On the Scalability of Ad Hoc Routing Protocols

Transcription

1 On he Scalabiliy of Ad Hoc Rouing Proocols César A. Saniváñez Bruce McDonald Ioannis Savrakakis Ram Ramanahan Inerne. Research Dep. Elec. & Comp. Eng. Dep. Dep. of Informaics Inerne. Research Dep. BBN Technologies Norheasern Universiy Universiy of Ahens BBN Technologies Cambridge, MA Boson, MA Ahens, Greece Cambridge, MA Absrac A novel framework is presened for he sudy of scalabiliy in ad hoc neworks. Using his framework, he firs asympoic analysis is provided wih respec o nework size, mobiliy, and raffic for each fundamenal class of ad hoc rouing algorihms. Proocols sudied include he following: Plain Flooding (PF), Sandard Link Sae (SLS), Dynamic Source Rouing (DSR), Hierarchical Link Sae (HierLS), Zone Rouing Proocol (ZRP), and Hazy Sighed Link Sae (HSLS). I is shown ha PF and ZRP scale beer wih mobiliy, SLS and ZRP scale beer wih respec o raffic, and HSLS scales beer wih respec o nework size. The analysis provides deeper undersanding of he limis and rade-offs inheren in mobile ad hoc nework rouing. Our analysis is complemened wih a simulaion experimen comparing HSLS and HierLS. An imporan conribuion of his paper is ha HSLS is an scalable, easy-o-implemen, alernaive o hierarchical approaches for large ad hoc neworks. I. INTRODUCTION Rouing proocols for ad hoc neworks have been he subjec of exensive research over he pas several years. Recenly, pracical applicaions such as inelligen sensor neworks have focused aenion on undersanding he issues and radeoffs in nework scalabiliy. An imporan quesion ha arises is : which rouing proocol scales he bes? The ypical answer is: i depends. Unforunaely, he neworking communiy lacks a ene for undersanding he fundamenal properies and limiaions of ad hoc neworks. Hence, a fundamenal undersanding of wha scalabiliy depends on, and how is currenly lacking. One reason for his shorcoming is a lack of sufficien research aimed a general principles and analyical modeling. Scalabiliy and oher performance aspecs of ad hoc rouing have been sudied predominanly via simulaions (e.g. [1], [2], [3]), versus heoreical analyses. Simulaion resuls, alhough exremely useful, are ofen limied in scope o specific scenarios. Thus, hey ofen fail o produce resuls ha provide he deph of undersanding of he limiaions of he proocols and heir dependence on sysem parameers and environmenal facors desired by researchers. The lack of much needed heoreical analysis in his area is due, we believe, in par o he lack of a common plaform o base heoreical comparisons on, and in par due o he absruse naure of he problem. This paper focuses on he developmen of principles and mehodologies for he analysis and design of scalable rouing sraegies for ad hoc neworks. Analyical models are developed and resuls are presened ha provide significan insigh ino he aforemenioned dependency and he general performance characerisics of he mos imporan classes of ad hoc nework rouing algorihms. The heoreical models developed esablish he basis for an unbiased analysis and comparison of he relaive scalabiliy of several proposed rouing proocols. The firs precise (asympoic) expressions reflecing he impac of nework size, raffic inensiy and mobiliy on proocol performance are developed in his paper. Analyical resuls are presened for a represenaive se of sae-of-he-ar proocols in he lieraure, including : no rouing Plain Flooding (PF), proacive Sandard Link Sae (SLS), reacive Dynamic Source Rouing (DSR) [7], hybrid Zone Rouing Proocol (ZRP) [9], hierarchical Hierarchical Link Sae (HierLS) [8], and limied diseminaion Hazy Sighed Link Sae (HSLS) [11], echniques. As such, he resuls provide researchers wih improved undersanding of he limis and rade-offs inheren in ad hoc nework rouing. A significan resul is ha, under he assumpions of his work, HSLS while being easier o implemen scales beer han HierLS and ZRP wih respec o nework size. This analyical resul is validaed wih simulaion analysis comparing HSLS and HierLS. Thus, anoher imporan conribuion of his work is o show ha HSLS is an scalable, more efficien alernaive han hierarchical approaches for rouing in large ad hoc neworks. 1 Despie limied prior relaed heoreical work, here have been noable excepions. In [4] analyical and simulaion resuls are inegraed in a sudy ha provides valuable insigh ino comparaive proocol performance. However, i fails o deliver a final analyical resul, deferring insead o simulaion. Thus, i is difficul o fully undersand he ineracions among sysem parameers. The presen work closes his gap and provides an undersanding of he dynamic ineracion among nework parameers. The asympoic capaciy of a fixed wireless nework was sudied in [5]; however, i did no include rouing overhead. In conras, we focus on oal overhead (defined laer), which includes rouing overhead. The impac of mobiliy on nework capaciy was sudied in [6]. They showed ha given no resricion on memory size and arbirarily long delays, mobiliy increases nework capaciy. This research, however, focuses on pracical scenarios, wherein, delay canno grow arbirarily large and mobiliy reduces he nework capaciy (degrading performance). The remainder of his paper is organized as follows: In Secion-II we characerize he (oal overhead) meric and he nework model used. Secions-III - VIII presen analysis of he asympoic performance of PF, SLS, DSR, HierLS, ZRP, and HSLS respecively. Comparison of proocol performance is discussed in Secion-IX, focusing on HierLS and HSLS under large nework size and including simulaion resuls. Finally, conclusions are presened in Secion-X 1 However, i should be noed ha HSLS requires more memory han HierLS. As nework size increases HSLS s memory requiremens may become he limiing facor.

2 II. MODELING PRELIMINARIES Ths secion presens he model assumpions and definiions employed in our analysis. A. Nework model The following noaion will be uilized in his paper: Le N be he number of nodes in he nework, d be he average in-degree, L be he average pah lengh (in hops) over all source desinaion pairs, be he expeced number of link saus changes ha a node deecs per second, be he average raffic rae ha a node generaes in a second (in bps), s be he average number of new sessions generaed by a node per second, and daa be he average daa packe size (in bis). This work uses he same se of assumpions, based on geographical reasoning, ha were presened and discussed in [11], [12], [14], [15], which are reproduced below for he sake of clariy. 2 a.1. As he nework size increases, he average in-degree d remains consan. a.2. Le A be he area covered by he N nodes of he nework, and = N=A be he nework average densiy. Then, he expeced (average) number of nodes inside an area A 1 is approximaely A 1. a.3. The number of nodes ha are a disance of k or less hops away from a source node increases (on average) as (dk 2 ).The number of nodes exacly k hops away increases as (dk). a.4. The maximum and average pahs (in hops) among nodes in a conneced subse of n nodes boh increase as ( p n). In paricular, he maximum pah lengh across he enire nework and he average pah lengh across he nework (L) increase as (p N ). a.5. The raffic ha a node generaes each second ( and s ), is independen of he nework size N (number of desinaions). As he nework size increases, he oal amoun of daa ransmied/received by a single node remains consan, bu he number of desinaions increases (raffic diversiy will increase). a.6. For a given source node, all possible desinaions (N ; 1 nodes) are equiprobable. The raffic from one node o a given desinaion decreases as (1=N ). a.7. Link saus changes are due o mobiliy. is direcly proporional o he relaive node speed. a.8. Mobiliy models : ime scaling. Le g 0=1 (x y) be he probabiliy disribuion funcion of a node posiion a ime 0 second, given ha i is known ha he node posiion a ime 1 will be (0 0). Then, he probabiliy disribuion funcion of a node posiion a ime < 1 given ha he node will be a he posiion (x 1 y 1 ) a ime 1,isgivenbyg =1 (x y x 1 y 1 )= 1 (; 1) 2 g 0=1 ( x;x 1 y;y 1 1; ). 1; Assumpions a.1 - a.8 represens a well-defined nework model, sill general enough o include mos of he ypical neworking scenarios. The reader is referred o [11], and [12] for a discussion on hese assumpions. I should be noed ha in he case any of he above assumpions does no hold for a paricular class of neworks, alernaive 2 Sandard asympoic noaion is employed. A funcion f (n) = (g(n)) [similarly, f (n) =O(g(n))] if here exiss consans c 1 and n 1 [similarly, c 2 and n 2 ] such ha c 1 g(n) f (n) [similarly f (n) c 2 g(n)] foralln n 1 [similarly, n n 2 ]. Also, f (n) =(g(n)) if and only if f (n) =(g(n)), and f (n) =O(g(n)). expressions may be derived by following he same mehodology se forh in his paper. B. Definiions: Toal Overhead and Scalabiliy B.1 Toal Overhead Tradiionally, he erm overhead has been used in relaion o he conrol overhead, ha is, he amoun of bandwidh required o consruc and mainain a roue. However, as shown in [11], a proocol s conrol overhead alone is no sufficien for assessing sysem performance, as i fails o accoun for he impac of sub-opimal roues. Wha is needed is a single meric ha is able o capure he rouing proocol impac on nework performance. For bandwidh-consrained sysems, he oal overhead inroduced in [11] and discussed below represens such a meric. Firs, he minimum raffic load of he nework mus be defined, as follows: Definiion 1: The minimum raffic load of a nework, is he minimum amoun of bandwidh required o forward packes over he shores disance (in number of hops) pahs available, assuming all he nodes have insananeous apriorifull opology informaion. The above definiion is independen of he rouing proocol being employed, since i does no include he conrol overhead bu assumes ha all he nodes are provided aprioriglobal informaion. I should be noed ha i is possible ha in fixed neworks a node is provided wih saic opimal roues, and herefore here is no bandwidh consumpion above he minimum raffic load. On he oher hand, in mobile scenarios his is hardly possible. Due o he unpredicabiliy of he movemen paerns and he opology hey induce, even if saic roues are provided so ha no conrol packes are needed, i is exremely unlikely ha he saic roues so forced remain being he opimal ones during he enire nework lifeime. Thus, since sub-opimal roues are presen, he acual nework bandwidh usage would be greaer ha he minimum raffic load value. This moivaes he following definiion of a rouing proocol oal overhead. Definiion 2: The oal overhead induced by a rouing proocol is he difference beween he oal amoun of bandwidh acually consumed by he nework running such rouing proocol minus he minimum raffic load ha would have been required should he nodes had apriorifull opology informaion. Thus, he acual bandwidh consumpion in a nework will be he sum of a proocol independen erm, he minimum raffic load, and a proocol dependen one, he oal overhead. Effecive rouing proocols should ry o reduce he second erm (oal overhead) as much as possible. The differen sources of overhead ha conribue o he oal overhead may be grouped and expressed in erms of reacive, proacive,andsub-opimal rouing overheads. All of hese sources of overhead has been considered in he pas, bu he oal overhead represens he firs meric ha successfully combines all of hem in a unified framework, allowing a racable model o be derived. The reacive overhead of a proocol is he amoun of bandwidh consumed by he specific proocol o build pahs from a source o a desinaion, afer a raffic flow o ha desinaion has been generaed a he source. In saic neworks, he reacive

3 overhead is a funcion of he rae of generaion of new flows. In dynamic (mobile) neworks, however, pahs are (re)buil no only due o new flows bu also due o link failures in an already acive pah. Thus, in general, he reacive overhead is a funcion of boh raffic and opology change. The proacive overhead of a proocol is he amoun of bandwidh consumed by he proocol in order o propagae roue informaion before i is needed. This may ake place periodically and/or in response o opological changes. The sub-opimal rouing overhead of a proocol is he difference beween he bandwidh consumed when ransmiing daa from all he sources o heir desinaions using he roues deermined by he specific proocol, and he bandwidh ha would have been consumed should he daa have followed he shores available pah(s). For example, consider a source ha is 3 hops away from is desinaion. If a proocol chooses o deliver one packe following a k (k > 3) hop pah (maybe because of ouof-daeinformaion), hen (k ;3)packe lengh bis will need o be added o he sub-opimal rouing overhead. The oal overhead provides an unbiased meric for performance comparison ha reflecs bandwidh consumpion. Despie increasing efficiency a he physical and MAC-layers, bandwidh is likely o remain a limiing facor in erms of scalabiliy, which is a crucial elemen for successful implemenaion and deploymen of ad hoc neworks. The auhors recognize ha oal overhead may no fully characerize all he performance aspecs relevan o specific applicaions. However, i can be used wihou loss of generaliy as i is proporional o facors including energy consumpion, memory and processing requiremens, and, furhermore, delay consrains have been shown o be expressed in erms of an equivalen bandwidh [13]. B.2 Scalabiliy This work is aimed a he sudy of he scalabiliy properies of rouing proocols for ad hoc neworks. However, currenly here is no a clear definiion of scalabiliy. Indeed, scalabiliy has a differen meaning for differen people. Thus, we need o define he exac meaning of his erm. Definiion 3: Scalabiliy isheabiliyofa neworkosuppor he increase of is limiing parameers. 3 Thus, scalabiliy is a propery. In order o quanify his propery, we use he concep of minimum raffic load (definiion 1) o define he nework scalabiliy facor as follows: Definiion 4: Le Tr( 1 2 :::) be he minimum raffic load experienced by a nework under parameers 1 2 :::(e.g. nework size, mobiliy rae, daa generaion rae, ec.). Then, he nework scalabiliy facor of such a nework, wih respec o a parameer i ( i ) is defined o be : def log Tr( 1 2 :::) i = lim i!1 log i The nework scalabiliy facor is a number ha asympoically relaes he increase in nework load o he differen nework parameers. For he class of mobile ad hoc neworks under sudy 3 The limiing parameers of a nework are hose parameer as for example mobiliy rae, raffic rae, and nework size, ec. whose increase causes he nework performance o degrade. On he remainder of his work only limiing parameers will be considered, and herefore he erm parameer will be used in lieu of he erm limiing parameer. (assumpions a.1 - a.8), he minimum raffic load Tr( N) is ( N 1:5 ), 4 and herefore = 0, = 1, and N =1:5. The nework scalabiliy facor maybeusedocomparehe scalabiliy properies of differen neworks (wireline, mobile ad hoc, ec.), and as a resul of such comparisons we can say ha one class of neworks scales beer han he oher. However, if our desire is o assess wheher a nework is scalable (an adjecive) wih respec o a parameer i, hen he nework rae dependency on such a parameer mus be considered. Definiion 5: The nework rae R ne of a nework is he maximum number of bis ha can be simulaneously ransmied in a uni of ime. For he nework rae (R ne ) compuaion all successful link layer ransmissions mus be couned, regardless of wheher he link layer recipien is he final nework-layer desinaion or no. Definiion 6: A nework is said o be scalable wih respec o he parameer i if and only if, as he parameer i increases, he nework s minimum raffic load doesno increase faser han he nework rae (R ne ) can suppor. Tha is, if and only if: log R ne ( 1 2 :::) i lim i!1 log i For example, i has been proved ha in mobile ad hoc neworks (N ) successful ransmissions can be scheduled simulaneously (see for example [5], [6]). The class of neworks under sudy in his work (i.e. resuling from applying power conrol echniques) are precisely he class of neworks ha achieves ha maximum nework rae. Thus, in order for mobile ad hoc nework o be regarded as scalable wih respec o nework size, we will need N 1. Unforunaely his is no he case, and as a consequence ad hoc neworks under assumpion a.1 hrough a.8 are no scalable wih respec o nework size 5. Wireline neworks, in he oher hand, if fully conneced may have N =1, and herefore hey are poenially scalable (in he bandwidh sense defined here) wih respec o nework size. Noe however, ha his scalabiliy requires he nodes degree o grow wihou bound, which may be prohibiely expensive. Similarly, since he nework rae does no increase wih mobiliy or raffic load, hen a nework will be scalable w.r.. mobiliy and raffic if and only if =0and =0, respecively. Thus, he neworks under his sudy are scalable w.r.. mobiliy, bu are no scalable w.r.. raffic. Noe ha similar conclusions may be drawn for scalabiliy w.r.. addiional parameers as for example nework densiy, ransmission range `, ec. ha are no being considered in our analysis. For example, as ransmission range increases (and assuming a infinie size nework wih regular densiy) he spaial 4 Each node generae bis per seconds, ha mus be reransmied (in average) L imes (hops). Thus, each node induce a load of L, which afer adding all he nodes resuls in a Tr( N)= NL. Since, by assumpion a.4 L is (p N ), he above expression is obained. 5 I has been shown in [6] ha if he nework applicaions can suppor infiniely long delays and he mobiliy paern is compleely random, hen he average pah lengh may be reduced o 2 ((1)) regardless of nework size and, as a consequence, ha nework scalabiliy facor wih respec o nework size N is equal o 1. Thus, hose ad hoc neworks (random mobiliy and capable of acceping infiniely long delays) are he only class of ad hoc neworks ha are scalable wih respec o nework size. This work does no consider ha class of neworks since hey have no pracical relevance.

4 reuse decreases and as a consequence nework rae decreases as rapidly as ` 2. Thus, ` should be lower han ;2 for he nework o be deemedscalable. Since he minimum raffic load will only decrease linearly w.r.. ` (pahs are shorening), ` = ;1, and herefore ad hoc neworks are no scalable w.r.. ransmission range. 6 Now, afer noicing ha mobile ad hoc neworks are no scalable wih respec o size and raffic, one may ask he meaning of regarding a rouing proocol scalable. The remaining of his subsecion will clarify his meaning. Definiion 7: Rouing proocol s scalabiliy is he abiliy of a rouing proocol o suppor he coninuous increase of he nework parameers wihou degrading nework performance. Thus, from he above definiion i is clear ha he rouing proocol scalabiliy is dependen on he scalabiliy properies of he nework he proocol is run over. Tha is, he nework own scalabily properies provides he reference level as o wha o expec of a rouing proocol. Obviously, if he overhead induced by a rouing proocol grows faser han he nework rae bu slower han he minimum raffic load, he rouing proocol is no degrading nework performance, which is being deermined by he minimum raffic load. To quanify a rouing proocol scalabily, he respecive scalabiliy facor is defined, based on he oal overhead concep (definiion 2), as follows: Definiion 8: Le X ov ( 1 2 :::) be he oal overhead induced by rouing proocol X, dependen on parameers 1 2 ::: (e.g. nework size, mobiliy rae, daa generaion rae, ec.). Then, he Proocol X s rouing proocol scalabiliy facor wih respec o a parameer i ( X i ) is defined o be : X def log X ov ( 1 2 :::) i = lim i!1 log i The rouing proocol scalabiliy facor provides a basis for comparison among differen rouing proocols. Finally, o assess wheher a rouing proocol is scalable, he following definiion is used: Definiion 9: A rouing proocol X is said o be scalable wih respec o he parameer i if and only if, as he parameer i increases, he oal overhead induced by such proocol (X ov ) does no increase faser han he nework s minimum raffic load. Tha is, if and only if: X i i Thus, for he class of nework under sudy, a rouing proocol X is scalable wih respec o nework size if and only if X N 1:5;iisscalable w.r.. mobiliy rae if and only if X 0;and i is scalable w.r.. raffic if and only if X 1. In he remainder of his paper we will derive asympoic expressions for he oal overhead (and herefore he rouing proocol scalabiliy facor) induced by a represenaive se of rouing proocols. The mehodology o be employed consiss of compuing each of he hree componens of oal overhead, namely proacive, reacive and sub-opimal rouing, separaedly and hen adding hem up. Besides he rivial resul ha Plain Flood- 6 This observaion is he main reason behind our focusing on neworks wih power conrol, where he ransmission range is kep in line so ha he nework degree is kep bounded. ing (PF) is he only proocol ha is scalable wih respec o mobily, and ha mos proocols are scalable wih respec o raffic, he more ineresingresul ha HSLS is scalable wih respec o nework size is found. III. PLAIN FLOODING (PF) In PF, each packe is (re)ransmied by every node in he nework (excep he desinaion). Thus, N ; 1 ransmissions are required for each daa packe, when he opimal value (on average) should have been L. Since here are N daa packes generaed each second, he addiional bandwidh required for ransmission of all hese packes is daa (N ; 1 ; L) N bps. Since L = (p N ), he PF s sub-opimal rouing- and oaloverhead per second is equal o ( (N 2 ;N 1:5 )) = ( N 2 ). In consequence PF =0,and PF N =2. SLS =0, SLS =1, PF IV. STANDARD LINK STATE (SLS) In SLS, a node sends a Link Sae Updae (LSU) o he enire nework each ime i deecs a link saus change. A node also sends periodic, sof-sae LSUs every T p seconds. There is no reacive overhead associaed wih SLS, and since he pahs deermined are opimal, here is no sub-opimal rouing overhead associaed wih i eiher. In SLS, each node generaes a LSU a a rae of per second, so in average here are N LSUs being generaed a any given second. Each LSU is reransmied a leas once per each node (i.e. N imes), inducing an overhead of lsu N bis (where lsu is he size of he LSU packe). Then SLS proacive and oaloverhead per second is lsu N 2 bps, ha is, ( N 2 );and =1,and SLS N =2. V. DYNAMIC SOURCE ROUTING (DSR) In DSR no proacive informaion is exchanged. A node (source) reaches a desinaion by flooding he nework wih a roue reques (RREQ) message. When a RREQ message reaches he desinaion (or a node wih a cached roue owards he desinaion) a roue reply message is sen back o he source, including he newly found roue. The source aaches he new roue o he header of all subsequen packes o ha desinaion, and any inermediae node along he roue uses his aached informaion o deermine he nex hop in he roue. The presen work focuses on DSR wihou he roue cache opion (DSR-noRC). A lower bound for DRS-noRC s oal overhead is derived nex. The DSR-noRC reacive overhead mus accoun for RREQ messages generaed by new session requess (a a rae s per second per node) and he RREQ messages generaed by failures in links ha are par of a pah currenly in use. If we only consider he RREQ messages generaed by new session requess, hen a lower bound can be obained. Each roue reques message is flooded o he enire nework, resuling in N ;1 reransmissions (only he desinaion does no need o reransmi his message). Thus, each message induces an overhead of size of RREQ(N ; 1) bis, and here are s N RREQ messages generaed every second due o new session requess. Thus, he DSR-noRC reacive overhead per second is ( s N 2 ). For he DSR-noRC sub-opimal rouing overhead a lower bound will be obained by considering only he exra bandwidh

5 required for appending he source-roue in each daa packe. The number of bis appended in each daa packe will be proporional o he lengh L i of pah i. Since his lengh is no shorer han L op i (he opimal pah lengh), using L op i insead of L i will resul on a lower bound. The exra bandwidh consumed by a packe delivered using a pah i (wih a leas L op i reransmissions) will be a leas (log 2 N )(L op i ) 2,wherelog 2 N is he minimum lengh of a node address. The average exra bandwidh per packe over all pahs is Ef(log 2 N )(L op i ) 2 )g (log 2 N )EfL op i g 2 = (log 2 N )L 2 bis. Thus, for each packe sen from a source o a desinaion here is an average sub-opimal rouing overhead of a leas (log 2 N )L 2 bis. Since N packes are ransmied per second, he sub-opimal rouing overhead induced over he enire nework is a leas N (log 2 N )L 2 bps. Recalling ha L =(p N ) (assumpion a.4), he DSR-noRC sub-opimal rouing overhead per second is found o be ( N 2 log 2 N ) bps. Combining he previous resuls, DSR-noRC oal overhead per second is ( s N 2 + N 2 log 2 N ). Also, DSR;noRC =1, 0 < DSR;noRC <=1, 7 and DSR;noRC N > 2. VI. HIERARCHICAL LINK STATE (HIERLS) In he m-level HierLS rouing, nework nodes are regarded as level 1 nodes, and level 0 clusers. Level i nodes are grouped ino level i clusers, which become level i +1nodes, unil he number of highes level nodes is below a hreshold and herefore hey can be grouped (concepually) ino a single level m. Thus, he value of m is deermined dynamically based on he nework size, opology, and hreshold values. Link sae informaion inside a level i cluser is aggregaed (limiing he rae of LSU generaion) and ransmied only o oher level i nodes belonging in he same level i cluser (limiing he scope of he LSU). Thus, a node link change may no be sen ouside he level 1 cluser (if hey do no cause a significan change o higher levels aggregaed informaion), grealy reducing he proacive overhead. HierLS relies on he Locaion Managemen service o inform a source node S of he address of he highes level cluser ha conains he desired desinaion D and does no conain he source node S. For example, consider a 4-level nework as shown in Figure 1. S and D are level 1 nodes; X:1:1, X:1:2, ec. are level 2 nodes (level 1 clusers); X:1, X:2, ec. are level 3 nodes (level 2 clusers); X, Y, V,andZ are level 4 nodes (level 3 clusers); he enire nework forms he level 4 cluser. The Locaion Managemen (LM) service provides S wih he address of he highes level cluser ha conains D and does no conain S (e.g. he level 3 cluser Z in Figure 1). Node S can hen consruc a roue oward he desinaion. This roue will be formed by a se of links in node S level 1 cluser (X:1:1), a se of level 2 links in node S level 2 clusers (X:1), and so on. In Figure 1 he roue found by node S is : 7 DSR s oal overhead does depend on mobiliy, since breakages of links forming exising roues will rigger roue discovery procedures ha will induce reacive overhead and/or cause roue degradaion. Similar o he lower bound derived in his secion, an upper bound for DSR s oal overhead may be derived by assuming ha each link breakage rigger a global roue discovery (regardless of he link being par of an acive roue or no). Such an upper bound would increase linearly wih he mobiliy rae, and herefore we obain he upper bound for DSR;noRC <= 1. S S X.1 X.7 X.2 X.1.2 X X.5 X.6 X.3 X.4 X.1.3 X.1.1 X.1.4 X.1.5 n n 1 2 X.1.7 X.1.6 Fig. 1. A Source (S) - Desinaion (D) pahinhierls. S ; n 1 ; n 2 ; X:1:5 ; X:1:3 ; X:2 ; X:3 ; Y ; Z ; D. When a node ouside node S level 1 cluser receives he packe, he node will likely produce he same high-level roue owards D, and will expand he high-level links ha raverse is cluser using lower level (more deailed) informaion. In Figure 1 his expansion is shown for he segmen Z ; D. The Locaion Managemen (LM) service can be implemened in differen ways, wheher proacive (locaion updae messages), reacive (paging), or hybrid. Typical choices are: LM1: Pure reacive. Whenever a node changes is level i clusering membership bu remains in he same level i +1cluser, his node sends an updae o all he nodes inside is level i +1 cluser. For example, (see Figure 1) if node n 2 moves inside cluser X:1:5, i.e. i changes is level 1 cluser membership bu does no change is level 2 cluser membership (cluser X:1), hen node n 2 will send a locaion updae o all he nodes inside cluser X:1. The remaining nodes will no be informed. LM2: Local paging. In his LM echnique, one node in each level 1 cluser assumes he role of a LM server. Also, one node among he level 1 LM servers inside he same level 2 cluser assumesheroleofalevel2lmserver,andsoonupolevel m. The LM servers form a hierarchical ree. Locaion updaes are only generaed and ransmied beween nodes in his ree (LM servers). When a node D changes is level i clusering membership, he LM server of is new level i cluser will send a locaion updae message o he level i +1LM server, which in urn will forward he updae o all he level i LM servers inside his level i +1cluser. Addiionally, he level i +1LM server checks if he node D is new in he level i +1cluser, and if his is he case i will send a locaion updae o is level i +2LM server, and so on. When a level i LM server receives a locaion updae message regarding node D from is level i +1LM server, i updaes is local daabase wih node D s new locaion informaion and forwards his informaion o all he level i ; 1 LM servers inside is level i cluser. Each of hese level i ; 1 LM servers forwards he locaion updae message o he level i ; 2 servers in is level Y V D Z

6 i ; 1 cluser, and so on unil all he level 1 LM servers (inside node D s level i +1cluser) are informed of he new level i locaion informaion of node D. When a node needs locaion informaion abou any node in he nework, he node pages is level 1 LM server for his informaion. LM3: Global paging. LM3 is similar o LM2. In LM3, however, when a level i LM server receives a locaion updae from a higher level i +1LM server, i does no forward his informaion o he lower level ( i ; 1) LM servers. Thus, a lower level (say level j<i) LM server does no have locaion informaion for nodes ouside is level j cluser. A mechanism for removing oudaed locaion informaion abou nodes ha lef a level i cluser need o be addedo he leveli clusers LM servers. Basically, a level 1 LM server ha deecs ha a node lef is level 1 cluser will remove he enry corresponding o his node from is own daabase, and will inform is level 2 LM server. The level 2 LM server will wai for a while for a locaion updae from he new level 1 cluser (if inside he same level 2 cluser) and if no such an updae is received i will remove he node enry and will inform is level 3 LM server, and so on unil arriving o a LM server ha already has informaion abou he new locaion of he node. When a node needs locaion informaion abou any node in he nework, he node pages is level 1 LM server for he informaion. If he level 1 LM does no have he required informaion, i (he level 1 LM server) pages is level 2 LM server, who in urn pages is level 3 LM server, and so on, unil a LM server wih locaion informaion abou he desired desinaion is found. Approach LM1, he easies o implemen, will induce greaer overhead and lower laencies for roue esablishmen. Approach LM2 poenially reduces he bandwidh consumpion (for reasonable values of s ) bu a he expense of complexiy (selecion and mainenance of LM servers) and an increase in he laency associaed wih roue esablishmen. However, he asympoic characerisic of HierLS are idenical under LM1 and LM2, as will be seen laer. Approach LM3 is he more complex o implemen. I will induce a significan amoun of reacive overhead, bu will reduce he amoun of overhead induced by mobiliy. In his paper, resuls for he HierLS oal overhead for all hree LM Techniques are presened in Table I. However, due o space consrains, only he derivaion for he oal overhead expression for a HierLS-LM1 (pure proacive LM echnique) will be presened nex. The reader is referred o [14] or [15] for he remaining derivaions. A. HierLS-LM1 proacive overhead A nework organized in m level clusers, each of equal size k (N = k m ) is considered. Noe ha k is predefined while m increases wih N. Under assumpion a.7, HierLS-LM1 s proacive asympoic overhead is dominaed by he locaion managemen funcion, ha induces an overhead ha grows a leas as fas as (sn 1:5 ) (explained below), where s is he node relaive speed. In he oher hand, mos of he LSUs updaes will correspond o level 1 links, and will be propagaed inside he level 1 clusers only. hus, LSU packes will induce a proacive overhead ha will only grow as fas as kn (his is, of course, a lower bound). HierLS-LM1 locaion managemen overhead expressions, can be obained by considering ha he ime a node akes o change is level m ; 1 cluser is direcly proporional o he diameer of his level m ; 1 cluser and inversely proporional o he node s relaive speed s. Since he level m ; 1 cluser size is N=k, hen he cluser diameer is (p N=k). Under approach LM1, he new locaion informaion will have o be forwarded o all he nodes inside he level m cluser (he enire nework). Thus, every node will send a locaion updae message o he enire nework (N ransmissions) each (p N=k=s) p seconds, inducing an overhead of (p ks N ) bis every second. Adding up all nodes conribuions, he proacive overhead per second due o level m ; 1 clusers membership change is (p ksn 1:5 ). Regarding he locaion updaes generaed due o level m ; i membership change, i can be seen ha a level m ; i cluser is k i;1 imes smaller han a level m ; 1 cluser, and consequenly alevelm;icluser s diameer is k i;1 2 imes smaller han a level m ; 1 cluser s diameer. Thus, he generaion rae of locaion updaes due o level m ; i membership changes is k i;1 2 imes larger han he rae induced by level m ; 1 changes. Also, since he new locaion informaion will have o be ransmied o all he nodes inside he curren level m ; i +1cluser, hen he number of ransmissions required for each packe decreases by a facor of k ;(i;1) wih respec o he number of ransmissions induced by level m ; 1 changes, which resuls in a ne reducion of k ; i;1 2. Then, he overhead due o all locaion updaes is : Loc Upd Cos = (p ksn 1:5 )[1 + k ; k ;1 + ] p = (p ksn 1:5 1 ) 1 ; 1=k Thus, he locaion managemen overhead is ( N 1:5 ) bps (by assumpion a.7, is proporional o s). Combining his value wih he lower bound obained for he LSU-induced overhead (( N )), i is concluded ha he HierLS-LM1 proacive overhead is ( N 1:5 ). B. HierLS-LM1 sub-opimal rouing overhead To esimae he sub-opimal rouing overhead, i is assumed ha each level i (beginning wih level 2) increases he acual roue lengh by a facor f i (f i depends on he value of k, he LSU riggering hresholds, and is ypically close o 1, for example f = 1:05 means a 5% increase in he roue lengh). Thus, if he opimal pah lengh is l, hen he acual pah lengh will be i=mf i=2 i l. Le f be he geomeric average of he se ff i g,hais,f = ( m f m;1 i=2 i) 1. Then, he sub-opimal rouing overhead induced by a packe ransmission is daa [f m;1 ; 1] l = daa [k (log k f )(m;1) ; 1] l = daa [ N k ; 1] l, where = log k f. There are N packes generaed each second, hus he average sub-opimal rouing overhead per second is daa ( N p k ; 1) L N.SinceLis ( N ), we finally ge ha he HierLS-LM1 sub-opimal rouing overhead per second is ( N 1:5+ ). C. HierLS-LM1 oal overhead Combining he previous expressions, he HierLS-LM1 oal overhead is found o be ( N 1:5 + N 1:5+ ). Also,

7 =1,and HierLS;LM1 N = 1:5 +>1:5 (HierLS is almos scalable w.r.. newrok size). HierLS;LM1 =1, HierLS;LM1 TTL = TTL = VII. ZONE ROUTING PROTOCOL (ZRP) ZRP is a hybrid approach, combining a proacive and a reacive par, rying o minimize he sum of heir respecive overheads. In ZRP, a node disseminaes even-driven LSUs o is k- hop neighbors (nodes a a disance, in hops, of k or less). Thus, each node has full knowledge of is k-hop neighborhood and may forward packes o any node wihin i. When a node needs o forward a packe ouside is k-hop neighborhood, i sends a roue reques o a subse of he nodes in he nework, namely he border nodes. The border nodes will have enough informaion abou heir k-hop neighborhoods o decide wheher o reply o he roue reques or o forward i o is own se of border nodes. The roue formed will be described in erms of he border nodes only, hus allowing border nodes o locally recover from individual link failures, reducing he overhead induced by roue mainenance procedures. The following lower bound for ZRP oal overhead (ZRP ov ) was obained: ZRP ov = 8 >< >: ( N 2 ) if = O( s = p N ) ( s N 5 3 ) if =( s = p N) and = O( s N ) ( s N 2 ) if =( s N ) Due o space limiaions, he derivaion of he ZRP oal overhead was lef ou of he paper. Once again, he reader is referred o [14] or [15] for he complee derivaion. Noe ha he asympoic expression provides us wih much more informaion abou he parameers ineracions han he scalabiliy facors, which are compued assuming ha jus one parameer is increased while he ohers remain fixed. For ZRP, ZRP = 0 (pure proacive mode), 0 < ZRP <= 1 (pure reacive mode, similar o DSR s), and ZRP N 1:66. Noe ha he informaionprovidedbyhe scalabily facors is incomplee, and i hinds he fac ha he exponenial raes of increase of ZRP s oal overhead wih respec o mobiliy and raffic always add up o a leas 1, as can be seen from he oal overhead s asympoic expressions. VIII. HAZY SIGHTED LINK STATE (HSLS) HSLS is based on he observaion ha nodes ha are far away do no need o have complee opological informaion in order o make a good nex hop decision. Thus, propagaing every link saus change over he enire nework may no be necessary. In a highly mobile environmen, a node running HSLS will ransmi - provided ha here is a need o - a LSU only a paricular ime insans ha are muliples of e seconds. Thus, poenially several link changes are colleced and ransmied every e seconds. The Time To Live (TTL) field of he LSU packe is se o a value (which specifies how far he LSU will be propagaed) ha is a funcion of he curren ime index as explained below. Afer one global LSU ransmission LSU ha ravels over he enire nework, i.e. TTL field se o infiniy, as for example during iniializaion a node wakes up every e seconds and sends a LSU wih TTL se o 2 if here has been a link saus change in e 2e 16 3e 4e 5e 6e 7e 8e 9e 10e 11e 12e 13e 14e 15e 16e 0... Fig. 2. HSLS s LSU generaion process (mobiliy is high). he las e seconds. Also, he node wakes up every 2 e seconds and ransmis a LSU wih TTL se o 4 if here has been a link saus change in he las 2 e seconds. In general, a node wakes up every 2 i;1 e (i = :::) seconds and ransmis a LSU wih TTL se o 2 i if here has been a link saus change in he las 2 i;1 e seconds. If a packe TTL field value (2 i ) is greaer han he disance from his node o any oher node in he nework (which will cause he LSU o reach he enire nework), he TTL field of he LSU is rese o infiniy (global LSU), and he algorihm is re-iniiaed. Nodes ha are a mos wo hops away from a node, say X, will receive informaion abou node X s link saus change a mos afer e seconds. Nodes ha are more han 2 bu a mos 4 hops away from X will receive informaion abou any of X links change a mos afer 2 e seconds. In general, nodes ha are more han 2 i;1 bu a mos 2 i hops away from X will receive informaion abou any of X links change a mos afer 2 i;1 e seconds. Figure 2 shows an example of HSLS s LSU generaion process when mobiliy is high and in consequence LSUs are always generaed. An arrow wih a number over i indicaes ha a ha ime insan a LSU (wih TTL field se o he indicaed value) was generaed and ransmied. Figure 2 assumes ha he node execuing HSLS compues is disance o he node farhes away o be beween 17 and 32 hops, and herefore i replaces he TTL value of 32 wih he value infiniy, reseing he algorihm a ime 16 e. The reader is referred o [11] and [12] for more deails abou HSLS. A. HSLS proacive overhead A highly mobile environmen (i.e. a LSU is generaed every ime inerval) is considered. All he differen LSUs (re)ransmissions due o LSUs generaed by a node, say X, will be added and hen averaged over ime. The value obained will be muliplied by he number of nodes in he nework o ge he proacive overhead. LSUs will be grouped based on heir TTL value a he ime hey were generaed, beginning wih he LSUs wih larger TTL values. Le MD x be he maximum disance from node X o any oher node in he nework. Le R x be he power of 2 such ha R x <MD x 2R x. For example, R x =16infigure 2, where MD x wasassumedobebeween17 and 32. Under HSLS, node X compues MD x each e seconds based on is own opology informaion, which is no necessarily up-o-dae, so MD x ime

8 is a ime-changing value ha is no being imely updaed. The above observaion, however, will have lile impac on he value of R x, which may be assumed roughly consan over ime. Le s consider wha happens a ime R x e (16 e in figure 2). A his ime node X sends a LSU o he enire nework and he algorihm is re-iniiaed. Thus, every R x e seconds node X induces N ransmissions, and herefore he bandwidh consumpion due o hese global LSUs is lsu N R x e,wherelsu is he average lengh of a LSU packe. The second larger TTL is R x, and LSUs wih his TTL are generaed Rx 2 e seconds afer a global LSU is sen (imes 8 e in figure 2). Recalling ha he imers are rese a ime R x e,we noice ha he inerval beween consecuive generaion imes is (R x e ; Rx 2 e)+ Rx 2 e = R x e. Thus, he generaion rae of 1 LSUs wih TTL equal o R x is R x e (he same as he generaion rae of global LSUs). These LSUs will no reach all he nodes in he nework bu only a fracion f x. From assumpion a.3, f x should be around (R x =MD x ) 2, i.e., f x 2 [0:25 1]. In pracical siuaions, due o boundary effecs (i.e. he number of nodes a a maximum disance MD x is small), we obain ha ypically f x is in he inerval [0:5 1]. Thus, he bandwidh consumpion due lsu fxn R x e. o LSUs wih TTL equal o R x is For he remaining TTL values, boundary condiions are no longer relevan. Thus, for TTL equal o R x =2 he generaion rae doubles (e.g. LSUs wih TTL equal o 8 are sen a imes 4 e 12 e in figure 2), and he number of ransmissions induced per LSU is reduced by a facor of 4 (because of assumpion a.3, and he fac ha he TTL values are reduced o a half); hus he oal effec is a reducion by a facor of 2 wih respec he bandwidh consumpion due o LSUs wih TTL equal o R x. The same argumen applies for TTL equal o R x =4 R x =8 ::: Finally, he oal bandwidh consumpion due o all he LSUs generaed by node X is equal o : X pro lsu N lsu f xn lsu f xn lsu f xn HSLS = ::: R x e R x e 2R x e 4R x e = lsu N R x e [1 + f x ( :::)] lsu N R x e [1 + 2f x ] Since he size of a LSU depends only on he node densiy (bounded on average), f x is bounded below 1, andr x is (p N ) (assumpion a.4); he proacive overhead per second induced by one node is ( N 0:5 e ). Since here are N nodes, he proacive overhead per second induced by he enire nework is ( N 1:5 e ). B. HSLS sub-opimal rouing overhead Due o space consrains, he complee derivaion was lef ou of he paper. Below, an insigh ino i is provided. The reader is referred o [14] or [15] for he acual derivaion. Le elap k be he maximum ime elapsed since fresh LSU informaion abou a desinaion k hops away was las received. HSLS induces a quasi-linear relaionship beween elap k and k. In general, e 2 elap k k e. Thus, he raio beween he ime 8 Assumpions a.3 and a.4 are asympoic condiions, and as such, are no applicable o small values of TTL. However, he conribuions of LSUs wih small TTL values in he proacive overhead of a large nework is no significan and a more exac analysis can be safely omied. elapsed since fresh informaion was received and disance is bounded by e, independenly of nework size or disance o he desinaion. Based on he mobiliy model assumpion a.8 (ime scaling), his will cause he probabiliy of a sub-opimal nex hop decision o be bounded 9, and he fracion of he increase of he sub-opimal roues (wih respec o he opimal ones) o also be bounded independenly of nework size. Then, for a fixed value of e,hslssub-opimal rouing overhead will increase as ( N 1:5 ). To invesigae he dependence of he sub-opimal rouing overhead on he ime e, a more precise mobiliy model need o be defined. Assuming a mobiliy model ha induces an exponenial residence ime on a given area, HSLS sub-opimal rouing overhead was found o be equal o : ((e ek 4 ; 1) N 1:5 ),wherek 4 is a consan. C. HSLS oal overhead There is no reacive overhead associaed wih HSLS. Thus, he HSLS oal overhead for he class of neworks analyzed in he previous subsecions is equal o : HSLS ov = N 1:5 [K 5 1 e + K 3 (e ek 4 ; 1) ] The value of e should be uned o opimize performance. For a momen, le s use he approximaion e x ; 1 x, wherex = e K 4. Thus: HSLS ov N 1:5 [ K 5 e + K 6 e ] Choosing he value of e ha minimizes p he above expression we ge e = ( p 1 ), x = ( p ),andhsls ov = ( p N 1:5 ). The previous expression would define he asympoic behavior of HSLS s oal overhead only if our approximaion e x ; 1 x is valid. Indeed, if grows asympoically faser han,hevalueofx goes o zero and he approximaion e x ; 1 x is valid. On he oher hand, if grows asympoically faser han, he approximaion will no be valid. In his case, since he exponenial funcion is he fases growing, i is desirable o mainain he produc e (and herefore he value of p) bounded and herefore we choose e =( 1 ). Thus, he HSLS oal overhead in his scenario becomes (N 1:5 ( + )) = ( N 1:5 ), where he las equaliy holds due o our assumpion ha grows asympoically faser han and herefore dominaes he previous expression. Thus, he HSLS s oal overhead is : HSLS ov = ( p N 1:5 ) if = O( ) ( N 1:5 ) if =( ) Also, i can be noed ha HSLS = 0:5, HSLS = 1, and HSLS N =1:5. Thus, HSLS is he only proocol ha is scalable wih respec o nework size. 9 Since he raio maximum displacemen speed imes elapsed ime over disance is bounded, so is he angular displacemen of he desinaion. The angular displacemen will deermine wheher he node chosen as he nex hop is he proper one or no.

9 IX. COMPARATIVE STUDY In he previous secions he scalabiliy facors of several represenaive rouing proocols have been derived. From hose resuls we concluded ha PF is he only proocol known o be scalable w.r.. mobiliy ( PF = 0), while all of he proocols were scalable w.r.. raffic. More ineresing was o find ha HSLS is he only proocol scalable wih respec o nework size ( HSLS N = 1:5). However, much more informaion abou he proocol parameer s ineracions may be derived from he asympoic oal overhead expressions, which are summarized in Table I. Table I presens our resuls for he oal overhead when he unable parameers are seleced o opimize performance (or a leas, opimize he lower bounds derived before). These resuls increase our undersanding of he limis and provide valuable insigh abou he behavior of several represenaive rouing proocols. The beer undersanding of hese limis will help nework designers o beer idenify he class of proocols o engage depending on heir operaing scenario. For example, if he designer s main concern is nework size, i can be noed ha HierLS and HSLS scale beer han he ohers. Similarly, if raffic inensiy is he mos demanding requiremen, hen SLS and ZRP are o be preferred since hey scale beer wih respec o raffic (oal overhead is independen of ); HSLS follows as i scales as ( p ), and PF, DSR, and HierLS are he las since heir oal overhead increases linearly wih raffic. 10 Similarly wih respec o he rae of opological change, we observe ha PF may be preferred (if size and raffic are small and he rae of opological change increases oo rapidly), since is oal overhead is independen of he rae of opological change. Provably nex will be ZRP and DSR since heir lower bounds are independen of he rae of opological changes. The bounds are no necessarily igh, and ZRP s and DSR s behavior should depend somewha of he rae of opological change. Finally, for SLS, HierLS, and HSLS we know (as opposed o DSR and ZRP where we suppose) ha heir oal overhead increase linearly wih he rae of opological change. I is ineresing o noe ha when only he raffic or he mobiliy is increased (bu no boh), ZRP can achieve almos he bes performance in each case. 11 However, if mobiliy and raffic increase a he same rae; ha is, =() and =() (for some parameer ), hen ZRP s oal overhead ((N 1:66 )) will presen he same scalabiliy properies as HSLS s ((N 1:5 )) and HierLS s ((N 1:5+ )) wih respec o, wih he difference ha ZRP does no scale as well as he oher wo wih respec o size. These and more complex analyses can be derived from he expression presened in his paper, when differen parameers are modified simulaneously accordingly wih he scenario he designer is ineresed in. 10 I is ineresing o noe ha HSLS scales beer wih raffic inensiies han HierLS (he only oher proocol ha scales well wih size). This resul may have an inuiive explanaion in he fac ha HierLS never aemps o find opimal roues owards he desinaion, even under slowly changing condiions. HSLS on he oher hand, may evenually obain full opology informaion and herefore opimal roues if he rae of opological changes is small wih respec o 1= e, as is he case when grows faser han. 11 Almos, because ZRP can no achieve he independence of oal overhead from mobiliy. PF does. Proo. Toal over. (bes) Cases PF ( N 2 ) Always SLS ( N 2 ) Always DSR ( s N 2 + N 2 log 2 N ) no Roue Cache HierLS ( N 1:5 + N 1:5+ ) LM1 or LM2 ( N log N + N 1:5+ ) LM3 ZRP ( N 2 ) = O( s = p N ) ( s N 2 ) =( s N ) ( s N 5 3 ) oherwise HSLS ( p N 1:5 ) = O( ) ( N 1:5 ) =( ) TABLE I ASYMPTOTIC TOTAL OVERHEAD EXPRESSIONS. HSLS has beer asympoic properies han HierLS, which means ha as size increases HSLS evenually ouperform HierLS. The idea of HSLS being much more simple o implemen ouperforming HierLS is couner-inuiive. A firs reacion o his resul will likely be o assume ha he consans involved in he asympoic analysis may be oo large, prevening HSLS from ouperform HierLS under reasonable scenario. Thus, he auhors relied on a couple of simulaion experimen o validae if, in effec, HSLS may ouperform HierLS even under moderae nework size and raffic load. A. A simulaion experimen: HSLS vs. HierLS-LM1 Table II shows he simulaion resuls obained by OPNET for a 400-node nework where nodes are randomly locaed on a square of area equal o 320 square miles (i.e. densiy is 1.25 nodes per square mile). Each node choose a random direcion among 4 possible values, and move on ha direcion a 28.8 mph. Upon reaching he area boundaries, a node bounces back. The radio link capaciy was Mbps. Simulaion were run for 350 seconds, leaving he firs 50 seconds for proocol iniializaion, and ransmiing packes (60 8kbps sreams) for he remaining 300 seconds. The HierLS approach simulaed was of he ype HierLS-LM1, and corresponds o he DAWN projec [10] modificaion of he MMWN clusering proocol [8]. The minimum and maximum cluser size were se o 9 and 35 respecively. The meric of ineres is he hroughpu (i.e. fracion of packes successfully delivered). The simulaion resuls presened are no a comprehensive sudy of he relaive performance of HierLS versus HSLS under all possible scenarios. They jus presens and example of a real-life siuaion where HSLS ouperform HierLS, and complemen our heoreical analysis. The heoreical analysis focuses on asympoically large nework, heavy raffic load, and sauraion condiions where he remaining capaciy deermines he proocol performance. The simulaion resuls, in he oher hand, refer o medium size neworks wih moderae loads, where depending on he MAC employed, oher facors may have more weigh over he proocols performance. Table II shows he hroughpu obained under wo differen MAC proocols: unreliable and reliable CSMA. For reliable