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  1. 1. Opportunistic Routing in Ad Hoc Networks: How many relays should there be? What rate should nodes use? Joseph Blomer, Nihar Jindal Dept. ECE, University of Minnesota, Minneapolis, MN, USA 55455, Email: {blome015,nihar}@umn.eduarXiv:1004.0152v1 [cs.IT] 1 Apr 2010 Abstract—Opportunistic routing is a multi-hop routing scheme algorithm, the key adjustable parameters are the transmission which allows for selection of the best immediately available relay. probability and the transmission spectral efficiency. In blind opportunistic routing protocols, where transmitters The transmission probability, denoted by p, determines the blindly broadcast without knowledge of the surrounding nodes, two fundamental design parameters are the node transmission proportion of transmitters to receivers in each slot. When p probability and the transmission spectral efficiency. In this is large, there is a large amount of interference between the paper these parameters are selected to maximize end-to-end nodes. This interference will cause only receivers close to performance, characterized by the product of transmitter density, transmitters to be potential forwarders. When the transmission hop distance and rate. Due to the intractability of the problem probability is low, there are fewer simultaneous transmissions as stated, an approximation function is examined which proves reasonably accurate. Our results show how the above design pa- and thus less interference and more available relays. Thus rameters should be selected based on inherent system parameters longer hops are possible, but in fewer numbers. As a result, the such as the path loss exponent and the noise level. trade-off with transmission probability is essentially between many simultaneous short hops, or fewer long hops. I. I NTRODUCTION When the spectral efficiency is high, a large signal-to- Ad hoc networks operate on the basis of multi-hop rout- interference ratio (SIR) is required to decode. Thus, only relays ing, which allows information to be communicated across which are close to the transmitter are likely to decode. On the the network over a series of hops. End-to-end performance other hand, a lower spectral efficiency allows nodes that are depends critically on the quality of the multi-hop routes used, farther away to decode. Therefore, the trade-off with spectral but choosing good routes in a dynamic network (e.g. one efficiency is between shorter hops at higher data rate or longer where nodes are moving quickly) is a particularly difficult hops at a lower data rate. task. Opportunistic routing (e.g [1] - [6]) is well suited to such The objective of this paper is determining the transmission dynamic settings, as it can be performed with low overhead probability and spectral efficiency that optimally balance these while also exploiting spatial diversity (in terms of fading and trade-offs. We study blind opportunistic routing in a spatial topology). model, as in [1]. In this model, end-to-end performance is In this paper we focus on a purely opportunistic routing characterized by the forward-rate-density, which is the prod- protocol, in which nodes broadcast packets in a completely uct of average transmitter density, average hop distance and blind manner (i.e. without any knowledge of surrounding transmission rate (spectral efficiency). This is a function of nodes). More specifically, the protocol is described as follows: measured network properties (e.g. path-loss exponent) and 1) In each time slot, each node randomly decides, with adjustable parameters (spectral efficiency and transmission some fixed probability, to either transmit or receive. probability). Our objective is to find the transmission prob- 2) A transmitting node will blindly broadcast a packet at ability and transmission spectral efficiency that maximize this some fixed spectral efficiency, whereas receivers listen metric. Studying this quantity directly is intractable, however for any transmission. we derive a reasonably accurate approximation and reinforce 3) All receiver nodes that successfully decode the packet the conclusions with Monte Carlo simulations. will send an ACK back to the transmitting node during an acknowledgement period. The ACK packets contain II. P RELIMINARIES absolute geographic information about the RX location. A. Network Model 4) The transmitter selects the successful receiver which Consider an infinite set of transmit/receive nodes Φ dis- offers the most progress towards the packet’s final des- tributed according to a homogeneous 2-D Poisson point tination and sends a message to that node electing it to process (PPP) with density λ m−2 . We consider a slotted become the next forwarder. transmission scheme. In each slot a node elects to become a This protocol allows for the instantaneous choice of the transmitter with probability p, independent across users and best next hop, at the cost of an acknowledgement period and slots. The set of locations of the transmitters (TX1, TX2, with the requirement for geographical information. Within this ..) denoted Φt and the set of locations of the receivers Φr
  2. 2. Transmitter A. Forward progress density Receiver Succ. RX for node 1 We assume that the transmitter located at the origin has a packet to be sent to a receiver a very large distance away (e.g., fig. 1). Specifically, we assume the final destination is to dest. located an infinite distance away along the x-axis. Thus, the 1 relay offering the most progress is the one with the maximum forward prog. x-coordinate amongst the set of receivers that successfully 2 decode TX0’s packet. The corresponding progress, for a given network realization, is:Fig. 1. Example network: TX node 1 has a message to be sent along dashed D(Φr , Φt , β)line. Among successful message receivers, RX node 2 offers the most forward max 1(Sj ≥ β|Φt ) |Xj | · cos θ (Xj ) (3)progress and will forward the packet in the next slot. Xj ∈Φr where 1(Sj ≥ β|Φt ) is the indicator function that the SIR at RXj (at point Xj ) relative to TX0 (at the origin) is greater thanthen form independent PPP’s of intensity λp and λ(1 − p) the threshold. The expectation of this function with respect torespectively. the node process and fading is d(p, β). 2B. Channel Model d(p, β) EΦ,|h| D(Φr , Φt , β) (4) This quantity is the expected progress (towards the destination) We consider a path-loss model with exponent α > 2 and made in each hop. The forward progress density f (p, β) thenRayleigh fading coefficients hi,j from TX i to RX j. Denoting is:the signal transmitted by TXi (located at Xi ) as ui , thereceived signal of RXj (at location Xj ) is given by: f (p, β) d(p, β)λp. (5) Finally, the forward progress-rate-density function K(p, β) is −α/2 Yj = hi,j |Xi − Xj | ui + z. (1) K(p, β) (λp) · E[D(Φr , Φt , β)] · log2 (1 + β) i = f (p, β) · R(β). (6) For the time being, consider the network is interference Assuming each node in the network wishes to send data tolimited, so thermal noise (z) is negligible. The case of a another node a distance L away, at most p · d(p, β)/L packetschannel with thermal noise is considered in Sec IV. can originate at each node in each slot [1]. Translating to Assuming the transmit powers are all equal, the SIR (signal bits/sec/Hz, this means that the maximum end-to-end data rateto interference ratio) from TX i to RX j is defined: (per node) is |hi,j |2 |Xi − Xj |−α p · d(p, β) · log2 (1 + β)/L [bps/Hz]. (7) Si,j 2 −α . (2) k=i |hk,j | |Xk − Xj | Thus, K(p, β) is directly proportional to end-to-end rate. The quantity K(p, β), which is deterministic, is a functionWe assume that all users transmit at rate equal to R(β) = of transmission probability p and SIR threshold β. We seeklog2 (1 + β); thus a communication is successful if and only the optimal values of β and p which maximize this product:if the received SIR is larger than a threshold value β. (p∗ , β ∗ ) = arg max K(p, β). (8) p,β III. P ROGRESS -R ATE P RODUCT A closed-form expression for K(p, β) cannot be found in general, so we rely on an approximation which is developed Although we are interested in the network-wide perfor- in the following section.mance of the protocol, by the homogeneity of the PPP thisis statistically equivalent to the performance of any transmit- B. SIR Cell-Based Approximationter. Thus, without loss of generality we can focus on TX0 The basis of our approximation to K(p, β) is the concept oflocated at the origin. From the perspective of TX0, all the an SIR cell [8], [9]. In the absence of fading, any RX within aother TX nodes form a homogeneous PPP with intensity λp particular region (i.e., the SIR cell) around a TX can decode,(by Slivnyaks Theorem [7], the distribution is unaffected by and the best forwarder is the RX in this region with the largestconditioning on the presence of TX0). The transmitters and x-coordinate. Therefore, the forward progress is determined byreceivers are ordered by their distance to the origin i.e. TX1 the SIR cell – which is completely determined by the interfereris closest to the origin, TX2 the next, etc. For the sake of locations and thus is a function only of the interferer processsimplicity, the SIR from TX0 to RXj is denoted Sj and is – and by the receiver locations (i.e., Φr ). To reach a tractablegiven by Sj = S0,j . approximation for K(p, β), we remove the randomness in the
  3. 3. √ x v0 /2 − xSIR cell and assume it is a deterministic region that is afunction of p and β, while retaining the randomness in Φr . We consider the average size of the SIR cell. This quantity √is clearly defined in the absence of fading. On the other hand, v0the SIR cell is not well defined with fading because the SIRdepends on independent fading RV’s (specific to each receiverlocation) in addition to the interferer locations. However, wecan derive a quantity analogous to the SIR cell area, denotedby v0 , by integrating the point-wise success probability overall space. From [9], the success probability for a RX a distancey from the TX is: t p0 (y) = PΦ [Sj ≥ β] Fig. 2. SIR cell is assumed to be a square V0 of side length v0 . The √ 2 probability that the forward progress is x is the probability there are no RX −πλp|y|2 β α in the dark shaded area. = exp . (9) G(α) Thus, v0 is given by: Thus, our approximation to the progress-rate-density is: f (p, β) ≈ ˜ ˜ fsq (p, β) = dsq (p, β)(λp) (14) G(α) v0 = p0 (y)dy = . (10) ˜ ˜ sq (p, β) = fsq (p, β) log2 (1 + β). (15) y∈R2 λpβ α 2 K(p, β) ≈ Kwhere The optimal values then are the solution to the equation α G(α) = 2 2 (11) ∂ ∂ ˜ 2Γ( α )Γ(1 − α ) Ksq (p, β) = 0. (16) ∂p ∂β ∞and Γ(z) is the gamma function Γ(z) = 0 tz−1 e−t dz. Although there is no closed form solution to this equation, We will then assume the SIR cell has area equal to its mean. the approximation nonetheless yields valuable insight. By rear-We now approximate the SIR cell by a square centered on TX0 ranging the terms in (12), we can interpret the forward progress √with side v0 . In other words, we assume that all RX within approximation as the product of the maximum possible relaythis square are able to decode (ref. fig. 2). V0+ is the positive distance (i.e., half the side-length of the SIR cell) and ahalf of that square with area v0 /2. Under these assumptions, fractional term that captures how much of the maximum isthe expected forward progress is: attained by the best receiver. √ ˜ v0 1 − e−c ˜ dsq (p, β) = 1− (17) dsq (p, β) 2 c max. relay dist. frac. of max. = E max xj + Xj ∈V0 The value v0 is decreasing in p: as p increases, the amount of interference between transmitters is increasing, giving each = E E max xj # {Xj ∈ V0+ } = j transmitter a smaller expected SIR cell size. The value of c is + Xj ∈V0 also decreasing in p as a result of decreased v0 and smaller ∞ receiver density. = E max xi # Xi ∈ V0+ =j × ˜ The value Ksq (p, β) can be written: + j=0 Xi ∈V0 ˜ Ksq (p, β) v0 j v0 (λ · (1 − p) · 2 ) e −λ·(1−p)· 2 1 −2 1 − e−c j! = log2 (1 + β) λpG(α)β α 1− (18) ∞ √ v0 j 2 c v0 (λ · (1 − p) · v0 j 2 ) = · e−λ·(1−p)· 2 which lends the intuition that the end-to-end performance is j=0 2 j+1 j! increasing with p until the 1 − 1−e fractional term becomes −c c √ too small. When the value of c (expected potential forwarders) v0 e−c − (1 − c) = , (12) is greater than 3, this fractional term is greater than 2/3. When 2 c p is large enough to cause c to go below 3, the performancewhere #(·) gives the cardinality of a set and drops off sharply. λ · (1 − p) · v0 (1 − p) G(α) IV. N UMERICAL R ESULTS c = = . (13) 2 p 2β 2/α For numerical results, Monte-Carlo simulations were per-Note that the value c is the expected number of receivers in formed. Locations of receivers and transmitters were realizedthe half-square V0+ ; it is also equal to half the average number using the properties of the PPP (c.f. [10]). SIR was calculatedof successful outgoing transmissions from a TX as in [4]. at each receiver and the largest progress offered was averaged.
  4. 4. A. Accuracy, Path-loss and Fading Fig. 3 shows the accuracy of the approximation of the 0.08 0dBspatial density of progress (5), (14) for p and β. While the 0.07 5dB 7.5dBapproximation is not absolutely accurate, it does correctly spatial density of progress 10dB 0.06capture the dependence on p and β. Fig. 4 shows the best p or β maximizing the progress- 0.05rate-density (15) for a fixed β or p, respectively. Again, the 0.04approximation helps to answer the essential question of whatparameter choices are optimal. 1 0.03 In Fig. 5 the jointly optimal (p, β) are plotted for different 0.02values of the path loss exponent. Note that the optimal value 0.01of p is essentially the same for α = 3 and 4, but thatthe optimizing SIR threshold increases with the path loss 0 0 0.05 0.1 0.15 0.2 0.25exponent. This increase is similar to [11], where the optimizing p transmission probabilitySIR threshold is found for a slightly different, but related,problem. Fig. 3. Spatial density of progress: approximation (thick curves) and simulation for λ = 1, α = 3, varying SIR threshold β.B. Sensitivity to parameter variation Fig. 6 shows the normalized approximation of Eqn. (15) forpath loss α = 3 defined by p* transmission prob. 0.15 Simulation ˜ Ksq (p, β) Approximation ˜ Ksq,norm (p, β) . (19) ˜ max(p,β) Ksq (p, β) 0.1This figure shows how the performance decreases with differ- 0.05ent values of p and β. Observe that there exist a wide range of −5 0 5 β SIR threshold dB(p, β) pairs that are near optimal (within 90%). If there exists 4a strong reason to choose a particular β (e.g., limited code Simulation β* SIR dB Approximationrates & modulations), or a particular p (e.g., power cycling, 2energy saving from different TX and RX powers), end-to-end 0performance does not suffer as long as the other parameter isappropriately chosen (to figs. 4, 6) −2 0.04 0.06 0.08 0.1 0.12 0.14 0.16 p transmission probabilityC. Robustness to Noise In order to simulate the effect of noise on the forward Fig. 4. Optimal p values giving best progress-rate-density for fixed values of SIR β (top). Optimal β values giving maximum progress-rate for fixed valuesprogress, we modify the SIR (2) to include noise power σ 2 . p (bottom). |hi,j |2 |Xi − Xj |−α Si,j = 2 −α + σ 2 (20) k=i |hk,j | |Xk − Xj |The metric for comparison is the average SNR to a node 10located at the average nearest-neighbor distance (dN N ). 8 simulated approximation −α α=4 E d2 N 2 N (λπ)α/2 6 SN RN N = = (21) SIR threshold β (dB) 4 σ2 σ2 2 α=3Fig. 7 shows the maximum forward-rate-density is decreasing 0with increasing channel noise. Also, the number of potential −2forwarders is decreasing with increased channel noise. Fig. 8 shows the effects of channel noise power on optimal −4p and β. The value of p∗ is increasing and the value of β ∗ −6 α = 2.1is decreasing with higher noise floor. The maximum possible −8 α = 2.1spectral efficiency is expected to drop with increasing noise −10 0 0.05 0.1 0.15 0.2power. As explored in IV-B, when β is fixed to some value p transmission probability(in this case as a result of the noise floor), the transmissionprobability can increase to compensate. Fig. 5. Optimal choice of p and β giving maximum progress-rate for fixed α path-loss values. 1 Simulation results are similar for a non-fading environment.
  5. 5. V. C ONCLUSION 10 We studied the case of opportunistic routing in an ad hoc 0.3 8 wireless network with the goal of maximizing the product of 0.7 6 0.4 forward progress density and the rate of data transmission. We 0.5 developed an approximation using the concept of SIR cells β SIR Threshold dB 4 and found for α = 3 the optimal spectral efficiency is near 0.8 0.9 0.6 0.9 2 0.5 1.3 bps/Hz and the optimal probability of transmission is near 9 0 0.06. The results give the intuition that having a low transmis- 0.4 −2 sion probability ensures a low amount of interference and a 0.3 −4 relatively high SIR threshold ensures only local transmissions −6 are received. 0.2 −8 0.6 R EFERENCES −10 0.05 0.1 0.15 0.2 0.25 0.3 p transmission probability [1] F. Baccelli, B. Blaszczyszyn, P. Muhhlethaler, An Aloha Protocol for Multihop Mobile Wireless Networks IEEE Trans. on Inf. Th., Vol. 52, No. 2 Feb. 2006.Fig. 6. Contour plot of normalized forward-rate-density function approxi- [2] O.B.S. Ali, C. Cardinal, and F. Gagnon, A performance analysis of multi-mation. hop ad hoc networks with adaptive antenna array systems. [3] S. Biswas R. Morris, ExOR: Opportunistic Multi-Hop Routing for Wire- less Networks SIGCOMM Aug. 2005. [4] S. Weber, N. Jindal, R.K. Ganti, M. Haenggi Longest edge routing on max progress−rate−density the spatial Aloha graph IEEE Globecom 2008. 0.08 [5] E.S. Sousa, J.A. Silvester Optimum Transmission Ranges in a Direct- Sequence Spread-Spectrum Multihop Packet Radio Network IEEE Jour. 0.06 on Sel. Areas Comm Vol 8. No. 5 June 1990 [6] M. Zorzi, S. Pupolin, Optimum Transmission Ranges in Multihop Packet 0.04 Radio Networks in the Presence of Fading IEEE Trans of Comm. Vol. 43 No. 7 July 1995 0.02 [7] D. Stoyan, W. Kendall, and J. Mecke, Stochastic Geometry and Its 10 15 20 25 30 35 Applications. 2nd ed. New York: Wiley, 1996. SNR to node at avg. NN distance (dB) [8] F. Baccelli, B. Blaszczyszyn, Stochastic Geometry and Wireless Networks, 2.5 Volume I: Theory Foundations and Trends in Networking, NOW publish- ing, Vol 3, Iss. 3-4 opt. No. RX 2 [9] F. Baccelli, B. Blaszczyszyn, Stochastic Geometry and Wireless Networks, 1.5 Volume II: Applications Foundations and Trends in Networking, NOW 1 publishing, Vol 4, Iss. 1-2 [10] Martin Haenggi, On Distances in Uniformly Random Networks. IEEE 0.5 10 15 20 25 30 35 Trans. on Inf. Th., Vol. 51, No. 10 Oct. 2005. SNR to node at avg. NN distance (dB) [11] N. Jindal, J.G. Andrews, S. Weber, Bandwidth Partitioning in Decen- tralized Wireless Networks IEEE Trans. on Wireless Comm. Vol. 7 No. 12 Dec. 2008Fig. 7. Simulated best progress-rate-density achievable for varying SNR tonode at mean nearest-neighbor distance (λπ)α/2 /σ2 0.3 2 p* β* p transmission probability β SIR Thresh. (dB) 0.2 0 0.1 −2 0 −4 5 10 15 20 25 30 35 40 SNR to node at avg. NN distance (dB)Fig. 8. Optimal values of p, β for varying SNR to node at mean nearest-neighbor distance (λπ)α/2 /σ2