On High Spatial Reuse Broadcast Scheduling in STDMA Wireless Ad Hoc Networks

Transcription

1 On Hgh Spatal Reuse Broadast Shedulng n STDMA Wreless Ad Ho Networks Ashutosh Deepak Gore Abhay Karandkar Informaton Networks Laboratory Department of Eletral Engneerng Indan Insttute of Tehnology - Bombay Mumba Inda. {adgore,karand}@ee.tb.a.n Abstrat Graph-based algorthms for pont-to-multpont broadast shedulng n Spatal reuse Tme Dvson Multple Aess (STDMA) wreless ad ho networks often result n a sgnfant number of transmssons havng low Sgnal to Interferene and Nose densty Rato (SINR) at ntended reevers, leadng to low throughput. To overome ths problem, we propose a new algorthm for STDMA broadast shedulng based on a graph model of the network as well as SINR omputatons. The performane of our algorthm s evaluated n terms of spatal reuse and omputatonal omplexty. Smulaton results demonstrate that the proposed algorthm performs sgnfantly better than exstng graph-based algorthms. Keywords: Ad ho Networks, Spatal Tme Dvson Multple Aess, Broadast Shedulng, Spatal Reuse, hysal Interferene Model. 1. Introduton In a wreless ad ho network, a prevalent sheme for hannel spatal reuse s Spatal Tme Dvson Multple Aess (STDMA), n whh tme s dvded nto fxedlength slots that are organzed ylally and multple enttes an ommunate n the same slot. An STDMA shedule desrbes the transmsson rghts for eah tme slot n suh a way that ommunatng enttes assgned to the same slot do not ollde. STDMA shedulng algorthms an be ategorzed nto lnk shedulng and broadast/node shedulng algorthms [1]. In lnk shedulng, the transmsson rght n every slot s assgned to ertan soure-destnaton pars. In broadast shedulng, the transmsson rght n every slot s assgned to ertan nodes,.e., there s no apror bndng of transmtter and reever and the paket transmtted must be reeved by every neghbor. In ths paper, we only onsder entralzed broadast shedulng for STDMA networks Related Work The onept of STDMA for multhop wreless ad ho networks was formalzed n [2]. A broadast shedule s typally determned from a graph model of the network [1]. The problem of determnng an optmal mnmum-length STDMA shedule for a general multhop ad ho network s N-omplete for both lnk and broadast shedulng [1]. In fat, ths s losely related to the problem of determnng the mnmum number of olours to olour all the vertes (or edges) of a graph under ertan adjaeny onstrants. In [1], the authors show that a wreless ad ho network an be sheduled suh that the shedule s bounded by a length proportonal to the graph thkness 1 tmes the optmum number of olours. However, the above work does not take nto aount Sgnal to Interferene and Nose densty Rato (SINR) omputatons when determnng an STDMA broadast shedule. In ths paper, we propose a suboptmal algorthm based on the graph model as well as SINR omputatons. We ntrodue spatal reuse as a performane metr and demonstrate that the proposed algorthm has low omputatonal omplexty and hgh spatal reuse ompared to exstng algorthms n the lterature. The rest of the paper s organzed as follows. In Seton 2, we desrbe our system model, dsuss the lmtatons of graph-based shedulng algorthms and formulate the problem. Seton 3 desrbes the proposed broadast shedulng algorthm. The performane of our algorthm s evaluated n Seton 4 and ts omputatonal omplexty s derved n Seton 5. We onlude n Seton System Model Consder an STDMA wreless ad ho network wth N stat nodes (wreless routers) n a two-dmensonal plane. Durng a tme slot, a node an ether transmt, reeve or reman dle. We assume homogeneous and bak- 1 The thkness of a graph s the mnmum number of planar graphs nto whh the gven graph an be parttoned.

2 logged nodes. Let: r j = (x j, y j ) = Cartesan oordnates of the j th node = transmsson power of every node N 0 = thermal nose densty D(j, k) = Euldean dstane between nodes j and k We do not onsder fadng and shadowng effets. The reeved sgnal power at a dstane D from the transmtter s gven by D, where s the path loss fator. A broadast shedule effetvely assgns sets of nodes to tme slots. Spefally, a broadast shedule for the STDMA network s denoted by Ω(C, B 1,, B C ), where C = number of slots n the broadast shedule B = set of broadast transmssons n the th slot := {t,1 {r,1,1, r,1,2,..., r,1,η(t,1)},, t,m {r,m,1, r,m,2,..., r,m,η(t,m )}} where t,j {r,j,1,..., r,j,η(t,j)} denotes a pont-tomultpont transmsson of the same paket from node t,j to all ts neghbors 2 {r,j,1,..., r,j,η(t,j)} n the th slot and η(t,j ) denotes the number of neghbors of node t,j. Note that M denotes the number of onurrent transmssons n the th slot and t,j, r,j,k {1,..., N}. The SINR at reever r,j,k s gven by SINR r,j,k = D (t,j,r,j,k ) N 0 + M l=1 l j D (t,l,r,j,k ) We defne the sgnal to nose rato (SNR) at reever r,j,k by SNR r,j,k := N 0 D (t,j, r,j,k ) 2.1. hysal and rotool Interferene Models (1) (2) Aordng to the physal nterferene model [3], the unast transmsson t,j r,j,k s suessful f and only f (ff) the SINR at reever r,j,k s greater than or equal to a ertan threshold γ, termed as the ommunaton threshold. D (t,j,r,j,k ) N 0 + M l=1 l j D (t,l,r,j,k ) γ (3) Aordng to the protool nterferene model [3], t,j r,j,k s suessful f: 1. the SNR at reever r,j,k s no less than the ommunaton threshold γ. From (2), ths translates 2 The set of neghbors of a gven node depends on the geographal loatons of the nodes and wll be made prese n Seton 2.1. to D(t,j, r,j,k ) ( N 0 γ ) 1 =: R (4) where R s termed as ommunaton range. 2. the sgnal from any unntended transmtter t,l s reeved at r,j,k wth SNR less than a ertan threshold γ, termed as the nterferene threshold. Equvalently D(t,l, r,j,k ) ( N 0 γ ) 1 =: R l (5) j where R s termed as nterferene range. Note that 0 < γ < γ, thus R > R. The physal model of our system s denoted by Φ(N, (r 1,..., r N ),, γ, γ,, N 0 ). A shedule Ω( ) s feasble f t satsfes the followng: 1. Operatonal onstrant: A node annot transmt and reeve n the same tme slot. Also, a node annot reeve from multple transmtters n the same tme slot. 2. Communaton range onstrants: (a) Every reever s wthn the ommunaton range of ts ntended transmtter. D(t,j, r,j,k ) R (6) (b) Every reever s outsde the ommunaton range of ts non-ntended transmtters. D(t,l, r,j,k ) > R l j (7) If node b s wthn node a s ommunaton range, then b s defned as a neghbor of a, sne b an deode a s paket orretly (subjet to (3)). Note that f node b s outsde node a s ommunaton range, then t an never deode a s paket orretly (from (3)). A shedule Ω( ) s exhaustve f every two nodes, d who are neghbors of eah other are nluded n the shedule twe, one wth beng the transmtter and d beng a reever, and ve versa Graph-Based Shedulng Broadast shedules are typally desgned by modelng the STDMA network Φ( ) by a dreted graph G(V, E), where V s the set of vertes and E s the set of edges. Let V = {v 1, v 2,..., v N }, where vertex v j represents the j th node n Φ( ). In general, E = E E, where E and E denote the set of ommunaton and nterferene edges respetvely. If node k s node j s neghbor, then there s a ommunaton edge from v j to v k, denoted by

3 v j vk. If node k s outsde node j s ommunaton range but wthn ts nterferene range, then there s an nterferene edge from v j to v k, denoted by v j v k. Thus, the mappng from Φ( ) to G( ) an be desrbed as follows: D(j, k) R v j vk E and v k vj E R < D(j, k) R v j v k E and v k v j E The subgraph G (V, E ) onsstng of ommunaton edges only s termed as the ommunaton graph. An STDMA broadast shedule s equvalent to assgnng a unque olour to every vertex n the graph, suh that nodes wth the same olour transmt smultaneously n a partular tme slot, subjet to: Any two vertes v, v j an be oloured the same ff:. edge v vj E and edge v j v E,.e., there s no prmary vertex onflt, and. there s no vertex v k suh that v vk E and vk E,.e., there s no seondary vertex v j onflt. These rtera are based on the operatonal onstrant. Graph-Based shedulng algorthms utlze varous graph olourng methodologes to obtan a nononfltng shedule,.e., a shedule devod of prmary and seondary vertex onflts. To maxmze the throughput of an STDMA network, graph-based shedulng algorthms seek to mnmze the total number of olours used to olour all the vertes of G( ) Lmtatons of Graph-Based Algorthms Observe that Crtera ) and ) are not suffent to guarantee that the resultng shedule Ω( ) s onflt-free. Due to hard-thresholdng based on ommunaton and nterferene rad, graph-based shedulng algorthms an lead to hgh umulatve nterferene at a reever [4] [5]. Ths s beause the SINR at reever r,j,k dereases wth an nrease n M, whle R and R have been defned for a sngle transmsson only. For example, onsder Fg. 1 R r,1,1 t r,1,1,2 r t r,2,1,2,2,2 Fgure 1: Graph-Based algorthms an lead to hgh umulatve nterferene. wth sx labeled nodes whose oordnates are 1 (0, 0), 2 ( 80, 0), 3 (90, 0), 4 (280, 0), 5 (200, 0) R and 6 (370, 0). The system parameters are = 10 mw, = 4, N 0 = 90 dbm, γ = 20 db and γ = 10 db, whh yelds R = 100 m and R = m. A graph-based shedulng algorthm wll typally shedule the transmssons 1 {2, 3}, and 4 {5, 6} n the same tme slot, say the th tme slot, sne the resultng graph olourng s devod of prmary and seondary vertex onflts. However, our omputatons show that the SINRs at reevers r,1,1, r,1,2, r,2,1 and r,2,2 are db, db, db and db respetvely. From the physal nterferene model, the transmsson t,1 r,1,1 s suessful, whle the transmssons t,1 r,1,2, t,2 r,2,1 and t,2 r,2,2 are unsuessful. Ths leads to low throughput. Hene, graph-based shedulng algorthms do not maxmze the throughput of an STDMA network roblem Formulaton We propose a new suboptmal algorthm for STDMA broadast shedulng based on the physal nterferene model. To evaluate the performane of our algorthm and ompare t wth exstng suboptmal STDMA broadast shedulng algorthms, we defne the followng metr: spatal reuse. Consder the STDMA broadast shedule Ω( ) for the network Φ( ). Under the physal nterferene model, the pont-to-pont transmsson t,j r,j,k s suessful ff (3) s satsfed. The spatal reuse of the shedule Ω( ) s defned as the average number of suessful pont-to-multpont transmssons per tme slot n the STDMA shedule. Thus Spatal Reuse = C =1 M j=1 η(t,j ) k=1 I(SINR r,j,k γ ) η(t,j) C (8) where I(A) denote the ndator funton for event A,.e., I(A) = 1 f event A ours; I(A) = 0 f event A does not our. Note that a hgh value of spatal reuse 3 dretly translates to hgh long-term network throughput. We seek a low omplexty STDMA broadast shedulng algorthm wth spatal reuse reasonably greater than unty. We only onsder STDMA shedules whh are feasble and exhaustve. 3. SINR-Based Broadast Shedulng Algorthm Our proposed SINR-based broadast shedulng algorthm s alled MaxAverageSINRShedule, whh onsders the ommunaton graph G (V, E ) and s desrbed n Algorthm 1. In hase 1 (Lne 3), we label all 3 Note that spatal reuse n our system model s analogous to spetral effeny n dgtal ommunaton systems.

4 the vertes randomly 4. Spefally, f G ( ) has v vertes, we perform a random permutaton of the sequene (1, 2,..., v) and assgn these labels to vertes wth ndes 1, 2,..., v respetvely. In hase 2 (Lnes 4-7), the vertes are examned n nreasng order by label 5 and the MaxAverageSINRColour funton s used to assgn a olour to the vertex under onsderaton. The MaxAverageSINRColour funton s explaned n Algorthm 2. It begns by dsardng all olours that onflt wth u, the vertex under onsderaton. Among the set of non-onfltng olours C n, t hooses that olour for u whh results n the maxmum value of average SINR at the neghbors of u. Intutvely, ths average SINR s also a measure of the average dstane of every neghbor of u from all o-oloured transmtters. The hgher the average SINR, the hgher s ths average dstane. We hoose the olour whh results n the maxmum average SINR at the neghbors of u, so that the addtonal nterferene at the neghbors of all o-oloured transmtters s kept low. Algorthm 1 MaxAverageSINRShedule 1: nput: hysal network Φ( ), ommunaton graph G ( ) 2: output: A olourng C : V {1, 2,...} 3: label the vertes of G randomly 4: for j 1 to n do 5: let u be suh that L(u) = j 6: C(u) MaxAverageSINRColour(u) 7: end for Algorthm 2 nteger MaxAverageSINRColour(u) 1: nput: hysal network Φ( ), ommunaton graph G ( ) 2: output: A non-onfltng olour 3: C set of exstng olours 4: C p {C(x) : x s oloured and s a neghbor of u} 5: C s {C(x) : x s oloured and s two hops away from u} 6: C n = C \ {C p C s } 7: f C n φ then 8: r olour n C n whh results n maxmum average SINR at neghbors of u 9: f maxmum average SINR γ then 10: return r 11: end f 12: end f 13: return C Randomzed algorthms are known to outperform determnst algorthms, esp. when the haratersts of the nput are not known apror [6]. 5 In essene, the vertes are sanned n a random order, sne labelng s random Smulaton Model 4. erformane Results In our smulaton experments, the loaton of every node s generated randomly n a rular regon of radus R. If (X j, Y j ) are the Cartesan oordnates of the j th node, then X j U[ R, R] and Y j U[ R, R] subjet to Xj 2 + Y j 2 R 2. Equvalently, f (R j, Θ j ) are the polar oordnates of the j th node, then Rj 2 U[0, R2 ] and Θ j U[0, 2π]. Usng (4) and (5), we ompute R and R, and then map the STDMA network Φ( ) to the twoter graph G(V, E E ). One the broadast shedule s omputed by every algorthm, the spatal reuse s omputed usng (8). We use two sets of prototypal values of system parameters n wreless networks [7] and desrbe them n Seton 4.2. For a gven set of system parameters, we alulate the spatal reuse by averagng ths quantty over 1000 randomly generated networks. Keepng all other parameters fxed, we observe the effet of nreasng the number of nodes on the spatal reuse. In our experments, we ompare the performane of the followng algorthms: 1. BroadastShedule [1] (BS) 2. MaxAverageSINRShedule (MASS) 4.2. erformane Comparson In our frst experment (Experment 1), we assume that R = 500 m, = 10 mw, = 4, N 0 = 90 dbm, γ = 20 db and γ = 10 db. Thus, R = 100 m and R = m. We vary the number of nodes from 30 to 110 n steps of 5. Fgure 2 plots the spatal reuse vs. number of nodes for both the algorthms. spatal reuse R = 500 m, = 10 mw, = 4, N 0 = 90 dbm, γ = 20 db, γ = 10 db 0.5 BroadastShedule MaxAverageSINRShedule number of nodes Fgure 2: Spatal reuse vs. number of nodes for Experment 1.

5 In our seond experment (Experment 2), we assume that R = 700 m, = 15 mw, = 4, N 0 = 85 dbm, γ = 15 db and γ = 7 db. Thus, R = m and R = m. We vary the number of nodes from 70 to 150 n steps of 5. Fgure 3 plots the spatal reuse vs. number of nodes for both the algorthms. spatal reuse R = 700 m, = 15 mw, = 4, N 0 = 85 dbm, γ = 15 db, γ = 7 db 1 BroadastShedule MaxAverageSINRShedule number of nodes Fgure 3: Spatal reuse vs. number of nodes for Experment 2. From Fgures 2 and 3, we observe that spatal reuse nreases wth the number of nodes for both the algorthms. The MASS algorthm onsstently yelds hgher spatal reuse ompared to BS. The spatal reuse of MASS s 9-20% hgher than BS n Expt. 1 and 3-5% hgher n Expt. 2. Ths mprovement n performane translates to substantally hgher long-term network throughput. 5. Analytal Result In ths seton, we derve an upper bound on the runnng tme (omputatonal) omplexty of our algorthm. Let v denote the number of vertes of the ommunaton graph G (V, E ). Theorem 1. The runnng tme of MaxAverageSINRShedule s O(v 2 ). roof. Assumng that an element an be hosen randomly and unformly from a fnte set n unt tme (Chapter 1, [6]), the runnng tme of hase 1 an be shown to be O(v). In hase 2, the vertex under onsderaton s assgned a olour usng MaxAverageSINRColour. The worst-ase sze of the set of olours to be examned C n C p C s s O(v). Wth a areful mplementaton, MaxAverageSINRColour runs n tme proportonal to C n,.e., O(v). Thus, the runnng tme of hase 2 s O(v 2 ). Fnally, the overall runnng tme of MaxAverageSINRShedule s O(v 2 ). 6. Dsusson In ths paper, we have developed a broadast shedulng algorthm for STDMA multhop wreless ad ho networks under the physal nterferene model, namely MaxAverageSINRShedule. The performane of our algorthm s superor to exstng graph-based algorthms. A pratal expermental modelng shows that, on an average, our algorthm aheves 15% hgher spatal reuse than the BroadastShedule algorthm [1]. Sne shedules are onstruted offlne only one and then used by the network for a long perod of tme, ths mprovement n performane dretly translates to hgher long-term network throughput. Also, the omputatonal omplexty of MaxAverageSINRShedule s omparable to the omputatonal omplexty of BroadastShedule. Therefore, MaxAverageSINRShedule s a good anddate for effent SINR-based STDMA broadast shedulng algorthms. It would be nterestng to apply tehnques lke smulated annealng, genet algorthms and neural networks to determne SINR-omplant STDMA broadast shedules. 7. Referenes [1] S. Ramanathan and E. L. Lloyd, Shedulng Algorthms for Multhop Rado Networks, IEEE/ACM Trans. Networkng, vol. 1, no. 2, pp , Aprl [2] R. Nelson and L. Klenrok, Spatal TDMA: A Collson-Free Multhop Channel Aess rotool, IEEE Trans. Commun., vol. 33, no. 9, pp , September [3]. Gupta and. R. Kumar, The Capaty of Wreless Networks, IEEE Trans. Inform. Theory, vol. 46, pp , Marh [4] J. Grönkvst and A. Hansson, Comparson Between Graph-Based and Interferene-Based STDMA Shedulng, n ACM MOBIHOC, Otober [5] A. Behzad and I. Rubn, On the erformane of Graph-based Shedulng Algorthms for aket Rado Networks, n IEEE GLOBECOM 2003, vol. 6, Deember 2003, pp [6] R. Motwan and. Raghavan, Randomzed Algorthms. Cambrdge Unversty ress, [7] T.-S. Km, H. Lm, and J. C. Hou, Improvng Spatal Reuse through Tunng Transmt ower, Carrer Sense Threshold, and Data Rate n Multhop Wreless Networks, n ACM MobCom 2006, Los Angeles, CA, September 2006, pp