Deterministic Worst-case Performance Analysis for Wireless Sensor Networks

Transcription

1 Determnst Worst-ase Performane Analyss for Wreless Sensor Networks Humn She,, Zhongha Lu, Axel Jantsh, L-Rong Zheng Dept. of Eletron, Computer and Software Systems Royal Insttute of Tehnology (KTH), Sweden {humn, zhongha, axel, Dan Zhou ASIC & System State Key Lab., Dept. of Mroeletrons, Fudan Unversty, Chna Abstrat Dmensonng wreless sensor networks requres formal methods to guarantee network performane and ost n any ondtons. Based on network alulus, ths paper presents a determnst analyss method for evaluatng the worst-ase performane and buffer ost of sensor networks. To ths end, we ntrodue three general traff flow operators and derve ther delay and buffer bounds. These operators are general beause they an be used n ombnaton to model any omplex traff flowng senaros n sensor networks. Furthermore, our method ntegrates varable duty yle to allow the sensor nodes to operate at lower rates thus savng power. Moreover, t norporates traff splttng mehansms n order to balane network workload and nodes buffers. To show how our method apples to real applatons, we ondut a ase study on a fresh food trakng applaton, whh montors the food freshness n realtme. The expermental results demonstrate that our method an be ether used to perform network plannng before deployment, or to ondut network reonfguraton after deployment. I. INTRODUCTION As advanes n wreless ommunatons and eletrons, wreless sensor network (WSN) has beome a promsng tehnology wth a wde range of applatons, suh as health are, supply han management, strutural montorng, and mltary support []. In most of the applatons, t s essental to ensure that the performane of sensor networks s predtable even n the worst ase. Reently, network alulus has been developed for worstase performane analyss n paket swthng networks []. Wth network alulus, some fundamental propertes of paket-swthed networks, suh as delay bound and baklog bound, an be studed. Jens et al. [3] [4] extended ths theory to sensor network alulus, whh an be used as a tool for worst ase traff analyss n sensor networks. In [5], Ans et al. proposed a methodology for the modelng and worst-ase dmensonng of luster-tree sensor networks. In ths paper, we also apply network alulus to examne the worst-ase performane of sensor networks. However, our work dffers from the prevous work and makes sgnfant mprovement n the followng aspets. Frstly, varable duty yle (see seton -C) s onsdered n our approah, thus provdng a falty to make ompromses between lateny and power onsumpton aordng to applaton requrements. In [7], Sang et al. demonstrated that both energy savng and hgh performane an be aheved by ondutng varable duty-yle operatons n meda aess ontrol protools. Seondly, we appled the network alulus theory to analyze three general traff flow operators, whh an be used to haraterze any omplex traff flowng senaros. The method an be appled to networks wth any topologes as long as there s no loop. Thrdly, traff splttng routng an be appled n ths method. Wth traff splttng mehansms, a traff flow s splt nto several sub-flows and eah one s sent to the destnaton aross dfferent paths. Traff splttng an be useful n mprovng the bandwdth effeny, mtgatng ongeston, and nreasng delvery relablty [8]. In addton, n [3] [4] [5], ther works are based on a ommon assumpton that the serve rate s onstantly bgger than the nput data rate. The assumpton may not always be reasonable n sensor networks sne nodes should allow dfferent operaton rates for the best of power savng wthout ompromsng performane. Our method an norporate the rate adustment n nodes. We have desrbed the traff splttng mehansms n our prevous work [6], from whh we borrow many notatons used n ths paper. However, ths paper dffers from [6] on the followng aspets. In [6], the work manly foused on analyzng traff splttng mehansms; n ths paper our man ontrbuton s proposng a determnst analyss method ntegratng varable duty yle and three general traff flow operators. Moreover, the topology of sensor network examned n [6] s a regular D mesh. In ths paper, ths lmtaton s lfted. In ths paper, we apply and extend the network alulus theory to the worst-ase performane analyss of sensor networks. We ntrodue three general traff flow operators and derve ther haratersts usng network alulus theory. Based on the traff flow operators, we present a determnst performane analyss method whh ntegrates varable duty yle operatons and traff splttng mehansms. In order to show how the analyss method works, a ase study of desgnng a sensor network for montorng food freshness n real-tme (see detals n seton 4-A) s onduted. The numeral results ndate that varable duty yle operatons and traff splttng mehansms have sgnfant effets on

2 mprovng the performane of sensor networks. Thus, requrements of dfferent applatons an be satsfed by seletng approprate network parameters suh as duty yle, work perod and splttng oeffent. Therefore, the analyss method provdes a way for a sensor network desgner to perform network plannng pror to deployment, as well as to reonfgure network after desgn. The rest of ths paper s organzed as follows: seton ntrodues the system model of sensor networks and bas knowledge of network alulus theory. In seton 3, three general traff flow operators and the determnst analyss method are presented. We present the wreless sensor network for fresh food trakng and gve numeral results n seton 4. Fnally, onlusons are drawn n seton 5. II. MODELS A. Sensor Netowrk System Model We onsder a stat wreless sensor network onsstng of multple sensor nodes and one snk node. These sensors are randomly sattered n a feld that needs to be sensed (Fg. ). Sensor nodes perodally send ther aqured data to the snk through mult-hop routng. A sensor node has the ablty to sensng the envronment and generatng messages, as well as relayng messages for other nodes. burst tolerane (n unts of data) and the rate (n unts of data per unt tme), respetvely. Havng α(t) as an arrval urve allows a soure to send σ bts at one, but not more than ρ bts/s over the long run. Smlarly, the output flow from a node ould also be modeled by a umulatve funton denoted by R (t), whh s defned as the traff departng from the node n tme nterval [, t]. The relaton between the nput flow R(t) and output flow R (t) s expressed as (Fg. ), R ( t) nf ( R ( s) + β ( t s)); t () s t where β(t) s defned as the serve urve [] provded by the sensor node, whh s a wde sense nreasng funton wth β(). Assume an arrval flow R(t), onstraned by arrval urve α(t), traverses a sensor node that offers a serve urve β(t). Then, the delay bound D(t), buffer bound B(t), and output flow R (t) an be derved aordng to the followng lemmas. The proofs of these lemmas an be found n []. Lemma. Delay bound D ( t) nf{ τ : α( t) β ( t + τ )} (3) Lemma. Baklog bound B( t) sup{ α( s) β ( s)} s Lemma 3. Output flow: The output flow R (t) s onstraned by the arrval urve, α ( t) sup{ α( t + s) β( s)} s (4) (5) α(t) β (t) Fg. A typal sensor network In sensor networks, there are typally two knds of traff flows, whh are upstream traff flows (from sensor nodes to the snk) and downstream traff flows (from the snk to a sensor node). Typally, rtal messages are sent from sensors to the snk,.e. upstream. The methods used to analyze upstream traff flows and downstream traff flows are smlar. Therefore, our efforts onentrate on analyzng upstream traff flows. B. Traff Model To haraterze the traff generated by the sensor nodes, we model the arrval flow at a node usng ts umulatve traff R(t), defned as the number of bts omng from the flow n tme nterval [, t] (R()). We assume that the umulatve traff flow R(t) s onstraned by a wde-sense nreasng funton α(t), that s, R( t) R( s) α ( t s); t, t s () α(t) s alled the arrval urve of R(t) [] (Fg. ). In ths paper, we assume an affne arrval urve for all the sensor nodes, whh s defned as α(t) ρ t + σ, where σ and ρ represent the Fg. Traff Model: ) The relatonshp between nput traff flow R(t), outp ut traff flow R (t), arrve urve α(t) and serve urve β(t); ) Delay bound and baklog bound. The baklog s the amount of bts that are held nsde the sensor node. The requred buffer sze of a sensor node s determned by the maxmum baklog. The delay at tme t s the tme that would be experened by a bt arrvng at tme t f all bts reeved before t are served before t. Graphally, the delay bound and baklog bound are the maxmum horzontal and devaton dstane between arrval urve α(t) and serve urve β(t), respetvely (Fg. ). C. Varable Duty Cyle Wakng up the nodes all the tme s mpossble n wreless sensor network sne merely turnng on the rado wll soon deplete the node energy. To save energy, all the exstng sensor networks employ a low duty-yle operaton wth a perod sleep and wakeup. In ths paper, we also assume sensor nodes have two work modes whh are atve mode and sleep mode. Let the work perod of all the sensor nodes be T,

3 and duty yle of sensor node be λ. Duty yle s defned as the perentage of tme that the sensor node s atve n a perod. It an be expressed as a rato or a perentage. For example, a sensor node wth a seond work perod, whh onssts of.s atve tme and.9s sleep tme, s sad to have a duty yle of. or %. Assume the tme for node to proess the pakets s τ. Further, let C denote the ahevable lnk apaty. Therefore, sensor node provdes a rate-lateny serve urve β(t), whh s defned as, β ( τ + t ) λc[ t (( λ ) T + )], (6) where λc and (-T + τ denote the serve rate and delay, respetvely. The expresson [x] + s equal to x when x>, and otherwse. III. A DETERMINISTIC ANALYSIS METHOD In ths seton, we present a determnst method for worstase performane analyss of sensor networks. The method s desgned to analyze the delay bound, baklog bound, and delvery apablty whh s measured by data delvery rato (see examples n seton 4-B). A. Analyss of Traff Flow Operators We defned three knds of traff flow operators: traff passng operator, traff mergng operator, and traff splttng operator (Fg. 3). These operators are general and an be used to desrbe any ombned traff flowng senaros. F S F F F S F a) b) ) Fg. 3 Traff flow operators. F, F, and S denote the nput flow, output flow, sensor node, respetvely. a) Traff passng: one nput flow and one output flow; b) Traff mergng: multple nput flows and one output flow; ) Traff splttng: one nput flow and multple output flows. ). Traff passng For the traff passng operator, the sensor node has one nput lnk and one output lnk (Fg. 3-a). As we mentoned above, the nput traff flow s onstraned by arrval urve α(t) ρ t + σ, and serve urve s defned as equaton (6). Based on Lemma,, 3, we derved the delay bound, baklog bound, and output flow, respetvely []. The delay bound s expressed as, σ D + ( T + τ (7) λc And the baklog bound s, B σ + ρ( T + ρτ (8) The output flow s onstraned by, θ ( t ) ρ' t + ( σ + ρ( T + ρτ ), (9) where ρ ' mn( ρ, λc). In the ase that λc <ρ, the baklog wll nrease endlessly f the nput traff flows nto the node ontnuously. To avod F M ths, measures should be taken ether to nrease the serve rate or onstran the nput data rate. Otherwse, pakets wll be dropped when the buffer s full. From the expresson of the serve urve, we an see that the serve rate an be nreased by nreasng the duty yle. ). Traff mergng For the traff mergng operator, multple traff flows merge nto one traff flow at the sensor node (Fg. 3-b). In ths ase, t s mportant that a serve dsplne should be appled to alloate the bandwdth. The serve dsplnes are used to ontrol the order n whh pakets are served, and determne how pakets from dfferent onnetons nterat wth eah other. In [], Zhang desrbed several serve dsplnes for paket-swthng networks, for example, Delay Earlest-Due-Date, Vrtual Clok, Far Queung. However, dfferent serve dsplnes ft for dfferent applatons. The serve dsplne for sensor networks should be as smple as possble sne the hardware resoure n a sensor node s very lmted. Therefore, we take the followng two dsplnes for bandwdth alloaton n sensor networks. The frst one s alled rate-proportonal alloaton strategy, and the other one s alled weght-proportonal alloaton strategy. In rateproportonal alloaton, the bandwdth s alloated proportonal to the data rate of eah flow; whle n weghtproportonal alloaton, eah flow s assgned a weght value w, and the bandwdth s alloated aordng to the weght values. In fat, the prevous alloaton strategy an be regarded as a speal ase of the latter when all the w equals. As shown n Fg. 3-b, we assume there are N nput traff flows, eah of whh s denoted as F. The output s an ensemble of traff flows. Let F denote the output flow orrespondng to F, and C denote the bandwdth alloated to the traff flow F. Assume the arrval urve and serve urve are the same as those n seton 3.-A. Then, C n the two alloaton strateges s alulated as expresson (). Based on Lemma,, 3, the delay bound, baklog bound, and output flow are derved as expresson (), (), (3), respetvely []. C N w ρ λc w ρ The delay bound of flow s, () σ D + ( T + τ () C And the baklog bound s expressed by, B N N σ + (( T + τ ) ρ () The output flow F s onstraned by, θ ( t ) C t + σ + ρ ( T + ρ τ (3)

4 3). Traff splttng In order to balane network workload, a traff flow may be splt nto multple flows as shown n Fg. 3-. Let the splttng fator be γ (... M ) and γ, where M denotes the number of output paths. We assume the node has nfnte nput and output apaty. The arrval urve and serve urve are the same as those n seton.-b. Then, we derved the delay bound, baklog bound, and output flow aordng to Lemma,, and 3 []. The delay bound s, γ σ D + ( T + τ (4) λc And the baklog bound s, B σ + ρ( T + ρτ (5) Output traff flow F s onstraned by, θ mn( γ ρ, λc ) t + γ ( σ + ρ( + ρτ ) (6) For the traff splttng operator, there ould be ases that ρ>λmc, whh means the output bandwdth an not satsfy the requrements. To avod ths, ether the nput data rate should be onstraned or the serve rate should be enhaned. Otherwse, the data loss rate wll be nreasng. B. The Determnst Analyss Method In ths seton, we present the determnst analyss method as a whole. It works as follows:. Aordng to the topology of the sensor network and the routng algorthm, obtan the routng paths of eah traff flow.. Based on the general traff flow operators and analyss methods proposed n seton 3., onstrut traff flowng senaros usng the three general operators. Then ompute the output flow, delay bound and baklog bound for eah traff flow startng from the soure node. 3. Calulate the end-to-end delay bound. There are two ways to ompute the end-to-end delay bound. The frst method s summng up the per-hop delay together. The other method was proposed by Lenzn et al. []. The man dea of ths method s to derve an equvalent serve urve for a gven traff flow based on the network alulus theory. And then the end-to-end delay bound s alulated usng the equvalent serve urve. Both approahes an be appled n our analyss method. C. An Example We show how the determnst analyss method works through an example. Assume there are two flows F and F, whh are soured from node a and b, respetvely (Fg. 4). F and F denote the orrespondng output traff flow of F and F at node. Assume the traff models are F ~(σ, ρ ) and F ~(σ, ρ ), where σ and σ desrbe the burstness, and ρ ρ denote the data rate [9]. Let the duty yle and work perod of node (a g) be λ and T, respetvely. Let the lnk apaty be C. Fg. 4 An example of traff flows At node, there are two nput lnks and one output lnk. Then, the output flow, delay bound and baklog bound an be alulated aordng to the method desrbed n seton 3-A-. Assume the alloaton strategy s rate-proportonal alloaton strategy. Then, F F ~ ( σ + ρ( λ ) T + τ, λcρ /( ρ + )) ρ ~ ( σ + ρ( λ ) T + τ, λcρ /( ρ + )) ρ (7) (8) D σ ( ρ + ρ) /( ρλ C) + ( λ ) T + τ (9) D σ ( ρ + ρ) /( ρλ C) + ( λ ) T + τ () And the baklog of node s, B σ + σ + ( ρ + ρ)(( λ ) T + τ ) () At node d, there are one nput lnk and two output lnks. All the alulaton an follow the method desrbed n seton 3-A-3. Use the same method, at node e and f, the output flows, delay bound and baklog bound an be alulated reursvely aordng to the method desrbed n seton 3-A-. After the above alulaton, the burstness and data rate of the output of flow F at the snk an be derved as expressons () and (3), respetvely. σ σ + ρ T [( λ ) + ( λ ) + ( λ ) + 3 ] () d e τ ρ mn( λ C, λ C, F ( ρ)) (3) e where F ( ) denotes the data rate of flow F at node. ρ The end-to-end delay bound an be alulated by addng the ndvdual delay at eah node together. For flow F, the results an be omputed usng the same method. From the results, we an see that the output data rate s manly lmted by the bottlenek lnk. Therefore, f the delay bound and baklog bound an not satsfy the requrements of applatons, the duty yle needs to be dynamally adusted. IV. d APPLICATION AND NUMERICAL RESULTS A. Wreless Sensor Network for Fresh Food Trakng In European market, approxmately % of the whole argo of fruts and vegetables omng from dfferent parts of world s deterorated durng the transportaton proess. Ths leads to a loss of bllons of dollars per year []. Wth the rapd development of sensor network tehnques, the loss an be mtgated by deployng a sensor network to trak the freshness status of these knds of food n real-tme.

5 In the senaro of real-tme fresh food (e.g. meat, vegetable, fruts) trakng, sensors are deployed n the boxes flled wth fruts, vegetables and meat n a truk arrage (Fg. 5). Sne the possble auses of food deteroraton are mrobologal nfestaton and mproper envronmental ondton, four knds of sensors an be used n our applaton, whh are humdty sensor, temperature sensor, CO sensor, and O sensor. These sensors are responsble for olletng the orrespondng nformaton of food. All the data olleted by sensors are sent to a base staton, whh s put on the top of the truk. The base staton then transmts the data to a remote server through GPRS networks and Internet. Thus an expert at the remote server sde an read and analyze the data n real-tme. If somethng s wrong or abnormal atons have happened, he an send nstrutons to the base staton to take measures, suh as lowerng the temperature of the oolng system or sprnklng water onto fresh vegetables and fruts, to protet the food from beomng deterorated. In addton, there s a wred onneton between the base staton and the drver montor. So the drver an also read the nformaton olleted by the network and take proper measures f neessary. The sze of sensor networks appled n ths applaton depends on the sze of truks. For small truks, a -hop or 3-hop sensor network s enough. Whle for large truks, a network of more hops s needed. Fg. 5 A sensor network for real-tme fresh food trakng B. Numeral Experments and Results We have realzed our determnst analyss method usng Matlab. The parameters used n the numeral experment are as follows. We assume a sensor network was generated by randomly puttng a number of sensors n the food boxes, whh are loated n a truk arrage (Fg. 5). The base staton ats as the snk. Therefore, a wreless sensor network wth an rregular topology s set up. Aordng to Ma mote [3], we assume the lnk apaty C s 38.4 kbps and work perod T s.96s [4]. We assume the paket sze s 88 bts. The standard reportng frequeny of eah sensor s assumed to be. Hz,.e. the sensor node sends one paket n every ten seonds, leadng to a date rate of 8.8 bts/s. The burst sze s assumed to be the amount of data generated n two seonds. In the experments, the traff load s hanged by varyng the reportng frequeny from. Hz to Hz. Moreover, we assume there s no ollson n the network sne the effet of ollson s ndependent of traff mergng and splttng. The Ma mote s a mote module used for low-power wreless sensor networks (see Note that at a sensor node, ts nput data rate an be hgher than ts serve rate due to a lower duty yle onfguraton. If ths happens, data loss may our when the baklog buffer s full. Apparently, data loss s a bg onern. To apture ths n our experments, we defne data delvery rato as the amount of data reeved by the snk versus that of data sent by the soures. To study how duty yle mpats the performane of sensor networks, we ondut several numeral experments. The number of nodes s 3. Fg. 6 shows that the paket delvery rato dereases wth traff load nreasng. And the paket delvery rato an be enhaned by nreasng the duty yle. In Fg. 7, we an see that the end-to-end worst-ase delay nreases when the traff load nreases. Wth the same traff load, the delay dereases wth duty yle nreasng. Therefore, when the data delvery rato or the worst-ase delay an not meet the requrement of applatons, the duty yle needs to be nreased. Meanwhle, the power onsumpton an be more mportant than delvery rato and delay. In these ases, low duty yle operatons an be taken to save energy. As we mentoned n prevous setons, traff splttng mehansms play an mportant role n load balanng. In order to study the effeny of traff splttng strateges, we ompared the data delvery rato and average per-hop delay n dfferent splttng strateges, whh are no traff splttng (), averagely traff splttng () and randomly traff splttng (). Averagely traff splttng means that the traff s averagely splt at eah sensor nodes, whle randomly traff splttng means that the traff s splt wth random probabltes. In the followng smulatons, the duty yle of eah sensor s.4. Fg.8 shows that when the traff load s bgger than.5, the data delvery rato drops dramatally when the traff s not splt. Compared wth that n, the data delvery ratos n and are enhaned by 7.% and 4.8%, respetvely. In Fg. 9, the average per-hop delay n and s lower than that n non-traff-splttng ases, wth mprovement 9.65% and 7.8%, respetvely. To show the performane of dfferent network szes, we devse the followng experments. The traff load s set to be.6, and the duty yle s set to be.4. Fg. and Fg. show the delvery rato and the worst-ase delay sale wth the number of sensors, respetvely. In Fg., we an see that the delvery rato dereases as the number of sensors nreases. However, by adoptng the traff splttng strateges and, the data delvery rato s enhaned by.5% and 7.7%, respetvely. Moreover, Fg. shows that the end-to-end worst-ase delay s redued by 8.8% and 5.% n and, respetvely. To show the baklog varaton wth duty yles and traff splttng methods, we ondut two experments. The traff load s.5 for both fgures, and the duty yle s.8 for Fg. 3. B_mn, B_ave, and B_max denote the mnmum baklog, average baklog, and maxmum baklog, respetvely. These values are obtaned from baklog bounds at all sensor nodes. In Fg., we an see that the baklogs do not redue muh when duty yle nreases from.4 to.5. However, the

6 average baklog s muh smaller than the maxmum baklog wth the same duty yle. In ths example, adustng duty yle has smaller effet on redung the baklogs. But by applyng traff splttng strateges, the maxmum and average baklogs an be greatly redued. Delevery rato (n perentage) Delevery rato (n perentage) Delevery rato (n perentage) Baklog (bts) Traff load VS. Delvery rato Duty yle. Duty yle.4 Duty yle Traff load Traff load Traff load VS. Delevery rato Number of sensors Fg. 6 Data delvery rato wth varous duty yles Fg. 8 Data delvery rato n varous splt methods Network sze VS. Delevery rato Fg. Data delvery rato vs. network szes Baklog hangng wth duty yles Duty yle.4 Duty yle.8 Duty yle.5 B_mn B_ave B_max Fg. Baklogs wth dfferent duty yles Baklog (bts) Worst-ase delay (s) Average delay (s) Worst-ase delay (s) Traff load VS. Worst-ase delay Duty yle. Duty yle.4 Duty yle Traff load Traff load VS. Average delay Traff load Network sze VS. Worst-ase delay Number of sensors Baklog hangng wth splttng methods V. CONCLUSIONS Fg. 7 Worst-ase delay wth varous duty yles Fg. 9 Average delay n dfferent splt methods B_mn B_ave B_max In ths work, we proposed a determnst method for worst-ase performane analyss n wreless sensor networks. Three general traff flow operators are defned to model any traff flowng senaros. Based on the results from the bas operators, we presented the determnst method, whh adopts varable duty yle operatons and traff splttng strateges. The method s appled to analyze the performane of the sensor network for fresh food trakng. Wth the numeral results, we show that () nreasng duty yle and splttng traff flows an mprove data delvery rato and redue the delay; and () varable duty yle operatons have less sgnfant effet on redung the max and average baklogs, whle traff splttng mehansms an largely redue the maxmum baklog and average baklog. Hene, by adustng the duty yle and traff splttng mehansms, the performane requrements (suh as delay, baklog and data delvery rato) of dfferent applatons an be satsfed. Therefore, our proposed method not only provdes an effetve way for a desgner to estmate the worst-ase performane of sensor networks, but also an be used as a tool for network reonfguraton after desgn. For the future work, we ntend to ntegrate fault tolerane nto the analyss method. Furthermore, t s nterestng to explore the optmzed desgn spae wth gven buffer szes, performane requrements, energy onstrants, and serve strateges. ACKNOWLEDGMENT Ths work s supported by VINN Exellene Pak Center at Royal Insttute of Tehnology (KTH), Sweden. REFERENCES [] I. F. Akyldz, W. Su, Y. Sankarasubramanam, and E. Cayr, A survey on sensor networks, IEEE Comm. Magazne, pp. -4, [] J. Y. Le Boude, and P. Thran, Network Calulus: A Theory of Determnst Queung Systems for the Internet, Sprnger, LNCS 5, 4 [3] J. B. Shmtt, and Utz Roedg, Sensor network alulus A framework for worst ase analyss, IEEE/ACM Internatonal onferene on Dstrbuted Computng n Sensor Systems (DCOSS 5), USA, 5 [4] J. B. Shmtt, F. A. Zdarsky, and Utz Roedg, Sensor network alulus wth multple snks, In pro. of IFIP networkng, workshop on performane ontrol n wreless sensor networks, Portugal, 6 Fg. End-to-end worst-ase delay vs. network szes [5] Ans Koubaa, M. Alves, and E. Tovar, Modelng and worst-ase dmensonng of luster-tree wreless sensor networks, the 7th IEEE Fg. 3 Baklogs n dfferent splttng strateges Internatonal Real-Tme Systems Symposum (RTSS 6), Brazl, 6 [6] H. She, Z. Lu, A. Jantsh, L.-R. Zheng, D. Zhou, Traff splttng wth network alulus for mesh sensor networks, the 7 Internatonal Workshop on Wreless Ad Ho, Mesh and Sensor Networks (WAMSNet 7), Korea, 7 [7] Sang Hoon Lee, Joon Ho Park, and Lynn Cho, AMAC: Traffadaptve sensor network MAC protool through varable duty-yle operatons, In Proeedngs of IEEE Internatonal Conferene on Communatons (ICC 7), Sotland, 7 [8] S.-J. Lee, M. Gerla, Splt multpath routng wth maxmally dsont paths n ad ho networks, IEEE Internatonal Conferene on Communatons (ICC ), [9] R. Cruz, A alulus for network delay, part I: Network elements n solaton, IEEE Trans. on Info. Theory, vol. 37, pp. 4-, 99 [] H. Zhang, Serve dsplnes for guaranteed performane serve n paket swthng networks, Proeedngs of the IEEE, 995 [] L. Lenzn, L. Martorn, E. Mngozz, and G. Stea, Tght end-to-end per-flow delay bounds n FIFO multplexng snk-tree networks, Performane Evaluaton Journal(Elsever), vol.63, pp , 6 [] Bllerud AB, The Bllerud book of maths, [3] Crossbow Tehnology In., MICA Datasheet