Poisson-Voronoi Spanning Trees with Applications to the...

Poisson-Voronoi Spanning Trees with Applications to the Optimization of CommunicationNetworksAuthor(s): François Baccelli and Sergei ZuyevSource: Operations Research, Vol. 47, No. 4 (Jul. - Aug., 1999), pp. 619-631Published by: INFORMSStable URL: http://www.jstor.org/stable/223165Accessed: 15/09/2010 12:17

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POISSON-VORONOI SPANNING TREES WITH APPLICATIONS TO THE OPTIMIZATION OF COMMUNICATION NETWORKS

FRANPOIS BACCELLI INRIA-Ecole Normale Supgrieure, Departement de Mathematiques et d'Informatique, LIENS, 45 Rue d'Ulm, 75230 Paris, Cedex 05, France,

[email protected]

SERGEI ZUYEV Statistics and Modelling Science Department, University of Strathclyde, 26 Richmond Street, Glasgow G1 1XH, UK

[email protected]. uk

(Received January 1997; revision received October 1997; accepted December 1997)

We define a family of random trees in the plane. Their nodes of level k, k = 0, . . ., m are the points of a homogeneous Poisson point process 11k, whereas their arcs connect nodes of level k and k + 1, according to the least distance principle: If V denotes the Voronoi cell w.r.t. Hk+1 with nucleus x, where x is a point of rlk+ 1 then there is an arc connecting x to all the points of Hlk that belong to V. This creates a family of stationary random trees rooted in the points of IIm. These random trees are useful to model the spatial organization of several types of hierarchical communication networks. In relation to these communication networks, it is natural to associate various cost functions with such random trees. Using point process techniques, like the exchange formula between two Palm measures, and integral geometry techniques, we show how to compute these average costs as functions of the intensity parameters of the Poisson processes. The formulas derived for the average value of these cost functions can then be exploited for parametric optimization purposes. Several applications to classical and mobile cellular communication networks are presented.

1. INTRODUCTION

Consider the following distribution problem: A family of concentration points is given, characterized by their random coordinates in the plane. Another family of subscribers is also given, with a similar characterization via the set of their random coordinates. A typical example, which is the one considered throughout this paper, is when both point processes are homogeneous Poisson processes, although more general cases can also be considered along the same lines (e.g., periodic patterns or Gibbs fields).

Assume one has to connect each subscriber to exactly one concentrator. A typical situation is where the concentrators are all linked by a connected network, so when each subscriber is connected to one concentrator, it is actually interconnected to all subscribers as well.

For connecting each subscriber to one concentrator, one can use either a direct link or an indirect link via some distribution point. A distribution point is a location where several links originating from subscribers can be grouped into one further link to a concentrator with some economy of scale (see Figure 1).

The first problem we consider in this paper is that of the optimal use of such distribution points. We do not address the problems of a minimal spanning tree (e.g., a Steiner tree structure), which is a quite different combinatorial optimization question. The problem of finding the optimal configuration is known to be NP-hard, although there exist different algorithms allowing to approximate the optimal solution for not too large number of points (see, e.g.,

Hwang et al. 1992). We restrict ourselves to a parametric optimization that consists of introducing an independent Poisson process of distribution points and determining the intensity of this process that minimizes a cost function, which reflects the economy of scale alluded to above. The general assumption is that connections are always to the closest point (i.e., each subscriber is connected to the closest distribution point, and each distribution point to the closest concentrator); the Voronoi cells associated with the Poisson point processes, therefore, play a central role in the construction of these trees.

The paper is structured as follows. Section 2 gives the basic model and a precise definition of the cost function, together with the main characteristics of the optimal parametric model. Section 3 focuses on an extension of this to a hierarchical model with several layers, which was proposed in Baccelli et al. (1996), to describe the structure of certain communication networks, and which includes the basic model of Section 2 as a special case. Finally, Sections 4 and 5 give some extensions of the optimization method proposed in Sections 2 and 3 to related problems arising in communication network modeling, and also based on Poisson-Voronoi tessellations and trees: a non purely hierarchical model is considered in Section 4, a class of infinite spanning graphs in Section 5.1, and a localization problem in mobile communications in Section 5.2.

Although the models described throughout the paper are originating from communication problems, it is worth- while stressing that they are generic in that they could, in

Subject classifications: Probability: stochastic model applications. Networks/graphs: stochastic. Communications, transportation: network models. Area of review: TELECOMMUNICATIONS.

Operations Research 0030-364X/99/4704-0619 $05.00 Vol. 47, No. 4, July-August 1999, pp. 619-631 619 ?D 1999 INFORMS

620 / BACCELLI AND ZUYEV

Figure 1. Links in a three-level model.

/ ...~.

/ ,. _',' '

principle, be applied to other planar or spatial distribution problems where randomness is intrinsic.

2. THREE-LEVEL CONNECTION SYSTEMS

Consider three independent homogeneous Poisson processes I7Ii (i = 0, 1, 2) in the plane R2 with intensities A1, representing subscribers (process flu), distribution points (process 1I1), and concentrators (process 112), respectively.

Random trees are built from any realizations of these three point processes as follows. Each point of the process flo is connected to a point of process LI1 according to the least distance principle. Similarly, each point of LI1 is connected to the closest point of the process H2 (this closest point is almost surely uniquely defined under the above Poisson and independence assumptions).

A typical realization of such Poisson-Voronoi trees is shown in Figure 7.

There is a cost function associated with such random trees. This cost function is perhaps best described through a physical example. In this example, the connections represent links. Each link has an associated cost which consists of two parts:

1. The capacity cost, which represents the cost of the communication medium (e.g., optical fiber, copper, etc.).

2. The infrastructure cost, which represents the cost of the civil engineering (trench, poles, etc.) that supports the medium.

In our example, the capacity cost for connecting a point of fli to the closest point of li +1 is given by a function of the form ajj+jrai+l, where r is the distance between the points and aii+ ,, aii+1l are some nonnegative parameters. This example of nonlinear dependence in the variable r (more general cases, which include the immediate generalization to functions of the type 1p aiPi ra1,i+1, will be considered in the next section) stems from the fact that the type of medium to be used depends on the distance r.

The infrastructure cost is also assumed to be given by a function of the form bji+j rt ij+1, where bij+l jii+1 > 0. Note that the cables from a distribution point to a concentrator share the same civil engineering, and it is reasonable to assume that the cost of civil engineering does not depend on the number of cables which are grouped there.

The various elements of this cost structure are illustrated in Figure 1.

Our first question is the following: Given the intensities o, k2 of the subscriber and concentrator point processes

and a cost C1 associated with the introduction of a distribution point, what is the intensity A1 of the distribution point process that minimizes the average total cost?

More precisely, the cost function is the following:

def G G =?G = E0 [ E [C1 + b1,21Ydo"2 Lye

E H1f0(2) Lyi ER1n vo (I2

+ a1,2 ly I 1,2NY

+ E (bo,l xj -yyi I xjElo nVy, (HIi)

+ a 0,1 |x1 - yi I 01)]1* (1)

Here and below,

* E? denotes the expectation with respect to the Palm distribution Pk, with respect to the point process Hk;

* V.(flk) is the Voronoi cell constructed with respect to I1k, with nucleus x E Ilk;

* X. is the number of Hl0-points linked to the point x.

For more on definitions and properties of these objects and the ones used below, see the Appendix.

Using Neveu's exchange formula (27), which describes the relationship between the two Palm distributions Po and P2, the expression for the cost function simplifies to

-Ci + AE? b1,2 ZO 132 + a 1,2 ZO ai,2K

+ E (b 0,1 JxjI 30,1 + a 0,1 xj1 I01)1 , (2) xjE H0 onVo(H1)

where z0 denotes the point of the process El2 that is the closest to 0.

On the other hand, for all a > 0, if one denotes SkI(u) the open ball centered in u and with radius kL,

Eolzoa = E? IXI 1 {EI2(SkI(0)) = 0}H2 (dx)

= E x~a 1 {fl2(Skl(O)) = 0}fI2 (dx)

= A2 f |X|a P?[ElO(SkI(-X)) = 0] dx

= 2i7rA 2 ra+l exp {-A2-7Tr2} dr = ( (+i), (3)

where the second equality follows from the independence of El1 and El2, and the third from Campbell's refined formula (F(*) here is the Euler Gamma-function).

Denote by Ti, i = 1, 4, the ith term of the sum in the expectation in (2). In view of (3), we have

BACCELLI AND ZUYEV / 621

E'T1 = b1,22 Ur 2 + 1

To compute the expectation of the second term, note that when the origin is occupied by a point of the process 1I1 the variables Izol and X0 are independent because they are functions of two independent processes H2 and Hlo + Ij1, respectively. It is easily seen that E? X0 = ko/X1 (more complete results on geometry of random points in a typical Voronoi cell can be found in Foss and Zuyev 1996), and we obtain

E?T2 = a1,2EO1zo I Ca1,2 EOXN(O)

al,2,k rt 1+ 1 k1 (WX2) a12/2

2 2 + )

Similarly, using once more the above exchange formula (27), but this time between E? and Eo, we get

A1 E ? 3 + T41 - (,,, A, ) Poj /2 2 2

( X 1 ) ao,1/2 2 / + (7A/),2 r(a 2 + 1

Combining the above expressions, we get the following explicit expression for the cost function:

G = H1XA + H2 + H3A1 /2+H4X1ka,/2 (4)

where

H, = f31 2/2X 1.2/2+ l r( 2 + 1 +

Ia1T ,2/2Xa 1,2/2+l 2 + / )

H2 = ab1,2Ak r( ? + 1)

H =b(',j A" r (a + 1) H4 a

o,jki_/ ojx

We see that the cost function has a unique minimal value, attained in the point k*, which solves the equation

H3 3 o,1j k 1 /2+ ) + H4ao,iiAo1l /2+l) = 2H1.

In the particular case when I3o,, = ao, 1= a we obtain

E a(H3 + H4) 2/(2+a)

2H1 ,

and the minimal value of the cost function is equal to

a(* a+2 H12 [a) H aI (2+a) G(A)= a + I(2+a) [2(H3 + H4)] + H2.

For a = 1 and a12 = p1,2 = 1, we obtain = X1/3 ko(ao,l + b01j) 12/3

2 A/[ 2b1, + 4C1AXY21

and

G(A ) = - [3(b 1,2 23 + C1)"3 (O(ao + )

+k 2 4o/]

+ a1 0,k2 /

The upper curve in Figure 4 shows a typical example of the cost function of a three level system. A more detailed discussion can be found in Section 4.3.

3. THE HIERARCHICAL MODEL

3.1. The Model

Consider the following extension of the models introduced in Section 2: There are m : 3 independent homogeneous Poisson processes I71k, k = 0, 1 ..., m, in D82 with respec- tive intensities Akk

We will use a terminology that makes things more easy to name and manipulate. The point process LI0 represents the network subscribers, whereas the point processes LI1, H2, H3, etc. represent distribution points or stations of various levels in the hierarchy. Of course in practice, we should have Ao > Ak > ... > Am, although this is not mandatory in our mathematical treatment.

Associated with the realizations of these point processes, which can be seen as denumerable collections of nodes located in the plane, one then defines arcs between certain pairs of nodes, which are built following the least distance hierarchical principle: For all k = 1, . .. , m, each point of Hk-l is connected to the point of Hk that is the closest. Equivalently, with each station xj of level k, if one denotes VX(Hk) the Voronoi cell with nucleus xj constructed with respect to Hk, each point of Hk-l lying inside V4(Hk) has an arc connecting it to xj. These arcs represent connections or links in the communication network.

These sets of nodes and arcs define a random graph that can be decomposed into a denumerable family of connected subgraphs that are all random trees of level m + 1 at most; there is one such tree rooted in each point of Hm

(see Figure 1), and this tree has either subscribers or points of Hk, k = 1, . . ., m, as leaves. For instance, if the Voronoi cell Vx(Hm) is empty of points of Hm-, then x is both a root and a leaf of the tree rooted in x. The random trees in this family are strongly dependent but identically distributed. In that, this family is a collection of spanning trees allowing one to reach a certain subset of the set of subscribers from each node of level m via certain distribution points of levels m - 1,..., 1. So if the roots are interconnected, each subscriber can be connected via a finite set of arcs to any other subscriber of the plane.

3.2. Description of the Cost Function via the Communication Example

Roughly speaking, the cost of each such tree consists of the cost of all the connections and stations of lower levels linked to its root. In our communication example:


* Each point of I71k has a cost associated with the installa- tion and maintenance of the corresponding station of level k, which we denote Ck.

* Each arc from a node x; of I71k to a node y, of Ik+1 has two components in its cost:

1. The capacity cost, namely the cost of the communication medium allowing the interconnection between the subtree rooted in x; and the station located in yl; this communication medium should allow the network to cope with the set of all subscribers being leaves of this subtree. The capacity cost of this medium is assumed to be given by a function of the form Hkk+l(n)Akk+l(r), when the number XJ of subscribers of the subtree rooted in x; is n, and when the arc connecting x; to y, is of length r. Here H and A are nondecreasing real-valued mappings. A typical example is that with H(n) n, in which case this capacity cost is purely additive in J{X and A(r) = Const - r. How- ever, the capacity of a connection is often designed to take advantage of the so-called statistical multiplexing between subscribers, and the function H is often subaddi- tive. The function A(r) will be referred to as the unit capacity cost (the capacity cost of the additive case when the variable X Xi is equal to 1). The special case A(r) =

Const - r' will be particularly considered. 2. The infrastructure cost, which is given by a function

Bkk+l(r) that does not depend on SXX by assumption, although it may of course depend on k.

3.3. Definition of the Cost Function

The cost function is defined recursively through the costs of the subtrees rooted in the points of 1k for all k.

Let Pko denote the Palm probability of the point process 1k, and E5 expectation w.r.t. Pko. On the Palm probability

space of 11k, there is a station of level k at the origin, and we define the following random variables (where [I denotes a realization of the family of all point processes [I, . . . , IFm, and { 0x} the group of translations of all these point processes-see the Appendix):

* C(fl) is the cost of the station at the origin (i.e., Ck);

* G(II) is the cost of the subtree rooted at the origin; * X(fl) (=X0) is the number of subscribers in this

subtree; * sd(fl) the unit capacity cost of the arc from 0 to the

nearest point of level k + 1; * 2A(H) the infrastructure cost of this arc; * We(H,-) is the mapping H of the station at the origin (i.e.,

Hk,k+1( ))-

For k = 0, we obviously have G(FI) = C0 = C(fl) and JX(H) = 1. For k = 1,

G(HI) = C 1 + E C0 + Hoj (1)A o,1 (lx; 1) +Bo,, (1xj 1) Xj GrE0 nvo(l1)

=%(H) + { [G(Oxfl) + X(o0flX(Ofl))

* 4(Oxt) +2(O0fl)]H~o (dx).

More generally, on the Palm space of I71k,

G(FI) = T(fl)

+ [G(Ojx) + X(0xo1,X(0xo)}j 4(oxfl) VO (Ik )

+ 2(0xFI)]lk-1 (dx)-

Since under the Palm distribution Pk., k : 1, there is a.s. a station of level k at the origin, we can apply the exchange formula (see (27), with N1 = r1k-1 and N2 = rlk) to the Palm expectation of the last expression to obtain

Ek?G(r) = Ck + k

* EO1 [G(FI) + Xi(fl,X(FI))s4(FI) + 2(fl)]

(5) V k= 1, ... ,m.

This relation is actually valid for all stationary family of point processes.

Let Sr(x) denote the open disc of radius r E R + and with center in x E R2. Similarly to (3), we can write

E?_1A(r1)

= E-1 { Bk-1,k(0x) l{Fk(SkI(O)) = 0}frk (dx)

= E Bk-Ix k() {Flk(SkI(O)) = ?}Frk (dx)

= Ak Bk-1,k(1x)Pk01[k(SI(-X)) = 0] dx

=2 7TTAk rBk-1,k(r) exp {-Ak 7Tr 2} dr, 0

where we successively used our probabilistic assumptions: the independence between f1k and lk-1, Campbell's formula (25), and the fact that I11k is a Poisson process of intensity Ak-

Next, observe that the random variable X(HI) in (5) depends only on HO..., flk-1, whereas s(f) depends only on fk. Therefore, in view of the independence assumption,

= Ek[Hk-l,k(X(rl))]EAr1s(l).

We now consider the particular case where Hk-lk =

Identity. It follows from the definition of X and from (27) that

E?X(H) = E{ J(OXH)rlk-1 (dx)

- AI Ek?1X<(I) - -= A Ak ..- A? E&<(FI)

_A0

Ak

and therefore,


Figure 2. Cost function in a four-level hierarchical model.

26000X -

E?G(ri) = Ck + A' ElG(FI) + 2ir [AOAk-1,k(r) k k k

+ Xk-lBklk(r)]r exp {-kkrr2 } dr.

Thus we have proved the following theorem.

THEOREM 1. In the case Hkk+l = Identity for all k,

k

AkEkG = Xk-lEkl-1G + Wk = X (6) n=O

uk= 1, ... tn,

where WO = AOCO,

Wn = knCn + XoRkAn (An-l,( )) (7)

+ An-l RkA, (Bn-l,n )) Vf n = 1, m,

and where &k denotes the integral operator

def -,k,2 Qk,\(fO) = 2rA J rf(r)eAm2dr. (8)

Jo

Note that operator tA(f( )) can be viewed as the expectation of random variable f(pA) where pA follows Rayleigh distribution with the parameter 1/(2I7T).

Equation (6) above receives the following natural inter- pretation: The increment of the cost density between subtrees of level k - 1 and k is equal to the sum of the cost density of the interconnections between these two levels and the cost density of the k-level stations.

3.4. Example of Optimization

Consider a four-level system with the unit capacity cost and infrastructure cost depending linearly on the distance:

Ajj(r) = aijr and Bjj(r) = bi jr, respectively. It is readily seen that the cost function EOG is smooth in 2+, tends to infinity if A1 or A2 tends to 0 or to infinity and the stationary point is given by the following system of equations:

aEOG b 1,2 A~b0,l A3 A = C+ 2 43I2 '0,

aEAG b, + 3 A Oaa1,2 + Albl2 'k 3 C2 2 0.

Expressing A1 from the second equation and substituting it in the first one, we obtain

Al = FA2 -_ 1_,2

C, + b1,2 Aobol -,

2 +14 4 (QMA 3/2 AOa 1,2 )32

where

CF= b12(c2 +2

Denoting v- = A2, after some algebra, we get

Xoai2) , 2(9) ( 3-b 1,2) 4(Cl- + b1)2-

9

We see that in the LHS of (9) is a continuous function which increases from a negative value to infinity, whereas the function in the RHS is positive and decreasing for v > 0. Therefore, there exists a unique positive solution to this equation. Noting that the value in the LHS is exactly A3, we conclude that there is a unique positive pair (Al, A2) providing the minimal value of the cost function. A typical form of the cost function is shown in Figure 2.

4. CASE OF DIRECT CONNECTIONS

4.1. Model

In the hierarchical model considered in the previous sections, a k - 1-level station is always attached to a k-level station, even if there are higher level stations closer to it. In this section we let such a station be linked to the station of levels k ..., m (see Figure 3) which is the closest. We consider here only a three-level system and the case with H(n) = n; higher level models can, in principle, be ana- lyzed by the same method.

Figure 3. Three-level model with direct connections.

/ .'"' /~~~~~~~~~ /II IiDIA

U 4-4w


4.2. Cost Function

We introduce marks associated with the points of 11, namely functionals of configurations having a 0-level station located at the origin:

sd'(f) is the unit capacity cost of the link from the origin to the nearest first-level station if there is no closer second level station, and 0 otherwise;

s5"(11) is the unit capacity cost of the link from the origin to the nearest second-level station if there is no closer first level station, and 0 otherwise.

Infrastructure cost variables QA' and CA" are defined similarly. Let s5(1) = sI'(1) + sI"(1) and J(11) = ?'(I() + @"(11).

Now under P1,

G(1I) = CC + CO + Aoj(xjx1) + Bo,,(1xj1) xjElonvou (Hi +112)

= C1 + f [G(Oxr) + d' (Oxr) V Vo (HI1 +112)

+ J'(0xr1)]r1o (dx). (10)

Because of the well-known property of Poisson processes, Il, + r12 is also a homogeneous Poisson process with intensity A1 + A2. Therefore, by Campbell's formula (25), we have

E1 Co=A C O=A CoE?1 VO(I1 + I12)1= AOCO

Xj E1nonvo(i+112) A1 + A2

where we used the notation IBI to denote the Lebesgue measure of any (Borel) set B of R2 and a classical formula for the Palm-mean area of a Voronoi cell (see Meijering 1953).

In addition, we have VO(IH1 + 112) C VO(-11); but for X E V0(I1)\Vo(I1 + I12), there exists zj E I12 such that k - zjj < ki, so that I'(Ojl) = Q'(0xI) = 0. Therefore, we can replace the integration domain in (10) by VO(I11) and then apply (27) to write

E1?G = Cl + A+ A -+ AO Eg?[s'(ri) + Q'(ri)]. A1 Al + A2 A1

By the same arguments as in the proof of (6), we obtain

Eo?'(FL) = E { Boi(IxI) 1{F1R(SkI(O)) = 0}

*l{fI2(Sk1 (0)) = 0}fl1 (dx)

= 21rr1 { rB0,1(r) exp {-lrr2(A1 + A2)} dr

_ A1- A1 + A2 R~1+k2(Boj()), (11)

and a similar formula for EsL'(11). Combining all the above results, we obtain

E1?G =C1+ A,+A. +A,+A

kA1+A2 (A01(*) + Bo,10).* (12)

Consider now the Palm space of LI2. On this space,

G(11) = C2+f [G(OjxI) + 2A(Oxl) VM(2)

+ X(0x(r1)s(0xr1)]r1j (dx)

+ { [G(Oxr) + A"(O0xr) V(H1 +H2)

+ s"(0xr1)]r1o (dx). (13)

As in (10), the last integration over VO(Hl, + 112) can be replaced by an integration over V0(I12), and using (27), we get

E?G = C2 + A2 EI[G + @ + XN>] A2 1

+ AO EO[G + Q" + s"]. (14) A2

We have EOG = C0 and

Eo?s"(HI) = E Ao,2(Izt) 1{H1(Sj1z(0)) = 0}

1*{H2(Sjz1 (0)) = 0}112 (dz) 00

= 2TA2 { rA0,2(r) exp {-wr2(A1 + A2)} dr, Jo

EO "V(HI) = 2WrA2 { rBo,2(r)

* exp {-wr2(A1 + A2)} dr.

The expression for E?J is given by (6) with k = 2, but some difficulties arise when computing E??JSA because the variables XN and A are no longer independent. However, we can write

E??N>I = E? {VJ II 0 (dx) V(II +12)

{A12(IzI) 1{l{I2(SIzl (0)) = 0}fl 2 (dz)

= E1 A12(JzI)1{Hr1(SkI(X)) = 0}

* {HI2(SkI(x) U Sl1(O)) = O}llo (dx)112 (dz)

= AOE A 12 2(IzI)e-A1 Tl2

* {IH2(SkJ(x) U Sl1(0)) = O} dxfl2 (dz)

- AOA2{ A 122(IzI)e _A1XkI2P?[O zl2(S I(X)

U Sizi(0)) = 0] dxdz.

Notice that the quantity

P20[-ZH2(Sk1 (x) U S1zl(O)) = 0]

is 0 unless z ? SkI(x). Switching to polar coordinates, the last expression reads


2I7rAT A 2 {rA 1,2 (r)e -eAl 7i2

* P[(rO) I 2 (S kj (X) U Sr (O)) = 0] dxdr

= 27rAOA2 rA1,2(r)

* exp {-AlIrx12 - A2ISkI(x) U Sr(O)I} dxdr,

where the integration is over

A = {(x,r) E R 2 x OR such that (r, 0) 0 S lj (x)}

This integral can be simplified if we represent x in polar coordinates of the form (rp, p) and adopt p as new integration variable. Then

Ei?X> = 2irAoA2{ r3A 1,2 (r)dr

{{p exp {-r2(AXlrp2 + A2U(p))} dpdqp,

where U(p) is the area of the union of the discs Sp(p cos Sp, p sin Sp) and S1(O), and where the double integral is over the set 8 of couples (p,Sp) such that Sp(p cos Sp, p sin Sp) does not contain point (1,0), i.e., over

SD E [l1p > 1/2} arccos 2p'

2T - l{p > 1/2} arccos 2p]

Thus we finally get

Ei?X> = 4irAOA2 r3A1,2(r) dr

wr {

r -

{p > 1/2} arccos 2p)

exp {-r2(A1Ip2 + A2U(p))} dp, (15)

where

HI if p E [0, 1/2],

2( 4 + 1/4p_ 1 1 if p> 1/2 (16) + (2p2 1) arccos 2

as elementary geometrical considerations show. We combine all the above results in the following theo-

rem.

THEOREM 2. For the three-level hierarchical model with direct connections, we have

EG = C1 + A0 O + A1 + A2

* RA1+A2 (A0,1( ) + BOl )(17)

E2G =AC2+A 0+A + A2 Al +A2 +

+ A [E1G + RA2(Bl2()) + Ei4>] (18)

where EOXS is given by (15) and 9A (f(-)) is defined by (8).

In the case A1,2(r) = a 12Tl,2, the integration w.r.t. r in (15) leads to

Ei4X> = 2Tal,2AoA2F(2 + 2'2)

00 (

,T

* T - l{p > 1/2} arccos 2p)(AlITp2

+ A2 U(p)) -2-a1,2/2 d2p

al,2AoA2F(1 + 2 -Al 7T alr,2/2

- 1 _rc 1

. [A+l, 2/2 (Au/4 + A2) 1+al,2/2

+ 2 a1 2 +3ITa 12 AOA2 (2 + X {Ix/2 (19)

( IT- y) sin _ COS l+al,2 y dy

[IT(A1 + 2A2) + A2 sin 2y + 2A2(T - y) cos 2y]2+al2/2

where y = arccos i. Using

RA(ara) = a(TA) -a/2r(a/2 + 1),

an explicit expression for the cost is obtained after substi- tution of (19) into (18).

In particular, if for all i, j, Aij(r) = aijr and Bij(r) = bijr, we get

E?G = C2 + A1 C, + A0(2Al + A2) cO + 2 A3/ A2 +A2(Al + A2) 22

+ AO(aO2 + bO,2) + AoAl(aol + bol)

2(A1 + A2)3/2 2A2(A1 + A2)3/2

+ 2a1,2A0 [A3/2 (A1/4 + A2)'3l2 r2 rl

i/2 + 121T312a 1,2A A Jo (20)

(wT- y) sin yCOS2y dy [7r(A1 + 2A2) + A2 sin 2,y + 2A2(IT - y) cos 2 y]5/2

Unfortunately, there are no closed-form expressions of the optimal value of parameters for this model.

4.3. Numerical Examples and Comparison

Figures 4-6 show three numerical examples for the costs of the strictly hierarchical model and the model with direct connections for the following choice of the parameters:


Figure 4. Cost functions in the hierarchical model and the model with direct connections-Example 1.

14 00-

O \ /~~~~~~

80 0-

60 0-

0 a10

1 abda-1

ko A2 CO C1 Aij(r) Bij(r) Example 1 20 1 1 50 r lOOr Example 2 20 1 1 50 r lOr Example 3 200 1 1 10 r 5r

The second level station cost C2 is not relevant for the optimal A1 and can be taken equal to 0. The abscissa is the intensity A1, the curves having finite values in A1 = 0 represent the cost in the model with direct connections.

As we see in the first example, allowing direct connections improves the minimal cost of the system. The optimal value of the intensity of Il1 in the strictly hierarchical model is A1 2.943, with cost E?G 913. For the model with direct connections, the optimal value is A1 1.874, and the cost is E?G 824.

Note the difference in behavior of the cost functions for small A1. For A1 small, there are mostly direct connections between 0 and second-level stations, if they are allowed. The topology of the system becomes star-shaped as in two- level systems. The cost of the model with direct connections is E?G 1030 for A1 = 0. In contrast, in the strictly hierarchical model, the 0-level stations are connected to first-level stations even if they are far away, which makes the cost of the system explode as A1 vanishes.

In the second example, direct connections also improve the minimal cost. But the global minimum of the cost E?G 130 is attained in A1 = 0, i.e., when there are no distribution points at all.

In the last example, we see that the optimal architecture is strictly hierarchical except for small values of A1. The optimal intensity is A1 8.320 and the minimal cost is given by the hierarchical model: E?G 612.

The corresponding optimal topology in Example 3 is visualized on Figure 7 and was drawn with the help of the program ARC (see Zuyev et al. 1997).


140 0

100 0

600 _

200 1 115 20

1 arbd._1


100-

4 010- 2 0

E 0


5. RELATED PROBLEMS

The methods considered in the previous sections can easily be adapted to other applications. Two examples are considered below.

5.1. Infinite Spanning Graphs

In all preceding cases, we assumed that the points of highest level (e.g., the roots of the trees) were linked via some connected network, which was then ignored. The aim of this subsection is to take this additional structure into ac- count within our parametric setting, and to show that optimization problems similar to those considered above can be also addressed.

Consider the following problem that is an infinite volume analog of the classical minimal spanning graph problem in a bounded region (Steele 1988). Let HA be a homogeneous Poisson process with intensity A in R2.

Can we find a connected planar graph with vertices in all the points of the Poisson process and with minimal "connection cost density, " that is, such that the cost of the edges of the graph per unit area is minimal?

We have inserted quotes because such a quantity is only well defined for finite graphs, and it may not exist for the infinite volume case.

Suppose that for all realizations of a homogeneous Pois- son process Hl, with intensity -v, we are given a rule to construct a connected graph '6(HI,) with vertex set [IV. Let SN, N E R N be a family of compact sets, where S1 is a 1-connected figure with unit area containing the origin in its interior, and SN is homothetic to S1 with coefficient N12. Let LN(HV) be the total length of (the parts of) the graph's edges lying in SN and assume that the following almost sure limit

def C = lim LN(H )/(Nv)112

N--oc

exists and does not depend on S1. Examples of such graphs include the Delaunay triangulation, for which Cg = 32/ (37r) 3.395 (cf. Okabe et al. 1992, p. 324). Another example is the optimal spanning tree, for which Cg 0.656 (see Roberts 1968; the existence of the limit was proved in Steele 1988). As already mentioned above, there is a basic difference between this and the preceding case as minimal spanning trees are well defined only in the finite volume case.

Of course, the optimal spanning tree does much better than the Delaunay triangulation. Nevertheless, such optimal spanning trees are rarely used in large communication networks because of their lack of robustness to the addition or deletion of points and because of reliability ques- tions. In fact, operator networks have both hierarchical parts, which look very much like the trees of Sections 2 to 4, and "horizontal" components that are usually present only in the upper layers of the network and look like the Delaunay triangulation associated with this level, or a subset of it.

Consider now the following variant of the basic problem of Sections 2 and 3: We split the initial process H. into independent subprocesses H1, ..w. n,, with intensities Aj, ... A,1, such that i=l Ai = A, say by a Bernoulli thin- ning. Then the spanning graph consists of the set of Poisson-Voronoi hierarchical spanning trees rooted in the points of the process Hln (see Section 3) and the graph

((ln) at the highest level. Assume that the cost of a tree edge is its length and the cost of an edge of f is its length times a weight o-.

By Theorem 1, the edge-cost density of this spanning graph is

n Ak-i

F = IE~ + o-Cg, 4,n k=22 /Ak

and the problem now consists in finding the parameters n, Al, . .. .A, that minimize the value of F.

Numerical evidence shows that the optimal value n of the number of levels is closely related to the value of o-Cg. The larger o-Cg, the larger the optimal value of n, and the larger the optimal proportion of points connected through the trees. On the opposite, if o-Cg < 1 then n = 1 is optimal and the graph gJ(1.) itself is the optimal topology in the described class.

If 6(H1) is the Delaunay triangulation on the point set H', and if o = 1, then the optimal values of the parameters are the following: n = 3, Al 0.7121A, A2 0.2466A, and A3 0.0413A. In this case the minimal value of the cost is approximately 2.014VX, a typical configuration of this optimal spanning graph is shown in Figure 8.

Of course, we can consider more general cost functions both for the hierarchical components (i.e., the trees), like those introduced in ? 2 and 3, and for the horizontal components (i.e., the Delaunay graph).

Figure 7. A family of Poisson-Voronoi trees in a three- level system.

/

1-~ ~~~~~~~*

I~~~~~

V0

.1" ~ .


Figure 8. The optimal spanning graph with Delaunay triangulation in the highest level.

-fJ

I

e

5.2. Localization in Mobile Communication Systems

Consider two independent Poisson point processes 110 and Ill representing, respectively, the positions of mobile subscribers at time 0 and the positions of the base stations of a cellular network. The base station located in point x of J1- is an antenna allowing for Hertzian communication with any mobile located inside the Voronoi cell Vx(rI1). In the case when attenuation increases with distance in a translation-invariant and isotropic way, and if all stations have the same emission power, this cell is also the part of the plane where the reception level of the antenna located in x is best.

The mobility of the points of Ho (which we will discuss below) creates a well-known problem which is that of mobile localization (see, e.g., Tabbane and Tabbane 1996). The network has to know where each mobile is located at any given time. This is particularly true for mobiles that are not in communication. Should any of those be called, the network has to be able to find the cell where the mobile is located at the time of the call.

This is done by continuously updating the position of each mobile in some database, which is organized as follows. There is a third level of stations, which we will refer to as location stations. This level will be represented as an independent Poisson process H2. The database is updated each time a mobile crosses a 112-cell boundary. So, one always knows the 112-cell in which the called mobile is located. When such a mobile is called, it is then enough to broadcast a search message to all base stations in this r2

cell to find the mobile and to establish the communication as there is a set of links connecting any node x of H2 to the base stations in VxI(H2).

There are two types of operational costs associated with this localization scheme:

1. An update cost, which comes from the update messages sent by all mobiles to the database; we will represent this as a linear function of the number of mobiles cross- ing I12-cell boundaries per unit of time; let X denote the cost of one update message.

2. A search cost, which comes from the search messages sent to all called mobiles; let Y(n) denote the cost to broadcast the search messages for a mobile located in a T12-cell with n base stations.

The optimization problem considered here consists in finding the intensity of the location stations, which minimizes the localization costs.

We now give a precise definition of the cost function. Assume that the reference probability space carries

three independent point processes Ill0, I11 and 12, together with some marks associating a random motion in R2, with each point of Il0. Below, 11 denotes a realization of these point processes together with their marks.

We assume that these marks are such that the temporal point process, which gives the epochs when a mobile crosses 112-cell boundaries, is stationary.

A simple example is that when the marks are chosen independently for each mobile and represent the trajectory of the mobile together with its velocity v, which it keeps constant. It is natural to assume that the trajectory of a mobile is continuous and piecewise 1-differentiable. From M0ller (1989, Eq. (7.22)), the intensity of intersections of 112-cell boundaries with a straight line is equal to 4A2/ir-. Therefore approximating each differentiable part of the trajectory by a segment, we see that the mean number of crossings during time At equals 4vAt X2/ir + o (At). Thus the temporal intensity of the cell boundary crossings by a mobile is given by

4vAX112 = 7T

(see also Mecke 1981 on intersections of a stationary fiber process with curves).

On the Palm space of llo, let

* X(H) be the temporal intensity of the crossings of H2-

cell boundaries by the mobile located in 0. * QI(H) be the search cost of the mobile located in 0.

If we denote g the intensity of calls directed to mobile 0 (we assume that this is a constant), then the localization cost per unit of time and space is naturally defined as

EG = E { (X(Oxu) + g,@(OxH))Ho (dx), (21) [0,1] 2

that is, using Campbell's refined formula (25) and the independence assumptions


EG = AoEO[X(1l)] + Ao0EO[Q,(I-1)]

= A oX + A0otE[Y(I1(f'o (I12))], (22)

where -v is the mean velocity of the mobiles and VO(112) is

the Voronoi cell constructed with respect to 12 and containing the origin in its interior. In case Y(n) = Y n we have

EH 1 (O (I12))

= E E E l{yi E Vzj (I2)}1{0 E Vzj (12)} yi E1 Zj E2

= A1E E f 1{y E Vo(O Z12)} Zj EJ2

* A2EjE

VV(O(112)} dy

= X1X2E2? If2 Y E Vo(J12)}1{z E Vo(112)} dydz

= AlA2E?V0(112)12,

because VO(I12) and -VO(I12) have the same distribution. Using the fact that under our Poisson assumptions, E?2|V0(112)12 = 35/(8rr2A2) (see, e.g., Okabe et al. 1992, p. 324), we obtain the following explicit expression for the cost function:

EG = H1A'/2 + H2+H (23)

with

H1- = H2 = AOA19Y 7T8,T

and the minimum is reached for

* 2H2\2!3 _ 35X/lY\ 2/3 A2 = (H1) = 35 A lX)

REMARK 1. Because the location stations are used to broadcast search messages to the antennas and eventually to concentrate communications from or to mobiles connected to these antennas, they can also be seen as a higher level of stations allowing to reach the base stations according to the least distance connection principle. So, one can also naturally consider a cost associated with these connections that is similar to what was considered in the previous sections, namely a capacity cost and an infrastructure cost.

5.3. Simulation Approach

Quite often analytical expressions for the average cost functions of stochastic models of the considered type are difficult to compute, even in the case considered here where all point processes are Poisson processes. In many such cases, the stochastic gradient method introduced in Baccelli et al. (1995) can then be applied for estimating the optimum. It consists in finding efficient ways of calcu- lating or of estimating the gradients -sA EG and then using steepest descent type algorithms to find the optimum. For details and examples of applications of this technique in

telecommunication studies see Zuyev et al. (1997) and the references therein.

APPENDIX A: POINT PROCESSES REFRESHER

In this Appendix we remind some basic notion of Palm theory of point processes, and we derive formulas which have been used in the paper.

Throughout this section, X denotes the phase space of the point processes which is always here the d-dimensional Euclidean space R d (generalizations to the case of locally compact topological groups are not difficult). 2 denotes the Borel o-algebra of Rd, XV the set of o-finite counting measures on A, Q the o-algebra of subsets of X generated by the events {J E X: q1(B) = k}, where B EA and k = 0, 1, 2, ... Each Tj EN X can be represented as a countable sum 1i ni6~x of Dirac measures with multiplicities ni E a, where xi are all distinct (by definition Ax(B) = lB(X)). The set Ui{xi} = supp qj is the support of the measure qj and it has no accumulation point if qj is o-finite. A countable measure rq is called simple if all ni = 1.

By point process, we mean a measurable mapping N from some abstract probability space (fl,J,P) into [,fl]. We can, for instance, take [fl,f] equal to [N,2], and N is then the identity mapping. Enriching if necessary the phase space, one can always assume that all realizations of the point process are simple. The addition in X gives rise to measurable flows in [X,@] and [NX,], defined as O.B = B + x = {y + x: y E B} for B E 2 and (Oxq)(B) de

7l(O.B) for Tj EN X and all B E (m. In particular, if N = Ei~x then ON = Eixi, The point process N is stationary with respect to the flow 0. if the distributions of N and 0,N coincide for all x E X. When dealing with several point processes, it is sometimes convenient to introduce a shift (p, x E X on the probability space (Q,5;,P), which is assumed to preserve P, and to assume that this shift is compatible with the translations 0. in R0d, namely that

N(^px) = OxN(ow)

for all x and w. The intensity measure of N is the measure on R d defined by A(B) = EN(B), B E Z. If N is stationary then A(B) = XIBI, where JBI is the Lebesgue measure of B and A is the intensity of N.

In this paper we often operate with several point processes Nk in R2. The Palm probability of an event _E- 9 with respect to Nk is defined by

Pk("*)= AklB E{I (pxw) Nk (dX), (24)

where Ak is the intensity of the process Nk, and where JBI is the Lebesgue measure of B; it can be shown independent on the choice of B E Q3. The intuitive meaning is that of the conditional distribution "given there is a point of the process Nk at the origin 0." The expectation with respect to Pk is denoted by Ek. The refined Campbell formula for stationary marked processes follows by the standard monotone class argument from this definition:


E F(x,w)Nk (dx) = Ak EkF(x, fpx@) dx (25)

for any measurable function F: R d X fl -O- R (cf. Stoyan et al. 1995, Eq. (4.4.11)).

The particular case where Nk = 11k, k = 1,..., m, a homogeneous Poisson processes in the plane, and where the point processes Ilrk, k = 1,..., m are independent, which is considered in the paper, can of course be embed- ded within the above framework.

Let now g: fl -- R be a measurable function. The following identity expresses the relationship between the Palm and the original distribution of a marked stationary point process:

Eg(w) = AXkE g(px cl)) dx. (26) Vo (Nk)

The set VO(Nk) above is the Voronoi cell with nucleus 0 constructed with respect to the point set supp Nk (note that under Palm distribution Pk there is a.s. a point of Nk at the origin 0). By definition VO(Nk) consists of those points of R d that are closer to 0 than to any other point of supp Nk. Vo(Nk), which coincides with the intersection of the half-spaces {x: i| - i - xi|}, xi E supp Nk, VO(Nk), is

a convex polygon. Identity (26) is readily obtained by modification of the

proof of Theorem 12.3.11 of Daley and Vere-Jones (1988). Take in (25) the function F(x,w) equal to g(w) if x is the closest to the origin point of the process Nk and to 0 otherwise. In the left-hand side of (25) we get the expectation Eg(w). Then it can be verified that F(x, p-xw) =

(Xct01 _Vo(Nk) (x), where -VO(Nk) = {x: -x E VO(Nk)} and (26) follows from the change of coordinates x -> -x in the right-hand side of (25).

The next exchange formula plays the central role in the paper. It relates two Palm distributions and can be obtained by taking a(w, t) equal the indicator of the event "t is the closest to the origin point of the process Nj(w)" in the remark following the proof of Proposition 1 in Neveu (1976, p. 201). We will, however, give a self-contained proof of this formula.

PROPOSITION 3. Let N1 and N2 be two stationary point processes, both compatible with the same flow 00. Then for any measurable f: ? ? > R-

A1EYf(W) = A2E ff t(ipxo)N (dx). (27) Vo (N2)

PROOF. First we prove this formula for a bounded function f. Take a ball SE of the volume E centered in the origin and put

g(w_) = e 1 { t(-yw)N1 (dy)

in (26). We then can write

Eg(w) = AXE f(w)

= A2E 2 [ E f f( px+,&w)(GpxNj) (dy)j dx V(N2 ) - Se

= E 2 E f({pzo)Nj (dz) dx V(N2) Se+x

= A2Ef E 11(-Se + z) n VO(N2)l

-f( pzoj) Nj (dz) .

Note that the function E-11(-SE + z)nvo(N2)1 -

E-11(SE + z)fnVo(N2)1 under the integral above tends to 1 as E -* 0 for all z in the interior of VO(N2) and to 0 for all z in the interior of the complement of VO(N2). Because VO(N2) is convex, its boundary has Lebesgue measure 0 and thus (27) follows by the bounded convergence theorem. The case of unbounded integrable f follows by the standard monotone argument. LI

ACKNOWLEDGMENT

This work was supported by CNET through the research grant CTI 1B 104. The authors are grateful to Maurice Klein and Marc Lebourges for their contributions to the formulation of the problems considered in this paper.

REFERENCES

Baccelli, F., M. Klein, M. Lebourges, and S. Zuyev. 1996. Geometrie aleatoire et architecture de reseaux de communications. Ann. Thhecommunications 51 158-179. 7 ~~, and S. Zuyev. 1995. Perturbation analysis of functionals of random measures. Adv. Apple. Probab. 27 306-325.

Daley, D. J., and D. Vere-Jones. 1988. An Introduction to the Theory of Point Processes. Springer, New York.

Foss, S., and S. Zuyev. 1996. On a certain Voronoi aggrega- tive process related to a bivariate Poisson process. Adv. Apple. Probab. 28 965-981.

Hwang, F. K., D. S. Richards, P. Winter. 1992. The Steiner tree problems. Ann. Discrete Math. 53.

Mecke, J. 1981. Formulas for stationary planar fiber processes III-intersection with fibre systems. Math. Oper. Statist. 12 201-210.

Meijering, J. 1953. Interface area, edge length and number of vertices in crystal aggregates with random nucleation. Philips Res. Report 8 270-290.

M0ller, J. 1989. Random tesselations in Rd. Adv. Appl. Probab. 21 37-73.

Neveu, J. 1976. Sur les mesures de Palm de deux processus ponctuels stationnaires. Z. Wahrsch. verw. Gebiete 34 199-203.

Okabe, A., B. Boots, and K. Sugihara. 1992. Spatial Tessela- tions. Wiley, New York.

Roberts, F. 1968. Random minimal trees. Biometrika 55 255-258.


Steele, J. 1988. Growth rates of Euclidean minimal spanning trees with power weighted edges. Ann. Probab. 16(4), 1767-1787.

Stoyan, D., W. Kendall, and J. Mecke. 1995. Stochastic Geometry and Its Applications (second ed.). Wiley, Chichester, UK

Tabbane, N., and S. Tabbane. 1996. Database architectures and location strategies for mobility management in

mobile radio systems. B. Jabbari, P. Godlewski, and X. Lagrange, eds. In Multiaccess, Mobility and Teletraffic for Personal Communications. Kluwer, New York.

Zuyev, S., P. Desnogues, and H. Rakotoarisoa. 1997. Sim- ulations of large telecommunication networks based on probabilistic modeling. J. Electronic Imaging 6(1), 68- 77.

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