. INTRODUCTIONe study the penetration of a ray propagating in a non-
niform random medium. We consider the canonical sce-ario of an external source radiating a plane wave im-inging at an angle � on a half-plane random grid whereach cell can be independently occupied with probabilityj=1−pj, j being the row index, and we ask how deep theay can travel inside the medium before being reflectedack into the empty half-plane; see Fig. 1.Assuming grid cells to be large with respect to the
avelength, the propagation mechanism is described byeans of geometrical optics and only specular reflections
y occupied cells are considered. We analytically estimatehe probability of penetrating up to level k inside the lat-ice before escaping back into the empty half-plane, andalidate the result with numerical experiments for differ-nt obstacles’ density profiles. We also compare our solu-ion with the one given in Ref. 1.
Franceschetti et al.1 considered the same canonicalroblem described in the previous paragraphs in the casen which the probability q=1−p does not depend on theow index j. Such a uniform two-state random grid isnown as the percolation lattice (see Refs. 2 and 3). In thisontext, lattice cells sharing a common side are calledeighbors. Neighbors of occupied sites are called occupiedlusters, and similarly, neighbors of empty sites are calledmpty clusters. One peculiar feature of the percolationattice is that there exists a threshold probability pc
0.59275 at which the lattice appearance suddenlyhanges: for p�p an empty cluster of infinite size that
pans the whole lattice forms, and we say that the modelercolates, while for p�pc all empty clusters are of finiteize, and the model does not percolate. Franceschetti etl.1 were inspired by the possibility of modeling built-uprban areas as percolating lattices with p�pc, and stud-
ed the ray propagation process inside such lattices. Ourresent paper was motivated by their interesting results.Our formulation improves the one in Ref. 1 in several
ays: (i) it is not restricted to the uniform distribution ofmpty cells, but it describes propagation in random lat-ices with general occupation profiles qj; (ii) in the specialase when the occupation profile qj=q for all j, our solu-ion is more accurate than that in Ref. 1 for a wide rangef incidence angles and occupation probabilities; (iii) evenhen compared with an extension of the method in Ref. 1
o nonuniform lattices, it provides more accurate results;iv) the proposed analytical derivation is simpler.
The formula presented in Ref. 1 for the probabilityr�0�k� that the propagating ray reaches a grid level k
nside the lattice before escaping back into the emptyalf-plane was obtained using Martingale theory4 andas given as a function of the occupation probability qnd of the impinging angle �. Numerical experimentshowed that such a formula requires � to be not so farrom 45° and the lattice to be not too sparse, nor tooense, to provide a good approximation of the probabilityistribution sought. Our simpler derivation assumes thathe ray never crosses cells that it has already encounteredlong its path. This allows us to reduce the problem to aimple 1D random walk that does not depend on �. De-
006 Optical Society of America
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2252 J. Opt. Soc. Am. A/Vol. 23, No. 9 /September 2006 Martini et al.
pite this simplification, our solution approximates veryell the probability sought in a wide range of � and p val-es. We note that our assumption clearly does not hold inhe two limiting cases of �→90° or �→0. In the first case,he ray tends to revisit the same empty cells at each levelf the lattice multiple times, and in the limit it does notnter the lattice and Pr�0�k� becomes 0. In the secondase, the ray tends to be reflected back out of the lattice athe first hit on the horizontal face of an occupied cell andr�0�k� simply tends to �j=1
k pj. Moreover, if the lattice isense with obstacles, it is more likely that the ray revisitshe same sequence of cells over and over. However, whenis far from these two limiting values, and the lattice isot too dense, it is reasonable to assume that new cellsre encountered along the path most of the time. Further-ore, high-density lattices are of little interest in our con-
ext, due to the percolation phenomenon described earlierhat inhibits propagation at high-occupation densities.
The case when the source is internal to the lattice andhe distribution of occupied sites is uniform is consideredn Refs. 5 and 6. A different stochastic ray scenario wherehe obstacles are assumed small compared with the wave-ength, and to diffuse isotropically rather than reflect ac-ording to Snell’s law, appeared in Refs. 7 and 8. All of thebove studies arose in the context of modeling propaga-ion of electromagnetic waves in urban areas (see Refs. 9nd 10 for general surveys on this problem). Other appli-ations are in remote sensing and in optical devices,here the estimation of the penetration of waves in dis-rdered media is of interest (see Refs. 11–13).
The remainder of the paper is organized as follows. Inection 2 the propagation model and the mathematicalerivation of Pr�0�k� are presented. Section 3 presents aolution extending the method previously proposed in Ref.. Section 4 provides numerical validation and a compari-on between the two approaches. Final comments andonclusions are drawn in Section 5.
. MARKOV APPROACHet us model the propagation environment by means of aalf-plane infinite lattice of square cells of unitary length.
ig. 1. Example of ray propagation in a random lattice. Left-haeaching level k. Right-hand side, the ray goes beyond level k.
ach cell is either empty, with probability pj, or occupied,ith probability qj=1−pj, j being the row index of the lat-
ice (see Fig. 1). The electromagnetic source is assumed toe external to the lattice, and it radiates a plane mono-hromatic wave impinging on the lattice at a prescribedngle �. Since the scatterers are assumed to be large com-ared to the wavelength �, wave propagation is modeledn terms of parallel rays reflected by the obstacles accord-ng to the geometrical optics laws. Other electromagneticnteractions (i.e., refraction, absorption, diffraction at thedges, and the scattering due to the surface roughness)re neglected. As in Ref. 1, we consider the problem of de-ermining the probability Pr�0�k� that a ray reaches arescribed level k inside the lattice before being reflectedack and escaping in the above empty half-plane. We fo-us on the general case of a nonuniform random lattice,here the density qj of the occupied cells changes with
he level index j. The homogeneous arrangement consid-red in Ref. 1 is a particular case with qj=q=1−p at everyattice level.
We proceed by transforming the problem from a 2D rayropagation problem into a simple 1D random-walk prob-em, where the dependence on � is lost. We formally pro-eed as follows. First, we note that at each level the rayuns into one horizontal face of a square cell, indepen-ently of �, and in a number s of vertical faces propor-ional to � �s= �tan �� or �tan ���. Then we observe thathenever the ray hits a vertical face of an occupied cell, itoes not change its vertical direction of propagation.hus, focusing on the propagation depth it is as thougheflections on vertical faces never occur. Assuming thathe propagating ray never crosses cells that it has alreadyncountered along its path, we consider propagation inhe vertical direction occurring with steps that are inde-endent of each other.Focusing on reflections on horizontal faces, we have
hat a ray proceeding into a generic level j either changesirection of propagation, remaining in the same level, oreeps the same direction of propagation, entering a newevel. The former event takes place with probability qj+1 ifhe ray is traveling with positive direction, or with prob-bility q if the ray is proceeding with negative direc-
e, the ray is reflected back in the above empty half-plane before
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Martini et al. Vol. 23, No. 9 /September 2006 /J. Opt. Soc. Am. A 2253
ion. Accordingly, the ray enters a new level with prob-bility pj+1 or pj−1, depending on its direction ofropagation. Furthermore, if the ray traveling in the posi-ive direction changes direction of propagation an evenumber of times before entering a new level, then theepth level is increased by one; otherwise it is decreasedy one. This situation is formally described by the Markovhain14 depicted in Fig. 2, where states j+ and j− denote aay crossing level j traveling with positive or negative di-ection, respectively.
We now introduce the following notation. We writer�A�B�C� to indicate the probability that a ray intate A reaches state B before going into state C. Accord-ng to this notation and to the Markov chain of Fig. 2, therobability that a ray reaches a grid level k inside the lat-ice before escaping back into the empty half-plane can bexpressed as Pr�0+�k+�1−�. As a matter of fact, when aay reaches the state 1− it escapes from the grid, sincehere are no occupied horizontal faces between level 1 andevel 0. Moreover, a ray always enters a new level travel-ng in the positive direction; therefore a ray alwayseaches state k+ before state k−.
We state our main result as follows:Proposition 2.1
Pr�0+ � k+ � 1−� =p1p2
1 + p1p2�i=0
k−3 qk−i
pk−ipk−i−1
, k � 1. �1�
In Eq. (1) the following convention is used. Consider aeneric summation �i=m
n f�i�. When m=n+1 the value re-urned is 0, while for m�n+1 the value returned is�i=n+1
m−1 f�i�. Accordingly, in Proposition 2.1, the summa-ion returns 0 for k=2, and −�q2 /p2p1� for k=1.
Before proving Proposition 2.1, some observations areppropriate. First of all, we note that when propagationn uniform random lattices is considered our solution re-uces to
Pr�0+ � k+ � 1−� =p2
�k − 2�q + 1, k � 1, �2�
hich simplifies the previously proposed formula of Ref., being independent of the incident angle �. We also notehat for very sparse or very dense lattices we have, as ex-ected,
Fig. 2. Markov chain. The ray propagation in the nonuni
limq→0
Pr�0+ � k+ � 1−� = 1, �3�
limq→1
Pr�0+ � k+ � 1−� = 0. �4�
Finally, we note that our solution is derived assuminghat the propagating ray never crosses cells that it has al-eady encountered along its path. Clearly this assumptionoes not hold whatever the value of � and for all occupa-ion profiles. When � is far from 45°, the ray is more likelyo travel back through the same cells whenever a reflec-ion occurs (see the left-hand side of Fig. 3). On the otherand, when the obstacle density increases, the ray tendso travel over and over on the same sequence of cells (seehe right-hand side of Fig. 3). Accordingly, we expect theroposed solution to be more accurate as the obstacles areore sparse and the incidence angle � is closer to 45°.his is confirmed by the numerical experiments reported
n Section 4.To prove Proposition 2.1, we now state and prove some
reliminary lemmas.Lemma 2.2
Pr��j − 1�+ � j+ � 1−�
=pj
pj + qj Pr��j − 1�− � 1− � �j − 1�+�, j � 2. �5�
Proof of Lemma 2.2. According to the Markov chainepicted in Fig. 2, we can write
r��j − 1�+ � j+ � 1−�
= pj + qj Pr��j − 1�− � �j − 1�+ � 1−�
� Pr��j − 1�+ � j+ � 1−�, j � 2, �6�
hus
r��j − 1�+ � j+ � 1−�
=pj
1 − qj Pr��j − 1�− � �j − 1�+ � 1−�, j � 2. �7�
Now, since the events ��j−1�−� �j−1�+�1−� and�j−1�−�1−� �j−1�+� are mutually exclusive, Eq. (7) cane written as
andom half-plane lattice is modeled as a Markov process.
form r
P
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2254 J. Opt. Soc. Am. A/Vol. 23, No. 9 /September 2006 Martini et al.
To prove that Eq. (18) holds, we need an additional in-uction argument. The base case k=2 trivially givesr�1−�1−�1+�=1. Let us now assume that Eq. (18)olds, and let us compute Pr�k−�1−�k+� for k�2. We ap-ly Lemma 2.3, which in this case is stated as
r�k− � 1− � k+�
=pk−1 Pr��k − 1�− � 1− � �k − 1�+�
pk + qk Pr��k − 1�− � 1− � �k − 1�+�, k � 2. �19�
ubstituting Eq. (18) in the numerator of Eq. (19) we ob-ain
r�k− � 1− � k+�
=Pr�0+ � �k − 1�+ � 1−�
pk + qk Pr��k − 1�− � 1− � �k − 1�+�, k � 2. �20�
ow, we note that by Eqs. (15) and (17),
r�0+ � k+ � 1−�
=pk Pr�0+ � �k − 1�+ � 1−�
pk + qk Pr��k − 1�− � 1− � �k − 1�+�, k � 2, �21�
nd thus, by comparing Eq. (20) with Eq. (21), we can ar-ue that
Pr�k− � 1−�k−� =Pr�0+ � k+ � 1−�
pk, k � 2, �22�
hich concludes the proof.
. MARTINGALE APPROACHn Ref. 1 Franceschetti et al. presented an analytical deri-ation based on Martingale theory, obtaining a solutionor Pr�0�k� that depends on the ray incident angle � onhe lattice. Their method was restricted to the case of uni-orm random lattices; however, it can also be generalizedo nonuniform random lattices. The detailed derivationnd discussion of the range of validity of the approach in
his case is the subject of a companion paper.15 Next, weriefly summarize the main steps required for this gener-lization, and then we compare the results with our ap-roach presented in the previous section.With reference to Fig. 4, we define the following sto-
hastic process:
rn = r0 + �m=1
n
xm, n � 0, �23�
here rn is the row where the reflection n+1 takes placei.e., it is the vertical component of vector r̄n) and
xm = rm − rm−1, m � 1. �24�
e now express the probability of reaching level k insidehe lattice as
Pr�0 � k� = �i
Pr�0 � k�r0 = i�Pr�r0 = i�, �25�
here Pr�r0= i� is the probability mass function of therst jump r0 and Pr�0�k �r0= i� is the probability that aay goes beyond level k conditioned to the level where therst reflection occurs.As far as Pr�r0= i� is concerned, proceeding along the
ame lines of Ref. 1, yields
Pr�r0 = i� = q1, i = 0,
qe1
+ ��j=1
i−1
pej
+� , i � 0,� �26�
here pej
+ =1−qej
+ =pjtan �pj+1 is the effective probability
hat a ray, traveling with positive direction and angle �hrough level j, reaches level j+1.
We now consider the second term of Eq. (25), i.e.,r�0�k �r0= i�. Following the same procedure as in Ref. 1,
t can be shown that
ig. 4. (Color online) Martingale approach. The propagationrocess is modeled as the sum of many vectorial variables. Theth element of the stochastic process �rn ,n�0� is the verticalomponent of the vector r̄n. Under some assumptions, the process
m=1n xm behaves as a martingale with respect to the sequence
x � (Ref. 1).
m
Elu
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wt
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2256 J. Opt. Soc. Am. A/Vol. 23, No. 9 /September 2006 Martini et al.
Pr�0 � k�r0 = i� � 0, i = 0,
i/k, 0 � i � k,
1, i � k.�27�
quation (27) is derived assuming that the jumps xn fol-owing the first one are independent, identically distrib-ted, and zero mean, and using Martingale theory.4
Now, substituting Eqs. (26) and (27) into Eq. (25), afterome mathematical manipulations (see Appendix B), theollowing closed-form solution is obtained:
Pr�0 � k� = �i=1
k−1 i
kp1qei
+�j=1
i−1
pej
+ + p1�j=1
k−1
pej
+ , �28�
hich represents a generalization of the result in Ref. 1 tohe nonuniform case.
. NUMERICAL COMPARISONe now validate our proposed approach with numerical
xperiments, and we provide a comparison with theethod in Ref. 1 and its generalization presented in Sec-
ion 3.In the following we refer to our proposed approach as
he Markov approach (MKV), while we refer to the ap-roach in Ref. 1 and its generalization as the Martingalepproach (MTG).As a reference, the propagation depth has been evalu-
ted by means of computer-based ray tracing experi-ents. N=100 random lattices with the same scatterers’
ensity have been generated and M=500 rays have beenaunched from different entry positions for every grid. Bysing the same numerical procedure described in Ref. 1,he probability Pr�0�k� has been estimated from the col-ection of paths in the first Kmax=32 levels of the lattice.
We define the following error figures:
�k ��PrR�0 � k� − PrP�0 � k��
maxk
�PrR�0 � k�� 100, �Prediction Error�,
�29�
��� �1
Kmax�k=1
Kmax
�k, �Mean Error�, �30�
�max = maxk
��k�, �Maximum Error�, �31�
here the subscript R indicates the value estimated withhe reference approach, and the subscript P stands for theame value computed by means of either Eq. (1) or Eq.28).
In the remainder of this section, we first consider thease of a homogeneous grid, providing a comparison withhe result in Ref. 1. Then we consider the nonuniform gridase.
. Uniform Random Latticess a first test case, a sparse grid is considered with q0.05. In Fig. 5, we report the estimated Pr�0�k� as a
unction of the penetration index k, for different values of. It is evident that the MKV approach describes very well
he propagation in the random medium in this case. Theange of ��� is from 0.23% (for �=45°) to 1.11% (for �15°), while 0.74% ��max�1.42%. On the other hand, theTG approach does not perform very well, resulting in
alues 2.16% � ����20.37% and 3.87% ��max�25.14%.his is not surprising since the MTG approach is not ex-ected to work well for low-density media.1
A similar behavior is observed when q is increased to.15. In Fig. 6 it is shown that the MKV approach againives the best prediction of the propagation depth in thisase when �=45° ����=0.46% and �max=0.68% �. As ex-ected from the theory, results become worse as � di-erges from 45°. Nevertheless, the MKV approach outper-orms the MTG approach for all considered incidentngles, and the error is also more stable for different val-es of the penetration index k and of the angle �.When q increases even further and the grid becomesore dense (i.e., q=0.25 and q=0.35), prediction results of
he MKV approach become worse (see Fig. 7). In fact, thessumptions behind the method fail: The ray tends toravel over and over through the same sequence of cellsnd independence is lost. Nevertheless, the MKV ap-roach is more stable than the MTG approach with re-pect to both the incidence angle � and the lattice depth k,nd prediction results are still good for a wide range of in-ident angles.
. Nonuniform Random Latticese now consider the nonuniform grid case with various
bstacles’ density profiles. The profiles depicted in theeft-hand side of Fig. 8 are increasing linear profiles of theype
q�x� = q + �x − 1�, �32�
hile the profiles depicted in the right-hand side of thegure are double exponential profiles of the type
q�x� = � exp��x − L�, x � L,
exp��L − x�, x � L,�33�
being the lattice depth. The parameters’ values corre-ponding to the plots in Fig. 8 are q=0.05,L=Kmax/216, and and as described in Table 1.We first consider the case �=45° for different density
rofiles. Results for the linear profiles are depicted in Fig.. It is evident that the MKV approach outperforms theTG approach in all the considered cases. For thisethod, the values of ��� range from 0.29% (profile L1) to
.71% (profile L4). On the other hand, performance of theTG approach is very sensitive to the considered profileith ��� increasing with the slope of the occupation pro-le from 1.11% (profile L1) to 7.82% (profile L4).Results for the double exponential profiles are depicted
n Fig. 10. Similar observations hold in this case. For theKV approach, the values of ��� range from 0.31% (profileB1) to 1.09% (profile DE4), while for the MTG approach,e have 1.44% � ����10.25%. It is worth noting thatKV satisfactorily performs even in correspondence with
evel L=16 where the discontinuity in the occupancy pro-le of Eq. (33) occurs. On the contrary, when the MTG ap-roach is used, we can observe nonnegligible errorsround the level L=16 (see Fig. 10). This is because in the
Msetcop
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Martini et al. Vol. 23, No. 9 /September 2006 /J. Opt. Soc. Am. A 2257
TG approach, ray jumps following the first one are con-idered as a single mathematical entity, i.e., they are gov-rned only by Pr�0�k �r0= i� [see Eq. (25)]. On the con-rary, in the MKV approach, each single jump isonsidered. As a consequence, abrupt changes in the slopef Pr�0�k� due to discontinuities in the obstacles’ densityrofile are correctly detected and reconstructed.
ig. 5. Uniform random lattice with q=0.05. We plot Pr�0�k� veashed curves represent reconstructions obtained by the MKV a
ig. 6. Uniform random lattice with q=0.15 We plot the predicKV approach; right-hand side, MTG approach.
We now consider a second set of experiments, varyinghe incident angle �. We report the results relative to theorst cases, i.e., the most variable profiles L4 and DE4.imilar considerations also hold true for the remainingrofiles. First of all, by looking at Fig. 11, we observe thathe MKV approach outperforms the MTG approach for allonsidered values of �. As expected from theory, the per-
for different values of �. Crosses denote reference data; solid andh and the MTG approach, respectively.
ror �k versus k for different incidence angles �. Left-hand side,
rsus k
tion er
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2258 J. Opt. Soc. Am. A/Vol. 23, No. 9 /September 2006 Martini et al.
ormance of the MKV approach slightly weakens whenhe incidence angle � diverges from 45°. In the worst case=15°, we have ���=3.16% and ���=3.59% for the profile4 and the profile DE4, respectively. On the contrary, theTG approach provides reconstructions that are muchore sensitive to the incident angle �. In the worst case
=15°, we have ���=17.31% and ���=19.26% for the pro-le L4 and the profile DE4, respectively. Finally, we canompare the maximum and the average error values. TheTG approach shows a larger gap between the two val-
es, thus showing larger variance of the error at differentattice levels.
. CONCLUSIONSn this paper we have statistically described the rayropagation process inside nonuniform random lattices.e have assumed a far-external source scenario and
arge, lossless scatterers, whose density changes with theattice depth. Our approach is based on the key observa-ion that in evaluating the propagation depth it is ashough reflections on vertical faces of occupied cells never
ig. 7. Uniform random lattice with q=0.25 and q=0.35. We ploand side, MKV approach; right-hand side, MTG approach.
Fig. 8. Density profiles q�x� versus the lattice depth x. Left-ha
ccur, since they do not change the vertical direction ofropagation of the ray. This observation has allowed us toransform the problem from a two-dimensional ray propa-ation problem into a simple one-dimensional random-alk problem. By modeling propagation in terms of aarkov chain, we have derived a simple closed-form ana-
ytical formula. The solution estimates the propagationepth as a function of the obstacle distribution, and it isndependent of the incident conditions.
Numerical experiments have confirmed the effective-ess of our approach, which is accurate for a wide range of
ncident angles and obstacle densities. They have shownmprovement upon previous results as well, in particular,
rediction error �k versus k for different incidence angles �. Left-
e, linear profiles; right-hand side, double-exponential profiles.
t the p
foi
os
Fr
Fc and th
Ft
Martini et al. Vol. 23, No. 9 /September 2006 /J. Opt. Soc. Am. A 2259
or low-density propagation media. Our approach alsoutperforms generalizations of previous methods to thenhomogeneous case.
ig. 9. Linear density profiles. Estimated values of Pr�0�k� vepresent predictions obtained by the MKV approach and the M
ig. 10. Double-exponential density profiles. Estimated valuesurves represent reconstructions obtained by the MKV approach
ig. 11. Linear and double-exponential profiles worst cases. We che mean error ��� and the maximum error � versus � for the
max
Possible extensions of the present work can be aimed atvercoming limitations that the percolation model intrin-ically exhibits in describing wave propagation in disor-
k. Crosses denote reference data; solid and dashed line curvesroach, respectively.
�k� versus k. Crosses denote reference data; solid and dashede MTG approach, respectively.
r the two density profiles with the worst prediction error. We plotapproach and the MTG approach.
ersusTG app
of Pr�0
onsideMKV
dsps
atitnt
AIh
S
w
AIs
Br
2260 J. Opt. Soc. Am. A/Vol. 23, No. 9 /September 2006 Martini et al.
ered media. With care about trading off accuracy versusimplicity, we can think about introducing in our modelhysical phenomena such as absorption, scattering due tourface roughness and small obstacles, and diffraction.
Finally, we stress that the percolation model can findpplication in a wide range of applied problems arising inhe framework of wireless communications, remote sens-ng, and radar engineering. Our solution based on theheory of Markov chains may be of interest in all the sce-arios that are studied in percolation theory, providedhat the ray approach is justified.
PPENDIX An this appendix we show that Eq. (1) follows if Eq. (16)olds true. By substituting Eq. (16) into Eq. (15) we get
ATRNctNM
a8a
R
Pr�0+ � k+ � 1−� = Pr�0+ � �k − 1�+ � 1−�
�pk−1pk
pk−1pk + qkPr�0+ � �k − 1�+ � 1−�,
k � 2. �A1�
ince by assumption
Pr�0+ � �k − 1�+ � 1−�
=p1p2
1 + p1p2�i=0
k−4 q�k−1�−i
p�k−1�−ip�k−1�−i−1
, k � 2, �A2�
e can write
Pr�0+ � k+ � 1−� =p1p2
1 + p1p2�i=0
k−4 qk−1−i
pk−1−ipk−2−i
pk−1pk
pk−1pk + qk�p1p2�� �1 + p1p2�i=0
k−4 qk−1−i
pk−1−ipk−2−i��
=p1p2
1 + p1p2�i=0
k−4 qk−1−i
pk−1−ipk−2−i
pk−1pk�1 + p1p2�i=0
k−4 qk−1−i
pk−1−ipk−2−i�
pk−1pk + pk−1pk�i=0
k−4 qk−1−i
pk−1−ipk−2−i+ qkp1p2
=p1p2
1 + p1p2�i=0
k−4 qk−1−i
pk−1−ipk−2−i+ qk
p1p2
pk−1pk
=p1p2
1 + p1p2 �i=−1
k−4 qk−1−i
pk−1−ipk−2−i
=p1p2
1 + p1p2�i=0
k−3 qk−i
pk−ipk−i−1
, k � 2. �A3�
PPENDIX Bn this appendix we show how to obtain Eq. (28) by sub-tituting Eqs. (26) and (27) into Eq. (25).
Pr�0 � k� = �i=1
k−1 i
kp1qei
+��j=1
i−1
pej
+� + �i=k
�
p1qei
+��j=1
i−1
pej
+� .
�B1�
y expressing qei
+ in terms of pei
+ , the second term of theight-hand side of Eq. (B1) can be rewritten as
�i=k
�
p1qei
+�j=1
i−1
pej
+ = �i=k
�
p1�j=1
i−1
pej
+ − �i=k
�
p1pei
+�j=1
i−1
pej
+
= p1�j=1
k−1
pej
+ + p1 �i=k+1
�
�j=1
i−1
pej
+ − p1�i=k
�
�j=1
i
pej
+
= p1�j=1
k−1
pej
+ + p1 �i=k+1
�
�j=1
i−1
pej
+ − p1 �i=k+1
�
�j=1
i−1
pej
+
= p1�j=1
k−1
pej
− . �B2�
CKNOWLEDGMENTShis work was supported in Italy, in part by the Center ofEsearch And Telecommunication Experimentations forETworked communities (CREATENET, www.
reatenet.it) and by “Study and Development of Innova-ive Smart Systems for Highly Reconfigurable Mobileetworks,” Progetto di Ricerca di Interesse Nazionale—iur Project COFIN 200509984_001.Corresponding author Andrea Massa may be reached
t the address on the title page, by phone at 39-0461-82057, fax at 39-0461-882093, or e-mail [email protected].
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