Data MULEs: Modeling a Three-tier Architecture for Sparse Sensor Networks

Transcription

1 Data MULE: Modelng a Three-ter Archtecture for Spare Senor Network Rahul C. Shah, Intel Reearch Seattle Sumt Roy, Intel Corp. Suhant Jan and Waylon Brunette, Unverty of Wahngton IRS-TR January, 2003 DISCLAIMER: THIS DOCUMENT IS PROVIDED TO YOU "AS IS" WITH NO WARRANTIES WHATSOEVER, INCLUDING ANY WARRANTY OF MERCHANTABILITY NON-INFRINGEMENT, OR FITNESS FOR ANY PARTICULAR PURPOSE. INTEL AND THE AUTHORS OF THIS DOCUMENT DISCLAIM ALL LIABILITY, INCLUDING LIABILITY FOR INFRINGEMENT OF ANY PROPRIETARY RIGHTS, RELATING TO USE OR IMPLEMENTATION OF INFORMATION IN THIS DOCUMENT. THE PROVISION OF THIS DOCUMENT TO YOU DOES NOT PROVIDE YOU WITH ANY LICENSE, EXPRESS OR IMPLIED, BY ESTOPPEL OR OTHERWISE, TO ANY INTELLECTUAL PROPERTY RIGHTS Copyrght 2002, Intel Corporaton, All rght reerved.

2 Data MULE: Modelng a Three-ter Archtecture for Spare Senor Network Rahul C. Shah Intel Reearch Seattle, WA 9805 Sumt Roy Intel Corp. Hllboro, OR 9724 Suhant Jan, Waylon Brunette Unverty of Wahngton Seattle, WA 9805 Abtract Th paper preent and analyze an archtecture to collect enor data n pare enor network. Our approach explot the preence of moble entte (called MULE) preent n the envronment. MULE pck up data from the enor when n cloe range, buffer t, and drop off the data to wred acce pont. Th can lead to ubtantal power avng at the enor a they only have to tranmt over a hort range. Th paper focue on a mple analytcal model for undertandng performance a ytem parameter are caled. Our model aume two-dmenonal random walk for moblty and ncorporate key ytem varable uch a number of MULE, enor and acce pont. The performance metrc oberved are the data ucce rate (the fracton of generated data that reache the acce pont) and the requred buffer capacte on the enor and the MULE. The modelng along wth mulaton reult can be ued for further analy and provde certan gudelne for deployment of uch ytem. I. INTRODUCTION Advance n devce technology, rado trancever degn and ntegrated crcut along wth evoluton of mplfed, power effcent network tack have enabled the producton of mall and nexpenve wrele enor devce [], [2], [3], [4]. Thee mall and nexpenve devce can be networked together to enable a varety of new applcaton that nclude envronmental montorng, emc tructural analy, data collecton n warehoue, traffc montorng etc. Such network hould collect data (typcally nfrequently) from the enor for long perod of tme wthout requrng human nterventon. The enor mut be low n cot and work wthn a lmted energy budget. Therefore, n order to acheve network longevty, a prmary concern n uch network power management. Dependng upon the applcaton, enor may need to be pread over a large geographcal area reultng n a pare network. The enor dtrbuton can be homogeneou (unform pread of enor) or heterogeneou (land of enor eparated by large dtance). Senor at each cty nterecton are an example of a homogeneou dtrbuton whle enor for habtat montorng [5] are dtrbuted heterogeneouly. Poble approache to enure connectvty n uch pare network nclude: Intallng of multple bae taton to relay the data from enor node n ther coverage area. Deployng enough enor to effectvely form a dene connected network [6]. The bae taton approach trade off hgh communcaton power needed by the enor wth the cot of ntallng addtonal taton. On the other hand, deployng cheap node to form a dene, fully-connected ad-hoc network may not be cot-effectve ether. The propoed archtecture n th paper eek to retan the advantage of both approache -.e. acheve cot-effectve connectvty n pare enor network whle reducng the power requrement at enor. The key to makng th feable the ubqutou extence of moble agent [7] n many of our target cenaro that we term MULE (Moble Ubqutou LAN Extenon) [8]. In the cae of traffc montorng applcaton, th role erved by vehcle (car, bue) outftted wth trancever; n a habtat montorng cenaro, anmal can perform th role. MULE are aumed to be capable of hort-range wrele communcaton and can exchange data from a nearby enor or acce pont they encounter a a reult of ther moton. Thu MULE can pck up data from enor when n cloe range, buffer t, and drop off the data to wred acce pont when n proxmty. The prmary advantage of our approach the potental of large power avng that can occur at the enor becaue communcaton now take place over a hort-range. Promng new rado technologe lke Ultra-Wdeband (UWB) [9] whch operate at extremely low-power wth large burt data capacty are potentally uted for enor to MULE communcaton. The prmary dadvantage of th approach, however, ncreaed latency becaue enor have to wat for a MULE to approach before the tranfer can occur. Neverthele for many data collecton applcaton (that requre data for analy purpoe only on the order of hour or even a day) uch ncreaed latency acceptable. The propoed three-ter MULE archtecture thu utable for uch delay-tolerant cenaro where power budget at the enor are the over-rdng contrant. Note that the above argument doe not addre the ue of energy conumed durng rado ltenng. Th can be potentally hgh becaue a enor ha to contnuouly lten to dentfy when a MULE pae by. The ame ue occur n ad-hoc network alo, where a node ha to contnuouly lten becaue t may have to forward ome other node data. Many reearcher are workng on addreng th ue for ad-hoc network [2] [0]. We beleve that the dea can be extended to our archtecture alo and hope to addre th more fully n future. The relatve trength and weaknee of varou approache for data collecton n pare enor network are qualtatvely ummarzed n Table I. In the bae taton approach there

3 Performance Metrc Approache Latency Senor Power Data Succe Rate Infratructure Cot Bae Staton Low Hgh Hgh Hgh Ad-hoc network Medum Medum-low Medum Medum-hgh MULE Hgh Low Medum Low TABLE I PERFORMANCE OF DIFFERENT APPROACHES FOR DATA COLLECTION IN SPARSE WIRELESS SENSOR NETWORKS. are a few bae taton (ame a acce pont) that cover the entre geographcal area and each enor communcate drectly wth the nearet bae taton. In the ad hoc network approach, enough enor node are preent o a to form an ad hoc network. The enor then end ther data to the wred acce pont by mult-hop routng over th ad hoc network. Note that whle the MULE approach uffer from hgher latency, t ha both low enor power conumpton and low nfratructure cot; charactertc that may be mportant for many applcaton. The ue of moblty to mprove performance n ad hoc network ha been condered prevouly n dfferent context [], [2], [7], [3], [4]. The prmary objectve ha been to provde ntermttent connectvty n a dconnected ad hoc network. However, the applcaton of moblty to the doman of enor network relatvely new and ha not been addreed n detal; the ZebraNet project [5] and the Manatee project [5], [6] are alo explorng the dea of ung moblty n enor network. Thee project focu on enurng the data reache all acce pont, wherea the MULE archtecture tre to delver data to only one acce pont. The next ecton gve an overvew of the MULE archtecture. After that, the ret of the paper focue on modelng the ytem to obtan ntal nght nto the performance of uch an archtecture. The goal of modelng wa to undertand the calng of the ytem charactertc a the parameter - number of enor, MULE etc. change. The model choen wa very mple, whch enabled u to obtan cloed form analytcal reult for many quantte of nteret, ncludng data ucce rate (the fracton of generated data that reache acce pont) and buffer occupance at MULE and enor. Although latency an mportant performance metrc, t not analyzed n th paper due to lack of pace. In addton to detalng the analy, ytem mulaton reult are alo preented. Thee verfy the analy whle provdng ome more nght nto ytem performance. The paper fnally conclude wth the nght ganed from the modelng analy and mulaton reult and outlne future reearch drecton baed on th ntal work. II. THE MULE THREE-TIER ARCHITECTURE The MULE archtecture provde wde-area connectvty for a pare enor network by explotng moble agent uch a people, anmal, or vehcle movng n the envronment. The ytem archtecture compre of a three-ter layered Fg.. The MULE three-ter archtecture abtracton (Fg. ) that can be adjuted to dfferent type of tuaton and dtrbuton need: A top ter of WAN connected devce, A mddle ter of moble tranport agent and A bottom ter made of fxed wrele enor node. The top ter compoed of acce pont/central repotore, whch can be et up at convenent locaton where network connectvty and power are preent. Thee devce communcate wth a central data warehoue that enable them to ynchronze the data that they collect, detect duplcate, a well a return acknowledgment to the MULE (ack may be neceary to enure relablty of data for certan applcaton). The ntermedate layer of moble MULE node provde the ytem wth calablty and flexblty for a relatvely low cot. The key trat of a MULE are large torage capacte (relatve to enor), renewable power, and the ablty to communcate wth the enor and networked acce pont. MULE are aumed to be erendptou agent whoe movement cannot be predcted n advance. However a a reult of ther moton, they collect and tore data from the enor, a well a delver ack back to the enor node. In addton, MULE can communcate wth each other to mprove ytem performance. For example a mult-hop MULE network can be formed to

4 reduce the latency between MULE and acce pont. The bottom ter of the network cont of randomly dtrbuted wrele enor. Work performed by thee enor node hould be mnmzed a they have the mot contraned reource of any of the ter. Dependng on the applcaton and tuaton, a number of ter n our three-ter abtracton could be collaped onto one devce. For example, to reduce latence n the traffc montorng applcaton, the MULE can be equpped wth an alway-on connecton (uch a a cellular or atellte phone) whch would allow t to act a the top and the mddle ter. Another key advantage of the MULE archtecture t robutne and calablty a compared to centralzed oluton. No enor depend on any ngle MULE, and hence falure of any partcular MULE doe not dconnect the enor from the pare network. It only degrade the performance. Alo the MULE archtecture ealy calable a deployment of new enor or MULE requre no network confguraton and (mot mportantly) obvate the need for algorthmc calablty for key functon uch a routng of packet. To mprove relablty acknowledgment can be ued. One can chooe to ue an end-to-end or ter-to-ter acknowledgment ytem. In ter-to-ter acknowledgment ytem the MULE ack the enor and the acce pont ack the MULE n turn. It ha the lmtaton that the MULE may fal at any tme wthout delverng the data to the acce pont and alo becaue the MULE may not be truted agent (data ent by the enor may alo be encrypted for th reaon). One of the key challenge n mplementng an acknowledgment baed protocol n uch a cenaro would be to determne when to retranmt due to hgh varablty n end-to-end latency. In ummary, the beneft of our ytem nclude: Far le nfratructure than a fxed bae-taton approach. For applcaton wth few enor pread over a large area, the cot avng could be order of magntude. There no overhead aocated wth routng packet from other enor a compared to an ad hoc network approach. For large ad hoc network, th overhead can lead to a ubtantal ncreae n energy conumpton at a node. Gven a uffcent denty of MULE, the ytem more robut than a tradtonal fxed network. Snce enor only rely on MULE, and MULE are nterchangeable, the falure of any number of MULE doe not mean connectvty falure; t merely ncreae the latency and decreae the data ucce rate of the network. Sytem flexblty allow the ame tranport medum to be ued multaneouly by dfferent applcaton. The MULE ytem can be vewed a a moble tranport mechanm for connectng heterogeneou node. The drawback of our ytem are: Latency for th type of network hgh and lmt the type of applcaton th oluton would be applcable for. Determntc delay bound guarantee eem feable only f MULE travere fxed route. The ytem preuppoe a uffcent amount of phycal ap ap Fg. 2. m ap ap m ap m denote acce pont denote enor denote MULE denote poble MULE movement A two dmenonal grd wth the dfferent ytem component movement n the envronment, whch a property of many enor ytem. Whle no network guaranteed to uccefully delver data all the tme, our erendptou network can encounter unexpected falure uch a lo of a MULE or nablty to reach enor becaue of change n terran caung lmtaton n moblty. III. SYSTEM MODELING We now focu on a mple and fully dcrete (n tme and pace) model of the network that neverthele allow u to nvetgate ytem performance a the parameter are caled. Fgure 2 how a pctoral repreentaton of dfferent ytem component. We make the followng aumpton n our modelng: The underlyng topology on whch enor, MULE and acce pont are placed aumed to be a dcrete and fnte two-dmenonal grd. Further, for analytcal mplcty the planar topology aumed to be the urface of a toru (.e the grd wrapped n both the north-outh and the eat-wet drecton). Only a fracton of the grd pont are occuped by enor and acce pont. The acce pont are modeled to be unformly paced on the grd whle the enor are randomly dtrbuted. The network evolve ynchronouly wth a global clock. At every clock tck the followng event take place: Senor generate one unt of data Every MULE move on the grd The MULE moton modeled a a mple ymmetrc random walk on the grd. At every clock tck, a MULE move wth equal probablty to any of the four neghbor of t current grd poton. The MULE communcate wth the enor or accepont only when they are co-located at the grd pont. We aume uffcently large bandwdth between the MULE and the enor o a to tranfer all the data redng at the enor n one contact. Although over mpltc we beleve that for certan envronment wth le data to tranfer th a practcal aumpton. We gnore relablty ue and aume that the communcaton error-free.

5 MULE move ndependent of each other and do not exchange any data among themelve when they nterect (occupy the ame grd pont) Both enor and MULE have buffer to tore data. For the enor, generated data placed n t buffer f t ha pace otherwe the new data dropped. Smlarly any data tranferred from enor to MULE placed n the MULE buffer only f pace avalable, ele t dropped. Intally all buffer are empty. Baed on th model, we analyze the performance of the ytem a the grd ze, number of acce pont and the number of MULE are changed. The performance meaure that we focu on are: Data Succe Rate: It meaure the fracton of generated data at the enor that the ytem able to tranfer to the acce-pont. In an deal ytem all the data generated by the enor would be tranferred to the acce-pont. Th would yeld a data ucce rate of one. Buffer Szng: A mentoned earler both enor and MULE have buffer. Whle mall buffer could lead to hgh packet drop rate, reducng the data ucce rate, large buffer have an aocated penalty n term of energy conumpton, phycal ze and manufacturng cot. Thu we would lke to determne mnmum buffer ze that would enure hgh data ucce rate whle beng cot-effectve. The model preented above very mple and exclude many real-world apect uch a rado propagaton, lnk falure and bandwdth contrant. Another major concern the choce of moblty model for analy. We realze that a dcrete random walk not an accurate repreentaton of the moton of vehcle, people etc. However, the mplcty of th model enable u to obtan cloed-form reult for the quantte of nteret, gvng u nght nto ytem calablty. Alo a mentoned n a recent urvey [7], random walk a wdely ued moblty model whch ueful n modelng the unpredctable moton of entte. We hope to develop a more ophtcated tochatc model whch can ncorporate more generalzed moblty model uch a Smooth Random Moblty Model [8] or Brownan moton wth drft [9], [20]. However note that wth the ncreang complexty of moblty model the hope of cloed form analy dmnhe and one ha to rely prmarly on mulaton. Thu we beleve that a frt order analy wth our mple model provde u wth a ueful bae. IV. GLOSSARY OF NOTATION AND SYMBOLS Th ecton lt all the commonly ued ymbol and notaton n th paper: (X n ) n 0 A dcrete-tme Markov chan S State pace of the Markov chan p j The tranton probablty P {X n+ = j X n = }, j S π=(π : S) Statonary dtrbuton for the Markov chan A The cardnalty of a et A N N mule N AP N enor ρ enor MB SB AP H R H N mule R N mule R AP Z M (k) S The number of pont on the grd,.e. the grd N on a de The number of MULE n the ytem The number of acce pont (AP) n the ytem The number of enor n the ytem The rato of the number of MULE to the grd ze (N mule /N); (0 ) The rato of the number of acce pont to the grd ze (N AP /N); (0 ) The rato of the number of enor to the grd ze (N enor /N); (0 ρ enor ) The total buffer capacty on a MULE (n number of packet) The total buffer capacty on a enor (n number of packet) Acce pont The httng tme to a enor n the grd,.e. the tme taken by a MULE tartng from the tatonary dtrbuton to frt ht when there only one MULE n the ytem The nter-arrval tme to a enor n the grd,.e. the tme between conecutve MULE arrval to when there only one MULE n the ytem The httng tme to a enor n the grd by any mule when there are N mule n the ytem The nter-arrval tme at a enor n the grd by any mule when there are N mule n the ytem The tme taken by a partcular MULE to tart from the et of acce pont and return back to t The buffer occupancy for a enor wth SB = when a MULE vt t The buffer occupancy for MULE k on one excuron from the et of acce pont back to the et. If there only one MULE, then we ll drop the upercrpt for convenence The data ucce rate of the ytem, whch the fracton of generated data that reache the acce pont V. BASIC RESULTS The mplet cenaro cont of one acce pont (N AP = ) and one MULE (N mule = ) n the ytem. We aume that the MULE and the enor have nfnte buffer capacty. The AP at ome poton (the exact poton not crtcal) n the grd of ze N on a de. The MULE aumed to perform a mple ymmetrc random walk on the grd. The tate pace S cont of the pont on the grd canned n any order to form a vector of length N (.e., S = N). Th mple model allow u to apply the large body of relevant reult from dcrete-tme, fnte tate Markov chan. We rely

6 on the tatonary dtrbuton π = (π : S) to etmate average value of the quantte of nteret. The tranton probablte for the Markov chan wth tate pace S are: { /4 f (,j) ha an edge p j = () 0 otherwe Snce S π = and all tate are equprobable (.e π = π j, j S), we get, π = N (2) We next compute the followng: Average nter-arrval tme at a enor node, E[R ] Average length that the MULE travere before t return to the AP, E[R AP ] Average number of data ample the MULE pck up durng one traveral, E[M] The average tme t take for the MULE to return to the ame enor node the nvere of the tatonary probablty by Markov chan theory. Therefore, E[R ] = π = N (3) Snce a unt data generated every clock tck, th alo the average value of the buffer occupancy at the enor E[Z ] when the MULE vt t (becaue SB =, o the buffer occupancy the ame a the amount data generated). Note that th the average value of the enor buffer occupancy oberved only when the MULE vt the enor, not over all ntant of tme (the econd quantty not of much ue n analyzng the ytem and alo harder to characterze). Smlarly, the average number of tep the MULE take before returnng to the acce pont : E[R AP ] = π AP = N (4) The number of data ample the MULE pck up durng one traveral depend on three thng - the length of the traveral R AP, number of enor encountered whch depend on ρ enor and the buffer occupancy at the enor Z. Snce the three quantte are ndependent, the average mply gven by (nce MB = ), E[M] = E[R AP ] ρ enor E[Z ] = E[R AP ] ρ enor E[R ] = ρ enor N 2 (5) The above reult provde ueful prelmnary nght nto the performance of the ytem a the grd caled. Clearly, the tme between MULE vt to a enor grow lnearly wth the grd ze a hown n (3). Th ha two mplcaton. Frtly, the requred buffer at the enor need to cale wth the grd ze to prevent lo of data. Secondly, the latency for data ample Notce that whle we aume SB =, n realty the buffer capacty ha to be fnte but uffcently large to avod packet drop. Thu we ue E[R ] to provde an ndcaton of uffcently large. Fg. 3. A two dmenonal grd wth the quare repreentng the poton of the acce pont alo ncreae wth the grd ze. Both thee problem can be mtgated by havng multple MULE n the ytem, a cae condered n ecton VII. The econd nght that wth only one acce pont n the ytem, the length of MULE excuron from the AP to the AP grow lnearly a hown n (4). Smlar to the cae above, there are two mplcaton. The frt that the requred MULE buffer need to be large to prevent lo of data. In fact, the requred buffer ze grow a the quare of the grd ze a hown by (5) above (Agan we ue E[M] to get an dea of the buffer ze needed to avod packet drop). The econd mplcaton that the latency for the data when travelng from the enor to the acce pont grow lnearly. Th mean that the number of acce pont n the ytem need to cale wth the grd ze, a cae condered n ecton VI. VI. SCALING WITH NUMBER OF ACCESS POINTS In th ecton, we analyze the effect of multple acce pont n the ytem. We aume that the acce pont are paced at a dtance of K pont on the grd n both the x and y drecton. Therefore, K = N/N AP = /. We tll aume that only one MULE preent n the ytem. Reult : If the acce pont are regularly paced at a dtance of K pont on the grd n both the x and the y drecton, then the expected length of excuron for the MULE tartng from the et of acce pont tll t reache the et agan (could be the ame AP or another one), E[R AP ] = K = (6) Proof: Lookng at the ymmetry of the grd n Fg. 3, we can reduce the tate pace to a maller grd of ze K K a hown n Fg. 4. Th can be een to be the reult of foldng the entre grd onto the maller box contanng only one acce pont A (whch repreent all the acce pont). Th poble becaue from the perpectve of a MULE, all acce pont are equvalent. The reultant grd alo reman a toru (wrap around n the north-outh and eat-wet drecton).

7 Fg. 4. Folded veron of the two dmenonal grd to form a maller grd (The type of node and ther tranton probablte are alo hown) A n ecton V the tatonary dtrbuton for a node n th reduced grd (ze K K) can be hown to be: π = K Ung th tatonary dtrbuton, the return tme to the pont "A" can be calculated. Th alo the requred excuron tme of the MULE from the AP et to the AP et nce the pont "A" repreent all the acce pont of the orgnal grd. E[R AP ] = π A = K = Thu we ee that the MULE excuron length between the acce pont et ndependent of the grd ze a long a the number of acce pont cale a a fracton of the grd ze. A VII. SCALING WITH NUMBER OF MULES In th ecton, we analyze the cae when there are multple MULE n the ytem. The fracton of MULE n the ytem kept contant a the ze of the grd ncreaed,.e., N mule /N =. We frt calculate the average number of vt oberved at a enor per unt tme. We then calculate the expected nter-arrval tme for MULE to a enor. That wll extend the reult (3) obtaned n ecton V. A mentoned before, we aume that all the MULE are performng ndependent random walk, wth no communcaton among each other. Alo, note that every MULE tart n the tatonary dtrbuton, and ubequently perform a random walk, thu remanng n the tatonary dtrbuton. Now conder a enor and a partcular MULE M 0. Then the probablty that M 0 nterect the enor gven by, P {M 0 nterect enor} = N Defne: f one or more MULE nterect the enor at tme k Y k = 0 f no MULE nterect the enor at tme k B (7) (8) (9) Hence the probablty that no MULE nterect wth the enor gven by, ( P {Y k = 0} = ) Nmule N ( P {Y k = } = ) Nmule (0) N Therefore the expected number of MULE vt to a enor per unt tme, 2 [ n ] n lm n n E {Yk =} = lm P {Y k = } n n k=0 k=0 ( = ) Nmule N e (large N) () (mall ) (2) Reult 2: The average nter-arrval tme between MULE vt to a enor when there are N mule n the ytem gven by, E[R N mule ] = ( N )N mule (3) (large N) e (4) (mall ) (5) Proof: To fnd the average nter-arrval tme at a enor, we conder the Markov chan compoed of the product of the Markov chan of each of the MULE. Thu the new tate pace gven by, S = S S... S }{{} N mule tme In the modfed tate pace S, we are ntereted n the et of tate A whch repreent one or more MULE nterectng. Snce all the tate are equally lkely, the tatonary dtrbuton for the et A can be calculated a, π(a) = A S = S S A S = N N mule (N ) N mule N N mule = ( N )N mule (6) Thu, ung Kac formula [2], the average nter-arrval tme between MULE vt to a enor, E[R N mule ] = = π(a) ( N )N mule 2 Multple MULE nterectng the enor at the ame tme condered to be jut one nterecton.

8 Corollary 2.: Average buffer occupancy on a enor (wth uffcently large buffer capacty) can now be calculated a: E[Senor Buffer] = E[R N mule ] (7) Here we have ued the obervaton that the enor buffer occupancy at the tme of MULE vt exactly the ame a the nter-arrval tme between MULE. Hence the average value are alo the ame. Alo note that th jut the average buffer occupancy een at the tme of MULE arrval at the enor; not at all tme. Corollary 2.2: Average buffer occupancy on a MULE (wth uffcently large buffer capacty) can alo be calculated a: E[Mule Buffer] = ρ enor E[R AP ]E[R N mule ] ρ enor (8) Smlar to the prevou corollary, we ue the expected value of the nter-arrval tme at a enor a the expected value of the enor buffer occupancy when a MULE vt t. Agan mlar to the enor buffer occupancy, th the average buffer occupancy on the MULE a een at the tme of MULE nterecton wth an AP; not at all tme. Thu th the average amount of data that pcked up by the MULE durng one excuron between the AP et. It nteretng to note that the problem of ncreang buffer requrement at the enor a the grd ncreae whch we encountered n ecton V elmnated. A long a reman contant, the buffer requrement reman the ame. So far we have jut found the average value of the nter-arrval tme for MULE to a enor. We next need to obtan the probablty dtrbuton. However, we frt fnd the probablty dtrbuton for the httng tme at a enor a that needed for the reult on the nter-arrval tme. A. Httng tme dtrbuton at a enor For our purpoe, the httng tme for a enor defned a the frt tme a MULE ht when all the MULE tart from the tatonary dtrbuton. We frt fnd the probablty dtrbuton of the httng tme for a ytem wth a ngle MULE before evaluatng the general cae of multple MULE. [2] how that the mean of the httng tme for a ngle MULE Θ(N log N) for mple ymmetrc random walk on the urface of a toru. Furthermore, the dtrbuton of httng tme for an ergodc Markov chan can be approxmated by an exponental dtrbuton of the ame mean [2]. Therefore, ( P {H > t} exp t cn log N ) (9) where the contant c 0.34 a N (vald for N 25) [22]. Note that th reult ue the contnuou tme veron of the dcrete tme Markov chan, but the reult tll correct for the dcrete tme cae [2]. However, wrtng n contnuou tme mplfe the analy conderably, thu all the httng and return tme probablty dtrbuton reult wll be for the contnuou tme chan. Ung th we can now extend the reult for the cae when there are N mule (> ) n the ytem. Reult 3: The httng tme for a enor when there are N mule n the ytem, all of whch tart n the tatonary dtrbuton gven by: P {H N mule Proof: Let H (k) a ngle MULE k. Then, Thu, we obtan, ( t > t} exp (20) N 0.34 N mule log(n) denote the httng tme to enor for H N mule = mn k MULE H(k) (2) P {H N mule > t} = [P {H > t}] N mule [ ( t exp 0.34N log(n) ( ) t = exp N 0.34 N mule log(n) B. Inter-arrval tme dtrbuton at a enor ) )] Nmule To fnd the nter-arrval tme dtrbuton at a enor, we frt conder the cae when there only one MULE n the ytem. In that cae, the nter-arrval tme at the ame a the return tme R for the MULE. Unfortunately, there no cloed form reult for the dtrbuton, but can only be approxmated a π/ log t for t for an nfnte grd [23]. For maller tme and for fnte grd ze, th only provde a very looe upper bound on the tal probablty. To obtan a better characterzaton we derve a recurve equaton to compute P {R = t} (nter-arrval tme dtrbuton for a ngle MULE). Let the ntal poton of the MULE be at the grd poton 0. Defne L,j (t) to be the number of path tartng from and endng at j of length t, avodng the pont 0 at all the ntermedate tep. Alo, let the neghbor of a node k n the toru be denoted by the et N (k). Then, wthout lo of generalty, for any enor node, P {R = t} = L 0,0 (t)/4 t (22) In the above equaton, L 0,0 (t) denote the total number of vald path that return to 0 n t tep and 4 t denote the total number of poble path of t tep. The followng recurve equaton can now be ued to compute L 0,0 (t): L,j (t) = k N () k 0 L k,j(t ), t > { f j N () L,j () = 0 otherwe Reult 4: If the number of MULE n a ytem N mule, the nter-arrval tme at a enor can be wrtten a: P {R N mule > t} P {H N mule > t} P {R > t} (23) Proof: To fnd the nter-arrval tme dtrbuton at a enor, we conder only the moment at whch one MULE

9 nterect the enor. We gnore multple MULE at the enor whch a very unlkely event for low mule dente. At th tme ntant, the ret of the MULE are n the tatonary dtrbuton. Thu, R N mule = mn (R, H N mule ) nce the MULE at the enor ha to return to the enor, but for the (N mule ) remanng MULE, t dentcal to httng the enor tartng from tatonarty. The reult follow from th obervaton. C. Return tme dtrbuton to the acce pont et We now compute the dtrbuton of the excuron tme of a MULE between the acce pont et. A n ecton VI we conder the folded toru (Fg. 4) n whch all the acce pont are repreented a a ngle grd pont. Snce th pont repreent the et of all acce pont, we need to compute the return tme dtrbuton to th ngle grd pont. For th we can apply (22) to the folded toru to obtan the requred return tme dtrbuton. Thu, P {R AP = t} = L 0,0 (t)/4 t (24) wth L 0,0 (n) defned on the urface of the folded grd of Fg. 4. VIII. DATA SUCCESS RATE We now have the pece n place to calculate the data ucce rate. We defne the data ucce rate a the rato of the average amount of data delvered to the acce pont by tme t to the total data generated by tme t a t. Reult 5: The data ucce rate of the ytem gven by, [ ] E mn(ρ RAP enor = mn(rn mule, SB), MB) S = E[R AP ]N enor k MULE (25) Proof: We ue renewal reward theory [24] to derve data ucce rate. One excuron of the MULE from the acce pont et back to the et condered a a cycle. Therefore R AP the length of a cycle. Recall that the enor generate data at the contant rate of one packet per unt tme therefore the average data generated n the ytem per unt tme N enor. We now get the data ucce rate S a, S = E [ k MULE M (k)] E[R AP ]N enor Here, M (k) = Data pcked up by the MULE k n tme R AP R AP = mn(ρ enor Y (k), MB) = The mn-functon becaue the buffer capacty of the MULE bound the total amount of data a MULE can carry. Now, Y (k) the amount of data at a enor vted by MULE k at tme. Th gven by, Y (k) = mn(z, SB) Parameter Decrpton Grd ze Number of pont on the grd N # of enor = Nρ enor # of MULE = Nρ MULE # of acce pont = N Senor buffer ze Number of data ample each enor can hold MULE buffer ze Number of data ample each MULE can hold Event MULE moton Data Generaton MULE-Senor Interacton MULE-AP Interacton TABLE II INPUT PARAMETERS TO THE SIMULATOR Acton Change grd poton of the MULE Generate new data at the enor and tore t n the buffer. If the enor buffer full data dropped Tranfer all data from the enor to the MULE. If the MULE buffer full, all the extra data dropped Tranfer all data from the MULE to the AP TABLE III EVENTS DEFINED BY THE SIMULATOR Smlar to the prevou tep, the enor buffer capacty bound the amount of data that can be preent at a enor, hence the mn-functon. Alo, nce Z the amount of data generated and not yet pcked up at the enor, t ha the ame dtrbuton a the nter-arrval tme at a enor. Hence, puttng th all together, S = E k MULE [ mn(ρ enor RAP = mn(rn mule E[R AP ]N enor IX. SIMULATION SETUP ], SB), MB) A cutom event drven mulator wa wrtten to verfy the precedng analy and alo explore the condton under whch t hold. In th ecton we preent a bref decrpton of the mulator. The mulator a dcrete event drven mulator where tme meaured n abtract unt of clock-tck. The underlyng grd tructure the urface of a toru wth the ze N pecfed durng ntalzaton. Dependng on the value of ρ enor and, approprate number of enor and MULE are placed randomly on the grd n the begnnng. Buffer ze on both the enor and the MULE can alo be pecfed and are completely empty when the mulaton tarted. Fnally, the AP can be ether randomly placed on the grd or regularly paced 3, wth the number of AP dependng on the value of. All the nput parameter to the mulator are hown n Table II. A ummary of the varou event handled by the mulator gven n Table III. 3 Interetngly, mulaton howed mlar reult for both unform and random placement of acce pont.

10 0 3 =% =5% =0% =20% E[R AP ] (# of tep) Grd ze (N) Probablty (cdf) = % (Smulaton) = % (Eqn. 20) = 0% (Smulaton) = 0% (Eqn. 20) = 20% (Smulaton) = 20% (Eqn. 20) Tme (# of tep) Fg. 5. E[R AP ] whle calng the grd ze wth = %, 5%, 0% and 20% Fg. 7. Cdf of the httng tme (H N mule ) at a enor (20 20 grd) E[R ] (# of tep) Grd ze (N) =% =5% =0% =20% Fg. 6. E[R ] whle calng the grd ze wth = %, 5%, 0% and 20% The mulator alo aume a perfect rado channel,.e., there no lo of packet durng tranmon. The only way packet can be lot f the enor or MULE buffer overflow. However, the enor do not mantan any tate (uch a ack etc.) to mplement relablty. Alo there no MULE to MULE nteracton, even though they may occupy the ame grd pont. X. SIMULATION RESULTS In th ecton, mulaton reult are preented whch verfy all the major reult of the analy and alo provde certan nght. To verfy calng wth acce pont, E[R AP ] wa meaured for a varety of grd ze from to A expected, E[R AP ] remaned contant acro all grd ze (Fg. 5) when wa kept contant, verfyng (6). Fg. 6 how the effect of calng the number of MULE on the average nter-arrval tme to a enor. A expected E[R ] Probablty (cdf) Fg Tme (# of tep) = % (Smulaton) = % (Eqn. 23) = 0% (Smulaton) = 0% (Eqn. 23) = 20% (Smulaton) = 20% (Eqn. 23) Cdf of the nter-arrval tme (R N mule ) at a enor (20 20 grd) remaned contant for dfferent grd ze a long a the value of dd not change, n accordance wth (5). Fg. 7 plot the cumulatve dtrbuton functon of the httng tme H N mule for = %, 0% and 20% on a grd. The fgure verfe that ung the httng tme reult for the contnuzed chan vald for the dcrete tme cae alo. Smlarly, Fg. 8 plot the cdf of R N mule for a grd wth the ame value of. Fnally, Fg. 9 plot the cdf of R AP for a mule on a grd where = 0.25%, % and 4%. Fg. 0 and plot the data ucce agant the normalzed MULE and enor buffer repectvely. Normalzed MULE Buffer Actual value of the MULE Buffer = E[MULE Buffer] Normalzed Senor Buffer Actual value of the Senor Buffer = E[Senor Buffer] (26) (27)

11 Probablty (cdf) = 0.25% (Smulaton) = 0.25% (Eqn. 24) = % (Smulaton) = % (Eqn. 24) = 4% (Smulaton) = 4% (Eqn. 24) Tme (# of tep) Fg. 9. Cdf of the return tme (R AP ) for the acce pont et (20 20 grd) Data Succe Rate (%) =0.% (Eqn. 25) =0.% (Smulaton) =% (Eqn. 25) =% (Smulaton) =0% (Eqn. 25) =0% (Smulaton) Parameter: ρ enor = % = 0.5% Mule buffer = grd Normalzed Senor Buffer Fg.. Data ucce rate v. normalzed enor buffer ze for = 0.%, % and 0% (20 20 grd) Data Succe Rate (%) =0.% (Eqn. 25) =0.% (Smulaton) =% (Eqn. 25) =% (Smulaton) =0% (Eqn. 25) =0% (Smulaton) Parameter: ρ enor = % = 0.5% Senor buffer = grd 50% data 90% data ucce rate ucce rate MULE buffer MULE buffer enor 0.% 43, , 000 buffer % , 800 = 0% Senor buffer Senor buffer MULE 0.% buffer % = 0% 0 47 TABLE IV SAMPLE VALUES OF MULE AND SENSOR BUFFER SIZES FOR 50% AND 90% DATA SUCCESS RATES Normalzed Mule Buffer Fg. 0. Data ucce rate v. normalzed MULE buffer ze for = 0.%, % and 0% (20 20 grd) For Fg. 0, the enor buffer ze wa nfntely large. Note the teep drop-off of the data ucce rate wth the MULE buffer ze. Alo, more than 95% data ucce rate acheved when each MULE buffer greater than 0E[M]. Interetngly, the plot alo how that one can trade-off the number of MULE n the ytem wth the amount of buffer capacty on each MULE. Th evdent from the fact that the data ucce rate curve are roughly the ame for dfferent MULE dente, but reducng the number of MULE by a factor k ncreae the expected MULE buffer ze by k (and vce vera). Th wll obvouly mpact latency, a the enor wll have to wat longer (or horter a the cae may be) before a MULE come by to pck up the data. However, the analy of the latency left a future work. Smlarly, for Fg. the MULE buffer ze wa nfntely large. Agan, a teep curve wa obtaned for the data ucce rate. Alo, the data ucce rate aturate for each MULE denty when the enor buffer capacty reache roughly 0E[R N mule ]. However, the fgure how that we cannot trade-off a decreae n MULE denty by ncreang the buffer at each enor. Hgher MULE dente lead to hgher data ucce rate, n general, untl the enor buffer are uffcently large. Th can be een more clearly n Table IV whch how the actual value of the buffer ze needed to acheve data ucce rate of 50% and 90%. Thee are hown for both the cae of nfnte MULE and nfnte enor buffer. For SB =, the amount of MULE buffer needed to acheve a certan level of data ucce rate cale nverely a the mule denty. However, when MB =, the enor buffer need to ncreae by a rato greater than the decreae n the number of MULE. The reaon for th dfference le n the Law of Large Number. When M B =, the drop n the data ucce rate due to packet gettng dropped at the enor. Now a N mule reduce, the nter-arrval tme at a enor grow larger, and conequently, there are larger amount of data that are dropped due to enor buffer overflow. On the other hand, when SB =, data overflow occur at the MULE when the amount of data t pck up from all the enor exceed the MULE buffer capacty. However, due to the Law of Large Number, the probablty of the total amount of data on the

12 MULE exceedng the buffer threhold maller than n the fnte enor buffer cae. XI. CONCLUSION AND FUTURE WORK In th paper we have preented an archtecture to connect pare enor network at the cot of hgher latence. The man dea to utlze the moton of the entte that are already preent n an envronment to provde a low power tranport medum for enor data. After ntroducng the archtecture, the focu of the paper wa on preentng a mple analytcal model baed upon two-dmenonal random walk to provde nght nto varou performance metrc (data ucce rate and buffer ze). Our key obervaton are: The enor buffer requrement are nverely proportonal to. The MULE buffer requrement are nverely proportonal to both and. When the enor buffer large the buffer capacty on each MULE can be traded-off wth the number of MULE to mantan the ame data ucce rate. The change n the buffer capacty on each enor need to be greater than the change n the number of MULE to keep the ame data ucce rate. We plan to expand our work n couple of drecton. One to develop a more complete tochatc model to addre ome of the current mplfcaton uch a nfnte bandwdth, random-walk moblty model and error-free communcaton. Here we plan to ue dea from queung theory and renewal procee. An mportant ue that not addreed n th paper due to lack of pace latency. Latency ha two component - latency on the enor before a MULE pck up the data ample and the latency on a MULE before t encounter an acce pont. We have obtaned ome prelmmary reult and wll addre that n a future publcaton. From the protocol pont of vew, MULE-to-MULE communcaton and relablty ung acknowledgment are ome of the nteretng ue. Another lmtaton of current work the aumpton that the enor have to contnuouly lten n order to dentfy a MULE preence. Approache to ncreae the leep tme for enor, uch a reduced duty cycle, need to be explored alongwth ther effect on ytem performance. ACKNOWLEDGMENTS We would lke to thank Jaon Jenk (Unverty of Wahngton) and Anthony LaMarca (Intel Reearch, Seattle) for ther actve partcpaton n defnng the ytem archtecture. We are thankful to Prof. Krzyztof Burdzy (Dept. of Mathematc, Unverty of Wahngton, Seattle) and Prof. Davd Aldou (Dept. of Stattc, Unverty of Calforna, Berkeley) for provdng u wth nformaton and reference on nonnteractng partcle theory of Markov chan. The work wa partally upported by DARPA grant N REFERENCES [] D. Etrn et. al. Embedded, everywhere: A reearch agenda for networked ytem of embedded computer. In Computer Scence and Telecommuncaton Board (CSTB) Report, 200. [2] J. Rabaey et. al. Pcorado upport ad hoc ultra-low power wrele networkng. In IEEE Computer Mag., July [3] J. Kahn, R. Katz, and K. Pter. Next century challenge: Moble networkng for mart dut. In ACM/IEEE MobCom, 999. [4] UC Berkeley TnyOS. [5] P. Juang et. al. Energy-effcent computng for wldlfe trackng: Degn tradeoff and early experence wth zebranet. 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