denote the location of a node, and suppose node X . This transmission causes a successful reception by node X for any other node


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1 Fnal Report of EE359 Class Proect Throughput and Delay n Wreless Ad Hoc Networs Changhua He Abstract: Networ throughput and pacet delay are the two most mportant parameters to evaluate the performance of wreless ad hoc networs. Generally t s dffcult to acheve both hgh throughput and low pacet delay. In ths proect, the obectve s to acheve hgh throughput whle eepng the pacet delay under certan threshold. We wll frst loo at the throughput capacty theoretcally n moble ad hoc networs. Gupta and Kumar [1] show the average avalable throughput per node decreases as 1 / n or 1 / n lg n n a statc ad hoc networ, where n s the number of nodes. That means, the total networ capacty ncreases as at most n. Furthermore, Grossglauser and Tse [2] show moblty can mprove the capacty. However, delay s not guaranteed n ther schemes. Actually delay wll ncrease due to possbly more hops or queueng n order to ncrease the throughput. Bansal and Lu [3] show t s possble to acheve closeto optmal capacty whle eepng the delay small. In ther model, each sender can acheve an average throughput of W mn( m, n) c, where W s the maxmum avalable bandwdth, wth the pacet delay at 3 nlog n 2 d most, where d s the dameter of the networ and v s the velocty of the moble nodes. v Based on ths fact, the authors propose a routng algorthm that acheves the obectve through explotng the patterns n the moblty of nodes. The throughput acheved by ths algorthm s only a polylogarthmc factor off from the optmal.
2 1. Introducton Wreless networ s becomng more and more popular n nowadays. Comparng to the tradtonal wred networ, wreless networ set up the connectons through wreless channel. Generally there are two nds of wreless networs. One has a wred bacbone networ n whch the base statons are the boundary nodes, and the extended connectons between moble users and the base staton are wreless channels. Ths onehop wreless networ s very popular currently,.e., the cellular networs and WLANs. The other s wreless ad hoc networ, whch has more than one hop wreless channels n the connecton. Ths nd of topology s not wdely mplemented yet, but t s useful sometmes, especally n mltary applcatons and sensor networs. In our proect, we wll focus on the latter topology, the wreless ad hoc networ, wthout consderng any wred lns. As an extenson to the bacbone networ, wreless ad hoc networ conssts of nodes that communcate wth each other through wreless channels only. We can descrbe the system as follows. Our system conssts of only wreless nodes, n whch all nodes can communcate wth other nodes n the range of rado transmsson through wreless channel. Each wreless node can act as a sender, a recever or a router. As a sender, the node can send message to the specfed destnaton node through some route. As a recever, t can receve the message from other nodes. As a router, t can relay the pacet to the destnaton or next router n the route f necessary. Each node can buffer pacets when the pacets need to wat for transmsson. We are nterested n the capacty and delay of such nd of networ. In general these two parameters are the most mportant performance measurement for any wreless networ systems. The capacty represents the throughput (bts per second) of the whole system ncludng all nodes, and the delay represents the average tme duraton of a pacet transmttng n the networ from a source to the destnaton. As n any other queueng system, there are tradeoffs between the capacty and the delay. Intutvely n order to ncrease the capacty, we need to eep all nodes busy wth transmttng or recevng pacets durng all the tme, whch means the queue of each node s always nonempty, obvously ths wll lead to a longer delay. On the other hand, n order to reduce the delay, the optmal stuaton s, all nodes along the route can transmt the pacet mmedately to the next node untl t reaches the destnaton, whch means there s no pacet competng for transmssons n the queues, surely ths causes very low throughput. We wll see ths tradeoff n wreless ad hoc networs n the report. Furthermore, our obectve s to fnd a way that the networ can acheve a hgh throughput whle eepng the delay under certan threshold. Ths report wll address the problem. In the followng, secton 2 wll descrbe the methodology to model the problem step by step, secton 3 wll gve out the man results n the papers and explan ther meanngs to the capacty and the pacet delay, based on these a routng algorthm s proposed to reach our obectve. Fnally n secton 4 we conclude the capacty and delay n wreless ad hoc networs.
3 2. Methodology In ths part, we wll show the methodology to solve the problem step by step. Recall that our obectve s to acheve hgh capacty n wreless ad hoc networs wth eepng the pacet delay under a small threshold. We wll model the networs from smple to complex, from general to specfc step by step. In each model, we wll descrbe the scenaro, the transmsson model and the measurement metrcs n detals. Frst of all, we develop the models for wreless ad hoc networs wth statc nodes. Gupta and Kumar [1] propose two models for such nd of networ. For smplcty, the models scale the space so that n nodes are located n a regon of area 1 m 2. Each node can transmt at W bts per second over a common wreless channel. The channel s dvded to several subchannels, each wth capacty W 1, W2, m, WM bts per second, where M m= 1 W m = W. 2.1 Model 1 (model of arbtrary networs n [1]) Frst we defne the scenaros. In the networ the nodes and traffc patterns are arbtrarly located. Say, n nodes are arbtrarly located n a ds of unt area n the plane. Each node arbtrarly chooses a destnaton to send message at an arbtrary rate, and also arbtrarly choose a transmsson range or power level. Then we use two models to ndcate successful recepton of a transmsson over one hop: the protocol model and the physcal model. In the protocol model, let denote the th locaton of a node, and suppose node transmts over the m subchannel to a node. Ths transmsson causes a successful recepton by node f ( 1 + ) for any other node same subchannel. On the other hand, n the physcal model, let { ; Τ} smultaneously transmttng over the be the subset of nodes smultaneously transmttng over a certan subchannel. Assume node transmts wth power P, for successful recepton by node Τ. The transmsson from a node, P f the nequalty N + Τ P Τ causes a β s satsfed, where β s a threshold of sgnaltonterference rato (SIR) for successful receptons, N s the ambent nose power level, and > 2 ndcates the sgnal power decay wth dstance 1. r
4 Fnally n the model we need to defne the measurement metrcs. We defne btmeter as the product of the number of bts and the dstances over whch the bts are carred. Accordng to ths, the capacty s defned as the sum of all btmeter n the networ. From Model 1, we can compute the upper bound and lower bound of the capacty, thus get some nowledge about the actual capactes, whch s descrbe n detals n secton 3.1. Whle ths model s qute general, a further model wth more nformaton on the locaton and traffc pattern of the nodes wll gve us more useful results. 2.2 Model 2 (model of random networs n [1]) Smlarly frst we descrbe the scenaros. There are some tny dfferences from Model 1. n nodes are ndependently and unformly dstrbuted on the surface S 2 of a threedmensonal sphere of area 1 m 2. We adopt ths change n order to elmnate the boundary effects. Each node randomly and ndependently chooses a destnaton to send λ n bts per second. message wth the rate of ( ) Then for the transmsson model we adopt both a protocol model and a physcal model for ndcatng the successful recepton, ust le what we do n Model 1. The only dfference s that we ntroduce a common range r for all transmssons and the nequalty n the protocol model changes to r and ( 1 + )r. Fnally n order to compute the throughput of the networ, we defne a throughput of λ ( n) bts per second for each node s feasble f there s a spatal and temporal scheme for schedulng transmssons, such that every node can send λ ( n) bts per second on average to ts chosen destnaton node through the ntermedate nodes and some bufferng strategy n the ntermedate nodes. Based on ths model, smlarly we can calculate a lower bound and an upper bound of the capacty. However, because node relayng and bufferng are ntroduced, t s possble that some pacets wll have long delays. Furthermore, n both models we only consder the networ conssts of statc nodes. As we now, one of the bggest advantages of wreless networs s the moblty. In the next we wll extend the model to nclude moble nodes. Grossglauser and Tse [2] study nfluence of moblty on the capacty of wreless networs. 2.3 Model 3 (model wth moble nodes n [2]) Frst we descrbe the scenaros. The networ stll conssts of n nodes lyng n the ds of unt area, but dfferent from Model 1 and 2, all the nodes are moble. Denote the locaton of the th user at tme t as () t. Assume the traectores of dfferent users are ndependent and dentcally dstrbuted, and each node s both a source node for one sesson and a destnaton node for another sesson. Let d () represent the destnaton node of node. Durng the transmsson, we assume that each source node has nfnte
5 number of pacets to send to ts destnaton. Furthermore, ths SD assocaton does not change wth tme no matter how the nodes move. We wll use the same physcal model as n Model 1 and 2 for transmsson. Furthermore, n order to ensure a hgh longterm throughput for each SD par, a scheduler s ntroduced to determne whch nodes wll transmt pacets, whch pacets they wll transmt, and at whch power levels P () t the pacets wll be transmtted from node. As a measurement metrc, the throughput ( n) λ s a random quantty dependng on the random locatons and movements of the nodes. The capacty of the networ s consdered as the total throughput to all SD pars. Based on ths model wth moblty, we can compute the theoretcal results for the capacty n case of ether wthout relayng nodes or wth relayng nodes, whch are dscussed n detals n secton 3.3. Untl now we have satsfactory models for the wreless ad hoc networs wth both statc and moble nodes. However, as we have seen, n Model 1, 2 and 3, we only consder the capacty of the networs wthout any consderatons on the delay. In order to acheve hgh capacty, we assume pacets can be relayed and buffered n the ntermedate nodes, ths mght cause very large delay when the buffer length s long and the number of ntermedate nodes s large. So t s necessary and mportant to get some deas on the pacet delay n the networs. Bansal and Lu [3] set up a model to address ths problem, wth more assumptons on the moblty pattern and traffc pattern. 2.4 Model 4 (delay model n [3]) Frst, the ad hoc networ conssts of n statc nodes and m moble nodes lyng n a ds of unt area. The statc nodes are unformly dstrbuted over the unt crcular ds and never move. The moble nodes are randomly dstrbuted n the ds ntally, later they wll change postons and veloctes wth a moblty model. There are many models to do so, here a unform moblty model s used. Intally each moble node moves at speed v nsde the unt crcular ds. The drectons of movement are ndependent and unformly dstrbuted n [ 0,2π ). At subsequent tme the node pcs a drecton unformly dstrbuted n ( 0,2π ] and moves n that drecton for a dstance d at speed v, where d s an exponentally dstrbuted random varable wth mean µ. And so on. When the node reaches the boundary of the ds, t s reflected bac to the ds agan. Smlarly we use the physcal model for transmsson wth mnor modfcatons. At tme t, let S 1, S 2, m, S m be the senders wth postons 1, 2,, m and let R be the recever wth poston 0. If S use power P () t for transmsson, t causes a successful recepton by node R f N + P () t P () t 0 0 β.
6 The same performance metrc s used as n the frst 3 models. But besdes capacty, the pacet delay s also consdered. From ths model, Bansal and Lu proves t s possble to acheve a hgh throughput whle eepng the delay under some threshold, furthermore, a routng protocol s proposed to mplement the obectve, whch s descrbed n secton 3.4. Wth studyng the evoluton of the models of wreless ad hoc networs, we almost solve our proposed problem by addng more and more assumptons to the smple model. Ths s a very mportant methodology to do research. 3. Man Results We have descrbed the models to solve the problem step by step. In ths secton we wll lst the man results for dfferent models and explan the mportance n desgnng a wreless ad hoc networs. We won t go through the dervatons of those results, the readers can refer to the papers ([1], [2], [3]) f nterested n the detals. 3.1 Model 1 (model of arbtrary networs n [1]) Result 1 (man result 1 n [1]) Wth the transmsson model of Protocol Model, the transport capacty of networs n Model 1 s Θ ( W n ) btmeters per second gven that the nodes are optmally placed, the traffc pattern s optmally allocated, and the range of transmsson s optmally chosen. Result 2 (man result 2 n [1]) Wth the transmsson model of Physcal Model, cw n / whle c ' Wn 1 btmeters per seconds s not feasble for approprate c, 1 Wn Specfcally, 1 2 n + 8π / β btmeters per second s feasble, ' c. btmeters per second (n a multple of 4) s feasble when the networ s approprately desgned, wth an upper bound of β + 2 Wn btmeters per second. π β From these results, we can drectly conclude that for arbtrary networ model, the Θ W n. If the total capacty s capacty of wreless ad hoc networ s n the order of ( ) W equally dvded among all the nodes, then each node can acheve the capacty of Θ n btmeters per second. Furthermore, consder each source node transmts to the
7 destnaton about the same dstance of 1m apart, each node can obtan a capacty of W Θ bts per second. n 3.2 Model 2 (model of random networs n [1]) Result 3 (man result 3 n [1]) In case of both the surface of the sphere and a planar ds, the order of the throughput capacty s ( ) W λ n = Θ bts per second for the Protocol Model. The upper bound nlog n ' can be ndcated by the fact that for some c, lm Pr ' W ob λ ( n) = c s feasble = 0. n log n n Specfcally, there exsts constants c '' ''', c ndependent on n, or W, such that '' ''' c W c W λ ( n) = bts per second s feasble, and λ( n) = bts per 2 2 ( 1+ ) n log n nlog n second s not feasble, wth the probablty approachng one as n. Result 4 (man result 4 n [1]) Wth the transmsson model of Physcal Model, a throughput of ( ) cw λ n = Θ bts nlog n ' c W per second s feasble, whle λ( n) = bts per second s not feasble, for approprate c, n ' c, wth probablty approachng one as n. Specfcally, there exsts constants '' ''' c and c ndependent on n, N,, β or W, such that '' c W λ( n) = bts per second s feasble wth 1 2 n log n ''' c β probablty approachng one as n. If L s the mean dstance between two ponts ndependently and unformly dstrbuted n the doman (ether surface of sphere or planar ds of unt area), then there s a determnstc sequence ε ( n) 0, ndependent on 8 W 1+ ε ( n) N,, β or W, such that btmeters per second s not feasble 1 / π L( β 1) n wth probablty approachng one as n.
8 From these results, we can see for random networ model, the capacty s n the order of ( ) W λ n = Θ, less than the capacty n the arbtrary networ mode. That s nlog n because we add some lmtatons on the traffc pattern. Furthermore, from result 3, we can get some nsghts on what lmts the capacty. In the case of a ds on the plane, the nodes lyng n the center wll have more possbltes to relay pacets, socalled hot spots, but the order of throughput capacty s the same as on the surface of the sphere. That shows the cause of the throughput constrcton s not the formaton of hot spots, but s the pervasve need for all nodes to share the channel locally wth other nodes. 3.3 Model 3 (model wth moble nodes n [2]) Result 5 (Theorem III3 n [2]) Consder a schedulng polcy that s only allowed to schedule drect transmsson between the source and destnaton nodes. Say, no relayng s permtted. If c s any 1/ ( 1+ / 2) 2 / 2 β + L constant satsfyng c > π β, ( ) then Pr{ ( ) 1/ 1= / 2 λ n = cn R s feasble} = 0 for suffcently large n. From ths result, we can see the capacty per SD par goes to 0 as n f no relayng s permtted n the networs. That s because n each source node, hgh power s requred to transmt the pacets drectly to the destnaton node, whch leads to hgh nterference and lmts the capacty. It s possble to gan hgher capacty f we schedule nodes to communcate only wth close neghbors and relay pacets for destnaton nodes far away. Result 6 (Theorem III4 & Theorem III5 n [2]) Consder a schedulng polcy π allowng relayng nodes. For a gven SD par, there s one drect route and n2 twohop routes that go through one relay node. The networ can acheve a throughput of Θ () 1 per SD par,.e., there exsts a constant c > 0 such that lm Pr λ n = cr s feasble = 1 n { ( ) }. Comparng Result 5, 6 wth Result 1, 2 and Result 3, 4, we can see mmedately that f relayng s permtted n the networs, moblty can dramatcally mprove the capacty n from Θ ( n ) or Θ to Θ ( n). lg n
9 3.4 Model 4 (delay model n [3]) Result 7 (man result n [3]) In the wreless ad hoc networ wth n statc nodes and m moble nodes, whch are characterzed n Model 4, there exsts a constant c > 0, such that each sender can W mn( m, n) acheve an average throughput of c, where W s the maxmum avalable 3 nlog n 2 d bandwdth, whle the pacet delay s at most, where d s the dameter of the networ v and v s the velocty of the moble nodes. Ths result solve our problems proposed at the begnnng, n the followng a routng algorthm s descrbed to acheve ths obectve. Result 8 (routng algorthm n [3]) Step 1. Local leader electon A local lead s elected among the statc nodes wthn each regon of sze 1 / m 1/ m. Ths leader wll be responsble for communcatng all the messages of the statc nodes n ts regon wth the moble nodes. Step 2. Statc to moble phase A statc node S1 wantng to send messages to destnaton R frst transfers ts message to ts local leader S. S stores the message and wats for a moble node M1 such that M1 s close enough to S and movng approxmately along the drecton of R. when such a node s avalable S hands over the data from S1 to M1. Step 3. Moble to moble phase The moble nodes relay the pacets towards R amongst all possble moble nodes such that the pacet moves closer and closer to the destnaton. Step 4. Statc to statc phase When the moble node carryng the pacet s close enough to the destnaton, t hands off the pacet to some leader node. Ths pacet s then relayed among the statc leader nodes towards the correct leader node, whch can transmt the pacet to the destnaton node drectly. Wth ths routng algorthm, the wreless ad hoc networ can acheve closeto optmal capacty whle eepng the pacet delay small. Ths algorthm explots the moblty patterns of the nodes to provde guarantees on the pacet delay. The readers can refer to [3] f nterested n the detaled operatons and arguments of the algorthm.
10 4. Concluson In ths proect, we explore the throughput and delay n wreless ad hoc networs. Our obectve s to acheve hgh throughput whle eepng the pacet delay relatvely small. In order to solve ths problem, we start from the smplest model, compute the capacty only, then add more assumptons step by step, and fnally fnd out a routng algorthm whch can acheve our obectves. Ths s a very mportant methodology for any nd of research. W For wreless ad hoc networs wth only statc nodes, the capacty per node s Θ n bts per second for Arbtrary Networ model, and W Θ for Random Networ n lg n model. If moblty s consdered n the networ, the capacty can be dramatcally mproved to Θ () 1 per SD par. Furthermore, f more assumptons on the traffc pattern and moblty pattern are ntroduced, the proposed routng algorthm can guarantee the pacet delay and acheve a closeto optmal capacty, whch s only a polylogarthmc factor off from the optmal algorthm. Note that we have no consderatons on the energy lmtaton of the nodes n the networ, whch s another mportant constrant actually exstng n the wreless ad hoc networs. References: [1] Pyush Gupta and P. R. Kumar, The Capacty of Wreless Networs, IEEE Transactons on Informaton Theory, Vol. 46, No. 2, March [2] Matthas Grossglauser and Davd Tse, Moblty Increases the Capacty of Ad Hoc Wreless Networs, IEEE/ACM Transactons on Networng, Vol. 10, No. 4, August [3] Nhl Bansal and Zhen Lu, Capacty, Delay and Moblty n Wreless AdHoc Networs INFOCOM 2003; TwentySecond Annual Jont Conference of the IEEE Computer and Communcatons Socetes, IEEE, Volume: 2, 30 March  3 Aprl 2003.
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