The Behavior of Unbounded Path-loss Models and the ... - CiteSeerX

The Behavior of Unbounded Path-loss Models and the Effect of Singularity on Computed Network Characteristics Hazer Inaltekin∗ , Mung Chiang ∗ , H. Vincent Poor∗ , Stephen B. Wicker† , ∗

Department of Electrical Engineering, Princeton University, Princeton, NJ 08544 Email: {hinaltek, chiangm, poor}@princeton.edu

†

School of Electrical and Computer Engineering, Cornell University, Ithaca, NY, 14850 Email: [email protected]

Abstract This paper addresses an important question for the wireless networking community: how reliable is it to use the unbounded path-loss model G(x) = x−α , where α is the path-loss exponent, to model the decay of transmitted signal power in wireless networking research problems? G(x) is not valid for small values of x due to the singularity at 0. However, in many analyses, it is used along with a random uniform node distribution without concern for the fact that in a group of uniformly distributed nodes, some may be arbitrarily close to one another. The unbounded path-loss model is compared to a more realistic bounded path-loss model, and it is shown that the effect of the singularity on the total network interference level is significant and cannot be disregarded when nodes are uniformly distributed. A phase transition phenomenon occurring in the interference behavior is analyzed in detail. More concrete metrics are also examined by using the computed interference distributions. In particular, the effects of the singularity at 0 on bit error rate, packet success probability and wireless channel capacity are analyzed.

This research was supported in part by the National Science Foundation under Grants ANI-03-58807 and CNS-06-25637.

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I. I NTRODUCTION For the purposes of mathematical analysis, it is necessary to have simple but sensible abstractions of wireless networks and the wireless medium. This approach has been pursued for decades, and a common example is the use of the function G(x) = x−α to model the power gain of the wireless channel between a transmitter and a receiver, where x is the distance between the two terminals and α is the path-loss exponent. The key artifact of this model is the singularity at 0. Even though the function is a reasonable approximation for wireless channel power gain for large transmitter-receiver separation, it becomes increasingly invalid as transmitters and receivers move closer to one another. In particular, the singularity can lead to the physically impossible scenario in which the received signal power exceeds the transmitted signal power. This model has been used extensively in conjunction with uniform distribution of nodes over the network domain without concern for this issue in many influential works such as [1] and [2]. The hope is that the singularity at 0 does not affect the realism of the model significantly. This is hard to justify for dense wireless networks because separation between many transmitter-receiver pairs is so small that the unbounded path-loss model is not valid anymore. On the other hand, if a network is sparsely populated, the temptation is to say that effect of the singularity at 0 is negligible since nodes are sufficiently far away from each other that the unbounded path-loss model holds - i.e., this is really the case where the far-field assumption can be applied. In this paper, we examine the effects of the singularity at 0 in the unbounded path-loss model on network performance by comparing it with a bounded path-loss model. We will first concentrate on the network interference signal strength level when nodes are uniformly distributed over the network domain, and we will show that a phase transition in the behavior of the network interference occurs at a critical value α∗ of the path-loss exponent. When α ≤ α∗ , the network multiple-access interference behaves in a similar fashion under both bounded and unbounded path-loss models. When α > α∗ , the network multiple-access interference behaves in differently under different path-loss models. In particular, the tails of the interference probability density function (PDF) are radically affected by the singularity at 0 when α > α∗ . The interference PDF becomes heavy-tailed under the unbounded path-loss model. On the other hand, it decays to zero exponentially under the bounded path-loss model. These results hold regardless of how

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densely or sparsely populated a network is. We will finally make use of results obtained for the interference distribution to compare bounded and unbounded path-loss models for three important performance metrics: bit error rate (BER), packet success probability and wireless channel capacity. We analyze these metrics as functions of the separation between a transmitter-receiver pair. Our results indicate that using the unbounded path-loss model results in significant deviations from more realistic performance figures obtained by using a bounded path-loss model for all of these three metrics. These results indicate that unbounded path-loss models should be used with caution in wireless communications and wireless networking research problems. A. Related Work In [3], the authors analyzed the behavior of one-dimensional shot noise when the impulse response of the filter is h(t) = t−β , and the noise process is a one-dimensional Poisson point process. They obtained expressions for the moments and the moment generating function of the shot-noise. In [4], authors considered the problem of determining optimal transmission range in a wireless ad hoc network where nodes are distributed according to a two-dimensional Poisson process. To this end, they obtained the interference signal power PDF when the wireless channel power gain is modeled according to G(x) = x−α . In [5], the outage probability in a codedivision multiple-access (CDMA) cellular network was considered. These authors again used the unbounded path-loss model for the path-loss function. They also considered log-normal shadowing. In [6], they analyzed the sizes and shapes of cells in a wireless cellular CDMA network by using marked Poisson processes and the shot-noise process associated with the marked point process under various path-loss models. They obtain the outage probability for wireless cellular CDMA networks under their model as the complement of the covered volume fraction. In all of the above work, locations of nodes are modeled by using Poisson point proccesses. In contrast to them, we consider networks for which we have knowledge of the number of nodes but do not know their exact locations as in the mainstream work such as [1] and [2] in the context of wireless networking. Thus, we model nodes’ locations by using a uniform distribution rather than a Poisson process, which changes the proof techniques from those of these earlier works, and some of the key theorems such as Campbell’s theorem (see [7]) for Poisson processes cannot be

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directly applied. In addition, in all these above work except [6], only the unbounded propagation model is considered. This is mainly due to the analytical simplicity of the calculations under this model. In contrast, we obtain interference signal strength PDFs and other network parameters under both bounded and unbounded propagation models, and analyze the effect of the singularity at 0 by comparing them. Thus, to the best of our knowledge, our work is the first analytical study obtaining and comparing interference characteristics under unbounded and bounded pathloss models when nodes are uniformly distributed over the network domain. II. P ROBLEM F ORMULATION A. Physical Layer and Network Model We consider a disk shaped network domain B(0, n) centered at the origin 0 and having radius n ∈ N. Transmitters are uniformly distributed over the network domain. For the ease of mathematical exposition, network nodes are assumed to use the binary phase shift keying (BPSK) modulation scheme for information transmission without any power control algorithm at the physical layer. Each transmitter transmits with a fixed power P =

Eb , Tb

where

Eb is the transmitted signal energy per bit, and Tb is the bit duration. The pair of signals q q 2Eb b cos(2πf t) and s (t) = cos(2πfc t), for 0 ≤ t ≤ Tb , are used to transmit s0 (t) = − 2E c 1 Tb Tb binary symbols 0 and 1, respectively. At the receiver side, transmitted signals are coherently q decoded by using the basis function φ(t) = T2b cos(2πfc t), 0 ≤ t ≤ Tb . The physical layer model for decoding transmitted bits is depicted in Fig. 1. Transmitted signals are impaired by the interference coming from other transmitters and the white noise process W (t) as shown in Fig. 1. Therefore, the received signal R at the demodulator output in Fig. 1 contains the desired signal coming from the transmitted signal to be decoded, the noise signal W0 coming from W (t), and the multiple-access interference signal In coming from the transmitters transmitting concurrently. We place a test receiver node at the origin, and focus on the strength of the multiple-access interference signal In at the output of the demodulator at the test node. We will study the convergence properties of the interference signal strength In at the origin as the network radius n grows to infinity. Let λ > 0 be the density of transmitters, which is in units of number of nodes per unit area. We pick a transmitter density λ > 0, and keep it constant while growing the network size to infinity. Small values of λ correspond to sparse wireless networks, and the large values of λ correspond to dense wireless networks. The

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total number of interfering transmitters for any given value of the network radius n is given by dλπn2 e, where due denotes the smallest integer larger than u. B. Path-loss Models We classify path-loss models as being either bounded or unbounded. For the unbounded pathloss model (UPM), we focus on the signal strength decay function G1 (x) =

1 α x2

. Therefore,

the transmitted signal power decays according to the commonly used signal power attenuation function G(x) =

1 . xα

This model is not correct for small values of x due to a singularity at 0.

However, it fairly well approximates the signal attenuation for large values of x. In contrast to the unbounded path-loss model, a sound propagation model for characterizing signal strength attenuation must always be a monotonically decreasing function of the distance and be bounded by unity. In addition, received signal power under this model must behave asymptotically as 1 α 1+x 2

as x → ∞. A useful such model for the received signal strength is G2 (x) =

1 xα

, which we will consider as our bounded path-loss model (BPM). It is a reasonable way of

improving the inverse-power law model by removing the singularity at 0. Note that G2 (0) = 1, which means that when a transmitter and a receiver are co-located, the receiver receives exactly what the transmitter transmits. C. Interference Models We consider two interference models. In the first model, all interferers transmit the same symbols, and therefore interference signals are perfectly correlated. This is also implicitly assumed in other work in the literature (e.g., see [1] and [2]) while modeling the total interference power at a receiver node. Under this model, which we call Correlated Signals Interference Model Pdλπn2 e (n) (n) (CSIM), In can be written as In = k=1 Ik , where Ik is the interference coming from the k th transmitter at the demodulator output of the receiver when the network radius is equal to (n)

n. Ik

(n)

is equal to Ik

√

=Z

Eb (n) α 1+||Xk || 2

(n)

under the BPM, and Ik

√

=Z

Eb (n) α ||Xk || 2

under the UPM, (n)

where Z is a random variable taking values ±1 with equal probabilities of 21 , and Xk

is a

random variable representing the location of the k th transmitter, which is uniformly distributed over B(0, n). Z = 1 (Z = −1) means that all interferers transmit the binary symbol 1 (0). When (n)

(n)

we write ||Xk ||, we mean the distance of the k th transmitter to the origin. Z and Xk ’s are independent from one another.

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CSIM contains best-case and worst-case scenarios. At the best-case, the symbol to be decoded at the test receiver becomes equal to the symbol transmitted by other transmitters. In this case, the signal quality at the test receiver is enhanced by other transmitters. At the worst-case, the symbol to be decoded at the test receiver becomes different than the symbol transmitted by other transmitters. In this case, the signal quality is deteriorated by other transmitters. The average network performance under CSIM becomes equal to the statistical average of worst-case and best-case scenarios. Our second interference model is proposed to capture the possibility that different transmitters in a wireless network may transmit different symbols. Under the second model, which we call Uncorrelated Signals Interference Model (USIM), individual interference signals at the (n)

demodulator output of the test receiver are given as Ik (n)

Ik

√

= Zk

√

= Zk

Eb (n) α ||Xk || 2

Eb (n) α 1+||Xk || 2

under the BPM, and

under the UPM, where the Zk ’s are independent random variables taking (n)

vaues ±1 with equal probabilities of 12 . We assume that the Zk ’s are also independent of Xk . The total interference signal at the demodulator output of the test receiver is equal to the summation (n)

of Ik ’s. III. I NTERFERENCE D ISTRIBUTION - C ORRELATED S IGNALS I NTERFERENCE M ODEL In this section, we present interference PDF calculations under different path-loss models for CSIM. We will look at the asymptotic distribution of In as n → ∞. As a result of our analysis, we will conclude that a phase transition occurs at the critical value of α = 4, below which network interference behaves the same under both the UPM and BPM, and above which it has very different characteristics under the two path-loss models. We start with the case α ≤ 4. A. Interference Behavior for α ≤ 4: We first look at the interference asymptote as network size grows to infinity when α ≤ 4. In particular, we will show that interference goes to infinity in probability and the rate at which it α

goes to infinity is n2− 2 for α < 4 and log(n) for α = 4. The following theorem is the central result of this subsection. A similar result for marked Poisson processes first appeared in [8]. Here, we extend the results in [8] to the case where locations of nodes are uniformly distributed as well as providing rates of convergence of In to infinity.

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√ i.p. 4λπ In Theorem 1: Under both the UPM and BPM, if α < 4, 2− Eb , and if α = 4, α → Z 4−α 2 n √ i.p. i.p. In → 2Z Eb λπ, where → means convergence in probability. log(n) Proof: See Appendix A. A key conclusion from Theorem 1 is that the interference signal strength does not converge in distribution to a real valued random variable for either of the path-loss models when α ≤ 4. In fact, the interference magnitude goes to infinity in probability at the same rate under both models. The intuitive explanation for such behavior is that the interference is determined by the number of interferers rather than the properties of the propagation models when α ≤ 4. That is, if α ≤ 4, interference signal strength from any individual interferer decays at a rate smaller
than O n12 . However, in order to keep the density of the transmitters constant, we increase their number at rate O(n2 ). Since the decay rate of the propagation models is not fast enough, addition of the small but relatively large number of interference signals results in the convergence of the interference magnitude to infinity. The results of III-B in conjunction with Theorem 1 will prove the existence of phase transition phenomena in the interference behavior at α = 4. In particular, it will be shown that whenever α > 4, interference signal strength converges in distribution as the network size grows to infinity, and the limiting interference distribution behaves very differently under our two path-loss models. B. Interference Behavior for α > 4: We will now analyze the asymptotic distribution of the interference for the case α > 4. Our analysis will be classical in that we will first obtain the characteristic function of the interference, and then invert it to obtain the interference PDF.1 We give the derivation for the BPM in Appendix B. The derivation for the UPM is similar. Theorem 2: The characteristic function of In under the BPM and CSIM is given by    −4λπ α2 4 lim φIn (t) = φI (t) = < exp Eb t α C (t) , n→∞ α where C(t) =

R t√Eb 0

(1 − exp(˙ııu))

„ 1− √u t

« 4 −1

Eb 4 +1 α u

α

du and x} ∼ λπEbα x α as x → ∞. For the decay rate calculations of the interference distribution under the BPM, we cannot use such a Tauberian type theorem since the characteristic function under the BPM does not satisfy the required necessary conditions. Therefore, we do not have a formal way of showing the exponential decay rate of the interference distribution under the BPM. However, numerical analysis suggests that the interference PDF under the BPM does in fact decay to zero exponentially. For example, as we see in Fig. 4, there is a very close match between the tail of the interference 
2  signal magnitude PDF and the Gaussian curve f (x) = 0.487 exp x−1.78 . 1.72 2

Note that this characteristic function is of the form of that for a stable distribution [10].

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IV. I NTERFERENCE D ISTRIBUTION - U NCORRELATED S IGNALS I NTERFERENCE M ODEL Our aim now is to extend the analysis presented in the previous section to the second interference model (USIM). Recall that interference signals are added constructively and destructively with equal probability under USIM. A similar phase transition phenomenon also occurs under USIM. This time, the critical value for α is 2. We start with the case α ≤ 2. A. Interference Behavior for α ≤ 2: The main result of this subsection is that there exists a sequence of constants bn with bn → ∞ such that the scaled network interference

In bn

converges in distribution to a Gaussian random

variable. Theorem 5: Under both the UPM and BPM, if α < 2, √ In log(n)

In 1− α 2

n


b , and if α = 2, ⇒ N 0, 2λπE 2−α

⇒ N (0, 2Eb λπ), where ⇒ means convergence in distribution, and N (0, σ 2 ) denotes a

Gaussian random variable with zero mean and variance σ 2 . Proof: See Appendix C. B. Interference Behavior for α > 2: We now provide the characteristic functions of the network interference under USIM in Theorems 6 and 7 for both BPM and UPM. The derivations of these interference characteristic functions are similar to the derivation given in Appendix B under CSIM for BPM. Therefore, we do not give their derivations due to space limitations, and refer interested users to [11]. Theorem 6: The characteristic function of In under the BPM and USIM for α > 2 is given by



4 4λπ α2 Eb C (|t|) |t| α lim φIn (t) = φI (t) = exp − n→∞ α 4   α −1 R t√E du. where C(t) = 0 b (1−cos(u)) 1 − t√uEb 4 +1

 ,

(3)

uα

Theorem 7: The characteristic function of In under the UPM and USIM for α > 2 is given   4 4λπ α2 lim φIn (t) = φI (t) = exp − E C|t| α , (4) n→∞ α b R∞ −4 where C = 0 (1 − cos(u))u α −1 du. An important distinction between the characteristics of the interference signal strength under CSIM and USIM is that as the network size grows to infinity, it converges in distribution for

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the values of α ∈ (2, 4] under USIM, whereas it diverges to infinity under CSIM for any α ∈ (2, 4]. For example, if α = 4, the interference PDF becomes a Cauchy distribution with median zero under the UPM for USIM. Therefore, we conclude that positive and negative additions of interference signals contribute some more stability to the total amount of network interference. In Figs. 5 and 6, we show interference PDFs obtained after the numerical inversion of the characteristic functions in Theorems 6 and 7, and from Monte Carlo simulation. Again, we observe a very close match of simulation results and the numerical inversion. This verifies our calculations. The simulation setup used here is the same with the one used in III-B. As we see from these figures, the interference PDF appears to decay exponentially under the BPM. On the other hand, the decay rate of the interference is very slow under the UPM. V. E FFECT ON B IT E RROR R ATE , PACKET S UCCESS P ROBABILITY AND W IRELESS C HANNEL C APACITY In this section, we will analyze the effect of the singularity at 0 in the UPM on various performance metrics such as steady state BER, packet success probability and wireless channel capacity. Our aim is to understand the consequences of using the unbounded path-loss model on more concrete metrics by making use of the PDFs obtained in the previous sections. A. Steady State Bit Error Rate We will first analyze the steady state BER under bounded and unbounded path-loss models. BER is an important performance metric which in turn helps us to determine packet success probability and wireless channel capacity. We will analyze the BER as a function of the separation between a transmitter and a receiver in the infinite network limit. We assume that a transmitted information bit is successfully decoded if and only if the interference plus noise level at the demodulator output of the receiver is sufficiently small enough. Let b be a generic transmitted bit, and E(d) be the event that b is decoded erroneously at the receiver when transmitter-receiver separation is equal to d. Due to the symmetry of the problem, we have 1 1 (5) P{E(d)} = P{E(d)|b = 1} + P{E(d)|b = 0} = P{E(d)|b = 1}. 2 2 Recall that transmitted bits are also impaired by the white noise process W (t) with power spectral density

N0 . 2

Let W0 be the corresponding noise at the demodulator output of the receiver

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coming from W (t). Then, W0 is a Gaussian random variable with mean zero and variance

N0 . 2

We also assume that interference signal reduction by A times is possible at receiver nodes. A = 1 corresponds to the classical narrowband digital communication, and A > 1 can be thought of as corresponding to a broadband communication scheme such as CDMA ([13]). As a result,   p I P{E(d)} = P + W0 < − Eb Gi (d) A Z ∞ n o p x fI (x)P W0 < − Eb Gi (d) − I = x dx = A −∞ s  Z ∞ Eb Gi (d)2 x 1 fI (x)erfc  + √  dx, = 2 −∞ N0 A N0

(6)

where fI (x) is the probability density function of the multiple-access interference signal I coming from other terminals transmitting concurrently and Gi , i = 1, 2, is the path-loss function. Below, we plot the steady state BER as a function of transmitter and receiver separation d for both highnoise regime and low-noise regime. For the high-noise regime, we assume that the background noise power is comparable with the transmitted energy per bit. To this end, we set N0 = 0.5 and Eb = 1. For the low-noise regime, we assume that the background noise power is much smaller than the transmitted energy per bit. To this end, we set N0 = 0.01 and Eb = 1 for the low-noise regime. The shape of the BER curves as functions of d depends on the specific selection of the parameter A, the interference model and the path-loss model. Below, we describe results for two different values of A under different path-loss and interference models. It is also assumed that α = 6 and λ = π1 . Examining Fig. 7, we observe that BER approaches 0 when d goes to zero under the UPM. In fact, this is the typical behavior of the BER under the UPM for any value of A. The main reason for such behavior is the unrealistic singularity of the UPM at 0. Received signal energy per bit increases unboundedly as the transmitter-receiver separation is made smaller and smaller, which results in the convergence of the BER to 0 no matter how big the interference plus noise signal strength is. Especially for dense wireless networks where transmitters and receivers become arbitrarily close to one another with high probability, this has far more dramatic effects on the data transmission rate, which further leads to very unrealistic estimates for the number of successfully communicating transmitter-receiver pairs. If we look at Fig. 7 again, we observe that the BER is bounded away from 0 under the BPM

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since the received signal energy per bit can be at most 1 [unit energy]. Therefore, whenever the interference plus noise level is high enough at the demodulator output of the receiver, the detector decodes transmitted bits erroneously. For example, consider the high-noise regime without any multiple access interference. Then, transmitted bits fail with probability 0.023 when d = 0. If, in addition to the background noise, there is also multiple access interference deteriorating transmitted bits, this error probability becomes around 0.09 for A = 5, and 0.04 for A = 10 under CSIM and the BPM. Similar qualitative conclusions continue to hold in other figures as well. When the UPM is used to model the physical layer of a wireless network, the BER is underestimated for small values of transmitter-receiver separation. This is due to received power being unboundedly large when transmitters and receivers are arbitrarily close to one another. For moderate to high values of transmitter-receiver separation, we start to overestimate the BER under the UPM. This is because interference signal strength PDFs are more spread to the left and right under the UPM which results in theoretically higher interference signal levels at receiver nodes. For the low-noise regime where transmitted bits are mainly impaired by multiple access interference coming from other transmitters, it is also possible to obtain BERs close to 0 for small values of transmitter-receiver separation under the BPM but for completely other reasons. For small values of transmitter-receiver separation, a bit is decoded in error if and only if the multiple access interference is high enough when the background noise is so small compared with the transmitted energy per bit. However, the tails of interference PDFs decay exponentially fast under the BPM, and the area underneath of these tails becomes negligible after high enough interference values. Therefore, the BER becomes close to zero under the BPM when transmitterreceiver separation is small enough in the low-noise regime (see Fig. 8 and Fig. 10). B. Packet Success Probability Next, we would like to compare packet success probability as a function of transmitter-receiver separation under different path-loss models. We assume that digital data is encoded by means of an error control coding technique (see [14]) into packets of D ≥ 1 bits, and a packet fails if and only if there are more than L, 1 ≤ L ≤ D, bit errors inside the packet. For the purposes of calculating packet success probability, we consider two extreme regimes of node mobility. The first one is the fast-mobility regime where transmitter locations shuffle sufficiently enough

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during the transmission of a bit that bit errors can be assumed to be independent. The second regime is the slow-mobility regime where nodes can be assumed to be static over the course of a packet transmission. Note that bit errors are dependent in the slow-mobility regime since locations of transmitters do not change dramatically during a bit duration. Let us start with calculating packet success probability for the fast-mobility regime. Let BERi (d), i = 1, 2, denote the steady state bit error rates calculated in V-A under different interference models when transmitter-receiver separation is equal to d. i = 1 corresponds to the UPM, and i = 2 corresponds to the BPM. Let S(d) be the probability that a transmitted packet is decoded successfully at the receiver when the transmitter-receiver separation is equal to d. Then, for different path-loss models, the packet success probability in the fast-mobility regime is equal to P{S(d)} =

L   X D e=0

e

BERi (d)e (1 − BERi (d))D−e .

(7)

The change of packet success probability as a function of transmitter-receiver separation in the fast-mobility regime is depicted in Fig. 11 for Eb = 1, α = 6, λ =

1 π

and A = 5. The

results shown in Fig. 11 are for D = 100 and L = 10. To cover as many different cases as possible, we considered the high-noise regime for CSIM and the low-noise regime for USIM. As we observe in Fig. 11, there is a significant difference between packet success probability curves under different path-loss models. The UPM can overestimate or underestimate the packet success probability substantially depending on the specific selection of interference models and other network parameters. The packet success probability becomes close to 1 for small values of transmitter-receiver separation under both path-loss models as the received signal strength becomes more dominant than the total network multiple-access interference plus noise while decoding transmitted bits. For large values of transmitter-receiver separation, the packet success probability becomes close to 0 under both path-loss models since the received signal strength becomes negligibly small when compared with the total network multiple-access interference plus noise. For values of transmitter-receiver separation close to 0, the UPM overestimates the packet success probability since the received signal strength becomes arbitrarily large as the transmitter-receiver separation goes to zero. This behavior is more prominent for CSIM in the high-noise regime. With increasing values of transmitter-receiver separation, the gap between packet success probability curves decreases, and the UPM starts to underestimate packet success

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probability since the multiple-access interference signal PDF becomes heavy-tailed under the UPM. Therefore, the tail of the packet success probability curve under the BPM eventually becomes higher than the tail of the packet success probability curve under the UPM. For USIM in the low-noise regime, BER is very close to the zero in Fig. 10 for small values of transmitterreceiver separation, which further results in packet success probability being close to 1 under BPM for small values of transmitter-receiver separation in Fig. 11. Next, we consider the packet success probability in the slow-mobility regime where nodes’ locations can be assumed to be static over the course of a packet transmission. Calculations for the slow-mobility regime become a little bit trickier than the fast-mobility regime since bit errors are now dependent due to dependencies among transmitter locations from one bit to another bit. Let us start slow-mobility regime calculations with CSIM. For CSIM, slowly moving transmitters assumption translates into the fact that the magnitude of the total network interference |I| stays the same during a packet transmission. Therefore, given |I| = x ≥ 0, and assuming that the background noise impairing transmitted bits is independent from bit to bit, bit errors also become independent. Let pi,CSIM (x, d) be the probability that a transmitted bit is decoded in error when transmitter-receiver separation is equal to d, |I| = x and Gi , i = 1, 2, is used to model the signal strength decay. Then, o n p x pi,CSIM (x, d) = P W0 + Z < − Eb Gi (d) s A s   2 2 1 1 Eb Gi (d) x Eb Gi (d) x = erfc  + √  + erfc  − √  . (8) 4 N0 4 N0 A N0 A N0
P Therefore, P (S(d)||I| = x) = Le=0 De pi,CSIM (x, d)e (1 − pi,CSIM (x, d))D−e and R∞ P (S(d)) = 2 0 P(S(d)||I| = x)fI (x)dx since the PDF of |I| is equal to the twice the positive part of the PDF of I. For USIM, packet success probability calculations become harder since dealing with dependencies among bit errors does not become as easy as in CSIM. However, given the points of the node location process U = {Xi }∞ i=1 , a reasonable approximation for the multiple access interference under USIM becomes the Gaussian noise approximation with mean zero and variance Y = E [I 2 |U] for α ∈ (2, 6] since the total interference level at the receiver is equal to the summation of large number of small interference signals coming from other transmitters. For larger values of α, deviations between the Gaussian noise approximation and the actual

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interference PDFs becomes more prominent. Such an approximation becomes better for the BPM since all interference signals are small, and there are no dominant terms. Note that the variance of the Gaussian noise in this approximation is a random variable, and depends on the particular realization of the network configuration U. Note also for the BPM that Y

=

∞ X

Eb α

i=1

1 + ||Xi || 2


2 .

(9)

The characteristic function of Y can be obtained exactly by using techniques similar to those of Sections III and IV. However, without going through the same but tedious calculations, a P Eb reasonable approximation for Y is Y˜ = ∞ i=1 1+||Xi ||α since for values of ||Xi || less than 1, 1 α

becomes the dominant term in (9), and for the values of ||Xi || greater than 1, ||Xi || 2 becomes the dominant term in (9). The characteristic function of Y˜ can be readily obtained by replacing with α and removing the operator 0, irrespective of how sparsely or densely populated a network is. We also analyzed the effects of the singularity in the UPM for more concrete performance metrics such as bit error rate, packet success probability and wireless channel capacity. We have shown that using the UPM leads to significant deviations from more realistic performance figures obtained by using the BPM. Overall, the results of this paper suggest that we should exercise caution when using the classical unbounded path-loss model. The effect of the singularity in the UPM on the network performance is significant, and cannot be ignored in many situations.

17

R EFERENCES [1] P. Gupta and P. R. Kumar, “The capacity of wireless networks,” IEEE Trans. Info. Theory, vol. 46, no. 2, pp. 388-404, March 2000. [2] M. Grossglauser and D. Tse, “Mobility increases the capacity of wireless ad hoc networks,” IEEE/ACM Trans. on Networking, vol. 10, no. 4, pp. 477-486, Aug. 2002. [3] S. B. Lowen. and M. C. Teich, “Power-law shot noise,” IEEE Trans. on Info. Theory, vol. IT-36, no. 6, pp. 1302-1318, Nov. 1990. [4] E. S. Sousa and J. A. Silvester, “Optimum transmission ranges in a direct sequence spread spectrum multihop packet radio network,” IEEE J. Selected Areas Commun., vol. 8, no. 5, pp. 762-771, June 1990. [5] C. C. Chan and V. H. Stephen, “Calculating the outage probability in a CDMA network with spatial Poisson traffic,” IEEE Trans. on Vehicular Technology, vol. 50, no. 1, pp. 183-204, Jan. 2001. [6] F. Baccelli and B. Błaszczyszyn, “On a coverage process ranging from the Boolean model to the Poisson Voronoi tessellation with applications to wireless communications,” Adv. Appl. Prob., vol. 33, no. 2, pp. 293-323, 2001. [7] J. F. C. Kingman, Poisson Processes, Clarendon Press, Oxford, 1993. [8] D. J. Daley, “The definition of a multi-dimensional generalization of shot noise,” J. Appl. Prob., vol. 8, no. 1, pp. 128-135, March 1971. [9] R. Durrett, Probability: Theory and Examples, Duxbury Press, Belmont, CA, second edition, 1996. [10] W. Feller, An Introduction to Probability Theory and Its Applications, John Wiley and Sons, Inc., New York, vol. 2, second edition, 1970. [11] H. Inaltekin, Topics on Wireless Network Design: Game Theoretic MAC Protocol Design and Interference Analysis for Wireless Networks, PhD Dissertation, School of Electrical and Computer Engineering, Cornell University, Ithaca, NY, Aug. 2006. [12] W. Rudin, Real and Complex Analysis, McGraw-Hill, New York, third edition, 1987. [13] D. Tse and P. Viswanath, Fundamentals of Wireless Communications, Cambridge University Press, New York, 2005. [14] S. B. Wicker, Error Control Systems for Digital Communication and Storage, Prentice Hall, Upper Saddle River, NJ, 1995.

A PPENDIX A P ROOF OF T HEOREM 1 The following weak law for triangular arrays from [9] will be helpful during our calculations. Theorem 8 (Weak Law for Triangular Arrays): For each m, let Ym,k , 1 ≤ k ≤ m, be a set of independent random variables. Let bm > 0 with bm → ∞, and let Y¯m,k = Ym,k 11{|Ym,k |≤bm } .  2  Pm P 1 ¯ Suppose that m k=1 E Ym,k → 0 as m → ∞. If we let k=1 P {|Ym,k | > bm } → 0, and b2m   Pm i.p. i.p. Sm = Ym,1 + . . . + Ym,m and put am = E Y¯m,k , then Sm −am → 0, where → means k=1

bm

convergence in probability. Proof: See [9]. Proof of Theorem 1: We will give the proof only for the BPM. The proof for the UPM is very similar. Fix the transmitter density λ > 0. Let B(0, 1) denote the disk centered around the origin

18

with radius 1, and let {Xk }k≥1 be a set of independent random variables uniformly distributed (n)

over B(0, 1). Note that nXk is equal in distribution to Xk . Then, 2e √ dλπn Eb X 1 d In = Z α α α , − n 2 k=1 n 2 + ||Xk || 2 d

where = means equal in distribution. We will first consider α < 4. Put ξn,k = − α2 1 α , n +||Xk || 2 P 2 α dλπn e ξ¯n,k = ξn,k 11{ξn,k ≤n2 } and Sn = k=1 ξn,k . We concentrate on Snn2 . Observe that ξn,k ≤ n 2 ≤ n2 .   Therefore, ξ¯n,k = ξn,k for all n ≥ 1 and 1 ≤ k ≤ dλπn2 e. As a result, E ξ¯n,k = E[ξn,k ] = R1 R 2 1 α 2x −2 n 2 2 2 2 and dy = 2xn dx. Then, E[ξ ] = n dy. Now define α α dx. Set y = x n n,k − 0 n 2 +x 2 0 1+y α 4 Rx 1 α (x) 4 f (x) = 0 x1− 4 . By L’Hopital’s rule, we have limx→∞ fg(x) = 1. α dy and g(x) = 4−α 1+y 4 R 2 α α n 4 1 4 Thus, f (x) ∼ 4−α x1− 4 as x → ∞.4 This implies that 0 n2− 2 . Set an = α dy ∼ 4−α 1+y 4 Pdλπn2 e 4 E[ξn,k ]. Then, an ∼ 4−α λπn2 . k=1 We will now check the conditions for triangular arrays.  Pdλπn2 e α α • P {ξn,k > n2 } → 0 as n → ∞: P {ξn,k > n2 } = P ||Xk || 2 < n−2 − n− 2 = 0, k=1 α

•

where the equality follows from the fact that n− 2 > n−2 when n > 1 since α < 4. Pdλπn2 e  ¯ 2  1 E (ξn,k ) → 0 as n → ∞ : 4 k=1 n Z n2   α
−2 2 α−2 ¯ E (ξn,k ) = n 1+y4 dy 0

Z

n2

α 1 4 n2. α dy ∼ 4−α 1+y4 0 h i h i
2
2 Pdλπn2 e 8λπ α Thus, k=1 E ξ¯n,k = dλπn2 eE ξ¯n,1 ≤ 4−α n 2 +2 , ∀n large enough. Since α < h i
2 Pdλπn2 e 8λπ α 4, we have limn→∞ n14 k=1 E ξ¯n,k n 2 −2 = 0. ≤ limn→∞ 4−α

≤ n

α−2

Therefore, the conditions of weak law for triangular arrays are satisfied. We thus have Sn − an i.p. → 0 as n → ∞. n2 √

By observing that In = Z

Eb α Sn n2

n

In 2− α 2

4λπ → 4−α as n → ∞, we conclude that √ 4 Eb λπ i.p. →Z as n → ∞. 4−α

and

an n2

For α = 4, the same calculations are repeated with bn = n2 log(n). 4

When we write a f (x) ∼ g(x), we mean limx→∞

f (x) g(x)

= 1.

(14)

19

A PPENDIX B P ROOF OF T HEOREM 2 (n) d

Similar to the proof of Theorem 1, let Xk

= nXk , where the Xk ’s are independent and uniPdλπn2 e d 1 1 formly distributed over B(0, 1). Then, In = Z Eα2 b k=1 α . Let ξn,k = α . −α −α 2 +||Xk || 2 2 +||Xk ||− 2 n n n h α i α Note that ξn,k ∈ n 2 α2 , n 2 . Its PDF can be obtained as √

1+n

α

4 x−1 − n− 2 fn (x) = α x2


α4 −1

 ˙ t n,1 Let φn,1 (t) = E eııξ . Then, φn,1 (t) = φn,1 (t) =

4 t α

R n α2 t

α n2 α 1+n 2

ııu ˙

t

e



t u

−

 α4 −1

1 α n2

4 α

 α α n2 2 . , x∈ α ,n 1 + n2 

R n α2

α n2 α 1+n 2

e

ııxt ˙ (x

α −1 −n− 2

x2

(15)

4 −1

)α

dx. Put u = xt. Then,

u−2 du.

The characteristic function of the interference In when the network radius is equal to n is given by

"

√ ııtZ ˙

φIn (t) = E e

Eb α n2

Pdλπn2 e k=1

# ξn,k

   √ dλπn2 e  √ dλπn2 e 1 Eb Eb 1 ∗ = φn,1 t α φn,1 t α + 2 2 n2 n2     √ dλπn2 e Eb  , = < φn,1 t α n2 where φ∗n,1 (t) is the conjugate of φn,1 (t), and 0, and ∀ > 0, k=1 E |Ym,k |   P Pm 2 2 limm→∞ m k=1 E |Ym,k | ; |Ym,k | >  = 0. Then, Sm = k=1 Ym,k ⇒ N (0, σ ) as m → ∞, where ⇒ means convergence in distribution, and N (0, σ 2 ) is the Gaussian distribution with zero mean and variance σ 2 . Proof: See [9]. Proof of Theorem 5: We will give the proof only for the BPM. The proof for the UPM is very similar. Fix the transmitter density λ > 0. With the same definitions of the proof of Theorem 1, √ √ Pdλπn2 e d Zk Eb Zk we have In = Eα2 b k=1 α . We will first consider α < 2. Let ξn,k = α and α −α − n n 2 +||Xk || 2 n 2 +||Xk || 2 Pdλπn2 e Sn = k=1 ξn,k . We concentrate on Snn , and check the conditions for the Lindeberg-Feller theorem. We only check the first condition. The second one is automatically satisfied since 2 i 2Eb dλπn2 e R 1 √ Pdλπn2 e h x |ξn,k | ≤ Eb nα and α < 2. We have k=1 E n1 ξn,k = α 2 dx, so we n2 0 n− α 2 +x 2 ) ( R1 R 1 1 < wish to analyze limn→∞ 0 − α x α 2 . Note that − α x α 2 ≤ x1−α and 0 x1−α dx = 2−α (n 2 +x 2 ) (n 2 +x 2 ) R1 1 ∞. By using the dominated convergence theorem [12], we have limn→∞ 0 − α2 x α2 2 = 2−α , (n +x ) h i 2
Pdλπn e 2 b b which implies limn→∞ k=1 E n1 ξn,k = 2λπE . Therefore, Snn ⇒ N 0, 2λπE , which 2−α 2−α
In b ⇒ N 0, 2λπE implies 2−α . 2−α n

2

For α = 2, similar calculations are repeated with √Sn n

log(n)

.

21

Fig. 1.

Physical layer model for decoding of transmitted bits.

0.35

Interference PDF

0.3

Iteration no = 10000, E b = 1, α = 6, λ = π1 .

Simulation Numerical Inversion

0.25

0.2

0.15

0.1

0.05

0 !8

!6

!4

!2

0

2

4

Interference Signal Strength

6

8

Fig. 2. Interference PDF under the BPM from simulation and from numerical inversion of the characteristic function for CSIM.

22

Interference PDF

0.12

Iteration no = 10000, E b = 1, 0.1 α = 6, λ = π1 .

Simulation Numerical Inversion

0.08

0.06

0.04

0.02

0 !10

!8

!6

!4

!2

0

2

4

Interference Signal Strength

6

8

10

Fig. 3. Interference PDF under the UPM from simulation and from numerical inversion of the characteristic function for CSIM.

Tail of PDF of Interference Magnitude Under BPM

0.5

Interference Magnitude PDF ! " # $ .78 2 f (x) = 0.487 exp − x −1 1.72

0.45

0.4

E b = 1, α = 6, λ = π1 .

0.35

0.3

0.25

0.2

0.15

0.1

0.05

0

2

3

4

5

6

Magnitude of Interference Signal

7

8

The tail of the interference PDF under the BPM. It is very well approximated by the Gaussian curve f (x) = “ ` ´2 ” 0.487 exp − x−1.78 . 1.72 Fig. 4.

23

0.6

Interference PDF

0.5

Iteration no = 10000, E b = 1, α = 6, λ = π1 .

Simulation Numerical Inversion

0.4

0.3

0.2

0.1

0 !4

Fig. 5.

!3

!2

!1

0

1

2

Interference Signal Strength

3

4

Interference PDFs under the BPM from simulation and from numerical inversion of the characteristic function for

USIM.

0.3

Interference PDF

0.25

Iteration no = 10000, E b = 1, α = 6, λ = π1 .

Simulation Numerical Inversion

0.2

0.15

0.1

0.05

0 !8

Fig. 6. USIM.

!6

!4

!2

0

2

4

Interference Signal Strength

6

8

Interference PDFs under the UPM from simulation and from numerical inversion of the characteristic function for

24

Steady State Bit Error Rate

0.5 0.45

BER, BPM, A = 5

0.4

BER, UPM, A = 5 BER, BPM, A = 10

0.35

BER, UPM, A = 10

0.3

CSIM, N 0 = 0.5

0.25 0.2 0.15 0.1 0.05 0 0

Fig. 7.

1

2

3

4

5

6

7

Transmitter-Receiver Separation

8

9

10

Steady state BER as a function of transmitter-receiver separation under CSIM for the high-noise regime.

0.5

Steady State Bit Error Rate

0.45

BER, BPM, A = 5

0.4

BER, BPM, A = 10

0.35

BER, UPM, A = 10 BER, UPM, A = 5

0.3 0.25

CSIM, N 0 = 0.01

0.2 0.15 0.1 0.05 0 0

Fig. 8.

1

2

3

4

5

6

7

8

Transmitter-Receiver Separation

9

10

Steady state BER as a function of transmitter-receiver separation under CSIM for the low-noise regime.

0.5

Steady State Bit Error Rate

0.45

BER, BPM, A = 5

0.4

BER, UPM, A = 5

0.35

BER, BPM, A = 10

0.3

BER, UPM, A = 10 USIM, N 0 = 0.5

0.25 0.2 0.15 0.1 0.05 0 0

Fig. 9.

1

2

3

4

5

6

7

8

Transmitter-Receiver Separation

9

10

Steady state BER as a function of transmitter-receiver separation under USIM for the high-noise regime.

25

0.5

Steady State Bit Error Rate

0.45

BER, BPM, A = 5

0.4

BER, UPM, A = 5

0.35

BER, BPM, A = 10

0.3

BER, UPM, A = 10

0.25

USIM, N 0 = 0.01

0.2 0.15 0.1 0.05 0 0

3

4

5

6

7

8

Transmitter-Receiver Separation

9

10

1

Packe t Suc c e ss Probability

BPM UPM

0.8 0.6

CSIM, N 0 = 0.5 Fast-mobility

0.4 0.2 0 0

0.2

0.4

0.6

0.8

1

1.2

1.4

Transmitte r-Re c e ive r Se paration

1.6

1.8

2

1.4

BPM UPM

1.2 1 0.8

USIM, N 0 = 0.01 Fast-mobility

0.6 0.4 0.2 0 0

Fig. 11.

2

Steady state BER as a function of transmitter-receiver separation under USIM for the low-noise regime.

Packe r Suc c e ss Probability

Fig. 10.

1

0.5

1

1.5

Transmitte r-Re c e ive r Se paration

2

2.5

Change of packet success probability as a function of transmitter-receiver separation in the fast-mobility regime for

CSIM and USIM. D = 100, L = 10, Eb = 1, α = 6, λ =

1 π

and A = 5.

26

Simulation Gaussian Approximation

0.3

Gaussian Approximation

0.4

BPM, α = 3

0.25

Simulation

0.5

I nte rf e re nc e PDF Give n Node Loc ations

BPM, α = 4

0.3

0.2

0.15

0.2

0.1

1

I nte rf e re nc e PDF Give n Node Loc ations

0.35

0.1

0.05 0 !10

!5

0

5

I nte re f e rnc e Signal Stre ngth

0 !10

10

Simulation

I nte rf e re nc e PDF Give n Node Loc ations

Gaussian Approximation

BPM, α = 5

0.8 0.6

0

5

I nte rf e re nc e Signal Stre ngth

10

Simulation

0.7 0.6

Gaussian Approximation

0.5

BPM, α = 6

I nte rf e re nc e PDF Give n Node Loc ations

1

!5

0.4 0.3

0.4

0.2

0.2

0.1

0 !10

!5

0

5

I nte rf e re nc e Signal Stre ngth

0 !10

10

!5

0

5

I nte rf e re nc e Signal Stre ngth

10

Fig. 12. Approximation of multiple-access interference distribution by means of a Gaussian PDF with mean zero and variance ˆ ˜ E I 2 |U for random realizations of network configurations U under the BPM and USIM.

Simulation I nte rf e re nc e PDF Give n Node Loc ations

I nte rf e re nc e PDF Give n Node Loc ations

UPM, α = 3 0.15

0.1

0.05

Simulation

0.16 0.14

Gaussian Approximation

UPM, α = 4

0.12 0.1 0.08 0.06

1

Gaussian Approximation

0.2

0.04 0.02

0 !10

!5

0

5

I nte rf e re nc e Signal Stre ngth

0 !10

10

!5

0

5

I nte rf e re nc e Signal Stre ngth

10

Simulation Gaussian Approximation

0.12

UPM, α = 5

I nte rf e re nc e PDF Give n Node Loc ations

I nte rf e re nc e PDF Give n Node Loc ations

0.14

0.1 0.08 0.06 0.04 0.02 0 !10

!5

0

5

I nte rf e re nc e Signal Stre ngth

10

0.35 0.3

Simulation

UPM, α = 6

Gaussian Approximation

0.25 0.2 0.15 0.1 0.05 0 !10

!5

0

5

I nte rf e re nc e Signal Stre ngth

10

Fig. 13. Approximation of multiple-access interference distribution by means of a Gaussian PDF with mean zero and variance ˆ ˜ E I 2 |U for random realizations of network configurations U under the UPM and USIM.

27

Packe t Suc c e ss Probability

1

BPM UPM

0.8 0.6

CSIM, N 0 = 0.5, A = 5, α = 6 Slow-mobility 1

0.4 0.2 0 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Transmitte r-Re c e ive r Se paration

Packe t Suc c e ss Probability

1

BPM UPM

0.8 0.6

USIM, N 0 = 0.01, A = 2, α = 4 Slow-mobility

0.4 0.2 0 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Transmitte r-Re c e ive r Se paration

Fig. 14.

Change of packet success probability as a function of transmitter-receiver separation in the slow-mobility regime for 1 . π

CSIM and USIM. D = 100, L = 10, Eb = 1, λ =

BPM UPM

0.8 0.7

CSIM, N 0 = 0.5 Fast-mobility

0.6

1

Channal Capac ity [Bits pe r Channe l Use ]

1 0.9

0.5 0.4 0

0.5

1

1.5

2

2.5

Transmitte r-Re c e ive r Se paration

3

Channe l Capac ity [Bits pe r Channe l Use ]

1

BPM UPM

0.9 0.8 0.7

USIM, N 0 = 0.01 Fast-mobility

0.6 0.5 0.4 0

Fig. 15.

0.5

1

1.5

2

2.5

Transmitte r-Re c e ive r Se paration

3

3.5

4

Change of channel capacity as a function of transmitter-receiver separation in the fast-mobility regime for CSIM and

USIM. Eb = 1, α = 6, λ =

1 π

and A = 5.

28

BPM UPM

0.8 0.7

CSIM, N 0 = 0.5, A = 5, α = 6 Slow-mobility

0.6

1

Channe l Capac ity [Bits pe r Channe l Use ]

1 0.9

0.5 0.4 0

0.5

1

1.5

2

2.5

3

Transmitte r-Re c e ive r Se paration

Channe l Capac ity [Bits pe r Channe l Use ]

1

BPM UPM

0.9 0.8 0.7

USIM, N 0 = 0.01, A = 2, α = 4 Slow-mobility

0.6 0.5 0.4 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Transmitte r-Re c e ive r Se paration

Fig. 16. Change of channel capacity as a function of transmitter-receiver separation in the slow-mobility regime for CSIM and USIM. Eb = 1, λ =

1 . π