Analysis of Wired Short Cuts in Wireless Sensor Networks

Transcription

1 Analysis of Wied Shot Cuts in Wieless Senso Netwos ohan Chitaduga Depatment of Electical Engineeing, Univesity of Southen Califonia, Los Angeles 90089, USA Ahmed Helmy Depatment of Electical Engineeing, Univesity of Southen Califonia Los Angeles 90089, USA Abstact In this pape we investigate the use of wied shot cuts in senso netwos. This new paadigm augments a senso netwo with a vey limited wied infastuctue to impove its oveall enegy-efficiency. Enegy-efficiency is obtained by eduction in aveage path length. We have developed an analytical model to analyze the gain in path length eduction by using shot cuts. We have also conducted extensive simulations to validate ou analysis. Ou esults show that thee is an optimal wie length fo which the path length eduction is at its maximum, beyond which it deceases. The optimal length is only a small faction ( %) of the netwo diamete. In a netwo with 000 nodes unifomly distibuted on a dis the path length eduction satuates at 60-70% with 5-4 wies, depending on the location of the sin. Also, we find that esticting the nowledge about the wies to hops does not degade the pefomance fom the case when we have global nowledge of all wies. These esults show pomise of the new paadigm.. Intoduction In this pape, we popose and study a new paadigm fo senso netwos in which a patial wied infastuctue would augment the wieless senso netwo, and in which wies may act as shot cuts to ceate a small wold []. The initial wo on small wolds pointed out that fo egula elative gaphs (that ae highly clusteed) adding a few andom shot cuts deceases the degees of sepaation (i.e., the aveage path length) dastically, esulting in a degee of sepaation simila to that of andom gaphs [][3]. Ou ealie wo on small wolds in wieless netwos showed that a simila elationship exists between spatial gaphs (including wieless netwos) and small wolds [4]. The shot cuts need not be andom but can be limited to only a faction of the netwo diamete. In ou pevious wo we teated the shot cuts as logical contacts to which the path may tavese seveal hops [5] [6]. This wo investigates the use of wies as physical shot cuts to educe the aveage hop count of the netwo. This in tun can incease the enegyefficiency of the senso netwo as it educes the numbe of tansmissions in the netwo. In many applications lie emote suveillance, it may not be possible to augment the netwo with wies. Also, fo some netwos, the duation of deployment maes the use of wies infeasible due to the cost. But fo some applications lie ecological monitoing in which the sensos monito the envionment fo long duations, it can be economically feasible to do so. Such a netwo is being developed and deployed as a pat of a poject stated by eseaches at UCLA and patne univesities. The Netwo Infomatical Systems (NIMS) [7] infastuctue consists of a collection of steel cables, each attached to any two points - buildings, tees, o othe natual stuctue - that seve as suspension points. Nodes suspended on the cable collect data about the envionment though a ange of sensos which can be loweed o elevated, and also move, activate, and ecove fixed nodes set along the cable pathway. They also have the ability to doc when necessay to echage thei enegy souce, emoving enegy constaints that have hobbled othe senso netwos in the past. The cables can also be used fo communication puposes. In ou netwo model, we focus on the class of senso netwos in which the data is outed towads a single sin. Taffic fom the sensos is outed to the sin using geedy geogaphic outing. We assume that the nodes at the ends of the wies ae simply moe poweful sensos with much lage battey powe, o have a mechanism by which they can eplenish thei powe lie in the case of the NIMS infastuctue. Also, we concentate on netwos in which the data geneated by the nodes is low ate, which implies that Poceedings of the IEEE/ACS Intenational Confeence on Pevasive Sevices (ICPS 04) /04 $ 0.00 IEEE

2 Figue. The new paadigm: Senso netwo augmented by a few wies o cables to be used in communications as shot cuts we can safely assume that the wies ae not bandwidth limited fo the taffic they cay. Thus in ou model, we have not consideed the cost of the wied tansmission. Using analytical modeling and simulations, we have focused on the following questions: How many wies ae equied to achieve maximum eduction in the aveage path length? What is the maximum attainable eduction? What is the minimum investment that leads to significant eduction in path length? How should the wies be placed? What happens if we estict the infomation to some hops fom the wie (instead of global nowledge of all wies)? We have developed analytical model to obtain theoetical values fo the aveage path length eduction. The analytical expession fo the path length povides the eduction that can be obtained fo any position of the sin. The expession shows that fo ou model, thee is an optimal length of the wies fo which the aveage path length is at its minimum. We futhe validate ou analysis using extensive simulations. We compae the aveage path length obtained by using wies with the case having no wies. Ou esults show that the maximum eduction in aveage path length that can be obtained is 70% fo the sin placed at the cente and is ove 60% when the sin placed at the edge of the netwo. The length of wie needed fo the same eduction is highe when the sin placed at the edge. The maximum eduction is achieved with 4 wies when the sin is placed at the cente. Fo a simila expeiment, 5 wies give nea satuation esults fo the sin placed at the edge. Thus esults show that we need moe numbe of wies when the sin is at the cente than when at the edge, but the length of the wie equied fo is much lage fo the sin positioned at the edge. esticting the nowledge of the wie does not esult in deteioation of pefomance fo small wie lengths. Oveall, ou esults show the pomise of this new paadigm of augmenting a senso netwo with wies, in achieving a dastic eduction in path length (and hence significant impovement in enegy-efficiency). A elatively small numbe of wies ae needed, and the nowledge about locations of the wies need only be popagated a couple of hops away fom the ends of the wies. To ou nowledge no such wo has been done studying the effect of adding wies to wieless senso netwos. The est of the pape is oganized as follows. In section, we discuss elated wo to place ou contibutions in context. Section 3 discusses the poblem fomulation, evaluation metics consideed and the analytical model. In Section 4 we pesent ou evaluation methodology. Simulation esults and analysis ae discussed in section 5. Section 6 discusses possible futue extensions to this wo. Concluding comments ae pesented in Section 7.. elated Wo Pevious wo has shown that wieless netwos ae spatial gaphs that tend to be much moe clusteed than andom netwos and have much highe path length chaacteistics. It has been obseved that by adding only few andom lins, path length of wieless netwos can be educed dastically [], [3]. Ou ealie wo on small wolds in wieless netwos showed that a simila elationship exists between spatial gaphs (including wieless netwos) and small wolds [4]. In ou pevious wo, we had used this nowledge to use shot cuts as logical contacts to develop efficient esouce discovey techniques fo lage scale wieless netwos [5], [6]. The wo showed pomising esults fo the use of shot cuts as logical contacts. It futhe motivated eseach on the use of shot cuts as physical contacts (eithe using wies o lage ange wieless). In this pape we focus on the use of shot cuts as physical contacts fo a subset of senso netwos which have a single sin. ecently eseaches at UCLA and patne univesities have developed a new class of aeial, suspended obotic sensos able to monito thei own pefomance as they move themselves along a netwo of cables. The technology, nown as Netwoed Infomechanical Systems (NIMS) [7] can be used to monito a mountain steam ecosystem fom the gound to the teetops fo global change indicatos, o obseve coastal wetlands and uban ives fo biological pathogens. The NIMS infastuctue consists of a collection of steel cables. Each cable is attached to any two points building, tees etc that can seve as suspension points. The cables can also be used to move obots that can monito these sensos, and eplenish esouces lie powe. Such infastuctue Poceedings of the IEEE/ACS Intenational Confeence on Pevasive Sevices (ICPS 04) /04 $ 0.00 IEEE

3 A O S Figue.Figue showing how the wies ae placed fo a sin (S) placed at (0, -L) suppot emoves pevious limitations of on site senso netwos, as nodes can be eplaced, elocated and eplenished. Ou wo can complement the effots of the NIMS poject as it gives us an insight into the use of wies fo caying data. By undestanding the minimum investment equied and the fundamental limitations we can gain insight into how the wies could be used as shot cuts in senso netwos. Vaious enegy-awae outing potocols fo senso netwos have been developed. Some of the potocols use data centic outing in which the nodes fist exchange some metadata infomation befoe the actual data tansmission [8, 9]. In [0], the authos popose LEACH, in which the nodes ae oganized into clustes and the lifetime of the netwo is inceased by andomly choosing the cluste heads. These outing potocols exploe one of the dimensions of inceasing the enegy efficiency of senso netwos. We have exploed an othogonal dimension in which enegy efficiency is obtained by the use of wies. Ou wo is complementay whee adding shot cuts may wo in conjunction with those schemes to even futhe impove the enegy efficiency dastically. Base station epositioning has been suggested in [] fo inceasing the netwo lifetime. Ou wo can be extended to develop schemes fo wie placement when the sins o the senso nodes ae mobile, which can complement the wo in []. Howeve, using mobility fo enegy efficiency may not wo in some scenaios (lie ugged teains) whee it may not be feasible (fom a obotics pespective) to contol the mobility of the base station and ovecome natual obstacles. In such cases, it may be moe feasible to install wies. 3. Poblem Fomulation and Models 3. Netwo Model We conside a dis-shaped netwo topology. Sensos ae distibuted on the dis in a unifom andom manne. Evey node nows about its neighbos. The L B netwo is location-awae; that is, senso nodes have some mechanism by which they can find out thei location and thei neighbos locations (e.g., Hello messages). The netwo consists of only one sin. So the taffic fom all the nodes in the netwo is outed towads the sin. The location of the sin can be anywhee in the netwo. The sensos and the sin ae not assumed to be mobile. 3. Wie Model The two ends of a wie ae equipped with tansceives that enable only the ends of a wie to communicate with the wieless senso netwo and with the othe end of that wie. The ends of the wies have infomation about thei locations and they shae that infomation with nodes in the wieless senso netwo up to h hops away. Moe will be said about this in the outing model. The wies ae placed with one end away one hop fom the sin. The othe end of the wies lie on a cicle which has its cente at the sin and the adius of the cicle is the length of the wie. The nodes on the cicle ae equidistant fom each othe. One end of each wie is one hop fom the sin, and the othe end lies on a cicle of adius, whee is the length of the wie. When the sin is placed at an abitay position, the cicle with adius may not lie completely inside the topology. In that case, the wied nodes ae placed equally spaced out on the ac of adius lying within the topology. Figue illustates the placement of wies fo an abitay position of the sin. If we conside wies of length, then a cicle of adius centeed at the sin intesects the topology at points A and B. One end of each wie is still one hop fom the sin while the othe end of the wies is placed equidistant on the ac A-B. We assume that the bandwidth of the wies is lage enough so that thee ae no bandwidth constaints on the wie. Also, we ignoe MAC issues as well as the cost of sending pacets on the wie. Anothe assumption we have made in ou simulations and analytical models is that the communication between any two nodes A, B is symmetic, i.e. if A is in adio ange of B, then B is in adio ange of A. 3.3 outing Model The outing policy used is geedy geogaphic outing []. Thus when a node A has pacets fo a node B, it is assumed that it has the location of node B. Node A then tansmits the pacet to thei neighbo who is geogaphically closest to B. Evey node does the same until the pacet eaches node B. If a node X eceives a pacet fo node B, and it finds that none of its neighbos ae close to the destination than itself, then Poceedings of the IEEE/ACS Intenational Confeence on Pevasive Sevices (ICPS 04) /04 $ 0.00 IEEE

4 w ie w ie B A X Y Z B A l l/ S Figue 3.outing policy when all nodes have the nowledge of all wies. The decision is shown fo an abitay node A, whee w ie w ie is the wie and B is the sin the pacet is dopped. Modified geogaphic outing lie the one discussed in [3] deal with such cases, but we have not used such techniques fo simplicity. When thee ae no wies in the netwo, then the pacets ae outed to the sin using geedy geogaphic outing. When wies ae intoduced in the netwo, the pacets may eithe be outed to the wie, o diectly to the sin depending on the wie nowledge model. Fo ou study we conside two cases about the nowledge of the wies locations. In one, we assume that all nodes in the netwo have full nowledge of the wies. The second assumes that nowledge about wie locations is popagated only h hops away fom the wie end. So only nodes h hops fom the end points of each wie now about the wie. Depending on the value of h, some nodes may have the location of moe than one wie, wheeas some nodes may not have the location of any wie. Also note that in ou model, we assume that the sins (S) must be eithe the souce o the destination fo all communications (i.e., the taffic flows fom the sin to the sensos (in tems of queies fo example) o fom the sensos to the sin (in tems of epots)). Let us fist conside the outing decision fo the case when all nodes in the netwo now about all the wies. Let the souce and destination be A and B, espectively. Let thee be n wies, denoted by w i, in. Let the two ends of the wie be denoted by w ie and w ie, with w ie epesenting the end close to the destination. Let d(x,y) epesent the distance between any two points x and y. Fo each pacet, the souce computes:. d wi = d(a,w ie ) + d(b,w ie ) fo each wie w i, i = to n.. d(a,b) The souce then computes the minimum of the above n + distance. If the minimum is d(a,b), then the pacet is sent towads the destination diectly ove the wieless lins without using the wied shot cut. If d wi is minimum fo some i, then the pacet is sent fom A Figue 4.outing policy when wie infomation is esticted to hops fom the wie. In the above figue, S is the sin; A-B is the wie of length. We have two nodes, X and Y wanting to send pacets to S, both of which ae outside the coveage of the wie A-B. towads w ie. It then tavels ove the wied shot cut to w ie and then finally eaches B fom w ie ove one hop wieless. This outing policy is illustated in Figue 3. When we have patial infomation about a wie, we assume that the nowledge about the locations of the two edges of a wie is popagated to only h hops fom the wie. In that case, nodes tae the outing decision in the following manne:. The souce computes:. d wi = d(a,w ie ) + d(b,w ie ) fo each wie w i, i = to.. d(a,b) It may be possible that is zeo fo some nodes. In that case, the nodes send the pacet towads the sin (B) by sending it to the neighbo geogaphically closest to B. If is geate than 0, then the nodes compute (d wi ) minimum and compae it to d(a,b). If (d wi ) minimum < d(a,b) then the pacet is sent towads w ie (to use the wie) else it is sent towad B.. An intemediate node checs to see if the destination of the pacet is the sin. This is possible as we assume evey node nows the location of the sin, and that the netwo has a single sin. If it is, then the node does the same computation as the souce and may e-diect the pacet to go though a wied shot cut if the policy favos that oute. This is illustated in Figue The Enegy-efficiency Metic We define the enegy-efficiency metic in tems of the aveage enegy consumed in the netwo to get bit deliveed to its destination. If we conside the enegy consumed by the netwo to be the sum of adio enegy spent in tansmitting and fowading the pacets, then the enegy will be diectly popotional to the numbe of hops tavesed by the taffic. Altenatively this is epesented in ou wo by the Poceedings of the IEEE/ACS Intenational Confeence on Pevasive Sevices (ICPS 04) /04 $ 0.00 IEEE

5 A i C (0,0) C (0,-L) Figue 5.Figue showing the intesection aea of the topology with a dis with cente at the sin and adius i. The intesection points ae A and B. aveage path length (o degees of sepaation in the small wold teminology). Hence educing the aveage path length, educes the enegy consumed by the netwo, and subsequently inceases the enegyefficiency of the oveall senso netwo. Let us call aveage path length to go to/fom the sin to any othe node in the netwo in the pue wieless case as l(0). Also, let us call the aveage path length in case of using wies of length i metes as l(i). We define the path length atio PL(i)=l(i)/l(0). Hence, the eduction in path length due to wies of length i is -PL(i). Anothe aspect of enegy efficiency is load balancing. In pue wieless senso netwos, nodes nea the sin ae used moe often than othe nodes and ae susceptible to ealy enegy depletion, sometimes disconnecting the netwo. By using distibuted wies in the senso netwo the taffic concentation on the nodes nea the sin is alleviated. Hence we expect the lifetime (o useful time) of the netwo to be also extended in ou paadigm. We shall study this aspect in ou futue wo. 3.5 Analytical Model In this section, we develop the analytical model fo the aveage path length by adding shot cuts. Though this model we shall eason about the maximum attainable gain (as eduction in the aveage path length) and the optimal wie length fo the shot cuts. Let the cente of the topology be the point (0, 0). Let the sin be placed at location (0,-L). Let the adius of the topology be and the ange of the nodes be. We can daw concentic ings in the topology of vaying adii with cente at the sin. Let the adii of these concentic ings be i. Let us choose i as follows i = i., i = to L B () Let Aea i be the aea of the cicle of adius i that lies inside the topology (Figue 5). As the ings ae multiples of the ange of the nodes, the egions between these ings epesent the aea in which all nodes have the same hop count to the sin using puely wieless communication. Let us denote such egions by CH (egion of Constant Hop Count). Futhe, nodes in CH i ae i hops fom the sin. The expession fo the aea of CH i (ACH i ) is given by L ACH i = Aea i Aea (i-), i = to () ACH = Aea (3) As nodes ae distibuted unifomly in the netwo, the numbe of nodes in a CH is popotional to the aea of the CH. The aveage hop count fo the pue wieless case is given by: AHC L i i. CH i i (4) Now, let us intoduce wies in the topology. Let us assume that the wies ae of length, and we have infinite numbe of wies connecting the sin to nodes metes away. This means that thee ae infinite nodes on the othe end of the wies. All these nodes ae placed equidistant fom one anothe on a ing of adius. Let us divide the topology into 3 egions centeed at the sin, II and III. epesents the egion of the topology within adius / fom the sin. II epesents the egion of the topology lying within acs of adius / and. III epesents the egion of the topology outside the ac of adius. Now, since we ae using geedy geogaphic outing, the nodes in egion I will not use the wie as the sin is close to the nodes than the othe end of the wie. Thus pacets fom all nodes in this egion will be outed to the sin using wieless tansmissions. The nodes in egion II and III will use the wie to each the sin. Thus these nodes will send thei pacets using the wieless medium to the end of the wie away fom the sin. The pacets will then tavel ove the wie and be tansmitted to the sin fom the othe end. In the pesence of wies, we no longe have the fact that nodes in CH i ae i hops fom the sin. This is because now some nodes send the pacets to one end of the wie athe that the sin. The expessions fo the hop count of nodes in egions I, II and III ae given by: Poceedings of the IEEE/ACS Intenational Confeence on Pevasive Sevices (ICPS 04) /04 $ 0.00 IEEE

6 I: II: H i iach i i (5) ( i). ACHi i H (6) L ( i ). ACHi i III: H3 (7) Note that the expession fo H is simila to equation (4). This is because in egion I nodes do not use the wie. The expession fo II and III ae diffeent as the aveage hop count is now the aveage hops taen by pacets fom nodes in II and III to the wie and not the sin. The expession fo the aveage hop count in the pesence of wies is simply the sum of equations (5), (6) and (7): i L i i. ACHi ( i). ACHi ( i ). ACHi i i i AHC (8) The expession fo the AHC depends puely on ACH i. That in tun, depends on computing the values of Aea i. Fo an abitay position of the sin, Aea i does not yield a closed fom solution. This is because the acs centeed at the sin and having a adius i intesect the topology fo some values of i, while may lie completely inside the topology fo othe values of i. The detailed deivation of Aea i fo an abitay position of a sin is povided in [4]. Once we have that, we can compute the aveage hop count with and without wies fo diffeent lengths of the wie. We have plotted the path length atio fo vaying wie length and thee diffeent sin locations in Figue 6. In the next section, we will compae the esults obtained though simulation with the values obtained by using this model. The gaphs show that the path length atio deceases apidly with incease in the wie length, afte which the path length inceases. We see this happening fo each of the thee positions of the sin. The apid decease is shown by the fact that path length atio eaches 0.5 fo wie length of 400m. Fo the sin placed at the edge, the path length atio is at its minimum fo wie length equal to 0.5D, while it eaches the minimum at 0.375D fo the sin placed at the cente, D being the diamete of the topology. This esult is quite significant, as it puts an uppe limit on the ecommended length of the wie to be placed in the netwo. Path Length atio cente edge midw ay Wie Length (i) [metes] Figue 6.Path length atio obtained fo the eduction with the analytical model Thee is one case which yields an analytical expession fo the length of the wie equied fo minimum aveage path length. This is the case when the sin is placed at the cente. The minimum path length fo this case is found at the wie length of 0.756, whee is the adius of the 7 topology. This means that we get maximum eduction at wie length of 37.8 % of the diamete. In the following discussion, we povide the deivation of the above esult. Fo the sin placed at the cente, the dis centeed at the sin having adius i and the topology do not intesect as they have a common cente. Thus, the value of Aea i is simply Aea, i = to i i (9) Thus the value of the aea of CH i can be witten as: ACH i (i ), i= to (0) Substituting the value of ACH i in equations (5), (6) and (7) and simplifying we obtain the values of H, H and H3 as: 3 x x x 8 H () x x ( )(4x) H 6 () ( z x)( z x )( x z 0.5) H3 (3) 3 Whee z and x. By summing up (), (), (3) and obtaining the fist deivative with espect to x Poceedings of the IEEE/ACS Intenational Confeence on Pevasive Sevices (ICPS 04) /04 $ 0.00 IEEE

7 Path Length atio Theoetical Wie Length (i) [metes] Figue 7.Path length atio fo sin placed at cente. Gaphs have been plotted fo numbe of wies fom -48. and equating it to 0, we get x ( z ) 7. Fo 6 most pactical values of z, we can z appoximate x without any significant eo. In 7 tems of length, this is equivalent to saying. 7 Next, we discuss ou simulation envionment and ou simulation esults, and compae the simulation esults with the theoetical values geneated by the model in this section. 4. Evaluation Method 4. Simulation Setup We use a dis topology simila to the analytical model. To povide unifom andom distibution of nodes, the dis is divided into cells, with the density in each cell being constant. Within the cell, the nodes ae placed andomly. The cells ae used only fo placing nodes in the netwo such that the nodes cove the entie netwo. We have made no assumption of the nodes nowing othe nodes in thei cell. While computing the aveage path length in the pesence of wies, we assume that the cost of sending and eceiving pacets on the wie is negligible. The simulation was witten in C. Since we measue the path length eduction, we do not conside MAC laye issues. The paametes of the simulation wee as follows. The ange of all nodes was assumed to be the same, and was taen as =55m. The adius of the topology was =000m. The numbe of nodes is 000 nodes unifomly andomly distibuted as descibed above. The netwo is divided into 00m x 00m cells, with the node density in each cell taen as 0. The taffic was geneated andomly by the sensos and sent to the sin. Each expeiment geneated data pacets. Geogaphic outing is used as descibed in Section Expeimental desciption We have conducted vaious expeiments to investigate the diffeent dimensions of the poblem. Simulations have been conducted fo the following cases: () vaying the numbe of shot cuts by vaying the intewie angle, () vaying the length of the wies, (3) vaying the position of the sin, (4) limiting the infomation about wies locations to nodes hops fom the wie. In this case, a node fowads a pacet towads the sin if it does not have any wie infomation. Once a pacet eaches an intemediate node having the location of the wie, it maes a local decision if the pacet needs to be fowaded to the sin, o to the shot cut. 5. Analysis and esults 5. Sin placed at cente Figue 7 shows the aveage path length atio fo the sin placed at the cente. The nodes in the topology ae assumed to have full nowledge of all the wies. The cuve we get fom the analytical model has also been plotted. As we can see fom the cuves, the cuve fo the numbe of wies as 4 is vey close to the theoetical esult. This means that going beyond 4 adial wies (o inte-wie angle lesse than 5 o ) does not help in educing path length futhe. Thus we can say that the numbe of wies satuates at 4. A mathematical analysis explaining why satuation occus with 4 wies is given in [4]. The diffeence between the theoetical cuve and the actual esults can be attibuted to the node placement. We have placed nodes in cells in a gid lie manne, while the theoetical cuve assumes unifom density acoss concentic ings. The eduction with inceasing length shown in Figue 7 indicates that the eduction we get inceases apidly with inceasing length of the wie, but stats to satuate afte some point. The maximum eduction is found at 75 % of the adius if the topology, close to the value obtained using the analytical model. The In geedy geogaphic outing a node fowads the pacet to the neighbo closest to the destination. This technique may fail with local maxima [][3] in which the fowade itself is the closest node to the destination. Also, netwo patitions may be ceated due to andom node placement. We educe the effect of both of the above poblems by using the unifom node placement technique. In ou simulations, geedy outing failed to delive less than 5% of the taffic fo which we accounted by slightly inceasing the taffic sent (fom 00 pacets to 0 pacets). Thus, we ensue that ou simulation esults ae not affected by local maxima o patition poblems. Poceedings of the IEEE/ACS Intenational Confeence on Pevasive Sevices (ICPS 04) /04 $ 0.00 IEEE

8 Path Length atio Complete Wie Length (i) [metes] Figue 8.Path length atio obtained by limiting the wie infomation to hops. K is vaied fom to 6. maximum path length eduction we can get is aound 68 % (path length atio of 3 %). The cuve obtained mathematically shows the maximum eduction that can be achieved at 70 %. Thus the simulation esults obtained ae vey close to the analytical esult. Howeve, as the eduction impoves apidly with incease in wie length, we do not need to add a lage length of wie to get significant eduction. Fo example, if we have 3 wies, we can get eduction of 5 % by having the lengths of the thee wies as 300m o 30 % of the adius. Thus given a estiction on the total length of wie, we can use these cuves in Figue 7 and 8 to detemine if we need shot cuts of smalle lengths, but moe of them in numbe, o do we need smalle numbe of shot cuts, but each of those shotcuts will be of lage lengths. Obviously this decision is also affected by othe factos lie the capacity of the wies, taffic concentation aound the wies in addition to applicability of wie installation accoding to the teain of the senso netwo. We do not claim that this decision can be made solely on these cuves, just that they can be one of the factos in maing the decision. We plan to investigate these othe factos in the futue. Figue 8 shows the effect of limiting the wie infomation to hops fom the wies. The value of is vaied fom 8. Fo the expeiment, the numbe of wies was taen as 4, the satuation value obtained fom the pevious expeiment. Fo this case, we see that pefomance is the same fo small lengths of the wie. Beyond 300 metes, esticting the nowledge fo hop degades the pefomance. The cuve fo - hop nowledge stats deviating fom the cuve fo complete nowledge afte about 600 m. Fo all lengths, 3 hops give the same pefomance as the complete nowledge case. Note that this is a function of the numbe of wies we use, because the smalle the numbe of wies, the moe will be the degadation by esticting the infomation of the wies. Figue 9.The sin is placed at the cente. We have shown wie of length in the topology (S- C). When they have complete nowledge of the wies, we have egions, II and III. II educes to II when nowledge of the wies is esticted to hop. In the figue, SC =, SA = /, SB = -, with being the adio ange of the nodes. Thee ae two easons fo the gain to degade beyond a cetain length when we estict the nowledge of wies:. Fo small lengths, the coveage of the wied nodes in the azimuthal angle ovelaps. As the length inceases, the coveage no longe ovelaps and thee ae egions pependicula to the adial diection in which nodes do not have infomation of any wied nodes.. In the adial diection, we can divide the topology into thee egions, II and III. When we had complete nowledge, nodes in egions II and III used the wie. Now by esticting the nowledge, egion II, has educed fom II to II (Figue 9). Fo nodes in egion III, the diection of the sin is same as the diection of the wied nodes. So the pacets ae fowaded hop by hop towads the sin, until it eaches a node which has the infomation about the wied nodes, and the pacet is then ediected towads the appopiate wied node. If we have enough wies fo azimuthal coveage, this should happen fo almost evey pacet oiginating in egion III. Thus the ovehead of esticting the nowledge in this case is the exta distance the pacet tavels due to the ediection, and this ovehead is small when the numbe of wies is lage. The eal ovehead comes because of egion II tansfoming to egion II. Let s tae the case when we estict to hop. Pacets hops fom the wied nodes and x hops fom the sin (x > ) which ealie used to tae hops, now tae x hops. This ovehead becomes significant fo lage lengths, as the diffeence between II and II becomes lage. Even with the degadation fo lage lengths, fo lage numbe of wies and lengths up to 30 % of the adius of the topology, esticting the nowledge to hop Poceedings of the IEEE/ACS Intenational Confeence on Pevasive Sevices (ICPS 04) /04 $ 0.00 IEEE

9 Path Length atio Theoetical Wie length (i) [metes] Figue 0.Path length atio obtained by fo the sin placed at the edge. does not have any effect on pefomance. This is vey significant as now we have a case fo eeping a lage numbe of wies, with each wies being of small length. This way we conseve enegy by not popagating the nowledge of the wie moe than hop fom the wie. 5. Sin placed at edge Figue 0 shows the aveage path length eduction fo vaying numbe of wies when the destination is placed at the edge. The aveage path length eduction satuates at aound 60 % (path length atio of 40 %). This satuation occus at a lage value of wie length aound % of the diamete of the topology. The satuation value is about 0% highe than the case when the sin is placed at the cente. We see that the cuves fo the aveage path length atio ae almost identical fo numbe of wies 5o geate. This value is much lesse than that fo the sin placed at the cente (~ 4). This can be attibuted to the fact that the wies in this case need not cove the entie 360 degees, but only the ac of adius that lies within the topology. This can be achieved with fewe wies. Even fo this scenaio, the gain we achieve inceases apidly as the length inceases initially, afte which we get diminishing etuns. Anothe point of obsevation is the diffeence between the simulation esults and the analytical esult. This is because we place nodes in the netwo in a gid lie manne. The analytical model though, assumes that the nodes within the concentic ings ae popotional to the aea of the egion, which is not exact. Figue shows the effect of esticting the infomation of each wie to hops fom the wie. The value of is vaied fom -6. The gaph shows that fo small lengths esticting the wie infomation does not affect the gain we can achieve. As the length inceases, the diffeent cuves stat to banch out. This can again be undestood by dividing the entie Path Length atio complete Wie Length (i) [metes] Figue.Effect of esticting wie infomation to hops fo sin placed at edge. K is vaied fom -6. topology into egions I, II and III (Figue 9), centeed at the sin. esticting the nowledge to hops means that the II changes to II. This shifts the division between I and II fom being at / fom the sin to fom the sin, being the length of the wie. Fo lage length, the diffeence between II and II is much lage than that fo the sin placed at the cente. Consequently, we do not see the cuves fo =6 and =complete nowledge match fo vey lage lengths ( > ). To illustate this point, fo a length of 500m, esticting to hops is as good as complete nowledge. On the othe hand, fo a length of 00, even 6 hops ae not sufficient to give the same path length eduction as the complete nowledge case. As netwos will typically have wie lengths to be a small pecentage of the diamete of the netwo, we can estict nowledge of wies to a minimum of hops without pefomance degadation. 6. Futue Wo The focus of this pape has been on path length eduction. A lot of wo needs to be done to undestand the application of shot cuts to solving othe poblems. Possible extensions to this wo include: Enegy balancing and data extaction: In senso netwos, the data gatheed is tansmitted via wieless tansmission towads the sin. This means that the nodes close to the sin ae heavily involved in pacet fowading and thus they can lose enegy vey apidly. This can finally lead to the sin being patitioned fom the est of the netwo when all the nodes close to the sin have died, thus endeing the netwo useless. One solution of the poblem has been suggested in []. A vaiant of the above poblem is the data extaction poblem [5]. In this, the sensos do not tansmit thei data continuously, but gathe all thei data and send it in one shot en masse. In such a case, the poblem becomes of maximizing the data Poceedings of the IEEE/ACS Intenational Confeence on Pevasive Sevices (ICPS 04) /04 $ 0.00 IEEE

10 extacted fom the netwo. Fo both of the above cases, shot cuts can be used to distibute the fowading done by the wieless nodes between nodes fa away and close to the sin. This can lead to enegy balancing between nodes close to the sin and fa way fom it. We plan to study the use of shot cuts fo such applications in the futue. Mobile sins and sensos on wie: Mobility has not been consideed in this wo yet. This is because most senso netwos, once deployed, ae non-mobile. The NIMS poject has intoduced the concept of limited mobility in senso netwos by having mobile obots on cables. We plan to extend ou wo to addess mobility in the futue. We plan to conside both scenaios limited mobility of the sensos and mobile sins. 7. Conclusions In this pape we have investigated a new paadigm fo senso netwos. The paadigm uses wies in location awae senso netwos with a sin fo achieving path length eduction. We have developed analytical models to analyze the aveage path length eduction fo abitay position of the sin fo diffeing lengths of the wie. The analytical esults show that fo any position of the sin, thee is an optimal wie length fo which the path length atio is at its minimum, beyond which it inceases. Futhe, fo the sin placed at the cente, this occus fo a wie length which is a small faction of the netwo diamete (37.8 %). Simulations esults show that the maximum eduction in aveage path length that can be obtained is 70 % fo the sin placed at the cente and is aound 60 % fo the sin placed at the edge of the netwo. The length of wie fo significant eduction is highe when the sin placed at the edge. The maximum eduction is achieved with 4 wies when the sin is placed at the cente. Fo the same expeiment, 5 wies give nea satuation esults fo the sin placed at the edge. Thus esults show that we need moe numbe of wies fo the cente than the edge, but the length of the wie equied fo significant eduction is much lage fo the sin positioned at the edge. We have also investigated the effect of esticting the nowledge of the wies. esticting the nowledge of the wie to hops does not esult in deteioation of pefomance fo small wie lengths (less than 40 % of the adius). At wie lengths close to the adius of the topology, esticting the nowledge does show a decease in pefomance. The model developed in this pape can be applied to investigating othe issues in senso netwos lie the data extaction poblem. To ou nowledge no pevious wo has addessed these issues of using wies in senso netwos. efeences [] S. Milgam, The small wold poblem, Psychology Today 967, pp -6. [] D. J. Watts, Small Wolds, The dynamics of netwos between ode and andomness, Pinceton Univesity Pess, 999. [3] D. Watts, S. Stogatz, Collective dynamics of smallwold netwos, Natue 998, pp [4] Ahmed Helmy, Small Wolds in Wieless Netwos, IEEE Communication Lettes, Vol. 7, No. 0, Octobe 003, pp [5] A. Helmy, "Mobility-Assisted esolution of Queies in Lage-Scale Mobile Senso Netwos (MAQ)", Compute Netwos Jounal Elsevie Science - Special Issue on Wieless Senso Netwos, Vol. 43, Issue 4, Novembe 003, pp [6] A. Helmy et. al, "Contact Based Achitectue fo esouce Discovey (CAD) in Lage Scale MANets", IEEE/ACM IPDPS WMAN, Apil 003, pp [7] W. Kaise et al., Netwo Infomechanical Systems (NIMS), demo, ACM SenSys, UCLA, [8] W. Heinzelman et al., Adaptive potocols fo infomation dissemination in wieless senso netwos, in the Poceedings of ACM MobiCom 99, Seattle, WA, August 999. [9] C. Intanagonwiwat et al., Diect Diffusion: A scalable and obust communication paadigm fo senso netwos, in the Poceedings of ACM MobiCom 00, Boston MA, August 000. [0] W. Heizelman, A Chandaasan and H Balaishnan, Enegy Efficient Communication Potocol fo Wieless Micosenso Netwos, in Poceedings of the Hawaii Confeence on System Sciences, Jan 000. [] M. Younis, M. Bangad and K. Aaya, Base-Station epositioning Fo Optimized Pefomance of Senso Netwos, in the Poceedings of the IEEE VTC Wieless Ad hoc, Senso, and Weaable Netwos, Olando, Floida, Octobe 003. [] ahul Jain, Anuj Pui, and aja Sengupta., Geogaphical outing using patial infomation fo wieless ad hoc netwos, Technical epot M99/69, Univesity of Califonia, Beeley, 999. [3] Bad Kap and H. T. Kung, GPS: Geedy peimete stateless outing fo wieless netwos, In Poc. ACM MobiCom, August 000. [4] A. Helmy, ohan Chitaduga, Analysis of Shot Cuts in Wieless Senso Netwos, USC Technical epot Numbe 04-84, Mach 004. [5] N. Sadagopan, B. Kishnamachai, Maximizing Data Extaction in Enegy-Limited Senso Netwos, INFOCOM, Mach 004. Poceedings of the IEEE/ACS Intenational Confeence on Pevasive Sevices (ICPS 04) /04 $ 0.00 IEEE