IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 4, JULY 2005 1553 On Routing in Random Rayleigh Fading Networks Martin Haenggi, Senior Member, IEEE
Abstract—This paper addresses the routing problem for large loss probabilities increase with the number of hops (unless the wireless networks of randomly distributed nodes with Rayleigh transmit power is adapted). fading channels. First, we establish that the distances between To overcome some of these limitations of the “disk model,” neighboring nodes in a Poisson point process follow a generalized Rayleigh distribution. Based on this result, it is then shown that, we employ a simple Rayleigh fading link model that relates given an end-to-end packet delivery probability (as a quality of transmit power, large-scale path loss, and the success of a service requirement), the energy benefits of routing over many transmission. The end-to-end packet delivery probability over short hops are significantly smaller than for deterministic network a multihop route is the product of the link-level reception models that are based on the geometric disk abstraction. If the probabilities. permissible delay for short-hop routing and long-hop routing is the same, it turns out that routing over fewer but longer hops may While fading has been considered in the context of packet even outperform nearest-neighbor routing, in particular for high networks [16], [17], its impact on the network (and higher) end-to-end delivery probabilities. layers is largely an open problem. This paper attempts to shed Index Terms—Ad hoc networks, communication systems, fading some light on this cross-layer issue by analyzing the perfor- channels, Poisson processes, probability, routing. mance of different routing schemes under Rayleigh fading. In Section II, we introduce the link model and show that noise and interference can be treated separately. We also prove that I. INTRODUCTION the internode distances in a Poisson point process follow a NERGY consumption in multihop wireless networks is a generalized Rayleigh distribution; we define and determine the E crucial issue that needs to be addressed at all the layers path efficiency; and we introduce the five routing strategies we of the communication system, from the hardware up to the consider. In Section III, the energy consumption of those five application. In this paper, we focus on routing strategies that routing strategies is analyzed; Section IV discusses the perfor- employ hops of different length in a network whose nodes mance of those routing schemes under equal delay constraints, constitute a Poisson point process. The analysis is based on and Section V concludes the paper. a Rayleigh fading channel model, and the results demonstrate that the properties of the physical channel have a substantial impact on optimum protocol design at the network layer. II. RAYLEIGH NETWORK MODEL Often, a deterministic “disk model” is used for the analysis A. Rayleigh Fading Link Model of multihop packet networks [1]–[9], where the radius for a successful transmission has a deterministic value, irrespective We assume a narrowband Rayleigh block fading channel. i j of the condition of the wireless channel. Interference is taken A transmission from node to node is successful if the γ into account using the same geometric disk abstraction. The SINR ij is above a certain threshold Θ that is determined stochastic nature of the fading channel and thus the fact that by the communication hardware and the modulation and cod- γ the signal-to-noise-and-interference ratio (SINR) is a random ing scheme [10]. The SINR is a discrete random process variable are neglected. Using such models, it is easy to show given by α that, for a path loss exponent of , there is an energy gain R of nα−1 if a hop over a distance d is split into n hops of γ . = N I (1) distance d/n. However, the volatility of the channel cannot + be ignored in wireless networks [10], [11]; the inaccuracy R is the received power, which is exponentially distributed with of “disk models” has also been pointed out in [12] and is mean R¯. Over a transmission of distance d = xi − xj 2 with easily demonstrated experimentally [13], [14]. In addition, the α −α an attenuation d ,wehaveR¯ = P0d , where P0 is propor- “prevalent all-or-nothing model” [15] leads to the assumption tional to the transmit power.1 N denotes the noise power, and that a transmission over a multihop path either fails completely I is the interference power affecting the transmission, i.e., the or is 100% successful, ignoring the fact that end-to-end packet sum of the received power from all the undesired transmitters. Theorem 1: In a Rayleigh fading network, the reception probability pr := P[γ Θ] can be factorized into the reception Manuscript received February 6, 2003; revised November 23, 2003; accepted June 1, 2004. The editor coordinating the review of this paper and approving it for publication is J. C. Hou. This work was supported in part by the DARPA/IXO-NEST Program (AF-F30602-01-2-0526), NSF (ECS02-25265 1This equation does not hold for very small distances. So, a more accurate −α and ECS03-29766), and ORAU. model would be R¯ = P0 · (d/d0) , valid for d d0, with P0 as the average The author is with the Department of Electrical Engineering at the University value at the reference point d0, which should be in the far field of the transmit of Notre Dame, Notre Dame, IN 46556 USA (e-mail: [email protected]). antenna. At 916 MHz, for example, the near field may extend up to 1 m (several Digital Object Identifier 10.1109/TWC.2005.850376 wavelengths).
1536-1276/$20.00 © 2005 IEEE 1554 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 4, JULY 2005 probability of a zero-noise network and the reception probabil- ity of a zero-interference network. Proof: Let R0 denote the received power from the desired source and Ri, i =1,...,k, the received power from k inter- ferers. All the received powers are exponentially distributed, p r /R¯ −r /R¯ R¯ i.e., Ri ( i)=1 i exp( i i), where i denotes the aver- −α age received power R¯i = Pidi . The probability of correct reception is2
pr = P [R0 Θ(I + N)] (2) Θ(I + N) = EI exp − (3) R¯0 ∞ ∞ k r N x Θ i=1 i + Fig. 1. Part of a Rayleigh network with the source at the origin and the -axis ··· − pointing towards the destination node. R denotes the distance to the nearest = exp R¯0 neighbor within a sector φ around x and ψ is its argument. Hence, (R, ψ) are 0 0 the polar coordinates of the nearest neighbor within a sector φ and (X, Y ) are k its Cartesian coordinates. × p r r ··· r Ri ( i)d 1 d k (4) i=1 that the packet loss probability is inversely proportional to the k transmit power for high p . ΘN 1 r =exp − −α · α . (5) P d P d 0 0 i i 0 =1 1+ΘP0 di N B. Poisson Random Networks pr pI r We consider a Poisson point process of intensity λ in the N plane, where the probability of finding k nodes in an area A p is the probability that the SNR γN := R /N is above the r 0 is given by the Poisson distribution3 threshold Θ, i.e., the reception probability in a zero-interference network as it depends only on the noise. The second factor pI r (λA)k is the reception probability in a zero-noise network. P k A −λA . [ nodes in ]=e k (7) This allows an independent analysis of noise and interfer- ! ence. If the load is light, then the number of interferers k is Note that for infinite networks, the Poisson point process cor- relatively small and/or their distances are relatively big, which responds to a uniform distribution [12], [20], and for large implies SIR SNR, thus the noise analysis alone provides networks, the two distributions are equivalent for all practical accurate results. This has been demonstrated in [18] in the purposes. Henceforth, we denote a network whose nodes con- case of slotted ALOHA with transmit probability p. Indeed, stitute a two-dimensional Poisson point process as a Poisson for small p, the throughput is proportional to p, indicating that I random network. Without loss of generality, we will restrict there are no losses due to collisions, i.e., p ≈ 1 for all timeslots r ourselves to the case λ =1(unit density), since the product λA and all transmissions. For high load, a separate interference I can always be scaled such that λ =1. analysis has to be carried out to determine p . Note that power r For the routing schemes that we consider, we need to deter- scaling, i.e., scaling the transmit powers of all the nodes by I mine the distance from one node to its neighboring nodes that the same factor, does not change the SIR (p only depends r lie within a sector φ, i.e., within ±φ/2 of the source–destination on power ratios), but (slightly) increases the SINR. Since only N axis (Fig. 1). p is affected by absolute energy levels, we will focus on r Proposition 1 (Distance to Nearest Neighbor): In a Poisson zero-interference networks to identify energy-efficient routing random network with unit density, the distance R between strategies. a node and its nearest neighbor in a sector φ is Rayleigh In a zero-interference network, the reception probability over distributed with mean π/(2φ). a link of distance d at a transmit power P0, is given by pr := −α Proof: Let R be the distance to the nearest neighbor in a P[γN Θ] = exp(−(ΘN/P d )), therefore 0 sector φ. The probability that there is no neighbor in a sector dαΘN φ up to a distance r is the complementary cumulative distri- P0 = . (6) P R>r −r2φ/ − ln p bution [ ] = exp( 2), thus the probability density r 2 is pR(r)=rφ exp(−r φ/2), which is a Rayleigh distribution Note that for high probabilities, the packet loss probability with mean (π/(2φ))1/2 and variance 2/φ − π/(2φ)=(4− 1 − pr is tightly upperbounded by the normalized mean NSR π)/(2φ). The distribution of the argument ψ is uniform between ΘN/R¯0 =Θ/γ¯N [19]. Since − ln pr ≈ 1 − pr, we can also say −φ/2 and φ/2.
2A similar calculation has been carried out in [16, Appendix] for a network 3This can be generalized to higher dimensions if A is the Lebesgue measure with spreading gain and equal transmit powers for all nodes. of the subset considered. HAENGGI: ON ROUTING IN RANDOM RAYLEIGH FADING NETWORKS 1555
Definition 1 (Rayleigh Network): A Rayleigh network is a Integration with respect to y yields the marginal density Poisson random network where the physical channel is subject 2 to Rayleigh fading. φ φ 2π − x φ pX (x) = erf tan x e 2 (13) To compare different routing schemes, we need to define the 2 2 φ path efficiency. Definition 2 (Path Efficiency): The path efficiency is the ratio where√ erf(·) denotes the error function, i.e., erf(x):= of the Euclidean distance between the end nodes and the actual / π x −t2 t φ 2 0 exp( )d . It is easily verified that for small ,as distance traveled. X ≈ R 4 , this tends to a Rayleigh distribution. On the other As an example, the path efficiency (for long connections )in hand, for φ → π, tan(φ/2) →∞, the error function tends to a square lattice network with nearest-neighbor routing is √1, and the distribution approximates√ the one-sided Gaussian −x2π/ x /π Y 1 2exp( 2), 0, with mean 2 . has zero mean κ(φ)= (8) and becomes Gaussian for φ → π with variance 1/π. Note that | cos φ| + | sin φ| X and Y are not independent unless φ = π or φ =2π. E X where φ denotes the direction from transmitter to receiver The expected progress [ ] is relative to the orientation√ of the grid. The maximum is 1, π 2 φ the minimum is 1/ 2 (at φ = π/4+kπ/2), and the expected E[X]= sin (14) value is 2φ φ 2 √ √ 2 2 2 which corresponds, as expected, to the product of the mean η := E[κ]= arctanh ≈ 0.79. (9) π 2 path efficiency η(φ) (10) and the mean of the Rayleigh distri- bution π/(2φ). We will use η to denote the expected path efficiency. Next, we generalize Proposition 1. Definition 3 (Link Efficiency): The link efficiency is X/R = Proposition 3 (Distance to nth Neighbor): The probability cos(Ψ), where X is the x-coordinate of the position of the density of the distance to the nth neighbor in a sector φ is nearest neighbor (see Fig. 1). n φ r2φ In Rayleigh networks with long connections, the distances 2n−1 2 − pR (r)=r e 2 . (15) and angles to nearest neighbors are independent identically n 2 (n − 1)! distributed, so the expected path efficiency equals the expected link efficiency. Proof: Let Sk be the kth coefficient in the Poisson distri- 2 k Proposition 2 (Path Efficiency in Rayleigh Networks): In a bution: Sk := (r φ/2) /k!. The probability that there are less Rayleigh network with nearest-neighbor routing in a sector φ, than n nodes closer than r in the sector φ is the expected path efficiency for a long connection is n− 1 2 − r φ Pn := P[0 ...n− 1 nodes within r]= Ske 2 . (16) 2 φ φ2 η(φ)= sin ≈ 1 − (10) k=0 φ 2 24 Hence, where the approximation is the second-order Taylor expansion. p d − P Proof: The expected link efficiency is E[cos Ψ], and since Rn = (1 n) dr the hop distances are i.i.d., this is also the expected path 2 k−1 efficiency. Thus, we have n− n− r φ 1 1 k 2 2 − r φ = rφ Sk − rφ e 2 . (17) φ k 2 k k ! 2 2 φ =0 =1 η(φ)=E[cos Ψ] = cos ψ dψ = sin . (11) Sk−1 φ φ 2 0 The only term that is not cancelled in the two sums is the one at n − 1, leading to 2 To compare the transport capacity of different routing − r φ pR = rφ · Sn− e 2 (18) schemes, we have to determine the progress X. Changing to n 1 Cartesian coordinates, the joint probability pXY (x, y) for the Erlang distribution nearest neighbor’s position (in a sector φ π)is
2 2 which is identical to (15). − x +y φ pXY (x, y)=e 2 Since pR is a Rayleigh distribution for n =1, it can be n φ φ considered a generalized Rayleigh distribution. Similarly, in x 0, −x tan y x tan . (12) one dimension, the Erlang distribution is a generalized expo- 2 2 nential distribution. So, the transition from one dimension to two dimensions simply entails a multiplication by rφ (that 4This is based on the assumption that the angle towards the destination is comes from the inner derivative of the exponential part) in the uniformly distributed over [0, 2π). distributions of the node distances. 1556 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 4, JULY 2005
The mean of Rn is given by √ √ n 1 π 2 Γ + 2 2 E[Rn]=√ ≈ √ n − 1+ . (19) φ Γ(n) φ 4
To get this approximation, we take the√ first term from the series expansion Γ(n +1/2)/Γ(n)= n(1 − (1/8n)+ O(1/n2)) [21]5 and, noting that this is not accurate for small n, adjust it (by adding π/4 − 1 to n) such that the approximation is precise for n =1.Forn>1, this yields a very tight upper bound. The second moment is 2n/φ, hence, the variance is
n − π Fig. 2. Illustration of the different routing strategies. Strategy A follows the 2 2 4 Var[Rn]= − E[Rn] ≈ (20) solid line (multiple hops), B and D follow the dashed line, C follows the φ 2φ dotted line, and E follows the dash-dotted line. The sectors φ, φ ,andφ are represented by different grades of gray. using the same approximation. From the series expansion E nκ αE κ above, we know that Var[Rn]=(1/2φ)+O(1/n).Sowehave case, we get 1 =( ) 0, where is the path efficiency. The found sharp lower and upper bounds on the variance energy consumption is identical for
− π 2 log n 4 1 α = . Var[Rn] < ∀n ∈ N. (21) nκ (23) 2φ 2φ log( ) √ α κ / E E Since 4 − π ≈ 1, we can conclude that, interestingly, the vari- For =4and =1 2, we get 1 = 2,soevenforlarge ance is almost independent of n.6 path loss exponents, there is no benefit in dividing a one-hop diagonal connection into two nearest-neighbor hops.
C. Energy Consumption of a Route D. Routing Strategies Assume a connection from node 0 to node n in a multihop Here, we introduce the five routing strategies that will be network. The desired end-to-end reliability is PEE. The recep- tion probability of a chain of n nodes is analyzed in the next section. In order to implement these routing algorithms, it is assumed that all nodes know their own location n n and that the source node knows the direction towards the − Θ −Θ 1 pn = e γ¯i =e i=1 γ¯i (22) destination. The strategies are the following (see Fig. 2). i=1 A) Multihop: Transmit hop by hop over n nearest-neighbor hops in a sector φ, i.e., the next-hop node is the near- γ i where ¯i denotes the mean SNR at link . est neighbor that lies within ±φ/2 of the axis to the P Assume that the maximum transmit power max is sufficient destination. n P to reach node in a single hop with probability of EE.The B) Single hop: Transmit directly to the nth node in the route question is how many hops are optimum in terms of energy found in A. consumption. First, we consider the case of a one-dimensional C) Single hop: Transmit directly to the n th nearest neighbor d E chain of equidistant nodes with distance .Let 0 be the energy in the sector φ. required for a transmission over distance d with probability n α D) Single hop: Transmit directly to the th nearest neighbor P E −d N/ P EE, i.e., 0 := Θ ln EE. Covering the total dis- in a sector φ <φ. E nαE tance in one single hop requires an energy of 1 = 0. E) Single hop: Transmit directly to the nearest neighbor in n In the√ multihop case with hops, a reception probability a sector φ <φ. p = n P is required at each hop. Since ln P = n ln p , r EE EE r Note that “single hop” here does not mean that the (final) desti- the total energy in this case is En = n · nE . So, for α =2, 0 nation is reached in one hop, but it means that a certain progress there is no benefit in multihop routing. X is made in one hop rather than in n hops. So, relative to the For a regular rectangular lattice network where the 0 end-to-end connection, all the schemes are multihop, but A uses maximum-hop case corresponds to routing over nearest neigh- 2 short hops, whereas B through E use long(er) hops. For a fair bors, we still have En = n E , whereas for the single-hop 0 comparison, we shall choose n , φ , and φ for strategies C, D, and E, respectively, such that the expected progress X¯ is the 5Note that this series can also be derived by using the identity Γ(n + same as for A and B. As long as routing decisions are taken √ n 1/2)/Γ(n)=[(2n)! π]/[n!(n − 1)!4 ] [22] and applying Stirling’s locally, virtually all routing strategies fall into those categories; approximation. 6This is not the case for one-dimensional networks, where the variance whatever the specific properties of an algorithm are, the goal increases linearly in n (variance of the Erlang distribution with parameter n). will always be to make some progress in the right direction. HAENGGI: ON ROUTING IN RANDOM RAYLEIGH FADING NETWORKS 1557
III. ENERGY CONSUMPTION which yields
In this section, the energies required to deliver a packet over π n = (n2 − 1) + 1 (31) a progress X¯ := nη(φ)E[R(φ)] = nη(φ) π/(2φ) with prob- 4 ability P are determined for those five strategies, normalized EE independent of φ. The path efficiency for this single-hop trans- by ΘN/(− ln P ). EE mission is still η(φ), since the argument of the destination node is uniformly distributed in the sector φ.Wehave9 A. Analysis of the Five Routing Strategies α α 2 n √ α 2 Γ + 2 p n P E [R ]= Strategy A: With r = EE, the total expected energy n φ Γ(n ) consumption is n2E[Rα] α 2 α α 2 √ π 2 2 α ≈ n − α . E n2 2 . φ 1+ (32) A = φ Γ 1+ (24) 4 2 E[Rα] Plugging in (31) gives α 2 π √ α Strategy B: First, we need to establish the path efficiency. In α 2 E [R ] ≈ (n + α − 1) 2 . (33) strategy A, the angle to the next node is uniformly distributed n 2φ in ±φ/2. With increasing n, the argument Ψ from the origin to the nth node tends to be Gaussian distributed with variance Strategy D: The reduction of the sector angle from φ to φ V (φ, n). Since the support of the probability density function shall ensure that the transmission to the nth nearest neighbor of Ψ is always [−φ/2,φ/2],7 the variance decreases inversely within φ results in the same progress as n nearest-neighbor proportional to n, i.e., V (φ, n) ≈ φ2/12n. So, for large n, hops in φ, i.e., 8 we get nE [R(φ)] · η(φ)=E [Rn(φ )] · η(φ ). (34) ηB(φ, n)=E[cos Ψ] (25) R R φ φ φ √ ∞ Since and n depend on and , respectively, cannot 6n − n ψ 2 be expressed in a closed form. However, we can find a tight ≈ √ · ψ 6 ( φ ) ψ φ π cos e d (26) upper bound on the energy consumption (lower bound on φ ) −∞ by assuming that the path efficiency does not increase with 2 − φ decreasing angle =e 24n (27) 2 φ 4(n − 1) + π ≈ 1 − (28) φ φ . (35) 24n n2π
α 2 φ ≈ φ/n from which we find 1/ηB(n, φ) ≈ 1+(αφ /24n). Clearly, with high accuracy. With (32), we get the upper B ηB(n, φ) η(φ) with equality only for n =1. Since we are bound D transmitting over a distance nR(η/ηB), we get α ED := E [Rn] BD η α α α α α EB = n E[R ] πn2 2 Γ n + η 2 2 . B = φ n − π n (36) α [4( 1) + ] Γ( ) 2 2 α 2 α αφ (n − 1) ≈ n − . φ Γ 1+ 1 n (29) To get a lower bound, we assume that the path efficiency for φ 2 24 is 1. Solving nE[R(φ)]η(φ)=E[Rn(φ )] using (19) yields So, the increase of nα−2 compared with strategy A is partially η/η α 4(n − 1) + π 1 compensated for by the factor ( B) , i.e., the increase in path φ φ (37) efficiency. n2π η(φ)2 Strategy C: A direct transmission to the n th neighbor in the and we find sector φ shall result in the same expected progress as in strategy A, i.e., E[Rn ]=nE[R].From(19),wehave α BD ED BDη(φ) . (38) √ 2 π π For large n, E tends to the lower bound. Since B has the √ n − 1+ = n (30) D D φ 4 2φ same asymptotic behavior for large n as strategy C, strategy D outperforms C.
7Hence, after every convolution, the support needs to be scaled, which results in a reduction of the variance. 9Again, we use the series expansion Γ(n + α/2)/Γ(n )=n (α/2)[1 − 8 O /n The Gaussian approximation is very accurate√ even for small n.Forφ = (1 )] and adjust the constant such that the approximation is precise for α n α α E Rα π/2 and n =1, e.g., the precise value is 2 2/π ≈ 0.9003, whereas this = =1. Note that for =1, this is identical to (19). For =2, [ n ] −π2/96 ∝ n α E Rα ∝ n 2 approximation yields e ≈ 0.9023, so the error is only 0.2%. The ,andfor√=4, [ n ] . The absolute error in the approximation second-order Taylor expansions are identical, even for n =1. for α =2is π 2/4 − 1 ≈ 0.11, independent of n . 1558 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 4, JULY 2005
TABLE I NORMALIZED ENERGY CONSUMPTION OF THE FIVE ROUTING STRATEGIES.THE APPROXIMATIONS ARE SECOND-ORDER TAYLOR EXPANSIONS
Strategy E: Here, we determine an even smaller angle φ in n2 in (31). For n =5, for example, the 20th neighbor would such that the first neighbor within that sector has the same need to be identified, which may cause too much overhead. expected distance as n hops in the original sector φ.AsinD, the equation nE[R(φ)]η(φ)=E[R(φ )]η(φ ) does not have a C. Determination of the Sector Angle φ closed-form solution, and we have to bound EE.Fromη(φ ) >
η(φ) we have φ φ/n2 and thus From the energy analysis, it can be seen that it is desirable φ α to choose as small as possible to achieve a high path effi- 2 α α 2 ciency. On the other hand, we have to make sure that a sufficient EE
Fig. 3. Gain of the n-transmission schemes without (left) and with (right) channel state information over a single transmission in the first timeslot as a function of the number of available timeslots n and the desired reliability PEE. Points with gain smaller than one are not shown.
Fig. 4. Gain of the schemes without (left) and with (right) channel state information over a multihop scheme as a function of the number of available timeslots n and the desired reliability PEE. The path loss exponent α is 3. Points with gain smaller than one are not shown. there is a benefit from time diversity (selection combining). The other words, the best timeslot out of n is picked for a single required single-use reception probability PEE,1 is then given by transmission of the packet. Note that if the delay constraint n is not hard but an average constraint, then this scheme may choose 1 n PEE,1 =1− (1 − PEE) . (43) from 2n slots. The cumulative distribution of the maximum of n n fading random variables is Fn(x)=[1− exp(−x/R¯)] with −α Compared with the single-transmission case, this leads to an R¯ = P0d .From energy gain of n P − ΘN ΘN log EE,1 P =1− 1 − e R¯ =⇒ R¯ = G = (44) EE 1 n n log PEE − ln 1 − (1 − PEE) (45) since from (6), the necessary transmit power in the single- α transmission case is −(ΘNd / log PEE), whereas for the we see that the reception probability increases in the same α multitransmission case, it is −(nΘNd / log PEE,1). way as in the multiple transmission case but with only one If we assume a simple automatic repeat request (ARQ) transmission. So, this strategy has an n times higher gain than scheme, where a short ACK or NACK packet is sent immedi- the one without CSI. ately after the receipt of the data packet, this gain is almost dou- Fig. 3 shows the gains for both strategies (as a function of bled (and the average delay cut in half), since after a successful the number of transmissions n and the desired probability) transmission, the packet does not have to be retransmitted any over a single-transmission scheme. For small PEE and large n, longer. the gain drops below 1 for the scheme without CSI (see the left With CSI: A Single Opportunistic Transmission When the plot). Channel is Best: With CSI at the transmitter, the packet can Fig. 4 compares the single-hop retransmission-based strate- be scheduled to be transmitted when the fading level is best. In gies (n transmissions) with conventional multihop routing over 1560 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 4, JULY 2005 n hops for α =3. It can be seen that the single-hop schemes of- that minimizes the overall energy consumption is 1.6 dB above ten outperform multihop routing if the same delay is permitted. ΘNdα. The minimum energy value is then given by Since the factor n between the strategies with and without CSI − − P corresponds to an increase of the path loss exponent by one, we E P opt Ndα ln(1 L). 0 =Θ (51) find in general (ln 2)2
single α With E(P )=−d ΘN/ ln PL, the energy gain is GCSI(α)=GNCSI(α − 1). (46) 0 E P opt G α 0 − P P ,P> 1. So, the left plot in Fig. 4 shows the gain CSI for =4. = log2(1 L) log2 L L (52) E P single 2 Note that in this analysis, we did not take into account 0 the higher path efficiency of the single-hop scheme. So, the effective gain is slightly bigger. The reduction in transmit power is − ln 2/ ln PL.ForPL ≈ 1, this is approximately ln 2/(1 − PL) or ln 2 divided by the desired packet loss rate. The gain then is inversely proportional B. Multiple Transmissions in the Short-Hop Case to the packet loss rate times the logarithm of the packet loss If the delay constraints are not tight, then there is time for rate. For PL = 99%, for example, the reduction in transmit retransmissions even for the short-hop routing scheme. Indeed, power is ≈ 100 ln 2 ≈ 69, the gain is slightly bigger than most wireless link layer protocols are based on ARQ, i.e., 10 dB, and nt ≈ 6.64. So, the reduction in transmit power acknowledgments and retransmissions are employed to ensure a more than compensates for the higher number of transmissions. satisfactory reliability. A given desired link reliability PL can be Of course, the single-transmission scheme and the energy- achieved by a single transmission or multiple transmissions at optimum scheme are just the two extreme possibilities in the lower power. The following proposition addresses the question energy-delay tradeoff. which scheme is energy optimum. Again, to be fair, the long-hop scheme should be permitted Proposition 4 (Optimum Power Level): The energy- the same delay. For the case without CSI, if the short-hop P opt n n optimum transmit power level 0 to achieve a link reliability scheme routes over hops with an average of t transmissions PL > 1/2 is given by at each hop, the long-hop scheme may use nnt timeslots, i.e., it may transmit nnt times instead of just n times. The required α ΘNd reliability is P = P n, so we find that the energy consumption P opt = (47) EE L 0 ln 2 for n transmissions is the corresponding single-transmission reception probability is ΘNd α En = n (53) popt =1/2, and the necessary average number of transmissions 1 r − ln 1 − (1 − P ) n 10 n − − P EE is t = log2(1 L). Proof: The average number of transmissions to achieve where d denotes the distance over this long hop. Permitting PL is given by nnt transmissions at lower transmit power, we have log(1−PL) α −p pr So, the total energy consumption is which results in an energy gain of 1 log(1 − PL) − − P nnt E P n P P En 1 log 1 (1 EE) ( 0)= t 0 = − ΘN 0 (49) −α G = = . (55) − P0d E n 1 log 1 e nnt t n log 1 − (1 − PEE) which is minimized at Since we know that the optimum single-transmission reception α − − ΘNd probability is 1/2, this gain is smaller than 1 if 1 (1 P opt E P 1/n 0 = arg min ( 0)= (50) PEE) 1/2. Therefore, only for high PEE or small n is P0 ln 2 there a benefit in transmitting nnt times. In contrast, the scheme with CSI available at the transmitter always benefits from a popt / n − − corresponding to r =1 2, and, in turn, t = log2(1 larger number of timeslots. PL). In summary, for very loose delay constraints, short-hop Note that the optimum does not depend on PL. The factor schemes derive more benefit from retransmissions. Since there 1/ ln 2 ≈ 1.44 corresponds to 1.6 dB, hence, the transmit power is an optimum number of transmissions, the long-hop scheme cannot benefit from having a very large number of timeslots available with this simple retransmission scheme (unless CSI 10 log2(·) is the binary logarithm, while log(·) denotes any logarithm. is available). In this case, the available time should rather be HAENGGI: ON ROUTING IN RANDOM RAYLEIGH FADING NETWORKS 1561 exploited by using a stronger channel code, thereby lowering Hence, the robustness of the network is bigger. The avail- the SINR threshold Θ. ability of intermediate nodes also increases the network’s capability of route repair and maintenance. 7) Energy and delay balancing. Often, the data gathered by V. C ONCLUDING REMARKS sensor networks has to be delivered to a single point, the We have offered a fresh look at the routing problem in ran- observer or base station. Using nearest-neighbor routing, dom wireless networks by taking into account Rayleigh fading, the lifetime of a node close to the base station is much end-to-end packet delivery probabilities, node distributions, and shorter than the lifetime of a node far away due to traffic path efficiencies. The main result is that, from an energy point accumulation [27]. With long-hop routing, the energy of view, routing over many short hops is not as beneficial in consumption is better balanced. Instead of just a few a Rayleigh network as it seems to be for the deterministic critical nodes, there are many more, which results in an “disk model.” If we impose a delay constraint that is, for a fair increase of the network lifetime. comparison, loose enough such that it can be met by short-hop In conclusion, long-hop strategies have a clear advantage in routing, long-hop schemes may actually save energy. most of these categories, in addition to being competitive in Apart from energy and delay, there are other important terms of energy. practical issues that determine the overall performance of a routing scheme. 1) Interference. The different transmit power levels of the ACKNOWLEDGMENT different routing schemes will change the interference by which other ongoing transmissions will be affected. 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Zurich (ETHZ), Zurich, Switzerland, in 1995. In [21] R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics: 1996, he earned the Dipl. NDS ETH (postdiploma) A Foundation for Computer Science, 2nd ed. Reading, MA: Addison- degree in information technology, and in 1999, he Wesley, 1994. completed his Ph.D. thesis on the analysis, design, [22] G. Arfken, Mathematical Methods for Physicists, 3rd ed. Orlando, FL: and optimization of cellular neural networks. Academic, 1985. In 1995, he joined the Signal and Information [23] J. N. Laneman, D. N. C. Tse, and G. W. Wornell, “Cooperative diversity Processing Laboratory at ETHZ as a Teaching and in wireless networks: Efficient protocols and outage behavior,” IEEE Research Assistant. After a postdoctoral year at the Trans. Inf. Theory, vol. 50, no. 12, pp. 3062–3080, Dec. 2004 [Online]. Electronics Research Laboratory at the University of California in Berkeley, he Available: http://www.nd.edu/~jnl/pubs/it2002.pdf joined the faculty of the Electrical Engineering Department at the University of [24] S. C. Cripps, RF Power Amplifiers for Wireless Communications.Nor- Notre Dame, Notre Dame, IN as an Assistant Professor in January 2001. wood, MA: Artech House, 1999. ISBN 0-89006-989-1. Dr. Haenggi was awarded the ETH medal for both his M.Sc. and Ph.D. theses [25] M. Haenggi, “Twelve reasons not to route over many short hops,” in IEEE and he received an NSF CAREER Award in 2005. He is a member of the Vehicular Technology Conf. (VTC Fall), Los Angeles, CA, Sep. 2004. Editorial Board of the Elsevier Journal on Ad Hoc Networks. His scientific pp. 3130-3134 [Online]. Available: http://www.nd.edu/~mhaenggi/pubs/ interests include networking and wireless communications, with emphasis on vtc04.pdf ad hoc and sensor networks.