Modeling Mobility-Assisted Data Collection in Wireless Sensor Networks

Transcription

1 Modelng Moblty-Asssted Data Colleton n Wreless Sensor Networks Hsham M. Almasaed and Ahmed E. Kamal Dept. of Eletral and Computer Eng., Iowa State Unversty, Ames, IA 11, USA E-mal:{hsham,kamal}@astate.edu Abstrat Explotng moblty to enhane the performane of wreless sensor networks (WSNs), n terms of onnetvty, overage, and lfetme elongaton, has reently been the fous of several researh studes. Moblty was adopted n two dfferent ways; ether usng a network of moble sensor nodes or deployng a few supplementary speal moble elements, usually referred to as moble agents to help enhane onnetvty and overage. Dfferent modes of operaton (roles) were assgned to moble agents nludng beng a data relay, data olletor, and data snk. In ths paper we use a losed queueng network to model moblty and then evaluate data lateny under all those roles. The proposed model provdes powerful means to understand the effet of dfferent parameters, lke veloty and number of moble agents as well as ther movement strategy, on data lateny. I. INTRODUCTION A wreless sensor network mght beome dsonneted (or even parttoned) nto several slands for a varety of reasons. Node falures, random deployments, drft of nodes, hannel qualty degradatons, and non-unform energy onsumpton wthn the network are some of these reasons. To ope wth overage and onnetvty gaps wthn the network, researhers followed two man approahes. Frst, deployng more stat sensors ether preventvely to aheve ertan level of faulttolerane [1], or remedally to repar the network topology [2]. Seond, explotng supplementary moble agents to nrease the unformty of energy onsumpton n the network and/or brdge onnetvty gaps. Dfferent roles have been assgned to those agents nludng beng a relay, olletor, orsnk. The onept of moble olletors was ntrodued n [3], and were referred to as Data MULEs, to aheve onnetvty n sparse WSNs. MULEs move randomly, ollet data from nearby sensor nodes, and ultmately unload the arred data as they get lose enough to a base staton. On the other hand, by redung the amount of traff that stat sensors have to relay, snk moblty was proposed by many researhers to remedy the hot spot problem and to prolong network lfetme [4]. The thrd role that a moble agent an play s data relay. Moble relays have been studed n three dfferent modes; message ferryng [5], fragment brdgng [6], and supportve relayng [7]. Both fragment brdgng and message ferryng are used to support networks wth dfferent levels of dsonneton. Message ferres forward pakets between ndvdual nodes n sparse moble ad ho networks,.e., dsonneton at the node level. However, fragment brdgng ams at ahevng Ths researh was supported n part by the Natonal Sene Foundaton under grants CNS and ECS onnetvty between a number of dsonneted fragments, where a fragment s a group of onneted nodes,.e., dsonneton at the fragment level. In supportve relayng, the man objetve of a moble agent s to ooperate wth orgnal nodes n relayng traff to save them power and prolong ther lfetme wthout assumng any network dsonneton. The purpose of ths paper s to present a modelng approah that an be used to evaluate the system performane, n terms of end-to-end delay, under all dfferent roles of moble agents. In our prevous work, [6], we modeled the problem of fragment brdgng usng moble relays as a losed queueng network to aurately evaluate the end-to-end delay and understand the effet of dfferent system parameters. A smlar modelng approah s used study the use of the two other roles: moble snks and moble olletors. The rest of ths paper s organzed as follows. In Seton II, we desrbe the system model and how to buld a losed queueng network to model the moblty of moble agents n a lustered wreless sensor network. In Seton III, we present the mathematal formulaton and derve the dstrbuton of the loadng tme, whh s the tme before the data s loaded from a luster nto a moble agent, the trp tme of a moble olletor from a ertan luster to the base staton, and the endto-end delay under both the use of moble snks and moble olletors. Analytal and smulaton results are presented n Seton IV. We onlude our work n Seton V. II. SYSTEM MODEL We onsder a large-sale wreless sensor network that onssts of two types of nodes: stat sensors and moble agents. A moble agent (MA) takes one of two dfferent roles. Frst, a moble data olletor (MC) whh ollets nformaton, arres t, and ultmately unloads t at a entral base staton. Seond, a moble data snk (MS) whh s a fnal destnaton of data. In ths ase, a moble agent ould be a gateway to some ommand nodes or t ould be a ommand node by tself [8]. As ertan spots n the network mght be naessble, we propose that a small number of easly aessble loatons are hosen as potental stop-statons, orstatons for short, for a moble agent. In the ase of an MS, data wll be loaded at stop-statons therefore we all them loadng ponts. Onthe other hand, for an MC, a stop-staton ould be the pont where data s loaded or unloaded. Therefore, we all the loatons of stat base statons unloadng ponts. The term stop-staton wll be used to refer to both loadng and unloadng ponts. The /8/$25. 8 IEEE. 1

2 Fg. 1. A WSN lustered around predetermned stop-statons. network s lustered around stop-statons usng any lusterng tehnque [9] as shown n Fgure 1. When a moble agent stops at a loadng pont, data wll be transferred from all over the ntended luster n multhops to the agent. A. Assumptons and Defntons Before we proeed wth the modelng approah, we ntrodue some mportant assumptons: - End-to-End delay s domnated by the movement tme of moble agents, therefore we neglet the ommunaton delay between stat sensors and moble agents. For MCs, the end-to-end delay s omposed of two parts: (1) Loadng Tme: s the tme perod elapsed before an MC arrves at an unouped staton startng from the moment at whh the staton beame unouped. (2) Unloadng Tme: the tme t takes an MC to delver the data olleted at a ertan loadng pont. However, for the ase of moble snks, the end-to-end delay s only the loadng tme. - No ommunaton takes plae between moble agents. - For smplty, we assume that only one unloadng pont exsts. Staton () s always the unloadng pont n ase of MCs, and s treated as a loadng pont n ase of MSs. - MAs are homogeneous n terms of ommunaton and moblty apabltes and role of operaton. - The tme t that an MA has to spend at staton s supposed to be known. We start wth a suffently large value of t, then Algorthm-1 that we proposed n [6] an be used to redue t to ts mnmum aeptable value. B. Modelng Approah We assume a dsrete movement sheme n whh MAs stop for ertan tme wth a mean value of t at eah staton to ether load data from the ntended luster, or unload data at a stat base staton. An MA upon leavng ertan staton hooses the next staton to be j aordng to a preassumed dstrbuton qj. We refer to q j as the movement poly. Eah staton s modeled as an nfnte-buffer/nfnte-server queue wth a state-ndependent exponental serve tme of rate μ = 1 t. We refer to queues used to model statons as serve queues. To apture the movement tme between statons, we add another nfnte-buffer/nfnte-server queue, whh we all movement queue, k, between every par of serve queues Fg. 2. The losed queueng network model of the WSN shown n Fgure 1 and j that have (qj > ). Ths queue has a state-ndependent exponental serve tme of rate μ k = L d j, where d j s the Euldean dstane between statons and j, and L s the speed of MAs. If q uv s the probablty of movng from queue u to queue v, then q k = qj and q kj = 1. Fgure 2 shows a losed queueng network model for the network n Fgure 1. In [6], we used a losed queueng network to model the problem of fragment brdgng usng moble relays. Ths model s extended n ths paper to nlude the roles of beng a snk or a olletor node. Therefore, we refer the reader to the work n [6] for a onrete understandng of the bas foundatons of the model. Note that ths modelng approah depends only on the number of statons and MAs whh makes t hghly salable wth respet to the network sze. III. MATHEMATICAL MODEL We start ths seton by ntrodung some notatons and defntons. Then, we mathematally derve the dstrbuton of the loadng tme, unloadng tme, and the end-to-end delay. A. Notatons - K s the number of MAs (MCs or MSs). - M s the number of statons n the network feld. - M s the number of queues n the losed queueng network (ths nludes both serve and movement queues). - All MA s move at the same speed of L m/s. - d uv s the Euldean dstane between statons u and v, and t s assumed that d uv =d vu. - t u s the amount of tme that an MA has to spend at staton u. In prate, the tme an MA spends at staton u s fxed and s equal to t u. However, we assume that ths tme s exponentally dstrbuted wth a mean of t u n order to make our queueng network model tratable. - N ={n 1 ( N ),...,n M ( N )} s the state of the queueng system n whh the K ustomers are dstrbuted over the M queues suh that queue has n ( N ) ustomers n ths state. Note that M =1 n ( N )=K for any state ( ) N M+K 1 and the total number of states s. K - μ j s the state-ndependent serve rate of queue j /8/$25. 8 IEEE. 2

3 - quv s the probablty that an MA leavng staton u goes to staton v (ths probablty s assumed to be gven). - q uv s the probablty that an MA leavng queue u goes to queue v, where q =. - π( N ) s the steady-state probablty of state N obtaned usng the Convoluton Algorthm [1]. - n s used to refer to the number of ustomers at queue regardless of the system state. - E s the number of system states n whh n =. - p dle (t, N ) s the jont probablty that queue beame empty (.e., n =) at tme t= and remaned so n [,t] endng up n state N at tme t. In other words, ths s the probablty that queue s dle for a tme greater than t. - p dle (t)=[p dle (t, N 1 ),..., p dle (t, N E )] T. - U s a row vetor of ones suh that U =E. - 1 s a row vetor of length M n whh the th element s 1, and all other elements are set to zero. - X s a random varable that represents the Loadng tme of queue, that s the tme perod durng whh queue has no ustomers,.e., MAs. - Y s a random varable that represents the Unloadng tme for queue, that s the tme t takes a ustomer (an MC) that has just left queue to reah queue (). - F X (t)= Pr (X t) and F Y (t)= Pr (Y t) are the umulatve dstrbuton funtons of X and Y respetvely. - f X (t)= d dt F X (t) and f Y (t)= d dt F Y (t) are the probablty densty funtons of X and Y respetvely. - ψ s(t) and ψ (t) are the umulatve dstrbuton funtons of the end-to-end delay experened by data loaded from loadng pont by MSs and MCs respetvely. B. Loadng Tme Dstrbuton The frst step towards evaluatng the end-to-end delay s to evaluate the loadng tme dstrbuton. Note that the loadng tme represents the end-to-end delay n the ase of MSs, but just part of t n the ase of MCs. For queue, let A =[a xy] be an E E matrx suh that: M a j=1,j n j( N x )μ j f x=y xy = n j ( N y )μ j q jk f N y = N x + 1 j otherwse (1) and let p dle () be, as defned earler, a olumn vetor wth eah element p dle (, N j ) gven as, p dle (, N j )= ( N μ q k π j + 1 ) ( M N l=1 μ ln l j + 1 ) f n ( N j )= k=1,k n k ( N j )> otherwse (2) Then loadng tme dstrbuton F X (t) s the same as that derved for the dle tme n [6], and s gven as, F X (t) =1 U e A t p dle (). (3) Note that p dle (, N j ) s the probablty that queue beame empty at referene tme t =n state N j. Therefore, the term ( N μ q k π j + 1 ) ( M N l=1 μ ln l j + 1 ) k=1,k =,n k ( N j )> sums up all the probabltes of movng from state N = N j + 1 n whh n ( N ) > to state N j n whh n ( N j )=, and of ourse p dle (, N j )=f n ( N j ). C. Unloadng Tme Dstrbuton For a loadng queue, the unloadng tme dstrbuton s the dstrbuton of the tme t takes a ustomer to reah an unloadng queue after leavng queue,.e., the tme t takes an MC to delver the data t olleted from loadng staton. As we mentoned earler, we assume that only one unloadng statons exsts, and that s staton (). To evaluate ths dstrbuton, let us frst assume that the queueng network has only one ustomer,.e., K =1. Then, the unloadng tme of queue () s the tme t takes ths ustomer to reah queue () after leavng queue (). But ths s equvalent to the tme that passes before queue () reeves ts frst ustomer startng from the moment at whh the ustomer left queue (). Therefore, the dstrbuton of ths tme s the same as the loadng tme dstrbuton of queue () takng nto aount that the begnnng of the tme perod s when the ustomer had just left queue (). To sum up, the unloadng tme dstrbuton of queue () s the same as the dstrbuton of the loadng tme of queue () gven the followng two ondtons: (1) The queueng network has only one ustomer,.e. K=1. (2) The system state N at the referene tme t =s 1. Let g (,) dle (, N j ) be the jont probablty that queue () beame empty at the referene tme t =and queue () remaned empty n [,t] endng up n state N j at tme t. In a queueng network wth one ustomer, ths s the probablty that ths ustomer wll reah queue () after leavng queue n a tme that s greater than t. Smlar to p dle (, N j ) but wth the two ondtons mentoned earler (K=1, and N = 1 ) taken nto aount, g (,) dle (, N j ) s gven by, g (,) dle (, N j )= ( N q k π j + 1 ) k=1, k = n k ( N j )> f n( N j)=, n ( N j )= otherwse Let g (,) dle N j s n whh n =and n =,.e., (4) () be a olumn vetor over g(,) dle (, N j ) for all g (,) dle ()=[g(,) dle (, N 1 ),..., g (,) dle (, N E )] T, (5) then, the unloadng tme dstrbuton s gven by, F Y (t) =1 U e A t (,) g dle (). (6) D. End-to-End Delay Dstrbuton After dervng the dstrbutons of both loadng tme and unloadng tme, we an evaluate the end-to-end delay dstrbuton for both modes of operaton; Moble snks and Moble /8/$25. 8 IEEE. 3

4 olletors. As mentoned earler, the end-to-end delay usng moble snks s the same as the loadng tme, thus: ψ s (t) =F X (t). (7) Then, the average end-to-end delay of the data olleted at loadng pont by moble snks s gven by, ψ s = E[X ]= t= (1 F X (t)) dt (8) On the other hand, the end-to-end delay usng moble olletors s the sum of X (the loadng tme), Y (the unloadng tme), and t whh represents the perod of tme between the arrval and departure of an MC at staton. As the two random varables are ndependent and t s fxed, the dstrbuton of the end-to-end delay s gven as, ψ (t) =F X (t t ) F Y (t t ). (9) Smlar to (8), the average end-to-end delay of the data olleted at loadng pont by moble olletors untl t s unloaded at staton () s gven by, ψ = t= (1 (t)) dt = E[X ]+E[Y ]+t. (1) Both equatons (8) and (1) an be evaluated numerally as summatons over a suffently small nterval Δ. Therefore, ψ s = (1 F X (kδ)) Δ, (11) and, k= ψ = (1 F X (kδ t ) F Y (kδ t )) Δ (12) k= IV. RESULTS AND DISCUSSION We evaluate the followng performane metrs for the asestudy network n Fgure 1 through analyss and smulaton and under both roles Colletors (C) and Snks (S): - The auray of the losed queueng network model. - The effet of the speed of MA s on the end-to-end delay, usng two values 1.2m/s and 3.89m/s. - The effet of the movement poly,.e., q uv, on the end-to-end delay. We propose two poles; Poly (A), ompletely probablst, shown n Fgure 1, and Poly (B), partally determnst, whh s defned as follows: q,1 = q,2 =.5, q 1,3 = q 3, =1, q 2,4 = q 4, =1. - The effet of the number of MAs on the end-to-end delay, and for ths ase we use ether 1 or 2 MAs. All possble ombnatons of the suggested values mentoned above are summarzed n the senaros shown n Table I. a) Model Auray: Let L be the set of all loadng queues, and let ψ and ψ s be the average end-to-end delay of the data loaded from usng MCs and MSs respetvely obtaned by smulaton. Then, the mnmum, average, and maxmum error of a ertan poly usng MCs s defned as, Mnmum error = mn L Average error = 1 L ψ ψ (13) ) ψ (14) L ψ TABLE I THE SCENARIOS STUDIED IN THIS PAPER BASED ON DIFFERENT NUMBER, SPEED, ROLE, AND MOVEMENT POLICY OF MOBILE AGENTS. Error (%) Fg. 3. Senaro M K L (Poly, Role) m/s (A,S) m/s (B,S) m/s (A,S) m/s (B,S) m/s (A,S) m/s (B,S) m/s (A,S) m/s (B,S) m/s (A,C) m/s (B,C) m/s (A,C) m/s (B,C) m/s (A,C) m/s (B,C) m/s (A,C) m/s (B,C) ` Senaro The auray of the model n evaluatng the end-to-end delay. Maxmum error = max L ψ (15) Smlarly, the error usng MSs an be obtaned usng equatons (13), (14), and (15) by replang ψ and ψ wth ψ s and ψ s, respetvely. Fgure 3 shows the maxmum, mnmum, and average error for all the poles n Table I. The model aheves a hgh level of auray wth an error that s always less than 6%, and an average error that does not exeed 3%. b) Influene of the speed of MAs: Intutvely, nreasng the speed of MA s should result n an mprovement n the endto-end delay. We plot the gan, n terms of end-to-end delay reduton, aheved by nreasng the speed of MSs, Fgure 4, and MCs, Fgure 5, from 1.2m/s to 3.89m/s. Asts evdent from the examples, a hgh gan of up to around 58% was aheved by nreasng the speed. The nterestng thng, however, s the dfferene of the effet under the two roles of moble agents. In Fgure 4, approxmately the same gan was aheved under the same poly usng dfferent numbers of MSs. On the other hand, the effet of speed dffers dependng on the number of MCs for the same poly, about 1.5% for poly B, as Fgure 5 shows. The explanaton for ths s that the end-to-end delay n the ase of MSs s just the loadng tme whh s affeted by roughly the same rato, at the same loadng pont, when the speed s nreased regardless of the number of MSs. However, the end-to-end delay n the ase of MCs onssts of the loadng tme and the unloadng tme, and the latter s ndependent of the number of MCs and s dfferent for dfferent loadng ponts. ) Influene of the number of MAs: As Fgures 6 and 7 show, the gan aheved by nreasng the number of moble ψ /8/$25. 8 IEEE. 4

5 MS's, Poly A 2 MS's, Poly B 1 MS, Poly A MCs, Poly A 2 MCs, Poly B 1 MC, Poly A Fg MS, Poly B The gan of nreasng the speed of MSs to 3.89m/s Fg MC, Poly B The gan of nreasng the speed of MCs to 3.89m/s m/s, Poly A 1.2m/s, Poly A m/s, Poly B 3.89m/s, Poly A m/s, Poly B 3.89m/s, Poly A m/s, Poly B Fg. 6. The gan of nreasng the number of MSs from 1 to Loadng pont 3.89m/s, Poly B Fg. 7. The gan of nreasng the number of MCs from 1 to Change Movement Poly 6 Change Movement Poly Inrease Speed Inrease Speed 1 Inrease Num. of MS's 1 Inrease Num. of MCs Fg. 8. The gan aheved n the network {(K, L, poly, role) =(1, 1.2m/s, A, S)} by: hangng the poly to B, nreasng the speed to 3.89m/s, ornreasng K by 1. agents from 1 to 2 s the same for the same poly regardless of the speed of those agents. However, the gan for the same poly s dfferent dependng on the role of moble agents. The gan s hgher when usng moble snks than that aheved when usng moble olletors, beause the unloadng tme n the ase of MCs does not depend on the number of agents. d) Influene of the movement poly: So far, we saw that a sgnfant reduton of the end-to-end delay an be aheved by nreasng the number of moble agents or by usng faster ones. But, ths s the last thng a system desgner should do as t entals extra ost. In fat, t turns out that havng a better movement poly mght aheve a hgher gan than just nreasng the number of moble agents, as Fgures 8 and 9 show. Usng poly B nstead of poly A for the same number and speed of MAs results n an average gan of 44.8% for MCs and 16.% for MSs as shown n Fgures 9 and 8 respetvely, however, t ould not overrde the gan of nreasng the speed from 1.2m/s to 3.89m/s. V. CONCLUSIONS In ths paper, we modeled the movement of a lmted number of moble agents n a lustered WSN as a losed queueng network. The umulatve dstrbuton funtons of Fg. 9. The gan aheved n the network {(K, L, poly, role) =(1, 1.2m/s, A, C)} by: hangng the poly to B, nreasng the speed to 3.89m/s, ornreasng K by 1. the loadng tme, the unloadng tme, and the end-to-end delay were derved for both moble snks and moble olletors. The model an also be used to understand the effet of dfferent system parameters on the end-to-end delay. As the results mply, hangng the movement poly or nreasng the speed of MAs mght be more ganful than just addng more of them. REFERENCES [1] J. Bredn et al. Deployng sensor networks wth guaranteed apaty and fault tolerane. MobHo, May 5. [2] Yongguo Me et al. Sensor replaement usng moble robots. Computer Communatons, (13): , 7. [3] R. Shah et al. Data mules: Modelng a three-ter arhteture for sparse sensor networks. SNPA, May 3. [4] J. Luo and J. Hubaux. Jont moblty and routng for lfetme elongaton n wreless sensor networks. INFOCOM, 5. [5] W. Zhao, M. Ammar, and E. Zegura. A message ferryng approah for data delvery n sparse moble ad ho networks. MobHo, 4. [6] H. Almasaed and A. Kamal. Data delvery n fragmented wreless sensor networks usng moble agents. In ACM MSWM, 7. [7] W. Wang, V. Srnvasan, and K. Chua. Usng moble relays to prolong the lfetme of wreless sensor networks. In MobCom, 5. [8] M. Youns, M. Bangad, and K. Akkaya. Base staton repostonng for optmzed performane of sensor networks. IEEE VTC, 3. [9] A. Abbas and M. Youns. A survey on lusterng algorthms for wreless sensor networks. Computer Comm., (14-15): , 7. [1] J. Buzen. Computatonal algorthms for losed queueng networks wth exponental servers. Communatons of the ACM, 16(9), /8/$25. 8 IEEE. 5