Jont Rate-Routng Control for Far and Effcent Data Gatherng n Wreless sensor Networks Yng Chen and Bhaskar Krshnamachar Mng Hseh Department of Electrcal Engneerng Unversty of Southern Calforna Los Angeles, CA 90089 Abstract In wreless sensor networks, far and effcent rate allocaton s an essental mechansm to avod congeston collapse and system degradaton. Whle most pror work n ths context has focused on a statc tree, we consder the ont optmzaton of routng and rate allocaton n ths work. We formulate LP problems to obtan max-mn farness and sum-rate effcency. We show the tradeoff between farness and effcency n ths settng, and develop dstrbuted algorthms based on Lagrange dualty to acheve these obectves. 1 Introducton Due to the unpredctable nature of wreless communcaton, such as varyng lnk qualty and bandwdth, congeston s a common problem n wreless sensor networks. Ths s even true for small wreless sensor networks wth known perodc traffc. Moreover, for contenton-based MAC protocols, lke CSMA, concurrent data transmssons over dfferent lnks may nterfere wth each other and aggravate congeston, fnally causng congeston collapse [9]. In order to avod ths, each sensor node must approprately control ts own rate to mprove channel utlzaton and per-node end-to-end throughput, especally n a mult-hop network. There have been many papers on rate allocaton n wreless sensor networks to allevate these problems [5, 6, 7, 8, 9, 10]. Intutvely, t s mportant to mprove the effcency of data collecton n terms of bandwdth and also energy and lfetme [11]. However, for many applcatons, such as earthquake montorng, t s more mportant to collect data from all nodes n varous geographcal locatons n a balanced manner rather than ust gatherng a large amount of data from one locaton. Therefore, farness s key ( [9], [7], [8]). Not only t mght ncrease per-node based throughput, but also mply a longer lfetme of a wreless sensor network. In ths paper, we consder a general network wth a sngle snk and multple randomly dstrbuted wreless sensors. A carrer sense multple access (CSMA) MAC s used. Assumng no data compresson, each sensor generates nformaton data whch should fnally reach the snk, possbly n mult-hop. In order to study the tenson between farness and effcency, we formulate two separate optmzaton problems. One of our obectve s to acheve a network-wde optmal rate allocaton n terms of farness, whle the other s for effcency, measured by the amount of data extracted from the network wth some mnmal rate requrement. There are dfferent defntons of farness [1, 12, 13] n the context of resource allocaton. We dentfy max-mn farness as the most approprate one for our settng. The effcency s defned as the network throughput n terms of the sum rates of all nodes. In wred networks, addtve ncrease and multplcatve decrease (AIMD) [17] rate adustment strategy s wdely used for far and effcent data transmsson. However, due to the broadcast nature of wreless medum, the bandwdth consumpton conssts both useful data and nterference. Furthermore, the nterference at each node hghly depends on the topology. Due to these enormous dfferences between wred and wreless communcaton, the desgn of rate control mechansms for wreless networks s not trval. We need to solve the followng problems. Frst of all, we need to approprately capture the nterference model. In [1], Srdharan et al. propose a recever capacty model, whch has been expermentally proved accurate for a CSMA-based lnk layer. Therefore, we also apply ths model n our optmzaton problem formulaton. Also, n ths problem doman (usng IEEE 802.15.4), data rate s very low ( less than 250 kbps ). So, the rado communcaton bandwdth s the key constrant. The maorty of prevous work on far rate allocaton n wreless sensor network s based on a predefned routng tree [5, 6, 9, 10]. By ths means, rate allocaton and routng are separated. However, we beleve that the selecton of paths for routng can mpact the farness or effcency of rate allocaton. Dfferent from prevous work, we model networks as drected graphs.
We assume that a sensor can dynamcally adust ts data generaton rate and data transmsson rates on all ts outgong lnks. We follow a cross-layer desgn to explore how to determne routes (at the network layer) and set rates for sources (at the transport layer) to maxmze network utlzaton n a far manner. The rest of the paper s organzed as follows. In Secton 2, we present the network model and formulate the optmzaton problems nto lnear programmng (LP) problems. Then, n Secton 3, the tradeoff between farness and effcency has been studed. In Secton 4, we analyze the LP problems and propose dstrbuted algorthms for them. Further, we report the smulaton results of these algorthms. Fnally n Secton 5, we gve our concluson and future work. 2 Network Model and Problem Formulaton We model the topology of a wreless sensor network as a drected graph G(V,E), where V s the set of all nodes (ncludng the snk), and E s the set of lnks. An edge (, ) E represents a communcaton lnk from sensor to. Let r represent the data transmsson rate on lnk (, ). We assume G s a connected graph,.e., every sensor has at least one path to the snk. r src s the data generaton rate on sensor. We further defne O as the set of sensors havng lnks outgong from sensor and I as the set of sensors havng lnks ncomng to. N s the set of sensor s neghbors. For example, n Fgure 1 (a), N 3 = {0,4,5}, I 3 = {4,5} and O 3 = {0}. We consder a wreless sensor network used for envronment montorng. Each sensor node generates data and s able to relay data for other nodes. All sensed data by nodes are fnally transmtted to the snk. In many exstng work, rate allocaton and optmzaton are based on fxed routng trees, usually generated by the SPT (Shortest Path Tree) method. Here, by modelng the network as a graph, we can select paths for routng and explore farness and effcency n wreless networks. In Fgure 1, we compare the tree-based and non-tree-based routng. As we can see, wthout the routng tree, a sensor may take dfferent paths to transmt data to the snk. For a gven network, dfferent routng paths may lead dfferent network utlty and max-mn far rate. If we consder a network as a graph nstead of a tree, t can reduce the mpact of bottleneck and ncrease the farness. However, the network utlty may decrease. Thus, we beleve that ont routng and rate allocaton can mprove the farness of networks. As we earler mentoned, to model the nterference we use the recever capacty model [1]. In ths Fgure 1: An example to compare the farness and effcency of networks wth and wthout predefned routng tree. Sensor 4 and 5 can only send data to 3 n (b), whle n (a), they have multple paths. model, every node s consdered as a recever. Due to the broadcast nature of the wreless medum, the bandwdth of a recever s consumed by both useful data and nose (nterference). Here, we use nose to refer to the data receved but not targeted to the recever. Let s use Fgure 1 (a) to explan the model. Ths fgure shows a 6-node sensor network. A sold lne represents a communcaton channel, whch s half-duplex and symmetrc. For node 4, t has two communcaton lnks. r 4 src s the data generaton rate at node 4. When node 4 sends ts data, t takes ether lnk 4 1 or 4 3 or both. r 41 and r 43 are respectvely defned as the allocated rates on these two outgong lnks of node 4. And there s r 41 + r 43 = r 4 src. Snce node 4 does not relay data for other node, only r 4 src (or r 41 and r 43 ) s ts useful data. However, due to the symmetry of the lnk, the bandwdth of node 4 s also consumed by the nose from node 1 and 3. Therefore, we have the followng nequalty, where B 4 s the bandwdth of node 4. r 4 src +r 1 nose +r 3 nose B 4 (1) In the above nequalty (1), rnose 1 s the data sent by node 1, but not targetng to node 4. As we can see, the useful data of node 1, ncludng ts own sensed data rsrc 1 and the relayed data r 41, s sent out to node 0 (the snk). Therefore, rnose 1 = r1 src +r 41. Smlarly, we can obtan rnose 3 = r3 src+r 43 +r 53, where the data relayed by node 3 s r 43 +r 53. As we can see, the recever capacty model successfully capture the feature of wreless communcaton. Also, t has been found emprcally good n modelng the nterference for a CSMA MAC. Recevers n the network are bandwdth constraned and have a fnte recever bandwdth capacty gven by B, whch can be set as the saturaton throughput of the CSMA MAC [21]. We can obtan a general form of recever
capacty constrant at node as follows: r + r l B (2) O N l O where O r s all the data sent out by node and consdered as useful data. Whle N l O r l s all the data lstened by node, ncludng both useful data and nose. In order to prevent the loss of data durng transmsson, flow conservaton s needed. That s, the amount of data transmtted by a sensor s equal to the sum of all receved data and new data generated by the sensor. Flow conservaton s modeled as follows: r r l = rsrc (3) O l I To study the farness and effcency, we formulate two optmzaton problems. Max-Mn farness has wdely been accepted as a formulaton of farness n many settngs. We also dentfy t as the most approprate one n our problem doman. Thus, our frst optmzaton problem s to obtan the max-mn farness n a wreless sensor network. Let 0 represent the snk and V = V {0}. The problem s formulated as follows: (P1) max: s.t. r mn r + r l B V O N l O r r l = rsrc V O l I r 0 = rsrc I 0 V r mn r src B V For the second optmzaton problem, our goal s to maxmze the sum of rates generated by all nodes ( V r src), whle mantanng a mnmal requred rate (r req ). The max-mn rate can be used as the mnmal requred rate, but not necessarly. Ths problem s formulated as follows: (P2) max: s.t. V r src r + r l B V O N l O r r l = rsrc V O l I r 0 = rsrc I 0 V r req r src B V Fgure 2: Average Max-Mn rate of Tree-based routng and Graph-based routng wth dfferent network szes. For the above formulaton, we should note that we gnore the overhead and quantzaton effects assocated wth packetzaton of data. We compare the farness obtaned by our proposed approach (as P1) and the tradtonal tree-based routng ( wth SPT ). Networks wth dfferent szes ( from 6 nodes to 25 nodes ) have been evaluated. Fgure 2 shows the result. In order to avod randomness, every pont n the fgure s the average of max-mn rates of 100 randomly generated networks wth same sze. Ths fgure proves that ont routng and rate allocaton can sgnfcantly mprove the farness. In the followng secton, we dscuss more on the relatonshp between farness and effcency n wreless sensor networks. 3 Tradeoff between Farness and Effcency There are many dfferent defntons of farness. In the exstng work, researchers use dfferent flavors of farness for rate allocaton. Kun et al.[18] propose a congeston control algorthm for wreless sensor networks desgned to obtan proportonal farness of flows n the network. In [16], Tassulas et al. use a centralzed algorthm to obtan a stronger sense of farness, the lexcographc max-mn farness, n wreless ad hoc networks. Other forms of farness are also used. In our paper, we am to obtan the max-mn farness of rate allocaton. That t, the mnmum data generaton rate allocated to any node s the maxmum over all possble allocatons. Effcency also has many dfferent defntons. A wdely used one s the network throughput measured by the sum rate ( V r src). Obvously, ths knd of
effcency severely bases the rate allocaton n largescale, mult-hop sensor networks. It favors the nodes that can drectly communcate wth the snk and the nodes wth less nterference. Especally, by only consderng network throughput, under heavy traffc load, t s mpossble to successfully delver packets that traverse many hops. Farness-effcency tradeoffs are relatvely wellunderstood n wred networks [2]. Due to the nterference nherent n wreless networks, ths tenson between effcency and farness s even stronger. However, ths problem s not well studed n the area. Let the Max-Mn rate be an ndcator of farness. We defne r src V N 1 as the average data generaton rate n the network, whch s used as the ndcator of effcency. We combne the farness and effcency nto one obectve as follows: V max : α r mn +(1 α) r src N 1 s.t. r + r l B V O N l O r r l = rsrc V O l I r 0 = rsrc I 0 V r mn r src B V By changng the value of α ( α [0,1] ), we assgn dfferent weghts to farness and effcency. We choose network szes rangng from 6 to 45. For each network sze, 100 nstances of network deployment are randomly generated. Then each node s randomly set a bandwdth of ether 100 or 200. Ten bandwdth dstrbutons are generated for each network nstance (for networks wth sze less than 10, t s possble to have some duplcated nstances.). We solve the above LP problems and obtan the effcency-farness curve. Fgure 3 and Fgure 4, respectvely, show the farness and effcency wth dfferent α for dfferent networks. It s clear f we ncrease the weght to effcency, farness wll decrease. When α = 0, maxmal effcency s obtaned at the cost Max-Mn rate r maxmn = 0. In Fgure 3, the Max-Mn rates are very close when α = 0.9 and α = 1. Fgure 5 s the effcency-farness curve ( for networks wth 45 nodes ). It shows the possble regon of effcency and farness. The bold dot on the curve s obtaned by maxmzng the network utlzaton after obtanng the max-mn rate. By ths means, we can obtan the maxmal farness and effcent network utlzaton ( over 83% of the maxmal possble network throughput when α = 0 ) at the same tme. Compared to lexcographc max-mn, max-mn s a weaker knd of farness. However, due to the trade- Fgure 3: Average Max-Mn rate (the farness) vs. the network sze. Fgure 4: Average Data generaton rate(the effcency) vs. the network sze. off between farness and effcency, max-mn farness s more sutable to obtan far and effcent rate allocaton. In the next secton, we analyze and propose some dstrbuted algorthms to acheve our goals. 4 Dstrbuted Algorthms In wreless sensor networks, after the deployment of sensors, usually t s not easy to access sensors agan. Also, due to the autonomous property and unpredctable channel, dstrbuted algorthms are much desrable n wreless envronment. In ths secton, we propose several dstrbuted algorthms usng the shadow prce nterpretaton to solve the optmzaton problems we descrbed n secton 2. Especally, we focus on solvng the optmzaton problem of farness n
solved [4]. Thus, we add a quadratc regularzaton term ε (rmaxmn 2 + (,) L r2 ) to the obectve wth ε > 0. Frst, we rewrte the optmzaton problem for farness (P1) as follows: mn : s.t. r mn r + r l B V O N l O r = r l +r mn V O l I (N 1) r mn = r 0 Snk I 0 r mn 0, r 0 Fgure 5: Effcency vs. Farness. Ths plot s for a network wth 45 nodes. a dstrbuted manner. After obtanng the Max-Mn rate, the sum-rate maxmzaton problem wll become easer. Snce for the sum-rate optmzaton, t always favors the nodes close to the snk. In [12], Kelly et al. frst appled optmzaton theory to rate control algorthms. Ths method was quckly accepted by researchers. Later, more approaches of optmzaton have been ntroduced nto ths area, such as dualty and sub-gradent methods. The dual-based method and sub-gradent methods are rapdly used to analyze and desgn dstrbuted algorthms, especally for the emergng world of wreless sensor networks. Partcularly, due to the dynamcal and unpredctable feature of wreless networks, an optmzaton problem usually nvolves dfferent elements and dfferent stack layers. Therefore, Chang et al. [19] and Johansson et al. [20] proposed cross-layer optmzaton and ntroduced the dual decomposton technques. These works establsh the bass of our followng algorthms. In secton 4.1 and 4.2, we elucdate these algorthms n detal. 4.1 Partally Dstrbuted Algorthms for Farness Defne X = {0 rsrc B V,0 r B (,) E} as the doman and relax the flow conservaton and bandwdth constrants. We appled the sub-gradent method to solve the dual problem and obtaned the optmal dual varables. However, ths soluton may not be prmal feasble. For example, flow conservaton and bandwdth constrant could not be satsfed. Ths s a typcal phenomenon for problems wth non-strctly convex prmal obectve functons. By addng a small strctly convex regularzaton term to the prmal obectve functon, the problem can be Introduce dual varables λ and μ for the constrants, the Lagrange functon of the prmal P1 s: L(r mn, r, λ, μ) = r mn +ε r 2 mn +ε (,) L r 2 + λ ( r + r l B ) V O N l O + μ ( r r l r mn ) V O l I + μ 0 ((N 1) r mn I 0 r 0 ) wth r mn 0, λ 0, and r X. Prmal varables r mn, and r can be separated n above Lagrange functon. To obtan the optmal value at teraton k, for each prmal varable, we solve the followng problem: r (k) mn r (k) = arg mn r {ε r2 mn +( 1 μ (k) mn 0 V + (N 1) μ (k) 0 ) r mn} = arg mn r {ε r2 +(λ (k) + λ (k) l X l N + μ (k) μ (k) ) r } ε can mpact the fnal result of Max-Mn rate and the speed of convergence. If ε s too large, t may change the optmzaton problem. If ε s too small, the convergence speed wll be slow. In order to calculate r mn, the value of μ s needed. Thus, ths method cannot provde a fully dstrbuted way to calculate Max-Mn value for the network. There should be a centralzed server to collect the value of μ on every sensor at each teraton. In [3], Madan et al. propose a way to desgn fully dstrbuted algorthms n the context of optmzng network lfetme. By learnng from ther method, we reformulate our problem as shown n the followng part.
4.2 Fully Dstrbuted Algorthm for Farness In ths new formulaton, each sensor solves the followng optmzaton problem: (P1 ) mn : s.t. r src r + r l B V O l N O l r = r l +rsrc V O l I rsrc = rsrc N r 0, rsrc 0 We also need to consder the bandwdth constrant on the snk I 0 r 0 B 0. By addng the quadratc regularzaton term ε ( V (r src) 2 + (,) L r2 ) to make the obectve functon a strctly convex functon. Sum up all the optmzaton problems on all nodes n the network, the Lagrange of the global optmzaton problem s gven by: L( r src, r, λ, μ, ν) = rsrc +ε ( (rsrc) 2 + V V + λ ( r + r l B ) V O l N O l + μ ( r r l rsrc) V O l I + ν (rsrc rsrc) V N (,) L r 2 ) wth λ 0 and r X. We apply sub-gradent methods to solve the dual problem. For dual varables, there are λ μ ν by λ (k) μ (k) ν (k) = r + r l B O l N O l = r r l rsrc (4) O l I = r src r src At each teraton k, dual varables are updated = [(λ (k 1) = μ (k 1) = ν (k 1) α (k) λ )] + α (k) μ (5) (k) α ν Step sze α (k) 0 wth k. The Lagrange dual functon s separable n r. We have the sequence of the prmal teratons as follows: r (k) src r (k) = arg mn r src 0{ε (r src) 2 + ( 1 μ (k) + =0 = arg mn r {ε r2 +(λ (k) X + λ (k) l l N +μ (k) N (ν (k) ν (k) )) r src} (6) μ (k) ) r } (7) The fully dstrbuted Algorthm 1 s descrbed by Algorthm 1 Fully Dstrbuted Max-Mn Algorthm 1: Intalzaton 2: set the value of δ and ε, set the doman X 3: ntalze prmal and dual varables: r src, r, λ, μ, ν 4: k 1 and step sze α C 1, ntalze D(0),D(1) 5: whle D(k) D(k 1) δ and k T do 6: solve rsrc (k) = argmn r src 0{ε (rsrc) 2 + ( 1 μ (k) + =0 N (ν (k) ν (k) )) r src} 7: solve r (k) = argmn r X{ε r 2 + (λ + k N λ k +μ μ ) r } 8: compute Lagrange Dual D(k) 9: compute sub-gradents of dual varables 10: λ = O r + l N O r l l B 11: μ = O r l I r l rsrc 12: ν = rsrc rsrc 13: update α (k) = C1 C2 k+c 2 14: compute new prces accordng to (5) 15: k++ 16: end whle 17: f Flow conservaton and bandwdth constrants are satsfed then 18: return r src, r 19: else 20: modfy ε, goto lne 3 21: end f Step sze s updated as α (k) = C1 C2 k+c 2, where C 1, C 2 are constants and satsfyng lm k α k = 0, k=1 αk =. When we generalze ths algorthm to dfferent network nstances, t s hard to fnd a common ε. Thus, we set an ntal value to ε and check flow conservaton and bandwdth constrants at the end of the whle-loop. If any of the constrants s volated, ε wll be changed to scale down the flows. By usng the smlar method, we also desgned a fully dstrbuted algorthm for the sum-rate problem.
But as we mentoned before, the sum-rate problem becomes easer after the max-mn rate problem s solved. Also due to the space of the paper, we won t repeat the deducton of the algorthm for sum-rate here. 4.3 Performance Evaluaton We evaluate our algorthms for networks wth 20 nodes. Fgure 6 and Fgure 7 shows the convergence speed of partally dstrbuted algorthm and fully dstrbuted algorthm (Alg. 1). We fnd that the fully dstrbuted algorthm can converge faster than the partally dstrbuted algorthm. Fgure 8 compares the max-mn rate allocated for each sensor by dfferent algorthms. In the fully dstrbuted algorthm, each sensor computes a max-mn rate by ts local nformaton. These max-mn values generated by Alg. 1 are very close to the optmal Max-Mn. Mnmal addtonal refnements are requred to scale the fnal solutons n order to ensure that all constrants are satsfed feasbly. Fgure 7: Convergence speed of fully dstrbuted Max- Mn Algorthm. Fgure 6: Convergence speed of partally dstrbuted Max-Mn Algorthm. 5 Concluson and Future Work In ths paper, we formulate LP problems to obtan max-mn farness and sum-rate effcency. Then we have studed the tradeoffs between farness and effcency n wreless sensor networks. In a large-scale low-rate network, farness exhbts great mportance. Our study shows that we can mprove the farness by allowng freedom n routng path-selecton. Therefore, to acheve a far rate allocaton, we focus on desgnng a prcng-based fully dstrbuted algorthm for the ont rate and routng control of a wreless network. Fgure 8: Compare the max-mn rate obtaned by dfferent algorthms. Ths plot s for network wth 20 nodes. We sort these nodes by the max-mn values they generated. x-axs shows the rankng. We beleve that these algorthms show great promse for future development, and plan to work on extendng them towards mplementng a proof-ofconcept on a real test-bed. In the future, we also want to obtan a dstrbuted algorthm for a general formulaton whch combnes farness and effcency together. References [1] A. Srdharan and B. Krshnamachar, Maxmzng Network Utlzaton wth Max-Mn Farness n Wreless Sensor Networks, 5th Intl. Symposum on Modelng and Optmzaton n Moble, Ad Hoc, and Wreless Networks (WOpt), Aprl 2007.
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