MAXIMUM TRANSMISSION DISTANCE OF ... - CiteSeerX
1
M AXIMUM T RANSMISSION D ISTANCE OF G EOGRAPHIC T RANSMISSIONS ON R AYLEIGH C HANNELS Tathagata D. Goswami and John M. Shea Wireless Information Networking Group, 458 ENG Building #33 P.O. Box 116130 University of Florida [email protected],[email protected]
Abstract— We consider the use of multiuser diversity to maximize transmission distance in a wireless network. This differs from previous work on multiuser diversity, which focused mainly on increasing the data rate. We analyze the maximum achievable transmission distance in a geocasting scenario, in which any radio in a specified geographical area is an acceptable destination for a packet. Performance results are presented for Rayleigh fading and nonfading channels. To illustrate the benefits of multiuser diversity, we show that higher transmission distances can be achieved over fading channels than over nonfading channels if the density of radios is sufficiently high. We also illustrate one application of our results to protocol design.
I. I NTRODUCTION In wireless networks, more than one of the neighbors of the transmitter may be an acceptable receiver for a message. For instance, in multi-hop wireless ad hoc networks, radios act as routers to direct packets from the source to the destination. The goal is for the message to reach the destination in an efficient manner. Since there may be many alternative routes available between any source-destination pair, there are many possibilities for intermediate radios between the source and destination. If the channels are changing because of multipath fading, then the neighbors of a radio may change over time even when the positions of the radios have not significantly changed the topology of the network. Techniques to adapt to these channel changes by exploiting the alternative acceptable receivers are called multi-user diversity techniques. Originally multiuser diversity was proposed for application to cellular networks [1], [2]. Later, however it was investigated for ad hoc networks in [3], [4]. In an ad hoc network, the choice of next-hop receiver for a packet is usually determined by a routing algoritm according to various cost metrics. As early as 1984, Takagi and Kleinrock proposed routing protocols based on the locations of the nodes in the network [14]. The importance of geographic routing (geocasting) was well established in [5] where the authors implement an application layer scheme overlayed on the network layer to transport a message to all geographically distributed users. Geographic routing in mobile adhoc networks is based on a multihop packet forwarding mechanism according to which a radio forwards a source packet to one (greedy forwarding) or more (restricted directional flooding) neighbors of the source according to their location with respect to the source and destination [6]. Several geographic routing protocols have been proposed in the literature (cf. [7], [6] and references therein).
Our work is similar to that on Geographic Random Forwarding (GeRaF) [8] in which the authors investigate the use of geographic transmission to route packets in the presence of unknown sleep schedules. Under the GeRaF protocol, when a radio has a packet to send it broadcasts to all radios in the coverage area. Those radios which listen to the transmission will act as intermediate relay(s), depending on its/their distance from the destination. Finally the authors study the multihop performance of their protocol, in terms of the average number of hops it takes for a packet to reach the destination and the average number of available neighbors. In [8], the authors consider that two neighbors can communicate when they are within the coverage radius of each other. This model although simplistic, fails to capture the propagation in a practical wireless environment. In this work, we focus on a single transmission interval, and we anaylze the performance with random fading channels. Most previous work on geographic transmission focuses on the design of the routing protocol when a specific destination is known. In this work, we focus on a single-hop and analyze the statistics of the maximum distance that can be achieved from the transmitter to a group of receivers. This scenario may arise in several applications. In certain sensor networks [9], a sensing node near the middle of a monitored area must relay their information toward collection nodes around the edges of the area. Previous work on transmission range in wireless networks focussed more on adjusting transmission power in the absence of fading to improve throughput or control network connectivity(cf.[12], [13], [14], [15]). In fact, [14] showed that the expected progress per transmission is proportional to the transmission radius. In [16], the optimum transmission range was obtained from a graph theoretic point. In [17], the channel model used was based on some fixed parameters and did not account for random fading. In [18], a random graph theoretic framework was used to obtain the critical power required when the number of nodes is large. This paper is organized as follows. The system model considered is presented in Section II, and our analytical results are detailed in Section III. Performance results and an application to protocol design are presented in Section IV, and conclusions are given in Section V. II. S YSTEM M ODEL We study a broadcast communication environment in which a single source radio (transmitter) transmits information to
2
mean 1. Without loss of generality, the SNR at node i can be modeled as !"
%
Yi
$
!#
Fig. 1. Geographic region considered in the analysis: sector of angle θ of an annular ring with interior radius R1 and exterior radius R2 .
destination radios (receivers) that are distributed around the source radio according to a Poisson point process in a twodimensional plane at rate λ radios per unit area. We first consider a sector of angle θ of an annular ring with inner radius R1 , outer radius R2 , as shown in Fig. 1. The probability that there are l radios inside the sector is given by exp (−λA)(λA) , l = 0, 1 . . . , l!
Yi
A. Transmission Model
We consider transmission in the wireless environment using a fading channel model and a nonfading AWGN channel model. We consider the AWGN channel model to determine in which scenario(s) fading can actually improve the transmission distance when multiuser diversity is used. We assume that all radios in the system use identical, omni-directional antennas. A transmission over a single hop is considered successful if the signal-to-noise ratio (SNR) at the receiver is greater than or equal to a receiver sensitivity, ρ which we assume is identical for all receivers. We consider a slowly varying, flat Rayleigh fading channel. We also assume that the channel fading gains are constant over each period during which a message is transmitted from a source radio, and we assume that the fading gains are independent between nodes. Thus, the signal power received at an arbitrary mobile receiver depends only on the distance between the base station and that receiver and the fading gain at the receiver during that transmission. Let Xi denote the distance from the transmitter to node i and let αi denote the Rayleigh fading coefficient for node i. Then hi = |αi |2 is an exponential random variable with
= Xi−n .
(4)
III. A NALYSIS A. Maximum Transmission Distance We first consider transmission into a sector of an annular ring, as described in the previous section. A radio can successfully recover a message if Yi ≥ ρ, or Xi ≤ (hi /ρ)1/n . Again, without loss of generality, we let ρ = 1 and define Vi = X i I
(1)
where A = π(R22 − R12 )θ/(2π) is the area of the sector. Under the Poisson point process, the receivers within such an annular sector are distributed uniform in area, and hence the distribution function for the distance to an arbitrary receiver is given by x ≤ R1 0,2 2 x −R1 FXi (x) = R2 −R2 , R1 < x ≤ R2 (2) 2 1 1, x > R2 .
(3)
where n denotes the path-loss exponent. We assume n ≥ 2 which is reasonable outside a small neighborhood of the transmitting antenna. We compare the performance of the system with Rayleigh fading to the performance with nonfading AWGN channels. For this scenario, the SNR Yi at the receiver at the end of a transmission, depends only on the random distance Xi from the transmitter and can thus be modeled as
l
P [L = l] =
= hi Xi−n ,
1
0,
hin
(Xi ) ,
(5)
where I[A] (.) is the indicator function given by % 1, x ∈ A I[A] (x) = 0, otherwise. That is, Vi = Xi if the receiver can successfully recover the message and Vi = 0 otherwise. Then conditioned on N randomly located radios in the sector, the maximum distance M to a receiver that can successfully recover a message from the transmitter can be expressed as % max{V1 , V2 . . . VN }, N = 1, 2 . . . M= (6) 0, N = 0. Thus, from (6), the conditional distribution of M given that there are N radios in the network is given by % N [FVi (v)] , v ≥ 0 FM (v|N ) = (7) 0, v R1 . In the above derivation, γ(a, x) is the incomplete Gamma function * x γ(a, x) = ta−1 exp(−t)dt, 0
3
Γ(a) is the Gamma function * ∞ Γ(a) = ta−1 exp(−t)dt, 0
Note that this derivation can be trivially modified to obtain FVi (v) for the nonfading AWGN channel, i.e. (hi = 1), for which Vi = 0 if Xi > 1. Now suppose that the receivers around the transmitter are Poisson distributed in two-dimensional space at rate λ nodes per unit area and the transmission is intended only for radios in an annular sector of area A. Then for v ≥ 0, the distribution of the maximum transmission range, FM (v), can be easily obtained from (7) and (1) as , + N FM (v) = EN [FVi (v)] =
exp [(λAFVi (v) − λA)].
(8)
For v < 0, FM (v) = 0. We shall now extend our analysis of the maximum transmission range to infinite networks, assuming that the source is at the origin and that all the radios in the network are awake, i.e R1 → 0, R2 → ∞. Consider a sequence of random variables M1 , M2 , M3 . . . Mi . . . where MR denotes the maximum transmission range when the radius is R. Let FMi (t) denote the cumulative distribution function of the random variable Mi . Let FM ∗ (t) = lim FMi (t). i→∞
∗
Then if the limit exists, M is a random variable with distribution function FM ∗ , and the sequence of random variables {Mi } converges to M ∗ in distribution. 1) Fading channel model: Putting R1 = 0, R2 = R in the expression for FVi (v) derived in the Appendix and substituting the resulting expression in (8) yields ' & ' ( 2 n θ FMR (t) = exp λ γ 1 + , t + t2 exp (−tn ) 2 n ' ( -( 2 n 2 n −γ 1 + , R −R exp (−R ) n & ' (λθ 2 n = exp − Γ (9) , ,t n n where the last line follows after applying the properties of the incomplete gamma function. The normalized node density λ0 is defined as the expected number of radios within a circle of radius unity, which is given by, λ0 = λ θ2 . Thus, we can rewrite (9) as ' ( 2λ0 FM ∗ (t) = exp − Γ (2/n, tn ) (10) n 2) Nonfading channel model: In the AWGN channel, all the radios within a circle of radius unity are able to receive the message correctly. Hence the distribution function of M ∗ as R → ∞, can be easily derived to be, % . . // exp λ θ2 t2 − 1 , t < 1 FM ∗ (t) = 1, otherwise
(11)
Similar to the fading channel, we have replaced the node density λ with the normalized node density, λ0 = λ θ2 . Thus, we can rewrite (11) as, % . . // exp λ0 t2 − 1 , t < 1 FM ∗ (t) = (12) 1, otherwise The expected value of the maximum transmission range for both the fading and nonfading channels is given by, * ∞ ∗ E [M ∗ ] = [1 − FM (x)] dx. (13) 0
Since we have not found a closed form expression for this expected value, we have obtained this value numerically. Note that the distribution function for the nonfading channel given by (12) does not depend on the path loss exponent n whereas the distribution function for the Rayleigh fading channel given by (10) is dependent on n. Thus the expected value of the maximum transmission range for the nonfading channel does not depend on the exponential path loss. B. Outage probability and critical distance for the infinitely large network In many scenarios, it would be desirable that the message at least travel some minimum distance. I.e., we desire that the furthest receiver to successfully recover the message be at least some critical distance dc from the transmitter. We define the outage probability Pout , as the probability that the limiting transmission range M ∗ is less than dc . Thus, for dc > 0, we have (14) FM ∗ (dc ) = Pout
It is very important to note that the distribution of the limiting transmission range M ∗ for both the Rayleigh fading and the nonfading channel model has some mass at zero. I.e., FM ∗ (d = 0) '= 0. This is in part because there is some nonzero probability that there are no receivers in the network. Thus when there are no receivers in the network, there is some nonzero probability that the broadcast transmission remained at the transmitter. Hence, while choosing the value of Pout for dc > 0, we must be careful to obey the bound given by, ' ' (( λ0 2 Pout ≥ exp −2 Γ n n
for the fading channel model and Pout ≥ exp (−λ0 ) for the AWGN channel model. We next obtain the expression for dc for the fading and AWGN channel models. 1). Fading .channel model: For dc > 0 and Pout ≥ // exp −2 λn0 Γ n2 , by equating (10) and (14), we have, '
( 2λ0 n exp − Γ (2/n, t ) = Pout n
Taking log on both sides and after some simplification, we end up having to solve the following integral equation for dc , 2 n Γ( , dnc ) = log (1/Pout ) (15) n 2λ0 We mention here that (15) admits solutions for the path loss exponent n = 2, 4. We have tight bounds for n = 3, but omit
them because of length limitations. Case 1: n = 2 Putting n = 2 on both sides of (15), we have Γ(1, d2c ) . / ⇒ exp −d2c ∴ dc
1 = log (1/Pout ) λ0 1 = log (1/Pout ) λ0 0 & λ0 log = log (1/Pout )
Case 2: n = 4 Putting n = 4 on both sides of (15), we have
⇒
*
∞
d4c
⇒
1 Γ( , d4c ) 2 t
− 21
√
exp (−t)dt
π erfc
12 3 d4c ∴ dc
Expected value of Max. transmission range
4
1.5
1 fading (n=2) fading (n=3) fading (n=4) no fading
0.5
0 0
5
2 log (1/Pout ) λ0 Fig. 2. Expected Value normalized node density 2 = log (1/Pout ) λ0 2 log (1/Pout ) = 1.6 λ0 0 ' ( 1.4 2 √ log (1/Pout ) = erfc−1 1.2 λ0 π =
10 15 20 25 Normalized node density λ0
30
of the Maximum transmission range vs. the
where erfc(.) is the complementary error function given by * ∞ . / 2 exp −t2 dt erfc(x) = √ π x
2) Nonfading channel model: For 0 < dc ≤ 1 and exp (−λ0 ) ≤ Pout < 1, by equating (12) and (14), we have, // . . = Pout exp λ0 d2c − 1 4 1 ∴ dc = log Pout 1+ λ0
dc
1
0.8 0.6
fading no fading
0.4 0.2 0 0
5
10
15
20
25
30
Normalized node density λ0 Fig. 3. Critical distance of transmission vs. average number of radios for an outage probability of 0.05, for a path loss exponent n = 2
IV. R ESULTS In this section, we present results on the expected value of the normalized maximum transmission range, which is found by taking the expected value over the distributions given by (10) or (12). Here, normalized indicates that we have normalized the maximum transmission range for the AWGN channel to 1.0. So, 1 unit of normalized distance corresponds to the transmission radius of the AWGN channel. We also plot with respect to normalized node density, which is the expected value of the number of radios that lie within that normalized transmission radius. We have plotted the expected value of the normalized maximum transmission range given by (13) vs. the normalized node density in Fig. 2. For the AWGN channel, the transmission range achieved saturates to the maximum transmission range of 1 as the number of nodes in that transmission range increases. The expected value of the achieved transmission range for the fading channels is larger than for the AWGN channels because of multiuser diversity. With as few as six average neighbors, fading channels along with multiuser diversity results in an increase in the average transmission distance of 25%, 50%, and 87.5% over the distance on the AWGN channel for path-loss exponents of 4, 3, and 2, respectively.
To further evaluate the benefits of multiuser diversity, we consider transmission in fading and nonfading channels in terms of the critical transmission distance dc . The results in Figs. 3 and 4 show the normalized critical transmission distance for path loss exponents n = 2 and 4. We have plotted these values by considering an outage probability of 5%. When n = 2, for 10 radios in the network, 95% of the time, fading improves the critical transmission distance by a factor of 1.375. Similarly, the diversity gain when there are 10 radios and n = 4 is 0.92. Note that this indicates that if there are 10 radios in the network, then the nonfading channel outperforms the fading channel. Thus, there is a break-even point in terms of the normalized radio density, after which fading with multiuser diversity outperforms the nonfading channel. We have plotted the break-even points for a range of values of the pathloss exponent n in Fig. 5, for an outage probability of 0.05. Fig. 5 also indicates the region in which transmission in fading channel is better than the nonfading channel to overcome pathloss and vice versa. Fading offers the best performance in the region above the curve, and nonfading in the region below.
5
fading no fading
1.2 1
dc
0.8 0.6 0.4 0.2 0 0
5
10
15
20
25
30
Normalized node density λ0 Fig. 4. Critical distance of transmission vs. average number of radios for an outage probability of 0.05, for a path loss exponent n = 4 30
Normalized node density λ
0
25
20
15
region with interior and exterior radii given by dc1 and dc2 , respectively. We have compared our protocol with a simpler “dumb” protocol that transmits to all receivers within a circular transmission radius Rd from the transmitter. For a fair comparison between the two protocols, we limit this transmission radius Rd such that the average number of radios within dc1 and dc2 are the same as that within a circle of radius Rd . We compare these two protocols based on the expected value of the maximum transmission range and the probability that not even a single receiver within the transmission range receives a message from the transmitter correctly. We provide results for α = 0.95 and path loss exponent n = 4. Average Max. transmission range
1.4
1.2 1 0.8 0.6 0.4
α protocol dumb protocol
0.2
10
5
10
0
15
20
25
Normalized node density λ0
5
2
3
4
5
6
7
8
9
10
Path loss exponent n
Fig. 6. Expected Value of the Maximum transmission range vs. the normalized node density for a path loss exponent n = 4
Routing Protocol We demonstrate the utility of our analytical results in protocol design by considering the design of a sleeping protocol. We do not claim that this protocol is in any sense optimal, but it does demonstrate how our results can be useful. Consider the receivers distributed in an infinitely large plane around the transmitter according to a Poisson process with normalized node density λ0 , i.e λ0 is the average number of radios inside a circle of unit radius. Our protocol limits the number of radios that must turn on to those that are most likely to be at the maximum reception distance. To do this, we sacrifice some reliability in the sense that limiting the set of radios that turn on may occasionally decrease the maximum transmission distance that can be achieved or may cause the message to not be successfully received by any radio. We state that our protocol is α-reliable if we limit the receivers that turn on according to, 1−α (16) 2 1+α FM ∗ (dc2 |M ∗ > 0) = . 2 Then we limit the radios that turn on to those within an annular FM ∗ (dc1 |M ∗ > 0) =
Prob. not recd. message correctly
Fig. 5. Region marking zones of operation for fading and nonfading channel for an outage probability of 0.05
0
10
α protocol dumb protocol
−1
10
−2
10
−3
10
−4
10
0
5
10
15
20
25
Normalized Node density λ Fig. 7. Probability that not even a single receiver in the transmission range received the message correctly, for a path loss exponent n = 4
The expected value of the maximum transmission range is shown in Fig. 6. Clearly, the α-reliable performs better than the dumb protocol in transmitting the message further. This is because for the dumb protocol, too many radios turn on that are usually short of the maximum transmission distance. The values of P (Rx) for various λ0 is plotted in Fig. 7 on a semilog scale. We see that we are introducing some cost in our α-reliable protocol in that we increase the probability that none
6
of the receivers successfully recovers the message. However, as the receivers in these cases are close to the transmitter, this is generally not a significant loss.
Here, γ(a, x) is the incomplete Gamma function: * x γ(a, x) = ta−1 exp(−t)dt 0
and Γ(a) is the Gamma function: * ∞ Γ(a) = ta−1 exp(−t)dt
V. C ONCLUSION AND F UTURE W ORK In this paper, we derived the distribution function of the maximum transmission distance achievable for a geographic transmission over a channel exposed to exponential path loss and Rayleigh fading. We used this distribution function to obtain the critical transmission range, given some outage probability. We provide expressions for the critical transmission range for path loss exponents n = 2, 4 and provide tight bounds for n = 3. To obtain the scenario(s) where fading is beneficial we also provided results for transmission in a nonfading channel. Our results indicate that when there are a large number of radios in the network, then fading is beneficial to transport the message from the source to the destination. We have also obtained the region of operation, in terms of the normalized node density, when fading overcomes signal loss due to exponential path decay. Though not the focus of this paper, we have provided an example of a very simple routing protocol that utilizes the maximum transmission distance as a metric.
0
R EFERENCES
A PPENDIX
FV (v)
= P (Vi ≤ v) = P
5
Xi I
1
Xi ≤hin
≤v
8 69 8 8 hi 1 ≤ v = Ehi P Xi I 8 Xi ≤hin 8 (& ' 1 8 = Ehi P Xi ≤ min{v, hin }88hi 8 (& ' 1 8 n8 + Ehi P Xi > hi 8hi 7 5
Substituting into (2) yields, 1 2 3 min hin , v 2 − R12 I FV (v) = Ehi R22 − R12 7 9 +Ehi I
−Ehi
75
1
min v,hin
>R2
6
1
R1 <min v,hin
≤R2
+1
2
hin − R12 I R22 − R12
1
R1 R2
69
[1] R. Knopp and P. A. Humblet, “Information capacity and power control in single-cell multiuser communications,” in Proc. 1995 IEEE Int. Conf. Commun., vol. 1, (Seattle), pp. 331–335, June 1995. [2] P. Black, M. Grob, R. Padovani, N. Sindhushyana, and S. Viterbi, “CDMA/HDR : a bandwidth efficient high speed wireless data service for nomadic users,” IEEE Communications Magazine, vol. 38, pp. 70– 77, July 2000. [3] P. Larsson, “Selection diversity forwarding in a multihop packet radio network with fading channel and capture,” in ACM SIGMOBILE Mobile Computing and Communication Review, pp. 47–54, Oct. 2001. [4] W. H. Wong, J. M. Shea, and T. F. Wong, “Cooperative diviersity slotted ALOHA,” in Proc. 2nd Int. Conf on Quality of Service in Heterogeneous Wired/Wireless Networks (QShine), (Orlando, Florida), pp. 8.4.1–8, Aug. 2005. [5] J. C. Navas and T. Imielinski, “Geocast - geographic addressing and routing,” in Proc. of the 3rd Annual ACM/IEEE Intl. Conf. on Mobile Computing and Networking,(MOBICOM ’97), 1997. [6] M. Mauve, J. Widmer, and H. Hartenstein, “A survey on position-based routing in mobile ad hoc networks,” IEEE Network Magazine, vol. 15, pp. 30–39, Nov. 2001. [7] I. Stojmenovic, “Position-based routing in ad hoc networks,” IEEE Commun. Mag., vol. 40, pp. 128–134, Jul 2002. [8] M. Zorzi and R. R. Rao, “Geographic random forwarding (GeRaF) for ad hoc and sensor networks : Multihop performance,” IEEE Trans. Mobile Computing, vol. 2, pp. 337–348, Oct.-Dec. 2003. [9] I. F. Akyildiz, W. Su, Y. Sankarsubramaniam, and E.Cayirci, “A survey on sensor networks,” IEEE Commun. Mag., vol. 40, pp. 102–114, Aug. 2002. [10] D. C. Verma, Content Distribution Networks, An Engineering Approach. New York: Wiley, 2002. [11] R. R. Choudhury and N. H. Vaidya, “Mac-layer anycasting in ad hoc networks,” SIGCOMM Comput. Commun. Rev., vol. 34, no. 1, pp. 75– 80, 2004. [12] L. Kleinrock and J. A. Silvester, “Optimum transmission radii for packet radio networks or why six is a magic number,” in Proc. Conf. Rec. Nat. Telecommun. Conf, 1978. [13] L. Kleinrock and J. A. Silvester, “Spatial reuse in multihop packet radio networks,” in Proc. IEEE, vol. 75, pp. 116–134, Jan. 1987. [14] H. Takagi and L. Kleinrock, “Optimum transmission ranges for randomly distributed packet radio terminals,” IEEE Trans. Commun., vol. 32, pp. 246–257, Mar. 1984. [15] T. C. Hou and V. O. K. Li, “Transmission range control in multihop packet radio networks,” IEEE Trans. Commun, vol. 34, pp. 38–44, Jan. 1986. [16] M. Sanchez, P. Manzoni, and Z. Haas, “Determination of critical transmission range in adhoc networks,” in Proc. of Multiaccess Mobility and Teletraffic for Wireless Communications Workshop (MMT’99), 1999. [17] J. Deng, Y. S. Han, P. Chen, and P. K. Varshney, “Optimum transmission range for wireless ad hoc networks,” in Proc. IEEE Wireless Communications and Networking Conf.(WCNC ’04), 2004. [18] P. Gupta and P. R. Kumar, “Critical power for asymptotic connectivity,” in Proc. IEEE Conf. on Decision and Control, 1998.
Now we shall consider the various intervals in which v lies. For v ∈ (0, R1 ], R1 > 0, R2 > R1 . & ' ( ' ( 2 n 2 n 1 2 n n γ 1 + , R , R − γ 1 + − R {exp (−R FV (v) = 1 − exp (−R2n ) − 2 ) − exp (−R )} 1 1 2 R2 − R12 n 2 n 1 For v ∈ (R1 , R2 ], FV (v)
= 1−
exp (−R2n )
& ' ( ' ( 1 2 n 2 n v 2 − R12 n 2 n n exp (−v ) + 2 γ 1 + ,v − γ 1 + , R2 + R1 {exp (−v ) − exp (−R2 )} + 2 R2 − R12 R2 − R12 n n
M AXIMUM T RANSMISSION D ISTANCE OF G EOGRAPHIC T RANSMISSIONS ON R AYLEIGH C HANNELS Tathagata D. Goswami and John M. Shea Wireless Information Networking Group, 458 ENG Building #33 P.O. Box 116130 University of Florida [email protected],[email protected]
Abstract— We consider the use of multiuser diversity to maximize transmission distance in a wireless network. This differs from previous work on multiuser diversity, which focused mainly on increasing the data rate. We analyze the maximum achievable transmission distance in a geocasting scenario, in which any radio in a specified geographical area is an acceptable destination for a packet. Performance results are presented for Rayleigh fading and nonfading channels. To illustrate the benefits of multiuser diversity, we show that higher transmission distances can be achieved over fading channels than over nonfading channels if the density of radios is sufficiently high. We also illustrate one application of our results to protocol design.
I. I NTRODUCTION In wireless networks, more than one of the neighbors of the transmitter may be an acceptable receiver for a message. For instance, in multi-hop wireless ad hoc networks, radios act as routers to direct packets from the source to the destination. The goal is for the message to reach the destination in an efficient manner. Since there may be many alternative routes available between any source-destination pair, there are many possibilities for intermediate radios between the source and destination. If the channels are changing because of multipath fading, then the neighbors of a radio may change over time even when the positions of the radios have not significantly changed the topology of the network. Techniques to adapt to these channel changes by exploiting the alternative acceptable receivers are called multi-user diversity techniques. Originally multiuser diversity was proposed for application to cellular networks [1], [2]. Later, however it was investigated for ad hoc networks in [3], [4]. In an ad hoc network, the choice of next-hop receiver for a packet is usually determined by a routing algoritm according to various cost metrics. As early as 1984, Takagi and Kleinrock proposed routing protocols based on the locations of the nodes in the network [14]. The importance of geographic routing (geocasting) was well established in [5] where the authors implement an application layer scheme overlayed on the network layer to transport a message to all geographically distributed users. Geographic routing in mobile adhoc networks is based on a multihop packet forwarding mechanism according to which a radio forwards a source packet to one (greedy forwarding) or more (restricted directional flooding) neighbors of the source according to their location with respect to the source and destination [6]. Several geographic routing protocols have been proposed in the literature (cf. [7], [6] and references therein).
Our work is similar to that on Geographic Random Forwarding (GeRaF) [8] in which the authors investigate the use of geographic transmission to route packets in the presence of unknown sleep schedules. Under the GeRaF protocol, when a radio has a packet to send it broadcasts to all radios in the coverage area. Those radios which listen to the transmission will act as intermediate relay(s), depending on its/their distance from the destination. Finally the authors study the multihop performance of their protocol, in terms of the average number of hops it takes for a packet to reach the destination and the average number of available neighbors. In [8], the authors consider that two neighbors can communicate when they are within the coverage radius of each other. This model although simplistic, fails to capture the propagation in a practical wireless environment. In this work, we focus on a single transmission interval, and we anaylze the performance with random fading channels. Most previous work on geographic transmission focuses on the design of the routing protocol when a specific destination is known. In this work, we focus on a single-hop and analyze the statistics of the maximum distance that can be achieved from the transmitter to a group of receivers. This scenario may arise in several applications. In certain sensor networks [9], a sensing node near the middle of a monitored area must relay their information toward collection nodes around the edges of the area. Previous work on transmission range in wireless networks focussed more on adjusting transmission power in the absence of fading to improve throughput or control network connectivity(cf.[12], [13], [14], [15]). In fact, [14] showed that the expected progress per transmission is proportional to the transmission radius. In [16], the optimum transmission range was obtained from a graph theoretic point. In [17], the channel model used was based on some fixed parameters and did not account for random fading. In [18], a random graph theoretic framework was used to obtain the critical power required when the number of nodes is large. This paper is organized as follows. The system model considered is presented in Section II, and our analytical results are detailed in Section III. Performance results and an application to protocol design are presented in Section IV, and conclusions are given in Section V. II. S YSTEM M ODEL We study a broadcast communication environment in which a single source radio (transmitter) transmits information to
2
mean 1. Without loss of generality, the SNR at node i can be modeled as !"
%
Yi
$
!#
Fig. 1. Geographic region considered in the analysis: sector of angle θ of an annular ring with interior radius R1 and exterior radius R2 .
destination radios (receivers) that are distributed around the source radio according to a Poisson point process in a twodimensional plane at rate λ radios per unit area. We first consider a sector of angle θ of an annular ring with inner radius R1 , outer radius R2 , as shown in Fig. 1. The probability that there are l radios inside the sector is given by exp (−λA)(λA) , l = 0, 1 . . . , l!
Yi
A. Transmission Model
We consider transmission in the wireless environment using a fading channel model and a nonfading AWGN channel model. We consider the AWGN channel model to determine in which scenario(s) fading can actually improve the transmission distance when multiuser diversity is used. We assume that all radios in the system use identical, omni-directional antennas. A transmission over a single hop is considered successful if the signal-to-noise ratio (SNR) at the receiver is greater than or equal to a receiver sensitivity, ρ which we assume is identical for all receivers. We consider a slowly varying, flat Rayleigh fading channel. We also assume that the channel fading gains are constant over each period during which a message is transmitted from a source radio, and we assume that the fading gains are independent between nodes. Thus, the signal power received at an arbitrary mobile receiver depends only on the distance between the base station and that receiver and the fading gain at the receiver during that transmission. Let Xi denote the distance from the transmitter to node i and let αi denote the Rayleigh fading coefficient for node i. Then hi = |αi |2 is an exponential random variable with
= Xi−n .
(4)
III. A NALYSIS A. Maximum Transmission Distance We first consider transmission into a sector of an annular ring, as described in the previous section. A radio can successfully recover a message if Yi ≥ ρ, or Xi ≤ (hi /ρ)1/n . Again, without loss of generality, we let ρ = 1 and define Vi = X i I
(1)
where A = π(R22 − R12 )θ/(2π) is the area of the sector. Under the Poisson point process, the receivers within such an annular sector are distributed uniform in area, and hence the distribution function for the distance to an arbitrary receiver is given by x ≤ R1 0,2 2 x −R1 FXi (x) = R2 −R2 , R1 < x ≤ R2 (2) 2 1 1, x > R2 .
(3)
where n denotes the path-loss exponent. We assume n ≥ 2 which is reasonable outside a small neighborhood of the transmitting antenna. We compare the performance of the system with Rayleigh fading to the performance with nonfading AWGN channels. For this scenario, the SNR Yi at the receiver at the end of a transmission, depends only on the random distance Xi from the transmitter and can thus be modeled as
l
P [L = l] =
= hi Xi−n ,
1
0,
hin
(Xi ) ,
(5)
where I[A] (.) is the indicator function given by % 1, x ∈ A I[A] (x) = 0, otherwise. That is, Vi = Xi if the receiver can successfully recover the message and Vi = 0 otherwise. Then conditioned on N randomly located radios in the sector, the maximum distance M to a receiver that can successfully recover a message from the transmitter can be expressed as % max{V1 , V2 . . . VN }, N = 1, 2 . . . M= (6) 0, N = 0. Thus, from (6), the conditional distribution of M given that there are N radios in the network is given by % N [FVi (v)] , v ≥ 0 FM (v|N ) = (7) 0, v R1 . In the above derivation, γ(a, x) is the incomplete Gamma function * x γ(a, x) = ta−1 exp(−t)dt, 0
3
Γ(a) is the Gamma function * ∞ Γ(a) = ta−1 exp(−t)dt, 0
Note that this derivation can be trivially modified to obtain FVi (v) for the nonfading AWGN channel, i.e. (hi = 1), for which Vi = 0 if Xi > 1. Now suppose that the receivers around the transmitter are Poisson distributed in two-dimensional space at rate λ nodes per unit area and the transmission is intended only for radios in an annular sector of area A. Then for v ≥ 0, the distribution of the maximum transmission range, FM (v), can be easily obtained from (7) and (1) as , + N FM (v) = EN [FVi (v)] =
exp [(λAFVi (v) − λA)].
(8)
For v < 0, FM (v) = 0. We shall now extend our analysis of the maximum transmission range to infinite networks, assuming that the source is at the origin and that all the radios in the network are awake, i.e R1 → 0, R2 → ∞. Consider a sequence of random variables M1 , M2 , M3 . . . Mi . . . where MR denotes the maximum transmission range when the radius is R. Let FMi (t) denote the cumulative distribution function of the random variable Mi . Let FM ∗ (t) = lim FMi (t). i→∞
∗
Then if the limit exists, M is a random variable with distribution function FM ∗ , and the sequence of random variables {Mi } converges to M ∗ in distribution. 1) Fading channel model: Putting R1 = 0, R2 = R in the expression for FVi (v) derived in the Appendix and substituting the resulting expression in (8) yields ' & ' ( 2 n θ FMR (t) = exp λ γ 1 + , t + t2 exp (−tn ) 2 n ' ( -( 2 n 2 n −γ 1 + , R −R exp (−R ) n & ' (λθ 2 n = exp − Γ (9) , ,t n n where the last line follows after applying the properties of the incomplete gamma function. The normalized node density λ0 is defined as the expected number of radios within a circle of radius unity, which is given by, λ0 = λ θ2 . Thus, we can rewrite (9) as ' ( 2λ0 FM ∗ (t) = exp − Γ (2/n, tn ) (10) n 2) Nonfading channel model: In the AWGN channel, all the radios within a circle of radius unity are able to receive the message correctly. Hence the distribution function of M ∗ as R → ∞, can be easily derived to be, % . . // exp λ θ2 t2 − 1 , t < 1 FM ∗ (t) = 1, otherwise
(11)
Similar to the fading channel, we have replaced the node density λ with the normalized node density, λ0 = λ θ2 . Thus, we can rewrite (11) as, % . . // exp λ0 t2 − 1 , t < 1 FM ∗ (t) = (12) 1, otherwise The expected value of the maximum transmission range for both the fading and nonfading channels is given by, * ∞ ∗ E [M ∗ ] = [1 − FM (x)] dx. (13) 0
Since we have not found a closed form expression for this expected value, we have obtained this value numerically. Note that the distribution function for the nonfading channel given by (12) does not depend on the path loss exponent n whereas the distribution function for the Rayleigh fading channel given by (10) is dependent on n. Thus the expected value of the maximum transmission range for the nonfading channel does not depend on the exponential path loss. B. Outage probability and critical distance for the infinitely large network In many scenarios, it would be desirable that the message at least travel some minimum distance. I.e., we desire that the furthest receiver to successfully recover the message be at least some critical distance dc from the transmitter. We define the outage probability Pout , as the probability that the limiting transmission range M ∗ is less than dc . Thus, for dc > 0, we have (14) FM ∗ (dc ) = Pout
It is very important to note that the distribution of the limiting transmission range M ∗ for both the Rayleigh fading and the nonfading channel model has some mass at zero. I.e., FM ∗ (d = 0) '= 0. This is in part because there is some nonzero probability that there are no receivers in the network. Thus when there are no receivers in the network, there is some nonzero probability that the broadcast transmission remained at the transmitter. Hence, while choosing the value of Pout for dc > 0, we must be careful to obey the bound given by, ' ' (( λ0 2 Pout ≥ exp −2 Γ n n
for the fading channel model and Pout ≥ exp (−λ0 ) for the AWGN channel model. We next obtain the expression for dc for the fading and AWGN channel models. 1). Fading .channel model: For dc > 0 and Pout ≥ // exp −2 λn0 Γ n2 , by equating (10) and (14), we have, '
( 2λ0 n exp − Γ (2/n, t ) = Pout n
Taking log on both sides and after some simplification, we end up having to solve the following integral equation for dc , 2 n Γ( , dnc ) = log (1/Pout ) (15) n 2λ0 We mention here that (15) admits solutions for the path loss exponent n = 2, 4. We have tight bounds for n = 3, but omit
them because of length limitations. Case 1: n = 2 Putting n = 2 on both sides of (15), we have Γ(1, d2c ) . / ⇒ exp −d2c ∴ dc
1 = log (1/Pout ) λ0 1 = log (1/Pout ) λ0 0 & λ0 log = log (1/Pout )
Case 2: n = 4 Putting n = 4 on both sides of (15), we have
⇒
*
∞
d4c
⇒
1 Γ( , d4c ) 2 t
− 21
√
exp (−t)dt
π erfc
12 3 d4c ∴ dc
Expected value of Max. transmission range
4
1.5
1 fading (n=2) fading (n=3) fading (n=4) no fading
0.5
0 0
5
2 log (1/Pout ) λ0 Fig. 2. Expected Value normalized node density 2 = log (1/Pout ) λ0 2 log (1/Pout ) = 1.6 λ0 0 ' ( 1.4 2 √ log (1/Pout ) = erfc−1 1.2 λ0 π =
10 15 20 25 Normalized node density λ0
30
of the Maximum transmission range vs. the
where erfc(.) is the complementary error function given by * ∞ . / 2 exp −t2 dt erfc(x) = √ π x
2) Nonfading channel model: For 0 < dc ≤ 1 and exp (−λ0 ) ≤ Pout < 1, by equating (12) and (14), we have, // . . = Pout exp λ0 d2c − 1 4 1 ∴ dc = log Pout 1+ λ0
dc
1
0.8 0.6
fading no fading
0.4 0.2 0 0
5
10
15
20
25
30
Normalized node density λ0 Fig. 3. Critical distance of transmission vs. average number of radios for an outage probability of 0.05, for a path loss exponent n = 2
IV. R ESULTS In this section, we present results on the expected value of the normalized maximum transmission range, which is found by taking the expected value over the distributions given by (10) or (12). Here, normalized indicates that we have normalized the maximum transmission range for the AWGN channel to 1.0. So, 1 unit of normalized distance corresponds to the transmission radius of the AWGN channel. We also plot with respect to normalized node density, which is the expected value of the number of radios that lie within that normalized transmission radius. We have plotted the expected value of the normalized maximum transmission range given by (13) vs. the normalized node density in Fig. 2. For the AWGN channel, the transmission range achieved saturates to the maximum transmission range of 1 as the number of nodes in that transmission range increases. The expected value of the achieved transmission range for the fading channels is larger than for the AWGN channels because of multiuser diversity. With as few as six average neighbors, fading channels along with multiuser diversity results in an increase in the average transmission distance of 25%, 50%, and 87.5% over the distance on the AWGN channel for path-loss exponents of 4, 3, and 2, respectively.
To further evaluate the benefits of multiuser diversity, we consider transmission in fading and nonfading channels in terms of the critical transmission distance dc . The results in Figs. 3 and 4 show the normalized critical transmission distance for path loss exponents n = 2 and 4. We have plotted these values by considering an outage probability of 5%. When n = 2, for 10 radios in the network, 95% of the time, fading improves the critical transmission distance by a factor of 1.375. Similarly, the diversity gain when there are 10 radios and n = 4 is 0.92. Note that this indicates that if there are 10 radios in the network, then the nonfading channel outperforms the fading channel. Thus, there is a break-even point in terms of the normalized radio density, after which fading with multiuser diversity outperforms the nonfading channel. We have plotted the break-even points for a range of values of the pathloss exponent n in Fig. 5, for an outage probability of 0.05. Fig. 5 also indicates the region in which transmission in fading channel is better than the nonfading channel to overcome pathloss and vice versa. Fading offers the best performance in the region above the curve, and nonfading in the region below.
5
fading no fading
1.2 1
dc
0.8 0.6 0.4 0.2 0 0
5
10
15
20
25
30
Normalized node density λ0 Fig. 4. Critical distance of transmission vs. average number of radios for an outage probability of 0.05, for a path loss exponent n = 4 30
Normalized node density λ
0
25
20
15
region with interior and exterior radii given by dc1 and dc2 , respectively. We have compared our protocol with a simpler “dumb” protocol that transmits to all receivers within a circular transmission radius Rd from the transmitter. For a fair comparison between the two protocols, we limit this transmission radius Rd such that the average number of radios within dc1 and dc2 are the same as that within a circle of radius Rd . We compare these two protocols based on the expected value of the maximum transmission range and the probability that not even a single receiver within the transmission range receives a message from the transmitter correctly. We provide results for α = 0.95 and path loss exponent n = 4. Average Max. transmission range
1.4
1.2 1 0.8 0.6 0.4
α protocol dumb protocol
0.2
10
5
10
0
15
20
25
Normalized node density λ0
5
2
3
4
5
6
7
8
9
10
Path loss exponent n
Fig. 6. Expected Value of the Maximum transmission range vs. the normalized node density for a path loss exponent n = 4
Routing Protocol We demonstrate the utility of our analytical results in protocol design by considering the design of a sleeping protocol. We do not claim that this protocol is in any sense optimal, but it does demonstrate how our results can be useful. Consider the receivers distributed in an infinitely large plane around the transmitter according to a Poisson process with normalized node density λ0 , i.e λ0 is the average number of radios inside a circle of unit radius. Our protocol limits the number of radios that must turn on to those that are most likely to be at the maximum reception distance. To do this, we sacrifice some reliability in the sense that limiting the set of radios that turn on may occasionally decrease the maximum transmission distance that can be achieved or may cause the message to not be successfully received by any radio. We state that our protocol is α-reliable if we limit the receivers that turn on according to, 1−α (16) 2 1+α FM ∗ (dc2 |M ∗ > 0) = . 2 Then we limit the radios that turn on to those within an annular FM ∗ (dc1 |M ∗ > 0) =
Prob. not recd. message correctly
Fig. 5. Region marking zones of operation for fading and nonfading channel for an outage probability of 0.05
0
10
α protocol dumb protocol
−1
10
−2
10
−3
10
−4
10
0
5
10
15
20
25
Normalized Node density λ Fig. 7. Probability that not even a single receiver in the transmission range received the message correctly, for a path loss exponent n = 4
The expected value of the maximum transmission range is shown in Fig. 6. Clearly, the α-reliable performs better than the dumb protocol in transmitting the message further. This is because for the dumb protocol, too many radios turn on that are usually short of the maximum transmission distance. The values of P (Rx) for various λ0 is plotted in Fig. 7 on a semilog scale. We see that we are introducing some cost in our α-reliable protocol in that we increase the probability that none
6
of the receivers successfully recovers the message. However, as the receivers in these cases are close to the transmitter, this is generally not a significant loss.
Here, γ(a, x) is the incomplete Gamma function: * x γ(a, x) = ta−1 exp(−t)dt 0
and Γ(a) is the Gamma function: * ∞ Γ(a) = ta−1 exp(−t)dt
V. C ONCLUSION AND F UTURE W ORK In this paper, we derived the distribution function of the maximum transmission distance achievable for a geographic transmission over a channel exposed to exponential path loss and Rayleigh fading. We used this distribution function to obtain the critical transmission range, given some outage probability. We provide expressions for the critical transmission range for path loss exponents n = 2, 4 and provide tight bounds for n = 3. To obtain the scenario(s) where fading is beneficial we also provided results for transmission in a nonfading channel. Our results indicate that when there are a large number of radios in the network, then fading is beneficial to transport the message from the source to the destination. We have also obtained the region of operation, in terms of the normalized node density, when fading overcomes signal loss due to exponential path decay. Though not the focus of this paper, we have provided an example of a very simple routing protocol that utilizes the maximum transmission distance as a metric.
0
R EFERENCES
A PPENDIX
FV (v)
= P (Vi ≤ v) = P
5
Xi I
1
Xi ≤hin
≤v
8 69 8 8 hi 1 ≤ v = Ehi P Xi I 8 Xi ≤hin 8 (& ' 1 8 = Ehi P Xi ≤ min{v, hin }88hi 8 (& ' 1 8 n8 + Ehi P Xi > hi 8hi 7 5
Substituting into (2) yields, 1 2 3 min hin , v 2 − R12 I FV (v) = Ehi R22 − R12 7 9 +Ehi I
−Ehi
75
1
min v,hin
>R2
6
1
R1 <min v,hin
≤R2
+1
2
hin − R12 I R22 − R12
1
R1 R2
69
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Now we shall consider the various intervals in which v lies. For v ∈ (0, R1 ], R1 > 0, R2 > R1 . & ' ( ' ( 2 n 2 n 1 2 n n γ 1 + , R , R − γ 1 + − R {exp (−R FV (v) = 1 − exp (−R2n ) − 2 ) − exp (−R )} 1 1 2 R2 − R12 n 2 n 1 For v ∈ (R1 , R2 ], FV (v)
= 1−
exp (−R2n )
& ' ( ' ( 1 2 n 2 n v 2 − R12 n 2 n n exp (−v ) + 2 γ 1 + ,v − γ 1 + , R2 + R1 {exp (−v ) − exp (−R2 )} + 2 R2 − R12 R2 − R12 n n
