Achieving Fault Tolerance and Network Depth in Hierarchical Wireless Sensor Networks

Transcription

1 Proceedigs of the World Cogress o Egieerig ad Computer Sciece 8 WCECS 8, October -, 8, Sa Fracisco, USA Achievig Fault Tolerace ad Networ Depth i Hierarchical Wireless Sesor Networs Aurag D ad Sompraash Badyopadhyay Abstract Most sesor etwor deploymets are -tiered ad are preceded by a topology study to determie the optimal locatio of sesors ad relays. Sesors form the leaves i a -tiered etwor ad do ot participate i the routig. A plot of the best path from each of the leaves to the si reveals the etwor topology to be hierarchical i ature. The AODV routig algorithm was desiged for highly mobile odes ad is ot directly suitable i a hierarchical sesor etwor where the sesors ad relays are predomiatly static. The Hierarchical routig as implemeted by ZigBee s C sip does ot support fault tolerace ad has a restrictio o the etwor depth. I this paper, we preset a probability based routig algorithm that merges the structure of a hierarchical tree with the flexibility of AODV. As devices joi a etwor, they are give a address to satisfy the hierarchical property. Here, the address geeratio is made based o the expected umber of devices to joi the etwor. Whe a coditio occurs where the hierarchical property ca o loger be supported, a route table etry, similar to AODV, is created oly for this o-coformig ode. The mechaism of maig route table etries has iheret support for fault tolerace. The completeess of the algorithm is determied theoretically ad by a exhaustive simulatio i C++. We provide the results of the memory requiremet, the sesitivity to the etwor deploymet probability distributios ad the probability of address duplicatio. Idex Terms Hierarchical Routig, ZigBee. I. INTRODUCTION The wireless sesor etwors comprise of sesors ad actuators that are desiged to ru o batteries for prologed periods of time. These devices ad such etwors eed to support low data rates ad the computatioal capability is restricted []. The devices compute routes amog themselves for multi hop data dissemiatio to a particular device geerally desigated as a si. Further, the devices are typically deployed i harsh ad ucoditioed eviromets where upredictable evets lie battery depletio or evirometal disturbaces ca lead to failures, thereby restrictig effective operatios. Thus, importat aspects of a wireless sesor etwor are its capability to achieve a efficiet routig scheme ad the degree of robustess. The topology of a wireless sesor etwor is the logical route tae by a pacet movig from the source to the destiatio. The topology costructio, through the routig scheme, is the fuctio of the NTW (etwor) layer i the 7 layer OSI stac. I multi-hop wireless systems, the topology ca be of two forms a mesh or a hierarchical tree. I a mesh topology, wireless devices maitai routes to every device i Mauscript received August, 8. This wor was supported i part by the Ifosys Fellowship Grat. Aurag D is with the Idia Istitute of Maagemet Calcutta, Kolata, Idia 7 ( aurag@ .iimcal.ac.i). Sompraash Badyopadhyay is with the Idia Istitute of Maagemet Calcutta, Kolata, Idia 7 ( sompraash@iimcal.ac.i). its commuicatio rage while i a hierarchical etwor the device maitais a route to oly its paret ad its childre. The algorithms for routig ad fault tolerace are desiged to develop a efficiet path for the multi-hop routes durig ormal operatio ad i the evet of a ode/li failure. The widely studied ad implemeted routig protocol for wireless mesh etwors is AODV (Ad-hoc O-demad Distace Vector) []. AODV was developed for mesh etwors that have highly mobile odes where routes are broe easily. The hierarchical tree topology has received little attetio with the most otable implemetatio support comig from the ZigBee [] stac. Here a addressig scheme called C sip, is utilized to provide addresses to the odes. The hierarchical topology requires o routig tables to be geerated as the ode addresses are sufficiet to mae the routig decisios. The hierarchical topology has bee show to provide the most efficiet routig [9]. However, it has o support for fault tolerace. Further, the C sip algorithm faces a severe address wastage problem [], where exteded multi hop reach is restricted. These aspects are elaborated i sectio. II. MOTIVATION We study the deploymet of a wireless sesor etwor with sesors beig placed at particular poits i the regio to be moitored. The sesed parameters iclude temperature, vibratio ad (harmful) gases. The sesors are placed at pre-determied poits ad it is well uderstood by the sesor etwor user that the sesed parameters correspod to the poit of the sesor locatio rather tha the regio (for e.g. a vibratio sesor is placed o a specific asset ad is immobile). The sesed iformatio is to be trasported to the cetralized data repository located at the si. The sesors do ot participate i the pacet forwardig ad relays are placed at appropriate locatios to trasmit the data i a multi-hop fashio to the si. Such etwors are ow as -tiered ad are the most commo form of sesor etwor deploymet. wireless coectivity logical path sesor ode relay ode si Figure. A mesh etwor reduces to a hierarchical miimum spaig tree whe data commuicatio is required oly betwee the odes (sesors ad relays) ad the si. ISBN: WCECS 8

2 Proceedigs of the World Cogress o Egieerig ad Computer Sciece 8 WCECS 8, October -, 8, Sa Fracisco, USA The data propagates from the sesors to the si ad vice-versa. There is o requiremet for oe sesor to commuicate to aother sesor, or oe relay to commuicate to aother relay. If we plot the topology of the best path (the metric ca be hop couts, sigal stregth, etc) from the sesors to the si, it would result i a hierarchical structure as show i figure. This is aalogous to the miimum spaig tree. The AODV routig algorithm builds the route tables by a floodig the etwor with RREQ cotrol pacets. These cotrol pacets are costly sice all devices are battery powered ad pacet trasmissios are the biggest source of eergy dissipatio. I case of ode failures, RRER cotrol pacets are geerated ad subsequet RREQ pacets to establish the ew route. The C sip algorithm used by ZigBee allocates addresses to each ode such that routig ca be made without havig to geerate ay routig table. It earmars each ode to have a fixed umber of childre. This fixed allocatio restricts the total depth (umber of hops) that the etwor ca support. For example, if every relay is expected to support upto 9 relays as its childre, the maximum depth is. As show i figure (a), a device at depth caot joi the etwor. The ZigBee hierarchical routig algorithm also does ot support fault tolerace. Whe a ode i the etwor goes dow, all its child odes loo for a ew paret. If a ew paret is foud, the child ode s address chages. This i tur results i the chagig of the address of all the childre of the child ode ad cotiues recursively till the leaves of the etwor. While obtaiig a ew paret, if the depth of the ode icreases, devices that were part of the etwor earlier may ot be able to joi. This is show i figure (b). The li betwee devices umbered ad goes dow. Device coects to device 9 ad is reassiged a address of. Note that its depth has also icreased. As a cosequece the childre of, are reumbered as ad 6 while the sesors, earlier umbered,, 6 ad are o loger able to be part of the etwor although they are i wireless rage. This is due to the address wastage of C sip where it has earmared addresses for o-existet devices Node with address Node uable to joi although i wireless rage Figure (a). The odes give addresses accordig to C sip with each relay expected to support upto 9 childre. I such a case, depth larger tha is ot possible. (b). Brea i li betwee ad (show as X) results i devices beig assiged a ew address ad sesors o loger part of the etwor. As a further example, if the umber of child relays a relay is able to support is set to, the maximum depth is 6. Similarly, if the max child relay is set to, the maximum depth is. I this paper, we develop a routig algorithm that icludes both the properties of the hierarchical as well as AODV protocols. As devices joi a etwor, they are give address to satisfy the hierarchical property. Here, the address 6 X 9 9 geeratio is made based o the expected umber of devices to joi the etwor. As the etwor grows with more devices joiig, the expected umber of devices to joi falls. Thus the address space (ad wastage) is reduced as the etwor grows. Whe a coditio occurs where the hierarchical property ca o loger be supported, a route table etry, similar to AODV, is created oly for this o-coformig ode. This route table etry is also created whe a ode/li fails thus prevetig other odes from havig to chage their addresses or depth. Our routig algorithm ca be tued to the probability distributio of ode deploymets. For example, a dese etwor with low depth ca be tued to a uiform distributio. I case of a etwor with log chais (effectively large depth) a geometric distributio ca be used. We term our probability based routig algorithm as a hybrid of the hierarchical ad AODV algorithms. The aim is to achieve a routig algorithm suitable for -tiered wireless sesor etwor which has a low cotrol pacet overhead ad at the same time supports large etwor depth ad is fault tolerat III. RELATED WORK Numerous studies have bee made o achievig efficiet routig i wireless mesh etwors. Sigificat wor icludes floodig [], the AODV protocol [] ad the Locatio Aided Routig (LAR) protocol [7]. I hierarchical routig, the addresses of the odes play a critical role as they decide the routig decisio. A hierarchical address allocatio is made by the C sip algorithm of ZigBee []. Here, a device jois the etwor by requestig a address from a particular device, ow desigated as the paret. The paret gives a address accordig to the equatio: A = A C (d) () ew paret + where d is the depth of the child ad C sip is the maximum umber of devices that ca joi uder all the other childre of the paret. This scheme loos to reserve addresses i the evet a ode jois at a particular level. Sice this caot be predicted apriori, most reserved addresses are ever utilized restrictig devices to joi i a already dese area. This restricts the depth of the etwor. The C sip algorithm was maily expected to be used i a star etwor where typically there is o multi hop data trasmissio. However, this address wastage problem has bee thought about i the latest ZigBee specificatio made available to the geeral public o Jauary 8 []. The specificatio i additio to C sip, has icluded support for a stochastic addressig scheme where addresses are assiged i a radom fashio. Here, there is o guaratee for uiqueess of the address ad address coflict egotiatio mechaisms are defied. Stochastic addressig automatically implies AODV mechaism is used for routig. Wor o optimized deploymet ad fault tolerace has recetly gaied mometum. Sesor coverage problems [, ] have looed at placig eough sesors to cover the etire area of deploymet. Oce a iitial topology is obtaied, additioal routers (or relays) ca be placed to achieve fault tolerace [6, ]. A (-) fault tolerat etwor is oe where every ode is coected ad the etwor remais coected eve after - ode failures. Existig sesors are modeled as a uit dis graph with the sesors desigated as a vertex. The goal is have every vertex -coected. Thus these sip ISBN: WCECS 8

3 Proceedigs of the World Cogress o Egieerig ad Computer Sciece 8 WCECS 8, October -, 8, Sa Fracisco, USA optimizatio schemes augers well with the hierarchical topology, however o wor has bee made at developig a addressig ad routig scheme for wireless hierarchical etwors that support fault tolerace. IV. PROBLEM FORMULATION The sesor etwor comprises of oe si, multiple sesors that are fixed (immobile) ad form the leaves of the etwor. Routers are placed to relay the data from the sesors to the si ad vice-versa. Each of the sesors, routers ad the si are homogeeous ad have a fixed radius of trasmissio. The etwor grows from the si towards the leaves. The si is the first device i the etwor to come up ad taes the address of. Subsequet routers/sesors request a address from a already associated device i its rage. Our algorithm provides this address i a itelliget way. There is o apriori iformatio of which device would come up ext ad hece the algorithm is geeric ad supports ay structure of the topology. A sample topology is show i figure. 7 6 Figure. A sample hierarchical topology. The address of a device uiquely idetifies it i the etwor. Further, the address is used to mae routig decisios. For example, i figure, if ode umbered, receives a pacet with destiatio address, it must be able to decide to which of its childre (amog, 7 ad ) the pacet must be forwarded, to esure the pacet reaches the destiatio. I most routig protocols, typified by AODV, the ode eeds to maitai a list guidig it o the etwor topology. However, i the case of a hierarchical etwor, the ode addresses are assiged i a way to esure routig decisios ca be made. We defie this decisio of a ode as its routig strategy S. Note that the strategy, S, must be simple eough to ru o resource (computatioal) costraied devices i acceptable time. Further, the strategy must be the same for all the odes, i.e., referrig to figure, ode caot mae a routig decisio uless it ows the strategy followed by its childre. We ow defie the strategy, for our proposed algorithm, S as i () ad a ew requestig device is give a address accordig to (). S : ChildreAddresses > ParetAddress () A ew = A + E( ) () max A child has a address that is greater tha its paret s address. Where A max is the maximum address reachable through the paret, E ( ) is the expected value of the probability distributio ad is the umber of devices left to joi the etwor. The algorithm thus calculates the expected umber of devices for a particular distributio fuctio, rather tha earmar the worst case solutio as i C sip. Further, i cases where the umber of devices joiig exceeds the expected value, a exceptio list is maitaied which is the route etry oly for the o coformig device similar to the AODV etry. Thus, our addressig algorithm, maitais as much as possible, the hierarchical structure. However, i cases where it is ot possible, a exceptio is created ad saved as a list. The algorithm essetially is a merge of the structured hierarchical etwor ad the completely ustructured AODV mechaism. Notatio N Table. Notatios used. Meaig Expected umber of devices i a etwor. (user geerated) curret umber of devices i the etwor that have to yet joi (out of N) E() Expectatio for a give probability distributio geeratig the umber of devices expected to joi the give device p(x) probability of x devices joiig The Hybrid Routig Algorithm : Node Start-up (switched o). Perform Iitialisatio. : Perform Chael Sca. If o devices foud i rage, goto else set my_paret = device with strogest sigal stregth : Sed Associate_request to my_paret : If associate_respose received from my_paret set my_address = address from associate respose : If Associate_request received, the If I am ot PAN coordiator the sed ew_address_request to my_paret with P=my_address as parameter else Calculate E = um of devices expected to joi. set N = last child address+ E Sed Associate_respose with N as the ew address : If ew_address_request received from ay ode R If I am ot PAN coordiator the forward ew_address_request to my_paret else calculate E = umber of devices expected to joi uder P set N = last child address of P + E If N breas hierarchical routig, create route table etry with N as destiatio ad R as ext_hop reply with ew_address_respose; P, N as parameters 6: If ew_address_respose received from ay ode R If N breas hierarchical routig, create route table etry with destiatio as N ad R as ext_hop If my_address = P the sed associate_respose with N as address 7: If data pacet for routig is received If destiatio exists i route table forward to ext_hop else forward accordig to hierarchy 8: If pacet sedig to my_paret fails perform chael sca set my_paret=device with strogest sigal if device is ot reachable through me sed route_update to my_paret with my_address as parameter 9: If route_update received from ay ode R If address breas hierarchical routig, create/update route table etry with address as destiatio ad ext hop as R forward route_update to my_paret ISBN: WCECS 8

4 Proceedigs of the World Cogress o Egieerig ad Computer Sciece 8 WCECS 8, October -, 8, Sa Fracisco, USA Lemma : Defiig the routig strategy as i () ad the addresses as i (), ewly assiged addresses caot ivalidate the already setup routig scheme. Proof: A sample topology is show i figure. Assume a existig ode with a address A. We defie all the addresses of the odes (parets) coectig A from the si as P, where = to D, D= depth of A. Let a ew ode joiig the etwor get a address N. Let Q be the list of addresses of the odes coectig N to the si with = to D, D = depth of N. Select the ode commo to P ad Q that has the highest address. (There will be atleast oe ode commo to P ad Q ). Let this ode be deoted C. Let the ode uder C with the largest address be deoted as C. The routig to A will fail if N jois the etwor at C ad N < A. However, N > C ad C >= A by (). Thus the routig to A is preserved. Figure. Sample topology for Lemma I Lemma II: The upper boud o the umber of exceptios o a ode (hece the maximum requiremet of memory for route tables) with N devices i the etire etwor is give by: ( N ) N = p where if + < = E( N i) E( N ) p i= otherwise Proof: The worst case sceario for the exceptios is as show i figure. The maximum exceptios eeded whe E (x) = for all values of x, will be N, where N is the umber of devices expected i the etire etwor. No exceptios will be eeded for devices whose addresses are less tha E(N-). E(N-) C A E(N-) E(N-6) Figure. Sample topology for Lemma II ad III. Lemma III: The worst case address wastage i a etwor, expected to hold N devices, is give by: N i = E( N i) ( N ) Corollary: The worst case umber of devices that ca be supported by a etwor with a 6 bit address is give by: K where i = K i= C N E( K i) = P = {,,C,C} Q = {,,C } Maximum Exceptios occur whe maximum possible devices joi uder a sigle paret. 6 Proof: From figure, the worst case i terms of address wastage occurs whe all devices loo to joi the PAN coordiator. The third device would joi with a address space of E(N-) -. The fourth device would joi with a address space of E(N-) -. This address space would be wasted if o device jois the etwor at the secod, third, etc device ad istead joi at the first device. Thus the maximum address wastage is the sum of the address space. Similarly, the last device with a address equal to the sum of all the address 6 space must be just less tha to be successfully associated. Calculatig the value of N for such a sceario gives the worst case umber of devices. For example, if the distributio of the etwor formatio is cosidered to be uiform, the worst case umber of devices turs out to be 76 ad the probability of such a worst case etwor occurrig turs out to be 896 (76)! Support for Large Networ Depth:. I this sectio, we show pictorially how the algorithm hadles etwors of large depth. Cosiderig the example show i figure (a), the ew ode must be give a address to joi the etwor. However, o address will satisfy the hierarchical property. We therefore give it a address based o the expected umber of devices uder ode 7. The ew address creates route table etry oly at ode. Thus a sigle route table etry is sufficiet for routig. As show i figure 6, assume the ew ode is give a address of. A pacet destied for ode is routed as follows. At ode, the hierarchical strategy eq. () implies the pacet must be forwarded to ode. Similarly, the hierarchical property at ode implies the pacet must be forwarded to ode. However, the route etry is preset at ode directig the pacet to ode. The hierarchical property is satisfied at ode 7 as well thus reachig the pacet to ode. 9 Node with address New ode wishig to joi the etwor Figure 6. Support for large etwor depth Support for Fault Tolerace: Node is give a address based o expected umber of devices uder 7 A sigle route etry is eeded at ode The hybrid addressig topology has iheret capability to support fault tolerace. I the evet of a ode failure, a ode selects the ew paret that would eable it to reach the si. It presets its address to the ew paret. The ew paret i tur, checs for the hierarchical property ad creates/updates route table etries if eeded ad forwards the pacet to its paret. This ca be cosidered similar to the AODV style RREQ (route request) ad RREP (route reply) messages. A example is show i figure 7. The li betwee odes ad brea ad ode jois ode 9. Sice all of the addresses uder ode are less tha 9, a route table etry is required for each of the 7 devices at ode. However, o route table etries are eeded at ay other ode sice the hierarchical property is satisfied elsewhere. ISBN: WCECS 8

5 Proceedigs of the World Cogress o Egieerig ad Computer Sciece 8 WCECS 8, October -, 8, Sa Fracisco, USA Nodes cotiue to retai their earlier address. 7 route table etries correspodig to devices uder ode are made at ode Node with address Node which is uable to joi i ZigBee Figure 7. Support for fault tolerace V. IMPLEMENTATION AND EVALUATION We implemeted the routig scheme assumig a uiform distributio for ode geeratio. A ode ca joi ay of the previously joied odes with equal probability. Referrig to figure, deotes the umber of odes i the leg where the ew device is joiig ad deotes the umber of devices yet to joi. The expected umber of devices uder a particular ode is calculated i O() time. The appedix provides the implemetatio details. We compare the above address allocatio algorithm with the memory requiremet i AODV for storig the route tables. Ituitively, our scheme must show a sigificat improvemet sice the address allocatio happes i a itelliget way where the hierarchical structure is maitaied as much as possible. I cases where this is ot possible, the AODV style route table for oly the exceptio device is stored. A additioal overhead i our scheme is a sigle uit to store the maximum address uder the device as is eeded by (). Memory Requiremet Memory ( Avg )..... AODV (Avg) Hybrid (Avg) AODV (Max) Hybrid (Max) Number of odes Figure 8. Memory Requiremet for etwor topology geerated through a uiform distributio. The simulatio was carried out i C++ where the ode placemet is radomized. The placemet of a ew ode was studied i two settigs. I oe, a uiform probability distributio of ode placemet was doe. Here, a ew ode N would have equal probability to choose ay of the previously deployed odes as its paret. I the secod settig, a geometric probability distributio was used. Here a ew ode N has maximum probability of choosig the last deployed ode as its paret ad the probability of choosig a earlier deployed ode falls proportioately. The geometric distributio is a more realistic represetatio of the practical sceario. Oce a radom etwor topology is geerated, the memory eeded for AODV ad the hybrid scheme is 9 6 X 9 Memory ( Max ) calculated for the same topology. We calculate the etwor average (deoted as AVG) ad the ode average for the ode with the biggest route table (MAX). The Hybrid routig, as expected, clearly outdoes the AODV scheme o both the etwor average ad the ode average couts. The hybrid scheme requires aroud % less memory. Figure 9 gives the performace of the AODV protocol ad the Hybrid protocol for the geometric distributio of topology geeratio. Iterestigly, AODV suffers sigificatly compared to the Hybrid algorithm. This ca be explaied by otig that the topology would be of larger depth o most occasios. I a uiform distributio, the topology would be lower i depth ad wider i width. Therefore i uiform distributio there would be more odes that have o childre. Whe a ode has o childre, it eeds o route table. However, for a geometric distributio, odes with o childre are smaller i umber thus leadig to a larger average requiremet of memory. The hybrid algorithm is more comfortable with the geometric distributio as icrease i depth leads to a lower icrease i the umber of exceptios. memory ( avg ) AODV ( avg ) uiform AODV ( avg ) geometric Hybrid ( avg ) uiform Hybrid ( avg ) geometric umber of odes Figure 9. Memory requiremet for AODV ad Hybrid uder uiform distributio ad geometric distributio (p=.8) Address Clashes The hybrid algorithm assigs addresses i a determiistic maer; i.e. for the same topology geerated i the same order will lead to the same addresses beig calculated. Thus we could simulate our scheme to chec for the duplicatio of addresses. As see i figure, the scheme performs rather poorly o this frot. The geometric distributio has a approximate clash rate of oe i three. Number of clashes ( avg ) 8 6 Hybrid, geom dist Hybrid, uif dist Number of Nodes Figure. Number of address duplicatio for hybrid scheme. The poor address uiqueess property of our hybrid scheme immediately motivates us i icorporatig small stochastic elemets. A ew = A + E( ) + ε () max ISBN: WCECS 8

6 Proceedigs of the World Cogress o Egieerig ad Computer Sciece 8 WCECS 8, October -, 8, Sa Fracisco, USA where ε is a stochastic error which is a fuctio of the umber of devices i the etwor. Note, for Lemma I to hold, ε must ecessarily. We pla to explore such a scheme i our future wor. Table. Simulatio Setup. Simulatio Eviromet C++ Number or odes i etwor to, i steps of Locatio of odes pseudo radom uiform (Sesitivity Rus) distributio pseudo radom geometric distributio with p=.8 Number of rus for each etwor size VI. CONCLUSION Our proposed hybrid algorithm for addressig ad routig i hierarchical etwors borrows cocepts from C sip of ZigBee ad AODV protocol. By mergig the flexibility of AODV ad the structure of C sip, we achieve sigificat improvemets i terms of memory eeded for route tables over AODV ad flexibility ad support for fault tolerace over C sip. Ulie AODV based addressig, where ode addresses are give i a stochastic maer, we give addresses i a clever scheme where the hierarchical structure is maitaied as much as possible. Where addresses ca be maitaied i a hierarchy, there is o eed for routig tables. I cases, where the hierarchy breas, we maitai a route etry oly for the o-coformig ode. This exceptio is the overhead i our scheme. Simulatio results have show sigificat reductio i the routig tables (aroud %) ad much better scalability with etwor size tha AODV. The simulatio also shows the rather poor address uiqueess property of our scheme (aroud %). We have idetified a mechaism to abate this cocer through the use of small stochastic elemets. We loo to explore the iclusio of the stochastic compoets i our algorithm ad study the improvemet i the uiqueess property i our future wor. Wor has also bee made i a practical implemetatio. We see to mae a large scale implemetatio ad study the ease of deploymet. The impressive gais of our scheme ispire us to develop the algorithm ito practical implemetatios. VII. REFERENCES [] Ayildiz, I.F., Xudog Wag: A Survey o Wireless Mesh Networs, IEEE Commuicatios Magazie, September. [] A. Wheeler, Commercial Applicatios of Wireless Sesor Networs usig ZigBee, IEEE Commuicatios Magazie, April 7. [] C. Peris, E. Royer, Ad hoc O-Demad Distace Vector Routig, IEEE Worshop o Mobile Computig Systems ad Applicatios, 999 [] C.S. Raghavedra, A. Avizieis ad M.D. Ercegovac, Fault Tolerace i Biary Tree Architectures, IEEE Trasactios o Computers, 98. [] G. Lewis et al, A System for Simulatio, Emulatio, ad Deploymet of Heterogeeous Sesor Networs, SeSys. [6] ITU Report o Iteret of Thigs Executive Summary: thigs/ [7] J. L. Bredi, E. D. Demaie, M. Hajiaghayi, D. Rus Deployig Sesor Networs with Guarateed Capacity ad Fault Tolerace, MobiHoc. [8] K. Youg-bae ad N. Vaidya, Locatio-aided Routig (LAR) i Mobile Ad Hoc Networs, ACM MobiCom, 998. [9] M. B. Lowrie ad W. K. Fuchs, Recofigurable Tree Architectures Usig Subtree Orieted Fault Tolerace, IEEE Trasactios o Computers, October 987 [] R. Peg, S. Mao-heg, Z. You-mi, ZigBee Routig Selectio Strategy Based o Data Services ad Eergy-balaced ZigBee Routig, IEEE Asia-Pacific coferece o Services Computig,December 6. [] S. Toumpis, L. Tassiulas, Optimal Deploymet of Large Wireless Sesor Networs, IEEE Trasactios o Iformatio Theory, 6. [] W. R. Heizelma, J. Kuli, H. Balarisha, Adaptive Protocols for Iformatio Dissemiatio i Wireless Sesor Networs, ACM MobiCom, 999. [] W. You-Chiu, H. Chu-Chi, T. Yu-Chee, Efficiet Placemet ad Dispatch of Sesors i a Wireless Sesor Networ, IEEE Trasactios o Mobile Computig, 8. [] X. Ha, X. Cao, E. L. Lloyd, S. Chie-Chug, Fault Tolerat Relay Node Placemet i Heterogeous Wireless Sesor Networs, IfoCom 7. [] ZigBee Alliace: APPENDIX Calculatio of E() give the expected umber of devices i the etwor (N) ad umber of devices yet to joi (). E ( ) = Figure. Calculatio of E() whe a ew device jois p() = * * *... * = p() = * * * L * * * * L * M * * * L * = * p() + + p() = * * * L * * * * L * M p( ) = C + + * * L * * = C * * p() ( + ) M i = * * p( ) i * p( i) Cosidered as device K devices A ew device (show i grey) jois the etwor. ISBN: WCECS 8