Outage Probability in Multiple Access Packet Radio ... - CiteSeerX

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL.

43, N. 3, AUG. 1994

1

Outage Probability in Multiple Access Packet Radio Networks in the Presence of Fading Michele Zorzi, Student Member, IEEE, and Silvano Pupolin, Senior Member, IEEE

Abstract

— The outage probability in a mobile communications environment, where the interferers are randomly distributed on the ground, taking into account the background noise, the co-channel interference and the random effects of propagation (log-normal shadowing and Rayleigh fading), is evaluated. Although not feasible in closed form, the computation can be performed numerically in a very fast and accurate manner. A comparison to the results reported in the recent literature is presented, along with a sensitivity analysis and an example of application.

Keywords— Outage probability, Fading, Shadowing, Packet Radio, Mobile Communications

I. INTRODUCTION Self-interference is the major limiting factor in multiple-access radio systems. To combat the phenomenon, in order to increase the network throughput, several different modulation formats and random access strategies have been proposed in the literature [1]. Most authors have considered only one of the two aspects outlined above, namely the interference or the random access protocol, while in the real world they are strongly related, and a joint optimization could improve substantially the network throughput. Hereafter, we consider the effects of a random attenuation on the signal-to-interference plus noise ratio generated in a mobile radio environment, looking for a simple result to be applied to the comparison of the performance of multiple access protocols. In the analysis, we modeled the interfering signal as the sum of many mutually independent random processes, each representing the interfering signal generated by a specific radio. The statistics of each interferer depends on the power of the transmitter, the antenna gains, the distance from the receiver and the statistical characterization of the link attenuation. Typically, the following random effects should be included in the study [2; 3]: i) the random position of the interferers, ii) the random link attenuation due to the log-normal shadowing effect, and iii) the Rayleigh fading. Many authors have studied this problem with a different view; among them we recall Musa and Wasylkiwskyj [4], who performed the computation for a particular distribution of the interferers, by ignoring also log-normal shadowing and Rayleigh fading; Sousa and Silvester [5], who took into account a deterministic propagation law; Benvenuto et al. [6], who employed intensive simulation; Sowerby and Williamson [7], who obtained some results only by costly multiple numerical integrations; Gosling [8], who considered Rayleigh fading only, ignoring log-normal shadowing; Prasad [9] and Linnartz [10], This work was supported by MURST, Italy. Part of this work has been presented at IEEE MILCOM’92, San Diego, CA, 11-14 October 1992. Michele Zorzi is with the Dipartimento di Elettronica e Informazione, Politecnico di Milano - ITALY. Silvano Pupolin is with the Dipartimento di Elettronica e Informatica, Universit`a di Padova - ITALY.

who limited their analyses to fixed locations for the interferers, and to the absence of noise. In this paper, we take into account the superposition of several random features of both the channel characteristics and the population of users. In the following, we evaluate in an extremely simple and rapid manner the outage probability, taking into account different propagation laws, random propagation effects (fading and shadowing), background noise and possible spread spectrum modulations, and the obtained results are applied to the computation of a packet radio network throughput. Also, the interferers are distributed at random over some region, as is specified later. This accounts for the fact that many users, whose positions are not known a priori, may share the same communications resources, and this affects the transmission performance of each of them. In this paper, we apply the outage probability concept, generally used in cellular radio communications systems, to packet radio networks [11]. The motivation is related to the foreseeable transition from fixed-strategy resource assignment, used in cellular radio (e.g., TDMA and FDMA), to cellular packet communication networks [12]. Also, the contribution of this paper beyond the previous literature lies in the effort to take into account an extreme randomness in the communications channel (as in [10]), along with the randomness of the locations of the users, coming up with a mathematically tractable analysis and a final expression which is easy to compute. More interestingly, on the track of [5], we tried to combine the considerations about the link performance and the outage probability techniques with a network approach. We believe that this effort to combine link level and network level is very important in pursuing an effective analysis of such systems. In Section II the model for propagation and interference is given, while in Section III the performance measure is introduced and discussed. Section IV provides a more precise formulation of the problem and gives the obtained formulas. Section V reports computational results in different situations, along with a comparison with the results obtained by Sousa and Silvester [5] in the deterministic situation, and a sensitivity analysis about several parameters. Finally, Section VI reports a simple example of application. II. SYSTEM MODEL In a mobile radio communication environment the path loss is well described by means of a random variable, whose average value is given by the deterministic propagation law g(r) = PR =PT , given by:

gdB (r) = 10 log10 K ? 10 log10 r

;

[in dB]

(1)


43, N. 3, AUG. 1994

where typically assumes values between 3.5 and 4, and K is a constant dependent on the carrier frequency and the antenna heights and gains. Two different random effects are generally considered: Rayleigh fading and log-normal shadowing. Rayleigh (short-term) fading is generated by multiple paths due to reflections, and is very important in considering signals propagating within a town, where an extremely high number of reflections occur and the receiver-transmitter pair is rarely in visibility. Each signal at the receiver is modeled as the superposition of two orthogonal Gaussian components, so that its envelope turns out to be a Rayleigh random variable [2]. This results in an exponential distribution for the signal power [13]. We will focus on this model, even though it is not always valid, e.g., when diversity is employed at the receiver or when the channel is frequency-selective. Shadowing (long-term fading) is generated by obstacles, weather conditions and terrain roughness [14]: in a popular model, based on the available measurements, this random effect is described by a log-normal random variable, i.e., a r.v. whose representation in logarithmic units (e.g., in dB) is Gaussian. It must be noted that our model is only one of the many possible choices: many models have been proposed in the literature, and probably none of them can capture the exact nature of the physical phenomena which occur in this environment. However, given a specific set of environmental features, some models can fit better than others. The model we adopt in this study is considered fairly accurate for a dense urban environment, in which the mobiles are rarely in visibility and a log-normal model, whose mean value is given by (1), and whose standard deviation (also called dB spread) takes a value between 6 and 13 dB [10], fits the measurements fairly closely. The log-normal random variable, in turn, is the mean value of the exponential distribution of the signal power due to Rayleigh fading [15]. It should be noticed that (1) is a function of r and, as long as the transmitters are randomly distributed on the ground, the average value of the log-normal path loss is itself a random variable. All the terminals considered are transmitting at the same carrier frequency. Among them we focus on a pair of communicating transmit-receive radios, located at a distance r0 from each other. All the other transmitters represent the interferers for this pair and they are randomly distributed over a specified region. We assume that the terminals are distributed as a Poisson point process on the ground with parameter 0 , which represents the average number of terminals per unit area. Therefore, given a region R on the ground, the probability that k terminals are present in R depends upon its area, A, and can be written as:

P [k terminals in R] = e

?0 A (0 A)k

k!

:

(2)

If p denotes the probability of a terminal being active (i.e. transmitting), the distribution of the interferers can be modeled as a Poisson process as well, with parameter = p0 . In the following, we will use (2) (with 0 replaced by ) to indicate the distribution of the interferers. The system is slotted, with fixed packet length, i.e., packets either collide completely, or do not collide at all. The slots are assumed so short that both the topology and the interference can be considered fixed during its duration. Also, the interferers are assumed to cover the whole

2

plane, up to infinity. Finally, we assume that the interference can be modeled as a Gaussian random process [5]. III. PERFORMANCE ANALYSIS A. Outage probability The performance index we will consider throughout the paper is the outage probability, (b), defined as the probability that the short-term Signal to Noise Ratio (SNR) at the receiver is smaller than a given threshold b, called outage SNR [6]. More precisely, let P0 be the received power from the desired terminal, W be the power of the background noise, and PI be the power of the interference. The outage probability is given by [10]: P [outage] = P P P+0 W b ; (3 ) I where only the dependence of the threshold b is explicitly indi-

b

( )=

cated.

B. Packet success probability In the network analysis, one of the key parameters is the probability of packet success given the receiver is idle, Ps. The probability of a transmission being successful is an increasing function (from 0 to 1) of the SNR, and depends on the degree of error correction. The higher the level of coding, the sharper the transition. We can write:

Ps

1

Z = 0

Z =

s(x)fSNR (x)dx

1

1 ? FSNR (x)

0

s0 (x)dx;

(4)

where fSNR (x) and FSNR (x) are the probability density function (pdf) and the cumulative distribution function (cdf) of the SNR, respectively, and s(x) is the probability of packet success given that the SNR is equal to x. Using good long codes, s(x) approches a step function [16], and s0 (x) becomes a delta function, centered at the threshold SNR value b [5]. Under these conditions, (4) reduces to Ps = 1 ? FSNR (b) = P P P+0 W > b = 1 ? (b): I

(5)

In the following, we focus our attention upon the computation of the outage probability, (b). In doing this we will find the cdf of the SNR, which could be used also to compute the probability of packet success once a general s(x) is given. C. Bit error rate probability A modified version of (4) could be used to obtain the bit error probability. Indeed, let FBER (x) be the bit error rate (BER) versus the SNR, x. The average BER, Pe, turns out to be:

Pe

1

Z

=

Z

0

Z

0

= = 0

1 1

FBER (x)fSNR (x)dx fBER (x)FSNR (x)dx fBER (x) (x)dx;

(6)


with

(x) fBER (x) = ? dFBER dx :

43, N. 3, AUG. 1994

?

(7)

FBER (x) depends on the modulation format, and in many cases it can be approximated by a Gaussian distribution function, as happens for spread spectrum modulations [17], where

p FBER (x) = Φ(? x); Φ(y) =

Z

y

(8)

ep?t =2 : ?1 2

3

2

(9)

Pe =

1

e?x=2 dx: (x) p 8x

0

1

2 2 2

, called natural spread, is the standard deviation of log(li ), and is assumed independent of i. The double integral in (15) can be solved in closed form, yielding (see Appendix II for the details):

J () = Ue?2= ; where

2

#

Pk

=

? 1 ? e PT l0 bW

k Y

bli : i=1 1 + Gl0 1

(11)

The computation of (b) is done as follows: first, we require that the interferers be distributed within a radius a from the receiver; then, we compute the outage probability (b; a) by averaging (b; k; l) with respect to the number of active interferers, their positions and the shadowing; finally, we find (b) by computing the limit of (b; a) as a ! 1, to get (see Appendix I):

b

( )=

where

Z

? h i ? e 1 ? e?e ?J () ; d p 2 ?1 1

2 2 2

0 = bWr KPT

b = SNR0 ;

(17)

2= b =N G ;

(18)

N = r02 :

(19)

P0 b 1 G i=1 Pi + W ! Z 1 Z 1 k k X Y b = da1 : : : dak F0 G ai + bW fi (ai ) 0 0 i=1 i=1 b; k; l) = P

(16)

U = 2 cosec 2 e2(=) ;

(10)

First, the conditional outage probability, (b; k; l), where l = (l0 ; l1 ; : : :; lk ), is computed under the following conditions: a) k interferers are present; b) the link losses of both the signal and the interferers are fixed ( li ; i = 0; 1; : : :; k); c) all the radios transmit the same power PT . Each received power is an exponentially distributed random variable, with average P i = li PT , pdf fi (a) and cdf Fi(a). In order not to restrict ourselves, we take into account also the presence of a DS spread spectrum modulation, with processing gain G. Note that Schilling et al. [18] showed that, in the Gigahertz band, a channel few MHz wide can be adequately described by the same propagation model as for narrowband channels. Moreover, even when this is not strictly true, the narrowband propagation model can still be considered to be a worst-case model. Hence, the interference power turns out to be reduced by a factor G [19] (G=1 means no spreading at all). The power of the background noise after despreading is W . Note that W is relative to the narrowband information signal, with bandwidth B , and therefore it does not depend on the processing gain, G. Therefore, we have: (

(15)

0

IV. COMPUTATION OF THE OUTAGE PROBABILITY

"

(14)

2rdr e? x Z 1 : J () = dx p G 2 0 ?1 1 + b e?x rr Z

In this case, we obtain: Z

0 PT ; SNR0 = KrW

(12)

(13)

and

N is the average number of interferers which are closer to the

receiver than the transmitter. Based on the previous analysis, the computation of (b), not feasible in closed form, requires only one numerical integration (e.g., using the Gaussian Quadrature Formulas), and can be performed in a very fast and accurate manner. We remark that (b), although not explicitly denoted, depends upon the spatial density of interferers, , the distance between transmitter and receiver, r0 , the outage threshold, b, the processing gain, G, the natural spread of the shadowing, , the thermal noise level, , and the propagation loss factor, . Note also that these quantities are not necessarily independent of each other. In the following, some results will be assessed and the dependence of (b) on the above parameters will be investigated. V. RESULTS The outage probability has been computed for several different operating conditions. In the following, we discuss some different cases, and we study the dependence of on the relevant parameters of the system. A. Absence of noise

Fig. 1 reports the effect of the dB spread, (10= log 10) = (b) is plotted for b = 10 dB, various values of and = 4, corresponding to pure reflection, in the absence of noise and for conventional modulation (G = 1). In Fig. 2 the results for = 3 and 3.5 are also reported, for = 10 dB. In these figures, we note that for G = 1 the average number of admitted interferers at a distance less than the intended transmitter, N , must be roughly one order of magnitude lower than the required outage probability. Finally, we note that for the region of interest the curves are very well approximated by means of straight lines, so that simple expressions could be obtained to relate (b) to the system parameters. More specifically, since 1 ? e?x ' x, for x small, (12) asymptotically reduces to (as N ! 0)

dB .

2

2

b ' cosec e

( )

=)2

4(

b 2= N: G

(20)


43, N. 3, AUG. 1994

The effect of the threshold SNR, b, which is strongly related to the modulation format and the error correction coding, is also shown in Fig. 3. From the obtained results we get the following conclusions: the increase of the dB spread from 6 to 13 dB increases the outage probability by one order of magnitude; the reduction of the link loss factor from 4 to 3 increases (b) by one order of magnitude; the reduction of b by 10 dB will produce a decrease of (b) by half an order of magnitude. B. Presence of noise

Fig. 4 plots the behaviour of (b) for different values of the noise level, SNR0 . It can be seen that the outage probability, (b), exhibits a floor value, 0 (b), as the interference power (or, equivalently, N ) tends to zero. 0 (b) depends on the average signal-to-background noise ratio, SNR0 . For a given SNR0 , the floor value can be evaluated as

? h ? i e ' e =2; (b) = 1 ? e?e d p 2 ?1 where the last approximation holds for small . Z

1

2 2 2

2

(21)

On the other hand, when the interference level is high, (b) is substantially the same as in the absence of noise. We recall that the received power is a random variable, even when r0 , , and PT are fixed. C. Deterministic approaches The previous results are compared with those obtained by considering the following propagation models proposed in the literature: i) neither Rayleigh fading nor log-normal shadowing is considered, ii) only Rayleigh fading is considered. In case i) the outage probability, d (b), can be found by following an approach similar to that of Sousa and Silvester [5], and is given by: "

1=2 #

; for < 1 2 1? = 1; for 1: In case ii), the outage probability, R (b), turns out to be: R (b) = 1 ? e?? () ; d (b)

where

=

1 ? erfc

() = 2 cosec 2 :

(22)

(23)

(24)

These values are compared in Fig. 5, in which we took = 4. It can be seen that the model adopted in the present paper gives worse performance than the “more deterministic” models, as expected. In particular, the greater the natural spread , the greater the error. D. Sensitivity to the propagation constant

Let us consider the expressions of (b), given by (12). If we take as the independent variable, and fix the value of N , we can plot (b) vs. the propagation constant. The considered interval is between 2 and 4 (Fig. 6). We note that, for between 3.5 and 4 (the best approximation of the measurements), the value of (b) is not very sensitive to the exact value of . This means that the (b) can be studied fairly accurately by referring to the easiest case = 4, even when this is not exactly true.

4

E. Sensitivity to the spread

We could also plot (12) vs. 2 , for fixed N and (see Fig. 7). The behavior of the curves is almost linear. For the considered range of the dB spread, (b) presents a variation less than one order of magnitude. F. Spread Spectrum modulation A rough comparison will be done between Direct Sequence Spread Spectrum (SS) modulation and a conventional modulation. In the conventional system, each user transmits on a frequency band B , and suffers from the interference due to a density of users 0 . On the other hand, in a SS system with processing gain G, each user transmits on a frequency band G times larger, BG, but (since the number of users per unit bandwidth is kept constant for a fair comparison) suffers from an average number of interferers G times larger as well. As a result, in the SS system, the interference is reduced by a factor G [19], and the intensity of the interferers is (G) = 0 G. Eq. (18) can therefore be rewritten as

2= 2= b 2 = G r0 = Gb 0 Gr02 2= r02 0 G1?2= : = b (25) (b) is an increasing function of , and 1 ? 2= is always

Since positive, we can conclude that, from the spectrum efficiency point of view, a SS modulation with processing gain G is less efficient than a conventional modulation on G different channels. Note that in the latter case a channel frequency network management is required. VI. EXAMPLE OF APPLICATION

As an example of application of the above analysis, we report some results about the determination of the optimum transmission ranges in a CDMA multihop network. The chosen performance measure is the expected forward progress per slot (EFP), defined in dimensionless terms as:

p p Z 0 = 0 r0;

(26)

where is the local throughput of a node [5] (depending on (b)), and r0 is the distance from the receiver to the transmitter. The optimization of this index, following the approach in [5], has been carried out in [20], and yields the following results. If M = 0 r02 and p is the transmission probability (so that N = r02 = pM p ), for = 4 we get the optimal values p0 ' 0:271 and M0 = G=b, where depends on the propagation model (in particular on ). The maximum EFP turns out to be given by p (G=b)1=4, where = =. In Table I the optimal values p0, , and the corresponding values of Ps = 1 ? (b) and are reported vs. the adopted propagation model, in the absence of noise. The performance in the presence of noise has also been investigated (see Fig. 8). We notice the remarkable reduction of , due to the presence of fading as well as of shadowing, when compared with the deterministic case. VII. CONCLUSIONS The computation of the outage probability, (b), in a packet radio communications environment in the presence of interfer-


43, N. 3, AUG. 1994

ence, noise and fading has been considered. The expression of (b), although not known in closed form, lends itself to a fast and accurate numerical computation. This index can therefore be effectively used in the network design. A comparison of the obtained results with previous works suggests that the model adopted in this paper, namely Rayleigh fading and lognormal shadowing superimposed, gives results which are much worse than those presented in the recent literature, employing deterministic propagation models. Our opinion is that considering the deterministic propagation is not accurate, and very often gives too optimistic results. On the other hand, the model we adopted is probably too pessimistic, and gives worst-case results, which can be viewed as lower bounds to the system performance, whose actual value is somewhere in between these two “extreme” models. Further directions of research are the evaluation of packet radio network performance and the comparison of different kinds of systems. Also, with some changes in the analysis, we believe that this approach can be successful in analyzing cellular systems and other multiple access techniques.

? h i ? 1 ? e?e e?a [1?Ia ()] : d pe 2 ?1

Z =

5

1

2 2 2

2

Finally, the outage probability is given by:

b

( )=

a!1 b; a) =

fli (li ) = p

1

2li

e

?

1 2

? 2 log li ?log(Kr i )

; li > 0 :

(27)

Due to the nature of the Poisson point process, the locations of the interferers are uniformly and independently distributed on Da [5]. Hence,

? h i ? 1 ? e?e ?J () ; d pe 2 ?1

Z

(

lim

1

2 2 2

(32)

where

J () = =

=

2 alim !1 a [1 ? Ia ()] 2 Z 1 e? 2x2 Z 1

?1

dx p

2

0

"

2rdr 1 ?

#

1

1+

b x? ? r0 Ge r

; (33)

which is equal to (15). All the interchanges between limit and integrals are permitted, because of the absolute convergence of the integrals. Eq. (12) is therefore proved.

I. DERIVATION OF EQ. (12) Let Da be a circle of radius a. We start the derivation of (12) by computing the conditional outage probability (b; ajk in Da ), under the condition that k active interferers be present in Da . (b; ajk in Da ) is found averaging out li ; i = 0; 1; : : :; k , and ri; i = 1; : : :; k in (11). Recall that li is a log-normal random variable; its pdf, with log(Kri? ) the mean and 2 the variance of its logarithm, is given by:

(31)

II. DERIVATION OF (16) From the known results [21]

1

Z 0

2rdr 1 + (r)

=

1 2 2 2 cosec ;

e? e = e() =2 ; d p 2 ?1

Z

1

(34)

2 2 2

2

(35)

we obtain:

J () =

e? x 2 cosec 2 b 2= r2e?2(?x)= = dx p 0 G 2 ?1 2= (b; ajk in Da ) = 2 2 b " # cosec e2(=) r02 e?2= = Z 1 k Z a 2r dr Z 1 dl f (l ) Y G bW i i i li i : = dl0 fl (l0 ) 1 ? e? PT l ?2= ; bl 2 a = Ue 1 + Gli 0 0 0 i=1 (28) where U and are defined in (17) and (18), respectively. ? With the substitutions xi = log li ? log(Kri ); i = 1; : : :; k, REFERENCES and = log l0 ? log(Kr0? ), after a few passages (28) becomes: Z

1

2 2 2

2

0

0

0

b; ajk in Da ) =

(

e? h1 ? e?e? ?I ()k i ; d p a 2 ?1

Z

1

2 2 2

(29)

where

Ia () =

? Z a 2rdr 1 ? ; dx pe 2 b x 2 0 a 1 + G e ? rr ?1 1

Z

x

2 2 2

0

(30)

and is defined as in (13). Then, by averaging (29) with respect to the Poisson distribution of the number of interferers, we get:

b; a) = 1 e?a (a2 )k Z 1 e? h X k i ? ? p I () = 1 ? e?e d a k! 2 ?1 k=0 (

2

2 2 2

(36)

[1] F.A. Tobagi, “Multiaccess protocols in packet communication systems ”,vs. IEEE Trans. Commun., vol. COM-28, pp. 468-488, Apr. 1980. [2] W.C. Jakes, Jr., Microwave Mobile Communications, New York, Wiley, 1974. [3] IEEE Trans. Vehicular Tech., Special Issue on mobile radio propagation, vol. VT-37, pp. 3-70, Feb. 1988. [4] S.A. Musa and W. Wasylkiwskyj, “Co-channel interference of spread spectrum systems in a multiple user environment ”, IEEE Trans. Commun., vol COM-26, pp. 1405-1412, Oct. 1978. [5] E.S. Sousa and J.A. Silvester, “Optimum transmission ranges in a Direct-Sequence Spread-Spectrum multihop packet radio network ”, IEEE Journal on Selected Areas in Communications, vol. SAC-8, pp. 762-771, June 1990.


43, N. 3, AUG. 1994

[6] N. Benvenuto, S. Pupolin and G. Guidotti, “Performance evaluation of multiple access spread spectrum systems in the presence of interference ”, IEEE Trans. Vehicular Tech., vol. VT-37, pp. 73-77, May 1988. [7] K.W. Sowerby and A.G. Williamson, “Outage probability calculations for mobile radio systems with multiple interferers ”, Electr. Lett., vol.24, pp.1073-1075, Aug. 1988. [8] W. Gosling, “A simple mathematical model of co-channel and adjacent channel interference in land mobile radio ”, IEEE Trans. Vehicular Tech., vol. VT-29, pp. 361-364, Nov. 1980. [9] R. Prasad, “Throughput analysis of nonpersistent inhibit sense multiple access in multipath fading and shadowing channels ”, European Trans. Telecomm., vol. ETT-2, pp. 29-33, May 1991. [10] J.M.G. Linnartz, Narrowband land-mobile radio networks, Artech House, 1993. [11] Proc. IEEE, Special Issue on Packet Radio Networks, vol. 75, Jan. 1987. [12] D.J. Goodman, “Cellular packet communications”, IEEE Trans. Commun., vol COM-38, pp. 1272-1280, Aug. 1990. [13] F. Hansen, F.I. Meno, “Mobile fading - Rayleigh and Lognormal superimposed ”, IEEE Trans. Vehicular Tech., vol. VT-26, pp. 332-335, Nov. 1977. [14] A.G. Longley, P.L. Rice, “Prediction of tropospheric radio transmission loss over irregular terrain ”, Essa Tech. Report ERS, 79-ITS 67, U.S. Dept. of Commerce, July 1968. [15] H. Suzuki, “A statistical model for urban radio propagation ”, IEEE Trans. Commun., vol. COM-25, pp. 673-680, July 1977. [16] G.C. Clark, Jr., J.B. Cain, Error-correction coding for digital communications, Plenum Press, New York, 1981. [17] L.B. Milstein, et al., “Performance evaluation for cellular CDMA ”, IEEE Journal on Selected Areas in Communications, vol. SAC-10, pp. 660-669, May 1992. [18] D.L. Schilling, et al., “Broadband CDMA for personal communications systems ”, IEEE Commun. Mag., vol. 29, pp. 86-93, Nov. 1991. [19] M.K. Simon et al., Spread spectrum communications, Computer Science Press, 1985. [20] M. Zorzi, S. Pupolin, “Optimum transmission ranges in multihop packet radio networks in the presence of fading ”, to appear on IEEE Trans. Commun. [21] I.S. Gradshteyn, I.M. Ryzhik, Table of integrals, series and products, Academic Press, New York, 1965.

6

FIGURES AND TABLES Fig. 1: Outage probability vs. N , for various values of the spread: = 6 dB (a), 10 dB (b), 13 dB (c); b = 10 dB, = 4, no noise, G = 1. Fig. 2: Outage probability vs. N , for various values of the propagation loss factor: = 3 (a), 3.5 (b), 4 (c); b = 10 dB, = 10 dB, no noise, G = 1. Fig. 3: Outage probability vs. N , for various values of the threshold SNR: b = 0 dB (a), 10 dB (b), 20 dB dB (c); = 4, = 10 dB, no noise, G = 1. Fig. 4: Outage probability vs. N , for various values of the background noise level: SNR0 = 30 dB (a), 50 dB (b), 70 dB (c); b = 10 dB, = 4, = 10 dB, G = 1. Fig. 5: Comparison of (b) vs. N , for different propagation models: deterministic (a), Rayleigh fading only (b), fading and shadowing (c); = 10 dB; b = 10 dB, = 4, SNR0 = 50 dB, G = 1. Fig. 6: Outage probability vs. for various values of N : N = 0.00001 (a), 0.0001 (b), 0.001 (c), 0.01 (d); = 10 dB; b = 10 dB, no noise, G = 1. Fig. 7: Outage probability vs. 2 for various values of N : N = 0.00001 (a), 0.0001 (b), 0.001 (c), 0.01 (d); = 4; b = 10 dB, no noise, G = 1. Fig. 8: Maximum EFP coefficient, , vs. the noise level, , for different propagation models: deterministic (a) and random with = 0 (b), 6 dB (c), 9.375 dB (d), 13 dB (e). Tab. I: Optimal parameters vs. propagation models; = 4; no noise.