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

Transcription

1 On Hgh Spatal Reuse Broadcast Schedulng n STDMA Wreless Ad Hoc Networks Ashutosh Deepak Gore and Abhay Karandkar Informaton Networks Laboratory Department of Electrcal Engneerng Indan Insttute of Technology - Bombay Mumba Inda. {adgore,karand}@ee.tb.ac.n Abstract Graph-based algorthms for pont-to-multpont broadcast schedulng n Spatal reuse Tme Dvson Multple Access (STDMA) wreless ad hoc networks often result n a sgnfcant number of transmssons havng low Sgnal to Interference and Nose densty Rato (SINR) at ntended recevers, leadng to low throughput. To overcome ths problem, we propose a new algorthm for STDMA broadcast schedulng based on a graph model of the network as well as SINR computatons. The performance of our algorthm s evaluated n terms of spatal reuse and computatonal complexty. Smulaton results demonstrate that the proposed algorthm performs sgnfcantly better than exstng graph-based algorthms. I. INTRODUCTION In a wreless ad hoc network, a prevalent scheme for channel spatal reuse s Spatal Tme Dvson Multple Access (STDMA), n whch tme s dvded nto fxed-length slots that are organzed cyclcally and multple enttes can communcate n the same slot. An STDMA schedule descrbes the transmsson rghts for each tme slot n such a way that communcatng enttes assgned to the same slot do not collde. STDMA schedulng algorthms can be categorzed nto lnk schedulng and broadcast/node schedulng algorthms [1]. In lnk schedulng, the transmsson rght n every slot s assgned to certan source-destnaton pars. In broadcast schedulng, the transmsson rght n every slot s assgned to certan nodes,.e., there s no apror bndng of transmtter and recever and the packet transmtted must be receved by every neghbor. In ths paper, we only consder centralzed broadcast schedulng for STDMA networks. A. Related Work The concept of STDMA for multhop wreless ad hoc networks was formalzed n [2]. A broadcast schedule s typcally determned from a graph model of the network [1]. The problem of determnng an optmal mnmum-length STDMA schedule for a general multhop ad hoc network s N-complete for both lnk and broadcast schedulng [1]. In fact, ths s closely related to the problem of determnng the mnmum number of colours to colour all the vertces (or edges) of a graph under certan adjacency constrants. In [1], the authors show that a wreless ad hoc network can be scheduled such that the schedule s bounded by a length proportonal to the graph thckness 1 tmes the optmum number of colours. However, the above work does not take nto account Sgnal to Interference and Nose densty Rato (SINR) computatons when determnng an STDMA broadcast schedule. In ths paper, we propose a suboptmal algorthm based on the graph model as well as SINR computatons. We ntroduce spatal reuse as a performance metrc and demonstrate that the proposed algorthm has low computatonal complexty and hgh spatal reuse compared to exstng algorthms n the lterature. The rest of the paper s organzed as follows. In Secton II, we descrbe our system model, dscuss the lmtatons of graph-based schedulng algorthms and formulate the problem. Secton III descrbes the proposed broadcast schedulng algorthm. The performance of our algorthm s evaluated n Secton IV and ts computatonal complexty s derved n Secton V. We conclude n Secton VI. II. SYSTEM MODEL Consder an STDMA wreless ad hoc network wth N statc nodes (wreless routers) n a two-dmensonal plane. Durng a tme slot, a node can ether transmt, receve or reman dle. We assume homogeneous and backlogged nodes. Let: r j = (x j, y j ) = Cartesan coordnates of the j th node = transmsson power of every node N 0 = thermal nose densty D(j, k) = Eucldean dstance between nodes j and k We do not consder fadng and shadowng effects. The receved sgnal power at a dstance D from the transmtter s gven by D α, where α s the path loss factor. A broadcast schedule effectvely assgns sets of nodes to tme slots. Specfcally, a broadcast schedule for the STDMA 1 The thckness of a graph s the mnmum number of planar graphs nto whch the gven graph can be parttoned.

2 network s denoted by Ω(C, B 1,, B C ), where C = number of slots n the broadcast schedule B = set of broadcast 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 packet 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 concurrent transmssons n the th slot and t,j, r,j,k {1,...,N}. The SINR at recever r,j,k s gven by SINR r,j,k = D α (t,j,r,j,k ) N 0 + M l=1 l j D α (t,l,r,j,k ) We defne the sgnal to nose rato (SNR) at recever r,j,k by SNR r,j,k := N 0 D α (t,j, r,j,k ) A. hyscal and rotocol Interference Models Accordng to the physcal nterference model [3], the uncast transmsson t,j r,j,k s successful f and only f (ff) the SINR at recever r,j,k s greater than or equal to a certan threshold γ c, termed as the communcaton threshold. D α (t,j,r,j,k ) N 0 + M l=1 l j D α (t,l,r,j,k ) (1) (2) γ c (3) Accordng to the protocol nterference model [3], t,j r,j,k s successful f: 1) the SNR at recever r,j,k s no less than the communcaton threshold γ c. From (2), ths translates to D(t,j, r,j,k ) ( N 0 γ c ) 1 α =: R c (4) where R c s termed as communcaton range. 2) the sgnal from any unntended transmtter t,l s receved at r,j,k wth SNR less than a certan threshold γ, termed as the nterference threshold. Equvalently D(t,l, r,j,k ) ( N 0 γ ) 1 α =: R l j (5) where R s termed as nterference range. Note that 0 < γ < γ c, thus R > R c. The physcal model of our system s denoted by Φ(N, (r 1,...,r N ),, γ c, γ, α, N 0 ). A schedule Ω( ) s feasble f t satsfes the followng: 1) Operatonal constrant: A node cannot transmt and receve n the same tme slot. Also, a node cannot receve from multple transmtters n the same tme slot. 2 The set of neghbors of a gven node depends on the geographcal locatons of the nodes and wll be made precse n Secton II-A. 2) Communcaton range constrants: a) Every recever s wthn the communcaton range of ts ntended transmtter. D(t,j, r,j,k ) R c (6) b) Every recever s outsde the communcaton range of ts non-ntended transmtters. D(t,l, r,j,k ) > R c l j (7) If node b s wthn node a s communcaton range, then b s defned as a neghbor of a, snce b can decode a s packet correctly (subject to (3)). Note that f node b s outsde node a s communcaton range, then t can never decode a s packet correctly (from (3)). A schedule Ω( ) s exhaustve f every two nodes c, d who are neghbors of each other are ncluded n the schedule twce, once wth c beng the transmtter and d beng a recever, and vce versa. B. Graph-Based Schedulng Broadcast schedules are typcally desgned by modelng the STDMA network Φ( ) by a drected graph G(V, E), where V s the set of vertces 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 c E, where E c and E denote the set of communcaton and nterference edges respectvely. If node k s node j s neghbor, then there s a communcaton edge from c v j to v k, denoted by v j vk. If node k s outsde node j s communcaton range but wthn ts nterference range, then there s an nterference edge from v j to v k, denoted by v j v k. Thus, the mappng from Φ( ) to G( ) can be descrbed as follows: D(j, k) R c v j vk E c and v k vj E c R c < D(j, k) R v j v k E and v k v j E The subgraph G c (V, E c ) consstng of communcaton edges only s termed as the communcaton graph. An STDMA broadcast schedule s equvalent to assgnng a unque colour to every vertex n the graph, such that nodes wth the same colour transmt smultaneously n a partcular tme slot, subject to: Any two vertces v, v j can be coloured the same ff: ) edge v vj E c and edge v j v E c,.e., there s no prmary vertex conflct, and ) there s no vertex v k such that v vk E c and v j v k E c,.e., there s no secondary vertex conflct. These crtera are based on the operatonal constrant. Graph-Based schedulng algorthms utlze varous graph colourng methodologes to obtan a non-conflctng schedule,.e., a schedule devod of prmary and secondary vertex conflcts. To maxmze the throughput of an STDMA network, graph-based schedulng algorthms seek to mnmze the total number of colours used to colour all the vertces of G( ).

3 C. Lmtatons of Graph-Based Algorthms Observe that Crtera ) and ) are not suffcent to guarantee that the resultng schedule Ω( ) s conflct-free. Due to hardthresholdng based on communcaton and nterference rad, graph-based schedulng algorthms can lead to hgh cumulatve nterference at a recever [4] [5]. Ths s because the SINR at recever r,j,k decreases wth an ncrease n M, whle R c and R have been defned for a sngle transmsson only. For example, consder Fg. 1 wth sx labeled nodes whose Fg. 1. R c r,1,1 t r,1,1,2 r t r,2,1,2,2,2 Graph-Based algorthms can lead to hgh cumulatve nterference. coordnates are 1 (0, 0), 2 ( 80, 0), 3 (90, 0), 4 (280, 0), 5 (200, 0) and 6 (370, 0). The system parameters are = 10 mw, α = 4, N 0 = 90 dbm, γ c = 20 db and γ = 10 db, whch yelds R c = 100 m and R = m. A graph-based schedulng algorthm wll typcally schedule the transmssons 1 {2, 3}, and 4 {5, 6} n the same tme slot, say the th tme slot, snce the resultng graph colourng s devod of prmary and secondary vertex conflcts. However, our computatons show that the SINRs at recevers r,1,1, r,1,2, r,2,1 and r,2,2 are db, db, db and db respectvely. From the physcal nterference model, the transmsson t,1 r,1,1 s successful, whle the transmssons t,1 r,1,2, t,2 r,2,1 and t,2 r,2,2 are unsuccessful. Ths leads to low throughput. Hence, graph-based schedulng algorthms do not maxmze the throughput of an STDMA network. D. roblem Formulaton We propose a new suboptmal algorthm for STDMA broadcast schedulng based on the physcal nterference model. To evaluate the performance of our algorthm and compare t wth exstng suboptmal STDMA broadcast schedulng algorthms, we defne the followng metrc: spatal reuse. Consder the STDMA broadcast schedule Ω( ) for the network Φ( ). Under the physcal nterference model, the pont-to-pont transmsson t,j r,j,k s successful ff (3) s satsfed. The spatal reuse of the schedule Ω( ) s defned as the average number of successful pont-to-multpont transmssons per tme slot n the STDMA schedule. Thus Spatal Reuse = η(t C,j ) M k=1 I(SINR r,j,k γ c) =1 j=1 η(t,j) C where I(A) denote the ndcator functon for event A,.e., I(A) = 1 f event A occurs; I(A) = 0 f event A does R (8) not occur. Note that a hgh value of spatal reuse 3 drectly translates to hgh long-term network throughput. We seek a low complexty STDMA broadcast schedulng algorthm wth spatal reuse reasonably greater than unty. We only consder STDMA schedules whch are feasble and exhaustve. III. SINR-BASED BROADCAST SCHEDULING ALGORITHM Our proposed SINR-based broadcast schedulng algorthm s called MaxAverageSINRSchedule, whch consders the communcaton graph G c (V, E c ) and s descrbed n Algorthm 1. In hase 1 (Lne 3), we label all the vertces randomly 4. Specfcally, f G c ( ) has v vertces, we perform a random permutaton of the sequence (1, 2,..., v) and assgn these labels to vertces wth ndces 1, 2,..., v respectvely. In hase 2 (Lnes 4-7), the vertces are examned n ncreasng order by label 5 and the MaxAverageSINRColour functon s used to assgn a colour to the vertex under consderaton. The MaxAverageSINRColour functon s explaned n Algorthm 2. It begns by dscardng all colours that conflct wth u, the vertex under consderaton. Among the set of nonconflctng colours C nc, t chooses that colour for u whch results n the maxmum value of average SINR at the neghbors of u. Intutvely, ths average SINR s also a measure of the average dstance of every neghbor of u from all co-coloured transmtters. The hgher the average SINR, the hgher s ths average dstance. We choose the colour whch results n the maxmum average SINR at the neghbors of u, so that the addtonal nterference at the neghbors of all co-coloured transmtters s kept low. Algorthm 1 MaxAverageSINRSchedule 1: nput: hyscal network Φ( ), communcaton graph G c ( ) 2: output: A colourng C : V {1, 2,...} 3: label the vertces of G c randomly 4: for j 1 to n do 5: let u be such that L(u) = j 6: C(u) MaxAverageSINRColour(u) 7: end for A. Smulaton Model IV. ERFORMANCE RESULTS In our smulaton experments, the locaton of every node s generated randomly n a crcular regon of radus R. If (X j, Y j ) are the Cartesan coordnates of the j th node, then X j U[ R, R] and Y j U[ R, R] subject to Xj 2 + Y j 2 R 2. Equvalently, f (R j, Θ j ) are the polar coordnates of the j th node, then Rj 2 U[0, R2 ] and Θ j U[0, 2π]. Usng (4) and (5), we compute R c and R, and then map the STDMA 3 Note that spatal reuse n our system model s analogous to spectral effcency n dgtal communcaton systems. 4 Randomzed algorthms are known to outperform determnstc algorthms, esp. when the characterstcs of the nput are not known apror [6]. 5 In essence, the vertces are scanned n a random order, snce labelng s random.

4 Algorthm 2 nteger MaxAverageSINRColour(u) 1: nput: hyscal network Φ( ), communcaton graph G c ( ) 2: output: A non-conflctng colour 3: C set of exstng colours 4: C p {C(x) : x s coloured and s a neghbor of u} 5: C s {C(x) : x s coloured and s two hops away from u} 6: C nc = C \ {C p C s } 7: f C nc φ then 8: r colour n C nc whch results n maxmum average SINR at neghbors of u 9: f maxmum average SINR γ c then 10: return r 11: end f 12: end f 13: return C + 1 spatal reuse R = 500 m, = 10 mw, α = 4, N 0 = 90 dbm, γ c = 20 db, γ = 10 db 0.5 BroadcastSchedule MaxAverageSINRSchedule number of nodes network Φ( ) to the two-ter graph G(V, E c E ). Once the broadcast schedule s computed by every algorthm, the spatal reuse s computed usng (8). We use two sets of prototypcal values of system parameters n wreless networks [7] and descrbe them n Secton IV-B. For a gven set of system parameters, we calculate the spatal reuse by averagng ths quantty over 1000 randomly generated networks. Keepng all other parameters fxed, we observe the effect of ncreasng the number of nodes on the spatal reuse. In our experments, we compare the performance of the followng algorthms: 1) BroadcastSchedule [1] (BS) 2) MaxAverageSINRSchedule (MASS) B. erformance Comparson In our frst experment (Experment 1), we assume that R = 500 m, = 10 mw, α = 4, N 0 = 90 dbm, γ c = 20 db and γ = 10 db. Thus, R c = 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. In our second experment (Experment 2), we assume that R = 700 m, = 15 mw, α = 4, N 0 = 85 dbm, γ c = 15 db and γ = 7 db. Thus, R c = 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. From Fgures 2 and 3, we observe that spatal reuse ncreases wth the number of nodes for both the algorthms. The MASS algorthm consstently yelds hgher spatal reuse compared 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 performance translates to substantally hgher long-term network throughput. V. ANALYTICAL RESULT In ths secton, we derve an upper bound on the runnng tme (computatonal) complexty of our algorthm. Let v spatal reuse Fg. 2. Spatal reuse vs. number of nodes for Experment R = 700 m, = 15 mw, α = 4, N 0 = 85 dbm, γ c = 15 db, γ = 7 db 1 BroadcastSchedule MaxAverageSINRSchedule number of nodes Fg. 3. Spatal reuse vs. number of nodes for Experment 2. denote the number of vertces of the communcaton graph G c (V, E c ). Theorem 1: The runnng tme of MaxAverageSINRSchedule s O(v 2 ). roof: Assumng that an element can be chosen randomly and unformly from a fnte set n unt tme (Chapter 1, [6]), the runnng tme of hase 1 can be shown to be O(v). In hase 2, the vertex under consderaton s assgned a colour usng MaxAverageSINRColour. The worst-case sze of the set of colours to be examned C nc C p C s s O(v). Wth a careful mplementaton, MaxAverageSINRColour runs n tme proportonal to C nc,.e., O(v). Thus, the runnng tme of hase 2 s O(v 2 ). Fnally, the overall runnng tme of MaxAverageSINRSchedule s O(v 2 ).

5 VI. DISCUSSION In ths paper, we have developed a broadcast schedulng algorthm for STDMA multhop wreless ad hoc networks under the physcal nterference model, namely MaxAverageSINRSchedule. The performance of our algorthm s superor to exstng graph-based algorthms. A practcal expermental modelng shows that, on an average, our algorthm acheves 15% hgher spatal reuse than the BroadcastSchedule algorthm [1]. Snce schedules are constructed offlne only once and then used by the network for a long perod of tme, ths mprovement n performance drectly translates to hgher long-term network throughput. Also, the computatonal complexty of MaxAverageSINRSchedule s comparable to the computatonal complexty of BroadcastSchedule. Therefore, MaxAverageSINRSchedule s a good canddate for effcent SINR-based STDMA broadcast schedulng algorthms. It would be nterestng to apply technques lke smulated annealng, genetc algorthms and neural networks to determne SINR-complant STDMA broadcast schedules. REFERENCES [1] S. Ramanathan and E. L. Lloyd, Schedulng Algorthms for Multhop Rado Networks, IEEE/ACM Trans. Networkng, vol. 1, no. 2, pp , Aprl [2] R. Nelson and L. Klenrock, Spatal TDMA: A Collson-Free Multhop Channel Access rotocol, IEEE Trans. Commun., vol. 33, no. 9, pp , September [3]. Gupta and. R. Kumar, The Capacty of Wreless Networks, IEEE Trans. Inform. Theory, vol. 46, pp , March [4] J. Grönkvst and A. Hansson, Comparson Between Graph-Based and Interference-Based STDMA Schedulng, n ACM MOBIHOC, October [5] A. Behzad and I. Rubn, On the erformance of Graph-based Schedulng Algorthms for acket Rado Networks, n IEEE GLOBECOM 2003, vol. 6, December 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