IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 57, NO. 5, MAY 2009

1

Comments on “Probability Distributions for the Number of Radio Transceivers Which Can Communicate with One Another” Alberto Zanella, Member, IEEE, Mauro Stramazzotti, Flavio Fabbri, Enrica Salbaroli, Student Member, IEEE, Davide Dardari, Senior Member, IEEE, and Roberto Verdone, Member, IEEE

Abstract—We discuss the correctness of one of the main results of [1] about the probability density function of the distance between two audible nodes in an infinite 2-dimensional scenario. We prove that result [1, eq. (7)] is wrong and derive a more general expression, which is valid for an infinite d−dimensional area. It is worth noting that, although the results on the distribution of the distance between two nodes are wrong, the other results of [1], and in particular the fact that the number of audible nodes in an arbitrary area is a Poisson r.v. is correct. Index Terms—

I. I NTRODUCTION

I

N these years, the increasing interest towards wireless ad hoc and sensor networks has stimulated the research for connectivity models valid for decentralized architectures. Among the contributions appeared in the open literature in the past few years, the paper of Orriss and Barton [1] was important as it represented one of the first attempts to study the connectivity properties of networks composed of randomly located nodes in a propagation environment characterized by a distance dependent loss and log-normal shadowing. The following results were obtained in [1] for an infinite 2dimensional area: • a) the probability density function (p.d.f.) of the distance between a pair of audible nodes (a node is audible by others if the power loss does not exceed a given threshold); • b) the probability distribution of the number of audible nodes with respect to a node taken as a reference. This distribution was proven to be Poisson and its mean was evaluated; • c) the distribution for the number of audible nodes within a finite area of the plane. This distribution is still Poisson and its mean was evaluated. Results a) and b) were also extended to the case where the path-loss changes at a specified distance from the transmitting node [1, Section V]. In this letter, we revise the results obtained in [1]; more specifically, we prove that the p.d.f. expression for a) is wrong Paper approved by F. Santucci, the Editor for Wireless System Performance of the IEEE Communications Society. Manuscript received December 18, 2007; revised September 24, 2008 and November 27, 2008. The authors, with the exception of M. Stramazzotti, are with WILAB c/o DEIS (University of Bologna), Viale Risorgimento 2, 40136 Bologna, Italy (e-mail: [email protected], {flavio.fabbri, enrica.salbaroli, davide.dardari, roberto.verdone}@unibo.it). M. Stramazzotti is with Vodafone IT, Sottopassaggio M.Saggin n.2, 35131 Padova, Italy (e-mail: [email protected]). Digital Object Identifier 10.1109/TCOMM.2009.05.070065

and derive a correct one, which is valid for an infinite ddimensional area. We also provide the correct expression for the p.d.f. of the distance in the two-slope case. As far as the validity of the other results of [1] is concerned, results b) and c) are still correct since do not require the knowledge of a). II. R EVIEW OF EQ . (7) OF [1] The scenario considered in [1] is characterized by an infinite 2-dimensional area where the nodes are distributed according to a Poisson point process (PPP). Under this hypothesis, the probability that there is one node in an area of size δA is ρδA, where ρ is the density of nodes. Let us consider a node A which attempts to communicate with another one (say, B) located in the origin of a reference coordinate system. The channel model is affected by distance-dependent exponential path-loss component and log-normal shadowing. Nodes A and B are l1 -audible if the path-loss in decibels L does not exceed a given threshold l1 , that is L = k0 + k1 ln R + S ≤ l1 ,

(1)

where k0 and k1 are propagation constants, S is the shadowing term, which is assumed to be a Normal random variable with zero mean and variance σ 2 , and R represents the distance between the nodes. Let us also denote by C the event of l1 audibility between A and B.1 The steps followed in [1] to obtain fR|C (r|C) can be summarized as follows: • i): evaluation of the conditional density of R given S = s (see [1, eq. (6)]); • ii): evaluation of the joint pdf of R and S (conditioned on the event C); • iii): evaluation of the pdf of R (conditioned on the event C), by integrating out S from the joint p.d.f. of R and S (see [1, eq. (7)]). Let us consider step ii), which requires the evaluation of fR,S|C (r, s|C) = fR|S,C (r|s, C)fS|C (s|C).

(2)

To evaluate (2), the distribution of S|C is assumed in [1] to be normally distributed with zero mean and variance σ 2 . This would be true in the absence of any condition on the audibility, but this assumption is wrong under condition C. To prove this, we show that the mean value of S|C is nonzero. This can be 1 In this letter we have used the same terminology of [1], the only difference is related to fR (r) and fS (s), which denoted the p.d.f. of the distance and of the shadowing between two l1 -audible nodes, respectively. Here, fR (r) and fS (s) denote the p.d.f. of the distance between two arbitrary nodes (regardless of the audibility) and the p.d.f. of the corresponding shadowing sample. The p.d.f. of the distance between two l1 -audible nodes and the corresponding distribution of S are here denoted by fR|C (r|C) and fS|C (s|C).

c 2009 IEEE 0090-6778/09$25.00 

2

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 57, NO. 5, MAY 2009

checked by observing that in the presence of the events C and R = {the distance R is equal to r}, the distribution of S conditioned on C and R becomes

0

= 

fR|C(r|C)

0.015

0.01 σ=6

(4) 0

l1 −k0 −s k1 e

III. D ISTRIBUTION OF THE D ISTANCE B ETWEEN AUDIBLE N ODES Since the evaluation of fS|C (s|C) in (2) is not straightforward, the right expression for fR|C (r|C) and its generalization to an infinite d-dimensional area can be obtained using the following alternative approach. We assume that nodes are spatially distributed in a (d − 1)-dimensional sphere of radius Rs and denote by x = (x1 , . . . , xd ) the position of A with respect to B (whose position is assumed to be in the origin of the reference coordinate system). The distance between the two nodes is denoted by R = ||x||.2 . Let us consider R = r (event R). The probability that inequality (1) is verified is given by Prob {L ≤ l1 |R}

= =

Prob{C|R}  C(r)   k0 + k1 ln r − l1 1 √ erfc , (6) 2 2σ

where erfc(·) denotes the complementary error function. Note that eq. (6) is similar to [4, eq. (21)], the only difference is that here the path-loss is considered, instead of the signalto-noise ratio in [4]. Starting from (6) and by means of the Bayes theorem, we can derive the probability distribution of the distances (conditioned on the event C) as Prob{C|R}fR (r) Prob{C} C(r)fR (r) , =  Rs C(r)f (r)dr R 0

fR|C (r|C) =

indicates the euclidean norm of the vector x.

σ=3

0.005

where h(s)  0 fR|C (r|C)dr > 0 with limits h(−∞) = 1 and h(+∞) = 0. The expectation of S|C becomes  +∞ s2 K se− 2σ2 h(s)ds (5) E {S|C} = √ 2πσ −∞  +∞  l1 −k0 −s  l1 −k0 −s s2 Kσ ds. = −√ e− 2σ2 e k1 h e k1 2πk1 −∞ Since the three integrand functions in (5) are positive, the result of the integral is also positive. This proves that E {S|C} cannot be zero.

2 ||x||

0.02

s2

Ke− 2σ2 √ h(s), 2πσ

Simulation Analytical Eq. (7) of [1]

σ=0

0.025

2

s − 2σ 2

Ke fS|C,R (s|C, R) = √ u(l1 − k0 − k1 ln r − s), (3) 2πσ where K is a normalizing constant and u(x) is the unitary  1 if x ≥ 0 step function u(x)  . Starting from (3), the 0 if x < 0 p.d.f. of S|C can be calculated as  +∞ fS|C (s|C) = fS|C,R (s|C, R)fR|C (r|C)dr

k0=30dB, k1=17.37dB, l1=104dB

0.03

(7)

Fig. 1.

0

40

80 r (m)

120

160

The p.d.f. of distance between two audible nodes.

where fR (r) has the expression fR (r) =

d d−1 r , Rsd

0 ≤ r ≤ Rs .

(8)

Hence, by substituting (8) into (7) and letting Rs → ∞, we obtain fR|C (r|C)

=

C(r)rd−1 ∞ C(r)rd−1 dr 0

= C(r)d rd−1 e

2 2

σ d − kd (l1 −k0 ) − 2k2 1

e

1

.

(9)

Result (9) was also derived in [2, eq. (10)] for the particular case d = 1. In the 2-dimensional case, equation (9) becomes fR|C (r|C) = r e− k1 (l1 −k0 +σ 2

2

/k1 )

 erfc

k0 − l1 + k1 ln r √ 2σ

 .

(10) An equivalent expression was also obtained in [3, eq. (20)]. Note that [1, eq. (7)] is very similar to the previous expression, it differs from √ (10) only for the presence of an additional coefficient 2σ/k1 in the argument of the erfc function.3 The comparison between (10) and [1, eq. (7)] is shown in Fig. 1 for different values of σ, with k0 = 30 dB, k1 = 17.37 dB (which corresponds to a distance-loss exponent equal to 4) and l1 = 104 dB. To double check the validity of the analysis, Monte Carlo simulations have been added where 105 trials have been carried out to obtain each curve. The largest 95% confidence interval is 2 · 10−4 and has been obtained for r = 55.5 m and σ = 3. The figure shows that [1, eq. (7)] tends to underestimate the distance. In accordance with (10), we also report the p.d.f. of the distance between two audible nodes in the two-slope path-loss 3 The 1 erfc 2

Φ(x) used in [1, eq. (7)] can be easily written as  function  − √x . 2

ZANELLA et al.: COMMENTS ON “PROBABILITY DISTRIBUTIONS FOR THE NUMBER OF RADIO TRANSCEIVERS . . . ”

case fR|C (r|C) =

⎧ ⎪ ⎨ ⎪ ⎩

 r0 0

 r0 0

r d−1 C(r)  xd−1 C(x)dx+ r+∞ xd−1 C  (x)dx 0 

r C (r)  xd−1 C(x)dx+ r+∞ xd−1 C  (x)dx d−1

if r ≤ r0 if r > r0

3

in an infinite 2-dimensional scenario has been confuted and an alternative expression, valid for an infinite d−dimensional , area, has been presented. The validity of the other results of [1] based on expression [1, eq. (7)] has been also discussed.

0

(11)   ln r−l1 where C  (r) = 12 erfc k2 +k√3 2σ . Note that the integrals in (11) can be computed in closed form for the case d = 2. Equation (11) evaluated for d = 2 replaces the one found in [1, Section V] (just after [1, eq. (20)]). Now, let us briefly discuss the validity of results b) and c) of [1]. An extension of result b) was given in [4], where it was shown that the distribution of audible nodes is still Poisson even in the presence of arbitrary channel randomness. These results can be obtained in a more general way by using the Marking Theorem [5], valid under conditions which are more general than (1). Result c) can be seen as a special case of the Marking Theorem too. IV. C ONCLUSIONS The correctness of the result [1, eq. (7)] about the probability density function of the distance between two audible nodes

R EFERENCES [1] J. Orriss and S. K. Barton, “Probability distributions for the number of radio transceivers which can communicate with one another,” IEEE Trans. Commun., vol. 51, no. 4, Apr. 2003. [2] D. Dardari, “On the connected nodes position distribution in ad hoc wireless networks with statistical channel models,” in Proc. IEEE International Conference on Communications (ICC), pp. 4741-4745, June 2007. [3] S. Mukherjee, D. Avidor, and K. Hartman, “Connectivity, power, and energy in a multihop cellular–packet system,” IEEE Trans. Veh. Technol., vol. 56, no. 2, pp. 818-836, Mar. 2007. [4] D. Miorandi and E. Altman, “Coverage and connectivity of ad hoc networks presence of channel randomness,” in Proc. IEEE INFOCOM, 24th Annual Joint Conference, vol. 1, pp. 491-502, 13-17 Mar. 2005. [5] J. F. C. Kingman, Poisson Processes. Oxford University Press, 1993.