Link-Traffic Loop-Free Property in Optimal Routing for ... - IEEE Xplore

FrA03.5

43rd IEEE Conference on Decision and Control December 14-17, 2004 Atlantis, Paradise Island, Bahamas

Link-Traffic Loop-Free Property in Optimal Routing for Multi-Class Networks Hisao Kameda, Jie Li, and Eitan Altman Abstract— Communications networks where nodes are interconnected by a generally configured manner and wherein there are multiple classes of users, each of which has its distinct generalized link costs (communication delays), are considered. Individually and overall optimal routing problems for multiclass networks are formulated along with the discussions on mutual equivalence between both problems, on the existence and uniqueness of solutions, and on the relation between the formulations with path and link flow patterns. We show that a link-traffic loop-free property holds within each class for both individually and overall optimal routing in a wide range of networks, and obtain the condition that characterizes the category of networks for which the link-traffic loop-free property holds.

I. Introduction Optimal routing has been important in the field of computer networks. We have two typical approaches for optimal routing. (1) One arises in a context of minimizing the overall cost (overall mean delay) of all users (or packets) from the arrival (origin) node of each user to its destination node through a number of communication links over the entire network. The optimal routing policy with this framework is called the overall optimal routing policy. (2) A second approach is a distributed one in which one seeks for a set of routing strategies for all users such that no user can decrease its cost (expected delay) by deviating from its strategy unilaterally. This could be viewed as a result of leaving each user the decision on which path to route through. This approach is called the individually optimal routing or selfish routing [15], [16]. The situation where each user has unilaterally minimized its cost is called a Wardrop equilibrium [13], [17] or a Nash equilibrium where no user has any incentive to make a unilateral decision to change its route. Traditionally, most work has focused on overall optimal routing in computer networks (e.g., [2], [7], [8]). For computer networks, however, minimizing the cost of each user from its arrival (origin) node to its destination node is a major concern of the user. Especially in the Internet, a user may wish to have its minimum cost for a given traffic pattern that results from the routing decisions of all other users. Thus, individually optimal routing has attracted increasing attention of researchers and practitioners in modern Hisao Kameda and Jie Li are with the Graduate School of Systems and Information Engineering, University of Tsukuba, Tsukuba Science City, Ibaraki 305-8573, Japan. Email: {kameda,lijie}@cs.tsukuba.ac.jp. Eitan Altman is with INRIA, BP93, 06902 Sophia Antipolis Cedex, France. Email: [email protected].

0-7803-8682-5/04/$20.00 ©2004 IEEE

computer networks, and some research results have been obtained on individually optimal routing (or selfish routing) [1], [3], [12], [15], [16]. In most studies on optimal routing problems for communication networks in the literature (e.g., [1]–[3], [7], [8], [12]), the link cost is modeled as a simple function dependent only on the link flow itself. For convenience, we call it the traditional link-cost model. However, the link cost may not depend only the flow of the link itself but also on the flow through other links. For example, we consider a modern computer network implemented by a wireless network. In a wireless network, when a link connecting two nodes has more flow and, thus, uses more power, links neighboring the link may have less capacity. This is because there should be some mutual interference among the neighboring links, which may occur even in the case where electro-magnetic wave channels are best assigned to links, or more likely, in the case of ad hoc networks where nodes may move arround. Therefore, unlike the traditional link-cost model, in this paper, the cost on a link of a network is modeled by a function of the flows of all links of the entire network. We call it a general link-cost model. For convenience, we may call a network with multi-class users a multi-class network. We note, however, that, in these optimization problems, the cost to be optimized (the objective function) depends only on the link flow pattern while the instrument (the set of decision variables) is the path flow pattern. In this paper, we discuss individually and overall optimal routing problems in general link-cost models of multi-class networks, on which Dafermos has obtained some basic results [4]–[6]. Our treatment is, however, more general than hers in the following sense: Our model allows the origindestination scheme that would allow each user of a class to enter any origin and leave any destination both available to the class with/without fixing the arrival rate at each origin and the departure rate at each destination; the link loop-free property is discussed; the relation between the case where the instrument is the path flow pattern and the case where it is the link flow pattern. We confirm that, for multiclass networks under our assumptions, there is an associate overall optimal routing problem that corresponds to the individually optimal routing problem, and that both have the same solution. We discuss the existence and uniqueness of the solution to the overall optimal routing and, then, on the basis of the associate relation, of the solution to the individually optimal routing,

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both with paying attention to the relation between the case where the instrument is the path flow pattern and the case where it is the link flow pattern. Furthermore, we show that the link-traffic loop-free property holds for each class, that is, there is no link-traffic loop within each class, for the individually and overall optimal routing in a general link model of multi-class networks of a wide range. We obtain the condition that characterizes the category of networks where the link-traffic loop-free property holds for each class. The rest of this paper is organized as follows. In the next section, we provide the problem formulation. The relation between the individually and overall optimal routing problems is provided in Subsection II-C. Subsection II-D discusses the existence and uniqueness of individually and overall optimal routing solutions. Section III discusses the link-traffic loop-free property for the individually and overall optimal routing. Some examples are shown in Section IV. Section V concludes the paper. II. Problem Formulation Consider a computer network consisting of n nodes or vertices (host computers or routers) numbered, 1, 2, . . . , n, interconnected in an arbitrary fashion by a communication network. Let G = [N, L] be a network, where N is a set of nodes and L is a set of links whose elements are ordered pairs of distinct nodes. There are multi-class users in the network. Denote by C the set of user classes. Each class may have a distinct set of links available to the class. We call links available to class-k users ‘class-k links.’ Denote by Lk the set of ‘class-k links.’ Then, L = ∪k∈C Lk . By a path for a class connecting an ordered pair ω = (o, d), we mean a sequence of class-k links (v1 , v2 ), (v2 , v3 ), . . . , (vn −1 , vn ) that any class-k user can pass through where v1 , v2 , . . . , vn are distinct nodes, v1 = o, and vn = d. We call node o an origin, the node d a destination and the pair ω = (o, d) an origin-destination pair (or o-d pair for abbreviation). Each class may have a distinct set of origins and of destinations. We denote the path by (o, v2 , . . . , vn −1 , d). If vi is the same as node v j for some i and j such that j < i, we say that the path has a loop or cycle. We note, however, that in the optimal solutions such a loop within a path never exists. Therefore, a link appears in a path for a class at most once. On the other hand, although each path has no loop, the network may have a loop as to link flows as discussed in Section III. We have the following assumptions. A1 (1) If there exists a possible series of link connections for a class between an origin and a destination (an o-d pair), that must also be a path for the class between the o-d pair. and (2) If there exists a path for a class between an o-d pair, all possible series of link connections for the class between the o-d pair are also paths of the class between the o-d pair. A2 The rate of arrivals at each origin is given for each class, and the rate of departures at each destination is also given for each class. we do not restrict a particular

user of a given class to enter at a particular origin or to leave at a particular destination as long as the above rate restrictions are satisfied. Remark 1: As seen later in Sections III and IV, Assumption A1 presents the condition that characterizes the category of networks that have the link-traffic loop-free property within each class for overall and individually optimal routing. Even under the assumptions A1 and A2, we can model the situation where there are particular combinations of origins and destinations such that users arriving at an origin should depart the network only from the destination corresponding to the origin. This can be done as follows: Users of each particular origin-destination pair should belong to a distinct class that has only the origin-destination pair, but the characteristics of such classes may be mutually identical in the other aspects. In that case, there may exist a linktraffic loop over links of different classes, but our theorem shows that there exists no link-traffic loop within each class. In Assumption A2, it may look unnatural that the rate of the departure at each destination is given even though each user can leave the network at any available destination. Consider a network, named A, where each class-k user can leave the network at any available destination without fixing the class-k departure rate at each destination. We imagine another network, named A , where one class-k ‘final’ destination is added to the network A and that each of the class-k destinations is connected to the class-k final destination via a class-k zero-cost link in the network A for every class k. (Later in this section, we will describe zerocost links along with the definition of link-cost functions, Gkij .) Then, the imagined network A can be regarded as the one with multiple-origins and one common destination for class k and satisfies Assumptions A1 and A2. The optimal solutions of the networks A and A should be identical. In a similar way, we can consider a network where each class-k user can enter the network at any origin without fixing the class-k arrival rate at each origin but with the departure rate at each class-k destination being fixed. We can also consider a network where each class-k user can enter at any origin and leave from any destination without fixing the arrival and departure rates at any origin and destination for class k and with fixing the total arrival and departure rates for the class k, respectively. The former network is identical to the network where one ‘initial’ origin is added and connected to each origin via a zero-cost link for class-k. The latter network is identical to the network where one initial origin and one final destination are added. We therefore see that the assumption A2 is most general and covers all three kinds of networks each of which is identical to the three networks mentioned above, respectively.
For simplicity, we do not consider the case where a node is both an origin and a destination at the same time for the same class, in which case the optimal solution would be such that only the difference flow would either come in

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or go out of the network, and the node would appear to be either an origin or a destination in the optimal solution. The sets of all origin and destination nodes for class k are denoted by Ok and Dk , respectively. The set of all possible paths which originate from an o ∈ Ok for class k is denoted by Pko− . The set of all possible paths which are destined for a d ∈ Dk for class k is denoted by Pk−d . The set of all paths in the network for class k is denoted by Pk , each element of which must appear in a Pko− and in a Pk−d . With k every origin o ∈ Ok , let ro− (k ∈ C) be the nonnegative external class-k user traffic demand that originates at node k (k ∈ C) be o, and with every destination d ∈ Dk , let r−d the nonnegative external class-k user traffic demand that is destined for node d through a relevant set of paths. k For a path p ∈ Pko− , we denote the part of ro− which k flows through path p by y p . Similarly for a path p ∈ P−d . ykp is called the class-k path flow through the path p. We have the following relations.
k ykp = ro− , o ∈ Ok , (1) p∈Pko




k ykp = r−d , d ∈ Dk ,

(2)

p∈Pkd

ykp ≥ 0, p ∈ Pk , k ∈ C. (3)   k k Naturally, o∈Ok ro− = d∈Dk r−d . Denote the path flow k k pattern by y = [y ], where y = [ykp ]. By a feasible path flow pattern, we mean y which satisfies relations (1), (2), and (3). Denote by FS y the set of feasible path flow patterns. Clearly, FS y is convex, closed, and bounded. Denote by xkij the class-k user flow rate, also called the class-k flow, through link (i, j). Let x = [xk ] where xk = [xkij ]. Furthermore, let xi j = [xkij ], and then x = [xi j ]. For convenience, we call x the link flow pattern. We now consider the relation between path and link flows. Since a link appears in a path at most once, a class-k link flow is expressed by class-k path flows as follows.
δipj ykp , (i, j) ∈ Lk , k ∈ C, (4) xkij =

most of the link costs, Gkij (x) is a positive and differentiable function that is convex in x and, in particular, that Gkij (x) is strictly convex in xkij . For convenience regarding modeling, we also consider the possible existence of zero-cost links the flows through which do not influence other link costs, that   is, for some i , j , k , Gki j (x) = 0 for all x, and xki j does not affect any other Gkij . We denote by x s the vector that consists of the elements xkij such that each corresponding Gkij (x) is strictly convex in xkij , and by x−s the vector that consists  of the elements xki j each of which is the flow through the  class-k zero-cost link (i , j ). Denote the class-k cost of a path p by Dkp (x). Then, Dkp (x) is expressed as
δipjGkij (x), p ∈ Pk , k ∈ C. (5) Dkp (x) = (i, j)∈Lk

A. Overall Optimal Routing for Multi-Class Networks The overall optimal routing is to minimize the overall cost of all users from their arrivals at origin nodes till their departures from destination nodes. By using (4) and (5), the overall cost of users over all classes is expressed as 1

k k 1

k k D(x) = y p D p (x) = xi jGi j (x), R k∈C R k∈C k k 



δipj

 =

(i, j)∈L

(6)

y

Since Gkij (x) is convex in x and, in particular, strictly convex in xkij , D(x) is convex in x and strictly convex in x s and, then, is convex in y. Define the marginal link-cost gkij (x) as follows. gkij (x) = R

∂ D(x). ∂xkij

(7)

Consider the following Lagrangian function:



k φko− (ro−

k∈C o∈Ok

If link (i, j) is included in path p, we also express it as (i, j) ∈ p. From (4), we notice that a given feasible path flow pattern y induces a unique feasible link flow pattern x, while it is possible that more than one feasible path flow pattern y induces the same feasible link flow pattern x. The relation between feasible path flow patterns and feasible link flow patterns should be considered carefully. Let Gkij be the class-k link-cost of sending a class-k user from node i to node j through link (i, j). Unlike the traditional link-cost models, we consider that the link cost Gkij is a general function of all link flows x. We assume that, for



min D(x) subject to (4) and y ∈ FS y .

+

1, if link (i, j) is contained in path p, 0, otherwise.



k k = k∈C d∈Dk r−d . where R = k∈C o∈Ok ro− Thus, the overall optimal routing problem is expressed as follows.

p∈Pk

where

p∈P

L(y, φ) = RD(x)


 k − ykp ) + φk−d (r−d − ykp ) , p∈Pko−

d∈Dk

p∈Pk−d

where φko− and φk−d are Lagrange multipliers. Then, the path flow pattern y that satisfies the following set of relations derived from the Kuhn-Tucker condition as to the above Lagrangian function is a solution for overall optimal routing if such a flow pattern y exists,
gkij (x) = βko,d , for y p > 0, (i, j)∈p




gkij (x) ≥ βko,d , for y p = 0,

(i, j)∈p k

p ∈ P , o ∈ Ok , d ∈ Dk , k ∈ C, y ∈ FS y , where βko,d = φko− + φk−d .

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(8)

B. Individually Optimal Routing for Multi-Class Networks The link flow pattern x of individually optimal routing is a Wardop equilibrium [17] or a Nash equilibrium point in the sense of noncooperative game [11]. Thus, we define the individually optimal routing formally by using the equilibrium condition as follows. Definition I: For the multi-class network considered, a path flow pattern y is said to satisfy the equilibrium condition of the individually optimal routing if and only if the following relation holds.
Gkij (x) = Ako,d , for y p > 0, Dkp (x) = (i, j)∈p

Dkp (x) =




Gkij (x) ≥ Ako,d , for y p = 0,

(i, j)∈L

For convenience, we define 1 k ˆ Gi j (x) = k  Gkij (x)dxkij , (i, j) ∈ Lk , k ∈ C. (12) xi j k∈C |Lk | Then, we regard Gˆ kij as a new class-k link cost on link (i, j). Thus, 1

k ˆk ˆ xi jGi j (x). D(x) = R k∈C k (i, j)∈L

(9)

(i, j)∈p

p ∈ Pk , o ∈ Ok , d ∈ Dk , k ∈ C, y ∈ FS y . We can regard the flow pattern y as a solution to the individually optimal routing for the network if it satisfies the above equilibrium condition. Remark 2: The above definition I and the assumptions A1 and A2 imply the situation where it only holds that, for each combination of the origin and the destination, the paths used have equal costs that are not less than those of the unused paths. But, this situation may not reflect the freedom of each user of a class to choose one destination among those available to the class. In order that truly individual decisions may be realized, we may use the framework mentioned in the second and third paragraphs of Remark 1.
C. Relation between Individually and Overall Optimal Routing for Multi-Class Networks ˆ We construct a new overall cost function D(x) (i.e., a potential function) for the same network G = [N, L] as that of the overall optimization problem (6), such that ˆ ∂ D(x) , (i, j) ∈ Lk , k ∈ C. Gkij (x) = R ∂xkij

If the symmetric condition for matrix Λ holds, Relation (10) is satisfied by the following. 



1 k k ˆ G (x)dx (11) D(x) =  ij ij . R k∈C |Lk | k k∈C

  for xki j ∈ x−s , Gˆ ki j = Gˆ kij (i
i or j
j).



We recall that, 0 and xki j would not influence other Then, we have the following overall optimization problem. ˆ subject to min D(x) y

(4) and y ∈ FS y .

(13)

We call the overall optimization problem (13) an associate problem to the individually optimal routing problem. We note that it is another overall optimal routing problem. Consider the following Lagrangian function: +





k φˆ ko− (ro−

k∈C o∈Ok

ˆ φ) = RD(x) ˆ L(y,


 k k − yp) + − ykp ) , φˆ k−d (r−d p∈Pko−

d∈Dk

p∈Pk−d

where φˆ ko− and φˆ k−d are Lagrange multipliers. Then, the path flow pattern y that satisfies the following set of relations derived from the Kuhn-Tucker condition as to the above Lagrangian function is an overall optimal solution to the associate problem if such a flow pattern y exists.
Gkij (x) = Ako,d , for y p > 0, (i, j)∈p




(10)

Gkij (x) ≥ Ako,d , for y p = 0,

(14)

(i, j)∈p k

p ∈ P , o ∈ Ok , d ∈ Dk , k ∈ C, y ∈ FS y ,

According to the line of research given by Dafermos [4], [5], [13], we consider whether and under what condition ˆ such overall cost function D(x) exists. Since the linkk cost function Gi j (x) ((i, j) ∈ Lk , k ∈ C) is differentiable,   ∂Gkij /∂xklm ((i, j) ∈ Lk , (l, m) ∈ Lk , k, k ∈ C) exists. Then, we assume that the matrix of partial derivatives of link-cost  ∂Gk  ∂Gkij ∂Gklm ij , is symmetric (i.e., for functions, Λ =   = ∂xklm ∂xklm ∂xkij  all (i, j) ∈ Lk , (l, m) ∈ Lk , k, k ∈ C). We call it the symmetric condition for matrix Λ. Moreover, we consider a submatrix, Λ s , of Λ that contains the elements ((i jk), (i j k )) such that  both xkij and xki j are in x s . We note that the elements of Λ that are not in Λ s are all zero. We assume that Λ s is ˆ positive definite. Then, D(x) is strictly convex in x s , Λ is ˆ semi-positive definite, and D(x) is convex in x.

where Ako,d = φˆ ko− + φˆ k−d . We see that the above is equivalent to the condition (9) describing the solution of the corresponding individually optimal routing problem. Then, we have the following lemma. Lemma 1: A path flow pattern y is a solution to associate problem (13) if and only if it satisfies the equilibrium condition (9). On the other hand, if we regard gkij (x) as a new class-k cost on link (i, j), then we have the equilibrium condition (8) of individual optimization that is an associate condition to the overall optimal routing problem with Gkij (x) being the class-k cost on link (i, j). Corollary 1: A path flow pattern y satisfies the associate condition (8) if and only if it is a solution to the overall optimization problem (6).

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xkij

D. Existence and Uniqueness We first discuss the existence and uniqueness of the solutions to the overall optimal routing problem. Then, by noting that the individually optimal routing problem can be transformed into its associate overall optimal routing problem (13), we investigate the existence and uniqueness of the solution to the associate problem (13) and, then, to the individually optimal routing problem (9). Denote the set of feasible link flow patterns by FS x . That is, FS x = {x | There exists y such that x and y satisfy (4) and y ∈ FS y }. Clearly, the set {(x, y) | (x, y) satisfies (4) and y ∈ FS y } is convex, closed, and bounded. Then, by noting that the orthogonal projection of a convex set onto a subspace is another convex set (see, e.g., [14]), the set FS x is convex in x and a closed and bounded hyperplane. Note that the overall cost D(x) in the overall optimal routing problem (6) is continuous in x and, thus, in y, and that the feasible set FS y is closed and bounded. Then, there exists a solution of path flow patterns y to the overall optimal routing problem (6), according to the Weierstrass theorem (e.g., [9], [10]). Since the overall cost of a user, D(x), is continuous and convex in x and strictly convex in x s , we have the existence and uniqueness of a solution to the overall optimal routing as follows. Theorem 1: For the overall optimal routing problem (6), an optimal path flow pattern y exists and, in particular, the resulting x s is unique. The uniqueness of x s for the optimal routing is shown as follows. Suppose that x s is not unique and that both x1 = (x1s , x1−s ) and x2 = (x2s , x2−s ) give the minimum Dmin of D(x), for x1s
x2s . Then, from the convexity of the feasible region FS x , αx1 + (1−α)x2 , for some α (0 < α < 1), is also in the feasible region FS x , and D(αx1 + (1 − α)x2 ) = D(αx1s + (1−α)x2s , αx1−s + (1−α)x2−s ) < αD(x1s , αx1−s + (1−α)x2−s ) + (1−α)D(x2s , αx1−s + (1−α)x2−s) = αDmin + (1−α)Dmin = Dmin , where the inequality follows from the strict convexity of D(x) in x s and the second-last equality follows from the meaning of x−s . The above relation contradicts the assumption that Dmin is the minimum of D(x), and we see that the x that minimizes D(x) has a unique x s . For individually optimal routing problem for multi-class ˆ networks, we note that the new overall cost D(x) in the associate problem (13) is continuous in x and, thus, in y, and that the feasible set FS y is closed and bounded. Then, similarly as above, there exists a solution of path flow patterns y to the associate problem (13) according to the Weierstrass theorem. With Lemma 1 and by noting that ˆ D(x) is convex in x and strictly convex in x s , we have the existence and uniqueness of a solution to individually optimal routing as follows.

i

Fig. 1. routing

j

Link-traffic loop-free property in individually/overall optimal

Theorem 2: For the individually optimal routing problem, there exists a solution of path flow pattern y to the associate problem (13) and, thus, that satisfies the equilibrium condition (9), and, in particular, the resulting x s is unique. For the overall (respectively, individually) optimal routing problem, consider the following optimization problem ˆ (respectively, the problem with D(x) being replaced by D(x) in the following) that involves only the link flow pattern x and does not involve the path flow pattern y. min D(x) x

subject to x ∈ FS x .

(15)

Similarly as the above two problems involving the path flow pattern y, we see that there exists a solution of the link flow patterns x to the overall optimization problem (15) (respectively, to the individual optimization problem ˆ with D(x) being replaced by D(x) in (15)), according to the Weierstrass theorem. The optimization problem (15) ˆ (respectively, the one with D(x) being replaced by D(x) in (15)) is another nonlinear convex optimization problem, but clearly gives the solution x that is the same as the link flow pattern x that the solutions y to (6)(respectively, (13), thus (9)) result in. III. Link-Traffic Loop-Free Property In this section, we show a property that holds for the individually/overall optimal routing, called the link-traffic loop-free property, and establish the relation between the presentations by x and y. The link-traffic loop-free property is such that there exists no loop that consists of a sequence of links (v1 , v2 ), (v2 , v3 ), . . . , (vn −1 , vn ) where v1 , v2 , . . . , vn are distinct nodes while v1 = vn such that class-k link flows xkv1 ,v2 > 0, xkv2 ,v3 > 0, xkvn −1 ,vn > 0 (k ∈ C) (Fig. 1). From relations (1), (2), and (4), we have the following flow-balance relation with link flow pattern x.

k k ri− + xkli = r−i + xkil , i = 1, 2, . . . , n − 1, k ∈ C, (16) l∈Vik

l∈Vik

where Vik is the set of immediately neighboring nodes of node i for class k, i.e., Vik = { j|(i, j) ∈ Lk , or ( j, i) ∈ Lk }. Note that

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k k ri− = 0 if i  Ok and r−i = 0 if i  Dk .

The constraint with respect to i = n can be derived by summing up both sides of the above constraints for i = 1, 2, . . . , n − 1. Define the set of feasible solutions, FS I, as follows:

k + ri−

l∈Vi

subject to

x ∈ FS I.

(17)

The necessary and sufficient condition that a solution to the above individually/overall optimal routing problem satisfies is given as follows. Theorem 3: The link flow pattern x is an optimal solution to the individually (and overall) optimal routing problem (17) if and only if x satisfies the following set of relations (with Gkij (x) is to be replaced by gkij (x) for the overall optimal routing problem). αki − Gkij (x) = αkj , for xkij > 0, (i, j) ∈ Lk , k ∈ C, αki

− Gkij (x)

≤

αkj ,

for

xkij

= 0, (i, j) ∈ L , k ∈ C, k

(18) (19)

subject to x ∈ FS I, where αki (i ∈ N, k ∈ C) are Lagrange multipliers. Proof: We show the case of individual optimization. The case of overall optimization is shown in a similar way. To obtain an optimal solution to problem (17), we form the Lagrangian function as follows, ˆ H(x, α) = RD(x) +

n−1

k∈C i=1


 k
k k αki ri− + xli − r−i − xkil , l∈Vi

l∈Vi

where αki are Lagrange multipliers. ˆ Since function D(x) is continuous and convex in x (and strictly convex in x s ) and FS I is convex, closed, and bounded, there exists a solution (that has a unique x s ) to problem (17) similarly as Theorem 2. Thus, the link flow pattern x that satisfies the following Kuhn-Tucker condition is an optimal solution (that has a unique x s ) to problem (17) (see, e.g., [9]).

xkij ≥ 0, (i, j) ∈ Lk , k ∈ C,

xkil ,

l∈Vi

Rearranging the above relations, we have, αki − Gkij (x) ≤ αkj , for xkij = 0, (i, j) ∈ Lk , k ∈ C, x ∈ FS I. The above set of relations is equivalent to the set of relations (18), (19) and x ∈ FS I, and it is the necessary and sufficient condition for a policy x to be a solution to the individually optimal routing problem (17). Q.E.D. With Theorem 3, we proceed to have the link-traffic loopfree property in the individually/overall optimal routing. Theorem 4: (Link-traffic loop-free property) The classk link traffic in a solution to the individually (and overall) optimal routing problem is loop free for all k ∈ C. That is, there exist no class-k link flows such that xkv1 v2 > 0, xkv2 v3 > 0, . . . , xkvm−1 vm > 0, xkvm v1 > 0 (for all k ∈ C), where v1 , v2 , . . . , vm are distinct nodes in the solution to the individually (and overall) optimal routing problem and where at least one of the link involved is not a zero-cost link. Proof: We show the case of individual optimization. It is proved by contradiction. Assume that there exists link flow pattern x in the solution to individually optimal routing problem for multi-class users such that xkv1 v2 > 0, xkv2 v3 > 0, . . . xkvm−1 vm > 0, xkvm v1 > 0 for some k (∈ C), where v1 , v2 , . . . , vm are distinct nodes. According to Theorem 3, in the solution, we have αkv1 − Gkv1 v2 (x) = αkv2 , .. . k k αvm − Gvm v1 (x) = αkv1 .

(21)

Then, we have Gkv1 v2 (x) + Gkv2 v3 (x) + · · · + Gkvm v1 (x) = 0,

(20)

∂H = Gkij (x) + αkj − αki ≥ 0, ∂xkij ∂H xkij k = xkij (Gkij (x) + αkj − αki ) = 0, ∂xi j




αki − Gkij (x) = αkj , for xkij > 0,

Note that the set of FS I is convex, closed, and bounded. We note that FS I includes but may not be identical to FS x . We will touch on this possible difference later. Note the nonnegativity of class-k link flow xkij . We have the associate problem for the individually optimal routing (and the overall optimal routing problem) with following ˆ new constraint (with D(x) and Gˆ kij (x), respectively, are to be k replaced by D(x) and Gi j (x) for the overall optimal routing problem).

x

k xkli = r−i +

i = 1, 2, . . . , n − 1, k ∈ C.

FS I = {x | x satisfies (16) and x ≥ 0}.

ˆ min D(x)




(22)

which contradicts the fact that > 0 if > 0 for at least one of (non-zero-cost) links involved. The case of overall optimization is shown in a similar way as above by replacing Gkij (x) by gkij (x). Q.E.D. Gkij (x)

xkij

Since the constraint x ∈ FS I of the optimization problems may be weaker than the set of constraints x ∈ FS x , there may be the possibility that no path flow pattern may realize the link flow solution x. We can confirm, however, that there exists a path flow pattern y that results in any loop-free link-flow pattern x. Theorem 5: There exists a path flow pattern y satisfying the constraint y ∈ FS y that results in a link-traffic loop-free flow pattern x satisfying the constraint x ∈ FS I. Proof: Consider a loop-free link-flow pattern x that satisfies the constraint x ∈ FS I. Consider an arbitrary node i and a class k. Since the link-traffic loop-free property holds, node i has a group

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ika origin o

path is straightforward. Clearly, this assignment is relevant to (4) but does not violate them since x ∈ FS I must hold.

node i

(ii) The case where node i is neither an origin nor a destination: Each of the class-k intermediate paths at node j, j ∈ ika , that go through node i, will be assigned the xkij ratio  of the flow of the corresponding class-k k l∈ika xil intermediate path at node i. For example, if the class-k intermediate path (o, . . . , i) of a class-k intermediate path (o, . . . , i, j) has the flow yko... i , the class-k intermediate path (o, . . . , i, j) is to be assigned the flow

yko... i xkij yko... i j node j

Fig. 2.

Assigning flows to paths on the basis of link flows

of such nodes that, for each node in the group, there is a path that reaches it from node i. Starting from origin nodes to node i, we can have a set of ‘intermediate’ paths. Consider a path (o, v1 , v2 , . . . , vi , vi+1 , . . . , d) where o is an origin node, v1 , v2 , . . . , vi , vi+1 , . . . are intermediate nodes, and d is a destination node. Then, we call the sequence (o, v1 , v2 , . . . , vi ) an intermediate path at node vi of path (o, v1 , v2 , . . . , vi , vi+1 , . . . , d). Naturally, there may be multiple intermediate paths at each node including those coming from different origins. Furthermore, we also call the sequence (o, v1 , v2 , . . . , vi ) the intermediate path at node vi of the intermediate path (o, v1 , v2 , . . . , vi , . . . , v j ) for i < j and v j
d. The flow through each intermediate path will be the sum of the flows of the paths that go though the intermediate path if we can assign the amount of flow to each path properly. We can do this as follows, with satisfying the constraint y ∈ FS y at each step. The assignment to an intermediate path of node vi+1 is done by splitting the assignment of the intermediate paths of node vi , in a way proportional to the flow on the link (vi , vi+1 ) (in the case where node vi is neither an origin nor a destination, and other cases are also treated in a formal manner below). In that sense, we obtain a proportionally fair assignment. More precisely, given a link-traffic loop-free flow pattern, the path flow pattern for class k, k ∈ C, can be obtained as follows: Since the used links has the loop-free property, a ‘partial order’ relation among class-k nodes holds, that is, from the origins down to the destinations. Therefore, there must exist at least one origin that receives no class-k flows from any nodes but only sends class-k flows to other nodes. We call it a pure origin for class k. Similarly, there must exist at least one destination that sends no class-k flows to any nodes but only receives class-k flows from other nodes. We call it a pure destination for class k. Denote by ika the set of nodes that receive class-k flow directly from node i (Fig. 2). (i) The case where node i is a pure origin o. A node j directly connected to a pure origin o ( j ∈ oka ) has the classk link flow xko j . There must be only one class-k intermediate path from the origin at a node j ∈ oka , that passes through one class-k link to the node j from the origin, and, thus, the assignment of class-k flow to the class-k intermediate

yko... i j = yko... i 

xkij

l∈ika

xkil

.

Clearly, this assignment is relevant to (4) and (1) but does not violate them since x ∈ FS I must hold. The rest of the cases are the following (iii), (iv), and (v), and are discussed similarly as the above two cases (i) and (ii). (iii) The case where the node i is not a pure origin but an k . origin that has the external arrival rate ri− (iv) The case where the node i is not a pure destination but k a destination that has the departure rate r−i : (v) The case where node i is a pure destination for class k. We therefore see that, at every step of the above five assignments in cases (i), (ii), (iii), (iv), and (v), the set of constraints (4) and y ∈ FS y is satisfied. Then, starting from nodes that directly receives class-k flows only from origins that receive no class-k flows from other nodes, we can proceed the steps of assigning the amount of class-k flows to intermediate paths, and finally we can complete the assignment of class-k path flows, for all k ∈ C, that satisfy both the set of the constraints (4) and y ∈ FS y and the constraint x ∈ FS I. Q.E.D. We can confirm, therefore, that there exists a path flow pattern y that realizes any loop-free link-flow solution x for the individually/overall optimal routing. Corollary 2: For link-traffic loop-free networks, FS I = FS x . IV. Examples where Assumption A1 Does Not Hold In this section, we examine two examples wherein either (1) or (2) in the assumption A1 does not hold. We see that in both examples the link-traffic loop-free property does not hold. Therefore, we see that Assumption A1 is the condition that characterizes the category of networks for which the link-traffic loop-free property holds in individually and overall optimal routing in multi-class networks. 1) An example where (1) holds but (2) does not hold in the assumption A1: Consider a single-class network consisting of four nodes 1, 2, 3, and 4 (|C| = 1) and a single pair of origin 1 and destination 4, shown in Figure 3. Nodes 2 and 3 are connected by links (2,3) and (3,2).

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x23 = y1234 = r14 /2 > 0

V. Conclusion

2 r14 4

1 r14 3

x32 = y1324 = r14 /2 > 0 Fig. 3. A network with one origin and one destination that satisfies Assumption A1(1) but does not satisfy A1(2)

x23 = y123 = r13 > 0 r13

r13 1

2 r42

3

4

r42

x32 = y432 = r42 > 0

Fig. 4. A network with two origins and two destinations that satisfies Assumption A1(2) but does not satisfy A1(1)

In this paper, we have studied both individually and overall optimal routing problems for multi-class networks with generalized link-cost functions and network configurations. We have seen that there is an associate overall optimal routing problem which corresponds to each individually optimal routing problem with the same solution under some condition. We have discussed the relation between the formulations with path and link flow patterns. We have discussed the existence and uniqueness of the solution to overall optimal routing, and then, on the basis of the associate relation, to the individually optimal routing. Furthermore, we show that the link-traffic loop-free property holds for the individually and overall optimal routing in a wide range of networks. We have obtained the condition that characterizes the category of multi-class networks that have the link-traffic loop-free property for overall and individually optimal routing. We have discussed the relation between the problem formulation that involves both link and path flow patterns and what involves only the link flow pattern. References

We consider the case where (2) in the assumption A1 does not hold and we have only two paths (1,2,3,4) and (1,3,2,4) connecting the O-D pair (1,4). We assume that the cost of each link depends only on the flow of the link and that G12 (x) = G13 (x), G23 (x) = G32 (x) > 0, and G24 (x) = G34 (x) where x denotes the flow through each link. Let the arrival rate at the origin be r14 > 0. Then, the optimal path flows of the two paths are identical, and x23 = y1234 = r14 /2 = y1324 = x32 > 0, which means that the network has a link-traffic loop. On the other hand, if (2) in the assumption A1 is to hold, then paths (1,2,4) and (1,3,4) need to be additionally available; only the latter two paths are used in optimal routing, and the network has no link-traffic loop. 2) An example where (2) holds but (1) does not hold in the assumption A1: Consider a single-class network consisting of four nodes 1, 2, 3, and 4 (|C| = 1) shown in Figure 4. We consider the case where (1) in the assumption A1 does not hold and we have two distinct O-D pairs (1,3) and (4,2). Let the arrival rates at origins be such that r13 > 0 and r42 > 0. Thus, there are two origin nodes (i.e., nodes 1 and 4) and two destination nodes (i.e., nodes 3 and 2) in the network. It is clear that we have only one solution such that x23 = y123 = r12 > 0 and x32 = y432 = r42 > 0, which is the optimal solution to individually/overall optimal routing problem under the set of constraints (4) and y ∈ FS y . In this example, it is clear that we have a link-traffic loop, i.e., x23 > 0 and x32 > 0. On the other hand, if (1) in the assumption A1 holds, the solution under the constraint x ∈ FS x is such that x23 = r13 − r42 ≥ 0 and x32 = 0 if r13 ≥ r42 (under Assumption A2), which shows the freedom of link-traffic loops that holds under Assumption A1.

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