order to more cheaply forward traffc. There may also exst stuatons where the power source for moblty s renewable but separate from a non-renewable power source for communcatons. Such stuatons could exst n hybrd bo-electronc systems, the smplest example of whch s a network of people carryng small rados runnng on unrechargeable batteres. A more fancful example s a system of smple lvng organsms such as nsects that are outftted wth rado transmtters and whose moton s controlled by a neuro-electronc nterface. In lght of the fact that moblty s a capablty already perfected by nature, whle wreless communcaton s a human work-n-progress, ths type of technologcal separaton of dutes mght have ts merts. 163

2 Whle the prevous dscusson has motvated some of the potental applcatons of controlled moblty, there are stll few studes n the moble networkng lterature on mprovng communcaton performance through ths capablty. Although moblty has the potental to mprove network performance n many settngs, there may also exst scenaros n whch moblty wll be less effectve due to varous extenuatng factors ncludng hardware lmtatons and traffc patterns. The objectves of ths paper, therefore, are to 1) analyze when controlled moblty can mprove fundamental networkng performance metrcs such as power effcency and robustness of communcatons; and ) provde ntal desgn for such networks. One major ssue n usng moblty s how to effectvely control t. Desgnng moblty control algorthms for communcatons s challengng, because any scheme that would acheve the apparent potental of the dea should address the followng ssues. Frst, the precse nature of any effectve moblty control wll be applcaton dependent. It s clear that nodes wll need to move dfferently under dfferent traffc patterns, e.g., a sngle source-destnaton par (called a sngle flow), multple sourcedestnaton pars (called multflows), or multple sources and sngle destnaton (called concast). Is there a sngle self-adaptve moblty control scheme that can be appled to such a broad range of scenaros? Second, for scalablty and robustness purposes, there should not be a central entty who computes the movements of all the nodes. In other words, the moblty-control scheme should be a totally dstrbuted scheme. Is there a moblty control scheme such that although each node makes movement decsons for tself nformed by purely local nformaton, the collectve system acheves desrable global propertes? Thrd, the dstrbuted moblty-control scheme should be able to self-organze the nodes to optmze a performance metrc whle at the same tme satsfyng other constrants. For example, although one major objectve of a moblty-control algorthm could be to optmze data reportng power effcency after target detecton, t may be mportant that the network mantan connectvty and/or coverage throughout the operaton of the dstrbuted algorthm. Can we desgn a general moblty-control scheme possessng the flexblty to optmze communcaton performance whle smultaneously conformng to user-mposed connectvty/coverage requrements? The framework proposed n ths paper s the frst attempt to desgn and analyze a system addressng the above ssues. The foundaton of our system s self-organzng capablty s a dstrbuted descent prmtve. One nspraton for ths prmtve s the dstrbuted averagng algorthm used n [13, ]. The averagng algorthm of Rao et al. [] operates n vrtual space; our system subsumes such averagng as a specal case and operates n physcal space. Another nspraton of our prmtve s the rendezvous algorthm proposed by Ln, Morse and Anderson [17]. The objectve of the rendezvous algorthm s to have all nodes n an arbtrary connected network converge to a sngle pont n space by usng unform, dstrbuted, and locally nformed moblty control rules. In ths paper, we generalze elements of the two algorthms to desgn a powerful and self-organzng prmtve that can acheve dverse confguraton goals and that can be gracefully tuned to ensure desrable network propertes such as connectvty, coverage, and power effcency. We apply our moblty-control prmtve to a broad range of traffc scenaros, under dfferent applcaton requrements. For each scenaro, we present and formally prove the correctness of our algorthm. We perform extensve smulatons to evaluate the effectveness of controlled moblty. Our evaluatons show that there are many scenaros where moblty control can acheve substantal performance gans. For example, n a random network, we smulate a realstc scenaro n whch 1 Kbps voce stream data flows over a sngle 1 Km long greedly routed multhop path of moble nodes capable of movng at around.1 m/s. In under a mnute, our moblty control s able to gude the network to ts optmal routng confguraton n whch communcaton uses as lttle as 5% of the energy orgnally requred. Takng nto account the cost of moblty, total energy savngs are realzed after fve mnutes. The remander of ths paper s organzed as follows. In Secton, we dscuss related work and compare our approach wth prevous approaches. In Sectons 3, 4, 5 respectvely, we present algorthms for optmzaton of sngle source-destnaton par, multple sourcedestnaton par, and many-to-one traffc patterns. In each secton, we nclude extensve smulaton results to analyze when controlled moblty s effectve and demonstrate the effectveness of the presented algorthm. We conclude and dscuss future work n Secton 6.. RELATED WORK Although moblty has been extensvely nvestgated n the moble networkng communty, the focus so far has been on random moblty, e.g., [16, 14, 5], nstead of controlled moblty. For example, n [7], Grossglauser and Tse have shown that the random movement of users can be used to mprove network throughput. In [] Chakraborty, Yau and Lu have studed algorthms that try to predct user movements to reduce power consumpton. Controlled moblty s an actve research area n the control theory communty; for example see [1]. In the last few years much progress has been made n desgnng dstrbuted moble systems and understandng both natural and artfcal moble systems. The focus of these studes, however, s not on network communcatons. For example, n [3], Cortes et al. have shown that moblty can be purposefully controlled to mplement network coverage; n [15], Ladd et al. have shown that moblty can be used to mprove the accuracy of network localzaton; n DARPA s self-healng mnefeld project [5], moblty s used to mprove and mantan network coverage. However, none of these studes consders routng or power effcency, two of the fundamental ssues n networkng and communcatons. Some nspraton for ths work came from the averagng algorthm, whch s used n varous settngs, e.g., [13, ]. Wth the ntenton of provdng coordnates over whch to perform geographc routng, n [] Rao et al. let vrtual postons of nodes converge to the potental energy mnmzng confguraton of an equvalent network wth edges replaced by sprngs. In the same way n whch the converged vrtual confguraton of Rao et al. reflects the underlyng connectvty of the network, our resultng physcal confguraton reflects the connectvty of the porton of the network n actve use.e. the communcatng subgraph. However, there are several major dfferences between our work. Frst, our system operates n physcal space; thus we must guarantee that connectvty s preserved throughout the actual motons of the nodes. Our nodes also move to a mnmum potental confguraton, but ths tme wth sprng potentals assgned only to lnks actvely beng used for communcaton. Lastly, the more general potental functons we mnmze are equvalent to the communcaton energy usage of a confguraton. Because of ths, rather than averagng, we generalze to a weghted descent method that optmzes realstc transmsson cost models and weghts neghbors accordng to ther share of local communcaton volume. Furthermore, our algorthms make no assumptons about the global traffc pattern, wreless envronment, or hardware power usage propertes. Another nspraton of ths work s the rendezvous algorthm of Ln, Morse, and Anderson [17]. They descrbe dstrbuted local algorthms for gudng a system of multple nodes to a sngle pont. In ths paper, we combne deas from the rendezvous algorthm wth the generalzed averagng scheme to desgn a powerful and flexble tool that can acheve power optmzng confguratons and be 164

3 gracefully tuned to ensure desrable network propertes such as connectvty, coverage. There s a large lterature on power-effcent topology control and routng; for example see [, 19]. A major dfference between our scheme and the prevous work s that we leverage moblty whle the prevous work assumed that moblty cannot be controlled by the communcaton layers of the network. As a precedent to our work, there was also a prevous study [3] on the optmal postons of relay nodes for a sngle source-destnaton par. Under a lnk cost model of P (d) = a+bd α, where d s dstance, α a constant between and 6, and a and b other constants, Stojmenovc and Ln [3] show that over all multhop paths, straght paths are most energy effcent and further that there s a unque hop count for any dstance that mnmzes the cost of communcatons. However, the focus of [3] s not on reachng the optmal confguraton usng a dstrbuted algorthm. Also, the focus of [3] s on a sngle source-destnaton par whle we consder multple source-destnaton pars whle mantanng connectvty durng moblty control. 3. MOBILITY CONTROL FOR NETWORK WITH A SINGLE FLOW We frst present our moblty control algorthm for a network wth a sngle actve flow. Ths s a smple yet mportant applcaton scenaro. We make the followng assumptons. We assume that a path from the source to the destnaton consstng of nodes wth a moblty capablty s dscovered usng a routng protocol, e.g., a greedy routng protocol or one of the ad hoc routng protocols. We label the nodes from source to destnaton, 1,..., n + 1. We call nodes 1,..., n relay nodes. We assume that there s a lnk between two nodes ff ther dstance s less than a maxmum communcaton radus r. We also assume that the relay nodes know ther postons. Ths can be acheved through ether GPS [9] or some localzaton methods. Note that the above assumptons can be further relaxed; however, to make our moblty control scheme clear, we do not pursue these relaxatons. Fnally, our moblty control algorthm s orthogonal to the routng dscovery protocol. 3.1 Optmal Confguraton of Relay Nodes The objectve of the relay nodes s to move to a new confguraton to optmze network performance. We assume that the source and destnaton do not move as they are not relay nodes. Ths s reasonable n many applcaton scenaros n whch the source s reportng the results of some sensng task or gong about some other dutes, whle the relay nodes are n fact relay nodes because ther energy s well spent helpng the source to communcate wth ts destnaton. We expect that the source or the destnaton could also be movng, thus requrng moble trackng. For ths case, we expect that our moblty scheme s stll guaranteed to mantan a multhop communcaton lnk between them as long as the movng speed s below a threshold. Wthout connectvty/coverage constrants, the optmal confguraton of the relay nodes depends on the cost model of communcatons. One way to derve communcaton cost as a functon of lnk dstance s to use a lnk loss model, e.g., [4, 6]. If a node transmts to another node at dstance d away, takng nto account the loss rate of the lnk and mnmzng the expected energy cost to send one message, we have that the transmsson power functon s P (d) = mn ω{e [ω/s(ω, d)]}, where S(ω, d) s the success rate assocated wth transmttng a message at power ω over a dstance d. We assume that a message s successfully receved precsely n the case that the sgnal-to-nose rato at the recever s hgher than a certan threshold. Under varous realstc probablty dstrbutons on nose, we can prove that the power functon P (d) s a non-decreasng convex functon of d. As a result, the followng theorem becomes applcable: THEOREM 1. Assume that the energy cost functon P (d) s a non-decreasng convex functon. Then the optmal postons of the relay nodes must le entrely on the lne between the source and destnaton. Furthermore the relay nodes must be evenly spaced along the lne. PROOF. Let d be the dstance from node to node + 1, where =,..., n. Let D denote the drect lne dstance from the source to the destnaton. Snce P (d) s a non-decreasng convex functon, we have n = P (d) (n + 1)P ( n= d ) (n + 1)P ( D ), n+1 n+1 where the frst nequalty s due to the convexty of P (d), and the second one holds because P (d) s non-decreasng. 3. Moblty Control to Reach Optmalty: the Synchronous Scheme The prevous subsecton has establshed that the optmal confguraton of the relay nodes s lyng evenly on the lne from the source to the destnaton. We now ntroduce a unform dstrbuted algorthm that allows the relay nodes to move to ther optmal postons. x : current poston of node. x 1 and x +1 : postons of nodes 1 and + 1. g (, 1]: dampng factor. repeat send x to neghbors 1 and + 1 receve x 1 and x +1 set x = (x 1 + x +1 )/ move to x + g (x x ) untl (convergence) Fgure 1: The dstrbuted, synchronous moblty-control algorthm at relay node. Node 1 and + 1 are ts neghbors on the routng path. Fgure 1 shows a dstrbuted moblty control algorthm. The algorthm proceeds n globally synchronous rounds of maneuverng alternatng wth quescence. The key ngredent of the algorthm s the smple averagng step, whch we wll extend for more complex scenaros and call the target pont prmtve. Note that although a node computes the average of ts two neghbors, the node only moves toward ths pont, nstead of reachng t n one step. In other words, the movement s damped. In some confguratons, wthout ths dampng, oscllatons can occur that nflate the total dstance traveled by the nodes before convergence. Dampng s also useful as a tool for avodng node overreacton to ephemeral traffc by settng the tme scale over whch convergence takes place to be suffcently large. Next, we prove that our moblty control algorthm has the essental property that connectvty between communcatng neghbors s never broken. Ths property ensures that throughout maneuverng, the communcaton functonalty of the path s never compromsed and that each neghbor always has contact wth ts two neghbors necessary for computaton of a target pont. Furthermore, ths property can avod the cost of re-routng, whch can be a major source of overhead for many routng protocols n moble networks. X X X 1 X 1 X 1 X X 1 X Fgure : Illustraton of Theorem. 165

4 THEOREM. Connectvty between communcatng neghbors s not lost n the synchronous algorthm. PROOF. Wthout loss of generalty, suppose that node 1 at poston x 1 has as communcatng neghbors nodes and at postons x and x. Fgure shows the nodes before all nodes on the path move. Let x j denote the mdpont 1 (x + xj) of the lne between node and node j. Node 1 then moves to poston x 1 = x. We see that x 1 x 1 = 1 (x1 x) 1 r, where r s the communcaton radus. The new poston of node 1 s wthn half a communcaton radus away from x 1 and smlarly, x 1. Analogous statements must hold for ts neghbors as well: x x 1 r/, and x x 1 r/, so all new postons are wthn a dstance of r from each other. Snce nodes move along straght paths to ther target ponts, we have shown that connectvty s guaranteed throughout the maneuverng perod. We next establsh the convergence of our algorthm; that s, we prove our algorthm always termnates. Here we defne termnaton as arrval at a confguraton n whch all relay nodes are evenly spaced on the lne between the source and destnaton. THEOREM 3. Usng the dstrbuted synchronous algorthm, all nodes wll eventually be evenly dstrbuted on the lne between the source and destnaton. PROOF. Let x (k) denote the poston of node after k steps of the algorthm have completed. We wll assume unless otherwse stated that {1,,..., n}. The update equaton for x s gven by x 1(k) + x+1(k) x (k + 1) = x (k) + g[ x (k)] for k {, 1,...} and dampng factor g (, 1]. We let x = x + (xn+1 x) be the -th evenly spaced n+1 pont on the lne between x and x n+1. We wll show that ths s the locaton to whch node converges. Defne the error at step k as e (k) = x (k) x. Observng that x = 1 ( x 1 + x+1) = (1 g) x + g ( x 1 + x+1), we have that for {, 3,..., n 1}, e (k + 1) = x (k + 1) x = (1 g)x (k) + g (x 1(k) + x+1(k)) x = (1 g)e (k) + g (e 1(k) + e+1(k)). As for e 1 and e n, a smple calculaton reveals that and e 1(k + 1) = (1 g)e 1(k) + g e(k) e n(k + 1) = (1 g)e n(k) + g en 1(k). Defne the error vector e = (e 1, e,, e n) T. It follows that e(k + 1) = T e(k) = T k+1 e(), where 1 g g/ g/ 1 g g/ T = g/ 1 g g/ g/ 1 g n n and can be rewrtten as I + gm, where M = 1 1/ 1/ 1 1/ / 1 1/ 1/ 1 n n. Because M s symmetrc, ts egenvalues are real. From the Gerschgorn Crcle Theorem, ρ(m), where ρ(m) s the largest egenvalue of M. Smple calculaton reveals that nether nor s an egenvalue of M; thus < ρ(m) <. It follows from ths and the fact that g 1 that 1 1 g < ρ(t ) < 1 g 1,.e ρ(t ) < 1. It follows from a standard result n the theory of matrx products [1] that lm k T k =. Ths mples that lm k e(k) =, thereby establshng that the algorthm converges to a confguraton of nodes evenly spaced on the lne between x and x n Moblty Control to Reach Optmalty: the Asynchronous Scheme Whle the smplcty and functonalty of the synchronous algorthm s appealng, the globally synchronous mode of operaton s at odds wth the need for dstrbuted algorthms that do not requre any global nformaton. To remedy ths volaton of the localzed desgn requrement, we present an asynchronous algorthm, shown n Fgure 3, whch uses no global nformaton and requres only that each node eventually reach ts target pont n bounded tme. x : current poston of node. x 1 and x +1 : postons of nodes 1 and + 1. g (, 1]: dampng factor. repeat send x to neghbors 1 and + 1 repeat lsten() untl (L R == T rue) send movng to neghbors 1 and + 1 set L := F alse, R := F alse set x := (x 1 + x +1 )/ move toward x + g (x x ) repeat lsten() untl (arrve n bounded delay) untl (convergence) subroutne lsten(): upon receve x 1 do L := T rue upon receve x +1 do R := T rue upon receve movng 1 do L := F alse upon receve movng +1 do R := F alse L and R: nternal boolean state varables. movng : message sgnalng node startng to move. Fgure 3: The dstrbuted, asynchronous moblty-control algorthm at node, where =,..., n 1. The algorthm outlned n Fgure 3 defnes the operaton of all relay nodes other than the two nodes 1 and n respectvely connected to the source and destnaton. Node 1 has ts state varable L permanently set to T rue and node n has R permanently set to T rue. Other than ths, the operaton of nodes 1 and n s dentcal to that of all the other relays. A state transton dagram descrbng the asynchronous algorthm for nodes through n 1 s shown n Fgure 4. We omt the slghtly dfferent but straghtforward dagram for nodes 1 and n. The system starts n the state n whch nodes are statonary and nformed of the postons of ther neghbors. The state varables M, L, and R respectvely represent the state of movng and of havng fresh poston nformaton for the left and rght neghbors. In the dagram, we abuse notaton a bt and represent by x j the recepton of a message contanng the poston of node j, sgnalng that node j has stopped movng; by m j the recepton of a message ndcatng that node j s movng; and by \y the acton of sendng message y. Snce ths s an asynchronous protocol, one potental concern s that t could cause deadlock. The proposton below shows that f messages can be relably transmtted, then our asynchronous protocol s deadlock free. If messages can be dropped, then we can use 166

5 m 1 x +1 \x \x M M m +1 L R m x 1 x * x +1 L R x 1 x +1 1 m \m +1 m 1 m M M * M M x * L R x 1 x L R L R m +1 1 m +1 L R m m * 1 M m +1 x 1 M x +1 L R L R \x x 1 m +1 \x m 1 x +1 x 1 m +1 Fgure 4: State transton dagram for the asynchronous algorthm: =,..., n 1. a relable transport protocol to guarantee relablty; or f each node n quescent mode perodcally transmts ts poston and a transmsson arrves after fnte number of retransmssons, there wll stll be no deadlock. PROPOSITION 1. The asynchronous protocol s deadlock free, f messages are not dropped. PROOF. Assume the system s deadlocked. Then there s some node that s the last to stop movng, say by tme t. By assumpton, no nodes are movng at tme t. We wll denote the state of a node j by S j and refer to nodes by ther ndex. Wthout loss of generalty, upon node stoppng, R S 1, where 1 s the left neghbor of. It must be the case then that S 1 = { L, R} or else 1 would be able to move, volatng our assumpton of deadlock. Node 1 s left neghbor must have stopped movng strctly before 1 dd, or else L S 1. Snce 1 stops movng after, R S. But by the same argument as before, S = { L, R}. As ths process contnues as shown n Fgure 5, we reach the node k whose left neghbor s the source and conclude that S k = { L, R}. But by our algorthm desgn, L S k and we have a contradcton. Hence, the system cannot be deadlocked. S source X... L R L R L R L _ LR k 1 +1 Fgure 5: Illustraton of deadlock proof contradcton. Node must have at least one of L and R as F alse. We next prove that no node loses communcatng neghbors. THEOREM 4. Connectvty between communcatng neghbors s not lost n the asynchronous algorthm. PROOF. At the nstant that node starts movng towards the average of ts neghbors, ts neghbors must be statonary, or else t would have nvaldated at least one of ts state bts preventng tself from leavng. As node s movng, nether of ts neghbors can move, snce they nvaldated a state bt upon s departure. At the moment node stops, t s clear that t wll be wthn dstance r/ from ts neghbors. Fnally we prove that the asynchronous algorthm wll also converge to an evenly spaced straght confguraton. Ths proof uses a new proof technque smlar to that used n [17]. The translaton of the algorthm nto an manageable mathematcal model depends crucally on the nvaldaton of a state bt upon recevng a movng t... sgnal from a neghbor. Once the model s set up, the convergence result s a drect consequence of the assumptons that the source and destnaton are fxed and that nodes reach ther targets n bounded tme. {, j} j t 1 t t 3 t 4 t 1 t j1 t t j t j3 Fgure 6: Illustraton of event tme. tme tme tme Before we begn our proof, we frst establsh a prelmnary concept. We defne an event tme to be any real tme t k at whch some node begns to move, where k ndcates that t s the k-th tme that node starts to move. Deletng duplcates, we now arrange the set of all event tmes { t k : {1,,..., n}, k 1}, n ncreasng order and label the ordered tmes by t s, s {1,,...}. For {1,,...}, let s (k) denote the value of s for whch t s = t k.e., the ndex of the event tme at whch node moves for the k-th tme. Because s corresponds to an event tme at whch node starts movng, s s sad to be n the mage of s, Im{s }. Ths s llustrated n Fgure 6 where s (1) = 1, s () = 3, s j(1) =, s j() = 3, s j(3) = 4, Im{s } = {1, 3}, and Im{s j} = {, 3, 4}. THEOREM 5. Usng the asynchronous algorthm, all nodes wll eventually be evenly spaced on the lne between source and destnaton, gven that there s an upper bound on the tme t takes for a node to move to ts target pont. PROOF. (sketch) For the sake of brevty, we wll not nclude the complete proof but only a sketch. We set the dampng constant g = 1 for clarty. We wll frst defne a dscrete model that descrbes the poston of each moble node ndexed by event tme. At the event tme ndexed by s, every node s ether startng to move (f s Im{s }), n the process of movng, or statonary. We defne x (s) as the next restng poston of node after t s n the followng way. If at tme t s, s startng to move or n the process of movng, x (s) represents ts target pont. If s statonary at t s, we defne x (s) to represent the poston at whch t s restng. Ths state varable x (s) was carefully desgned n conjuncton wth the algorthm to guarantee accurate representaton of algorthm behavor as well as convergence. If node does not start movng at event tme s, then x (s) = x (s 1): The next restng locaton of s the target of ts last moton. Otherwse, the update equaton for x (s) s gven by and x 1(s) = 1 (x + x(s 1)), x n(s) = 1 (xn+1 + xn 1(s 1)), x (s) = 1 (x 1(s 1) + x+1(s 1)), for {, 3,..., n 1}. Note that the desgn of our algorthm ensures that f s Im{s }, nodes 1 and + 1 may not be movng at tme t s. Assumng s Im{s }, then wthout loss of generalty, f node 1 was movng at tme t s 1, then by desgn, node 1 must have come to rest at poston x 1(s 1) by tme t s. If node 1 was statonary 167

6 at tme t s 1, then at tme t s, ts poston wll be x 1(s 1), as by the defnton of event tme, t could not have moved n-between consecutve event tmes. We can see now that x (s), whch s the target of node s ncpent moton, s the average of ts neghbor s postons at the nstant t departs. Ths representaton s possble only because the algorthm dctates that each node nvaldate the poston of ether of ts neghbors who starts movng. Next,we normalze the dstance between source and destnaton n+1 to be one, and defne our equlbrum postons x =. We defne the error as e (s) = x (s) x, and set up the error equaton evolvng on the sequence of event tmes as we dd n the convergence proof of the synchronous algorthm. As before, we can put the error update equatons nto matrx form e(s) = M(s)e(s 1), where e = [e 1, e,..., e n]. In ths case however, nstead of the update matrx havng a sngle form, M(s) s now a matrx defned at an event tme n a way whch depends on exactly whch nodes begn to move at that event tme. Ths update matrx has the followng form. If node [1,..., n] does not start movng at event tme s.e., f / Im{s }, row appears as t does n the n n dentty matrx. If node does n fact move, row appears as t does n the followng n n matrx: 1/ 1/ 1/ / 1/ 1/ n n At event tme s, the error s gven by the product of s matrces of type M(s) appled to the ntal error vector e(). In our full proof, we show that after a fnte tme, the maxmal row sum or reduced nfnty norm of ths product has an upper bound strctly less than one precsely because the source and destnaton are fxed and every node reaches ts destnaton n fnte tme. Consequently, e(s) as s and the state varables and therefore the postons of all nodes converge to ther desred locatons evenly spaced on a lne between source and destnaton. 3.4 Mantanng Connectvty for Non-Communcatng Neghbors In the prevous two subsectons, we have shown that both the synchronous and asynchronous algorthms guarantee that communcatng neghbors are never dsconnected. However, t s possble that as a communcatng node moves towards ts optmal locaton, t becomes dsconnected from some of ts non-communcatng neghbors. In networks where preservng all connectvty s mportant, we can ntroduce a smple constrant on the moton of nodes to guarantee permanent connectvty to all nodes connected to t, ether communcatng or non-communcatng. We call the moblty control algorthm wthout ths constrant unconstraned moblty control, and the algorthm wth ths constrant constraned moblty control. Specfcally, the constrant s that a communcatng node does not move beyond the maxmum communcaton range away from any of ts non-communcatng neghbors. Ths means that the communcatng node moves to the pont closest to ts target pont that satsfes all constrants mposed by non-communcatng neghbors. 3.5 Evaluatons We now evaluate the performance of the moblty control algorthm Smulaton Setup We have mplemented a smulator to evaluate the performance of our moblty-control algorthms. The smulator generates nodes unformly at random, and then randomly chooses a source and a destnaton. Next, t runs the greedy geographc routng protocol to locate. a routng path. The nodes on ths routng path then move to decrease the energy usage of the path usng our synchronous moblty control protocol. Our statstcs regardng network power consumpton are obtaned by runnng our algorthm on 5 dfferent random network nstantatons for each combnaton of parameters. It s worth notng that n unformly random networks, the performance mprovements through moblty are mnmzed because the paths chosen by greedy routng already tend to approxmate ther optmal straght confguraton. Moblty n ansotropc networks or networks wth geographc routng holes wll yeld greater performance mprovements than we see here, so ths study serves to delneate the baselne of the potental performance enhancements offered by moblty. A key ngredent of the smulator s the communcaton cost model. We assume that the cost of transmsson of a sngle bt over a dstance d s P (d) = a + bd α, where α s between and 6, and a and b are constants. Ths s a commonly used power functon [1] where the values of a and b depend on the hardware and algorthms used for transmsson, recepton, decodng, and encodng. Typcal values, whch we adapt for use here, are a = 1 nj and b =.1 nj/m [8] for a path loss model of α =. For a general path loss model of α [, 6], n order to acheve the same recever sgnal-to-nose rato as for α =, we must transmt at a proportonately hgher power P d α. Therefore, we use parameters a = 1 nj, b =.1 nj/m α, and α [, 6]. By the results of [3], communcaton under ths cost model s acheved wth least power expendture n a multhop fashon wth hop length (1/(α 1)) 1/α m. Hence, n order to nterpret our smulaton results realstcally, we scale our smulaton dstances so that the hop lengths typcally used are of ths order of magntude. Another key ngredent of our smulaton setup s the cost of moblty. We choose to use a dstance proportonal cost model P m(d) = kd. A dstance proportonal cost model s reasonable for wheeled vehcles, where the energy used to accelerate can typcally be recovered upon brakng, neglectng losses due to frcton. It s possble that flyng, floatng, and swmmng vehcles may have to overcome larger fxed energy costs to ntate moton and less to mantan t, but we do not consder these detals here and abstract to the dstance proportonal cost model. So as not to overestmate the potental beneft of moblty, the values of k that we consder are kept conservatvely large; rangng from.1 J/m to 1 J/m. A one klogram wheeled vehcle wth rubber tres movng on concrete must overcome a.1 N force of dynamc frcton, or expend.1 J/m [4], so an energy cost of 1 J/m does not seem unrealstc Smulaton Results Before we report quanttatve results, we frst present several fgures to vsually llustrate the effectveness of the moblty control algorthm. Fgure 7 shows network confguratons before and after moblty control. In ths experment, we use both greedy routng and stngy routng to fnd an ntal routng path. Stngy routng s a form of routng that pcks the neghbor whch makes the least forward progress [11]. The proved convergence of our moblty control algorthm to the straght and evenly spaced lne s corroborated by these smulatons. Snce our algorthm converges from all ntal confguratons and requres only local nformaton, t s robust aganst both ncreases n network sze and hghly rregular paths such as those produced by stngy routng. The converged confguratons and Theorems 3, 5 have only shown that our moblty control algorthm wll move the relay nodes to the optmal confguratons. However, a moblty control algorthm could move the nodes along arbtrarly long curves, thus consumng much energy for moblty. We defne blowup as the rato between the dstance that a node actually travels between ts ntal and fnal postons and the straght lne dstance. We observe n Fgure 8 that the path blowup for greedy path optmzaton s small, ndcatng that our algorthm does not suffer from large oscllatons. In Fgure 8, MaxMove s a parameter that expresses the maxmum speed of 168

7 (a) unconstraned; greedy (b) constraned; greedy (c) unconstraned; stngy Fgure 7: Network confguratons before and after moblty control. The frst term of each subcapton ndcates whether moblty control s constraned or unconstraned. The second term ndcates the routng protocol used to fnd an ntal routng path. Percent Improvement Average Blowup MaxMove Unconstraned Constraned Fgure 8: Blowup of network path. node moblty by mposng an upper lmt on dstance traveled by a node per round. As one would expect, the path blowup of unconstraned optmzaton s greater than t s n constraned optmzaton; however, the value s stll small. Overall, the small blowup factor of our algorthm ndcates that our algorthm consumes close to optmal energy on moblty. Havng establshed the convergence and small blowup factor of our algorthm, now we evaluate whether moblty control can mprove the power effcency of a routng path. We evaluate ths under the cost model [8] P (d) = d 3 and P m(d) = kd, n Joules, where d s n meters, scaled from our smulaton as descrbed earler. Note that we are comparng the power usage of greedy routng paths before and after moblty throughout. Stuatons n whch greedy routng paths could not be found were dscarded n order to produce a baselne evaluaton of moblty, despte the fact that t s n precsely those dscarded cases that moblty wll perform the best neghbors 8 neghbors 1 neghbors neghbors Path Loss Exponent (a) unconstraned Percent Improvement neghbors 8 neghbors 1 neghbors neghbors Path Loss Exponent (b) constraned Fgure 9: Performance mprovement of moblty control under dfferent α values. We frst evaluate the effectveness of moblty control for a wde range of communcaton envronments. We control ths effect by varyng the value of α, the exponent of path loss and power dependency on dstance. Fgure 9 shows the performance mprovements of unconstraned and constraned moblty control. The x-axs s the exponent whle the y-axs s the percentage mprovement computed as 1 (E E m)/e, where E s the energy cost of the routng path before moblty control and E m s the energy cost of the path after moblty control. For constraned moblty control, as we ncrease α from to 6, the mprovement s ncreased from around 1% to around 5%, translatng nto a potental mprovement n lfetme of from 1% to 1%. The performance mprovement when there s no connectvty constrant s even hgher (note that the connectvty of communcatng neghbors s always mantaned). We observe that the typcal performance mprovement when there s no connectvty constrant almost doubles that wth the connectvty constrant. One concluson we can draw s that moblty control wll be more effectve n mprovng power effcency for larger values of α. Percent Improvement Unconstraned Constraned Average Number of Neghbors Fgure 1: Effect of node densty (number of neghbors) on performance mprovement. Network densty, expressed through average number of neghbors, also plays a role n the effectveness of moblty control. Fgure 1 shows the result for α = 3. We observe that wth ncreasng densty, the performance mprovement decreases. Ths s as expected gven that as densty ncreases, greedy routng fnds paths that more closely approxmate the straght lne. Thus one concluson we can draw s that moblty control wll ncrease n effectveness as the densty and regularty of networks decreases. As network densty ncreases, t s lkely that the requrement that relay nodes not lose connectvty wth ther statc neghbors wll become less essental. Ths may justfy the use of unconstraned moblty, whch produces robust energy savngs even for hghly dense networks. The prevous results evaluate only the total energy consumpton of a path. However, a path becomes dsconnected f any one of the nodes runs out of battery. From ths perspectve, moblty control has the further advantage that snce one of ts functons s to pro- 169

8 Frequency Dstance moved (a) unconstraned mean: 38.5 var: mn:.51 max: Frequency Dstance moved (b) constraned Fgure 11: Dstrbuton of dstance moved. mean: 9.16 var: mn: max: 41.1 duce paths wth equal hop length, the problem of premature path dsconnecton due to mbalanced power usage along a path s reduced. A potental problem wth ths clam however, s that nodes may move varyng total dstances, thus consumng unequal amounts of moblty energy. In other words, t could be possble that an unequal communcaton burden be reduced at the expense of unequal moblty energy cost. Fgure 11 shows that the dstances traveled by moble nodes are qute balanced. We do not observe the heavy tals that would ndcate some nodes spendng an nordnate amount of energy on moblty relatve to average. Next, we wll summarze the trade-off between moblty energy cost and communcaton energy cost. We assume that nodes communcate wth ther neghbors durng moblty perods as f they are aware of the maxmum dstance to ther neghbor over the perod and send at the exact power requred. In a realstc settng however, ths s not possble and nodes may do one of two thngs. Frst, they may send traffc to a neghbor durng moblty perods as f t s r/ + d unts away, where d s the separaton from the neghbor durng the prevous quescent perod. The alternatve s that nodes send the coordnate of ther target pont before they begn movng. In ths way, every par of neghbors can determne the maxmum potental separaton between them over each moblty perod and send at the approprate mnmum power level. As we wll see, the transent perod before the network converges s relatvely short, so such detals wll not affect the salent propertes of our smulaton results. The total energy usage was calculated usng our prevously descrbed transmsson power model wth a path loss exponent of α = 3 for a 1 Kbps flow 1 along a path of moble relays approxmately 1 Km long. Each maneuverng perod lasts 1 seconds and durng these perods, a node typcally moves no more than a meter, resultng n reasonable speeds of about.1 m/s. We ran a trace of the evoluton of the system under unconstraned moblty and another under constraned moblty. These were dstnct network nstantatons, resultng n the dsparate total power usage. Our results are shown n Fgure 1. The slope of the lnes s the energy used per MB. We can see that the slope of the lne correspondng to the statc network s always greater than the slope of the lnes correspondng to the moble networks, as expected. Note that also consstent wth earler smulaton results, the slope of the constraned moble traces shown n part (b) exhbts proportonately less decrease from the statc case than the unconstraned moble traces shown n (a). The hgher slopes of the moblty traces close to the orgn are due to the energy used on movement before convergence. We can see that the energy usage of the moble network s substantally hgher than that of the statc network n the early stages of ts development, meanng that moblty can ncur hgh energy penaltes f a flow s short-lved. However, there s a number of bts sent after whch the energy usage of the statc network permanently outstrps that of the moble network. For flows sendng more than ths number of bts, moblty saves energy. We plot ths crossng pont n part (c). Clearly, the value of the crossng pont depends on the cost of moblty. As moblty becomes more expensve, the crossng pont becomes larger. 1 1 Kbps s a rate approprate for mnmal voce streamng. Another feature to notce s that the crossng pont s always lower for constraned moblty than t s for unconstraned moblty. Ths s due to the fact that the nodes move less dstance and converge faster to ther fnal postons. So we can see dstnct advantages to constraned moblty besdes the ntended functonalty of preservaton of path connectvty wth statc neghbors. Frst, a performance mprovement s acheved for a smaller amount of traffc sent than n unconstraned moblty. Second, a smaller maxmum energy penalty for prematurely endng flows s ncurred than n unconstraned moblty. The dsadvantage to constraned moblty s of course that the energy savngs to be ganed are more modest than they are through unconstraned moblty. 4. MOBILITY CONTROL FOR NETWORK WITH MULTIPLE FLOWS Controlled moblty can also be appled to a network consstng of multple flows. For nodes that are on the path of only one flow, the averagng algorthm descrbed n the prevous secton s stll vald; that s, n each step a relay node moves to the average poston of ts two neghbors. However, n a network wth multple flows, some relay nodes wll be on the routng paths of multple flows; we call such nodes juncton nodes. Applyng the averagng algorthm from the prevous secton n the presence of juncton nodes can rase several ssues. 4.1 Issues of Multple Flows S 1 S D Fgure 13: Illustraton that movng to average can cause juncton nodes and j to dsconnect, where and j are on the paths from S 1 to D 1 and S to D. The frst ssue that arses n the applcaton of averagng to multflow networks s that juncton nodes may become dsconnected from ther communcatng neghbors. An example of ths s shown n Fgure 13. Nodes and j start out separated by the maxmum communcaton radus. As soon as one of them moves toward the average of ts neghbors, connectvty between them s lost. p q Fgure 14: Effect of α on the optmal confguraton; the pont p s the average of the four empty crcles, whle the pont q s the center of the mnmum enclosng crcle of the four empty crcles. The next ssue s that the optmal poston of a juncton node may not be the average of ts neghbors. Ths s the case because the optmal poston of a node servng as a relay between more than two neghbors depends on the power model. Consder the power model P (d) = d α. For α =, we can show that the energy mnmzng poston for a relay node between multple sources and destnatons carryng equal amounts of traffc s always the average of the postons of ts neghbors. However, ths s only a specal case. For α >, the average s no longer always an optmal soluton. j D 1 17

9 Total Power Used (J) w/o moblty 1. J/m.5 J/m.5 J/m.1 J/m Total Power Used (J) w/o moblty 1. J/m.5 J/m. J/m.1 J/m Bts Sent at Transton (MB) constraned unconstraned Total Bts Sent (MB) (a) unconstraned Total Bts Sent (MB) (b) constraned Moblty Power Constant (J/m) (c) crossng tme Fgure 1: Trade-off between moblty energy cost and communcaton energy cost. Fgure 14 llustrates ths pont wth an example of placng a sngle relay between four nodes. To mnmze total energy usage under a cost functon quadratc n dstance, the optmal pont s p; but as α the optmal pont approaches the pont q whch mnmzes the dstance of the maxmum dstance node. Ths pont q les at the center of the smallest crcle whch contans all the nodes.e., the pont mnmzng the l norm. Keep n mnd however, that for nonjuncton nodes, the average of the neghbors postons s the energy mnmzng pont for all convex energy cost functons, as we showed n the prevous secton. Another reason that the average s not n general the optmal target pont s that a sngle lnk to a neghbor may be on any number of paths, and further, there may smply be dfferent amounts of traffc along dfferent flows. We need to weght the mportance of neghbors by the amount of communcaton done wth them. Note that n a lne topology however, the weghts of neghbors of a relay are equal due to flow conservaton. Thus, n order to defne a general scheme capable of adaptng to these varyng optmalty condtons, we must devse an algorthm that yelds an optmal confguraton under varyng power usage envronments and whch allows flexble weghtng. 4. Moblty Control for Multple Flows To address the above two ssues wth multple flows, we generalze the averagng algorthm n two ways. Frst, for a general power cost functon and number of neghbors, nstead of havng nodes move toward a known mnmzer, as we were dong n the lne topology by movng toward the neghbors average, nodes must now descend along a local-mnmzaton drecton n order to decrease communcaton cost. Our descent drecton algorthm provably converges, but we omt the proof here. For a large class of power functons, the descent drecton can be computed easly usng only local nformaton; even further, such algorthms can be extended to be adaptve to channel condtons such as multpath and nterference. More precsely, assume that the lnk cost functon of a juncton node to ts -th neghbor s P (d ). Assume that the relatve amount of traffc to and from neghbor s w. For example, n Fgure 13, f each source sends the same amount of traffc to ts destnaton, the weght of lnk j s whle the weght of the other lnks s 1. Then the descent drecton can be computed as x = x n x o, where x o s the current poston of the juncton node, and x n can be computed as follows. Let x (k) denote the k-th dmenson of the vector x. We have x (k) n = n =1 n =1 w P d d x (k) w d P d, where x s the poston of the -th neghbor. For the partcular power functon P (d) = a + bd α, we have x (k) n = n =1 wdα n =1 wdα x (k) Note that when α = and all neghbors have the same amount of traffc, ths descent drecton s a constant pontng drectly to the average of the neghbors postons. In a lne topology wth any α >, the optmal confguraton of relays s at the average of ther neghbors but the descent drecton s no longer a constant. Ths results n a curved path leadng to the same eventual destnaton as averagng. Thus the prevous averagng algorthm s just a specal case of ths more general descent algorthm. For α >, nodes must locally follow the descent drecton wth suffcently small step sze to guarantee convergence. For lne topologes however, a small optmzaton s found n smply havng nodes move toward the average of ther neghbors regardless of the value of α, resultng n ther movng along a shorter path before convergence than usng the general descent drecton. To address the dsconnecton ssue, we mpose a parwse constrant on every movng node. The parwse constrant, shown n Fgure 15, holds between two potentally moble nodes that are neghbors. It makes explct the ntuton behnd the dsconnecton mpossblty proofs for the sngle path case by dctatng that n order to stay connected to a neghbor, a node must not move outsde the dsk of radus r/ centered at the mdpont of the lne between tself and that neghbor. A separate parwse constrant s enforced for every moble neghbor, thereby restrctng a node s moton to the ntersecton of a number of dsks. The less restrctve statc constrant that a node must stay wthn communcaton range r of all of ts non-moble neghbors may also be enforced. A node s potentally moble f and only f t s actvely communcatng. Ths parwse constrant n conjuncton wth the statc constrant guarantees that all pre-exstng connectvty s preserved throughout any moblty. S 1 S r/ Fgure 15: Illustraton of parwse constrant: nodes and j may not move outsde of ther respectve shaded regons. THEOREM 6. If the parwse constrant s satsfed, communcatng neghbors wll not become dsconnected. j. D 1 D 171