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Advances in Operations Research
Volume 2011 (2011), Article ID 645954, 21 pages
http://dx.doi.org/10.1155/2011/645954
Research Article

Optimal Routing for Multiclass Networks

1Graduate School of Systems and Information Engineering, University of Tsukuba, Tsukuba Science City, Ibaraki 305-8573, Japan
2INRIA, BP93, 06902 Sophia Antipolis Cedex, France

Received 30 September 2010; Revised 4 April 2011; Accepted 5 April 2011

Academic Editor: Yi Kuei Lin

Copyright © 2011 Hisao Kameda et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Overall optimal routing is considered along with individually optimal routing for networks with nodes interconnected in a generally configured manner and with multiple classes of users. The two problems are formulated, and we discuss the mutual equivalence between the problems, the existence and uniqueness of solutions, and 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 for a wide range of networks, and we show the condition that characterizes the category of networks for which the property holds.

1. Introduction

There are two typical approaches for optimal routing in networks. (1) One arises in the context of minimizing the overall cost (overall mean delay) of all users (e.g., packets) from the arrival (origin) node of each user to its destination node through a number of 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 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 the result of allowing each user the decision on which path to route. This approach is called the individually optimal routing. The situation where each user has unilaterally minimized its cost is called a Wardrop equilibrium [1, 2] or a Nash equilibrium where no user has any incentive to make a unilateral decision to change its route.

In computer and communication networks, most work has focused on overall optimal routing (e.g., [35]). For networks in general, however, minimizing the cost of each user from its arrival (origin) node to its destination node is a major concern of the user. Thus, individually optimal routing has attracted increasing attention of researchers and practitioners in computer and communication networks, and some research results have been obtained [69].

In most studies on optimal routing problems for communication networks in the literature (e.g., [35, 7, 8, 10]), the link cost is modeled as a simple function dependent only on the link flow itself. We call this the traditional link-cost model. In this paper, however, the cost on a link of a network is modeled by a function of the flows of all links in the entire network. We call this a general link-cost model. For example, in a wireless communication network, where, when a link connecting two nodes has more flow and, thus, uses more power, neighboring links may have less capacity. This paper studies optimal routing problems in general link-cost models for generally configured networks with multiple classes of users. We call a network with multiple-class users a multiclass network. We note, however, that, in these optimization problems, the cost to be optimized 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 on which Dafermos has obtained some basic results [1113]. Our treatment is, however, more general than hers in the following points. (1) Our model allows 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. (2) The link-traffic loop-free property is discussed. (3) The relation between the case where the instrument is the path flow pattern and the case where it is the link flow pattern is discussed. In particular, we note that, by definition, a path flow pattern determines a unique link flow pattern whereas it may not be sure whether for a link flow pattern there exists a path flow pattern that induces it, that is, whether a given link flow pattern is realizable.

We confirm the necessary and sufficient condition that, for an individually optimal routing problem under our assumptions, there exists an overall optimal routing problem associated to it, and that both have the same solution. We discuss the existence and uniqueness of the solutions to the overall and individually optimal routing. Furthermore, we show that the link-traffic loop-free property holds for each class, for the individually and overall optimal routing in general link-cost models of multiclass networks. We pay much attention to the relation between the cases where the sets of the control variables are, respectively, the path and link flow patterns. We show the condition that characterizes the category of networks where the link-traffic loop-free property holds for each class. Some examples are discussed. In contrast, note that, even in the networks where the link-traffic loop-free property holds for each class in overall and individually optimal routing, it does not always hold in noncooperative optimal routing by a finite (but plural) number of decision makers, where the decision makers strive to optimize unilaterally the cost of the users under its control. Such counter-examples are given in [14, 15] (with the definition of class in those papers changed to be the same as the one in this paper). Note, in passing, that overall optimal routing may have only one decision maker and that individually optimal routing has infinitely many infinitesimal decision makers.

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 for multiclass networks is provided in Section 2.3. Section 2.4 discusses the existence and uniqueness of individually and overall optimal routing for multiclass networks. Section 3 discusses the link-traffic loop-free property for the individually and overall optimal routing for multiclass networks. Some examples are shown in Section 4. Section 5 concludes the paper.

2. Problem Formulation and Solutions

Consider a network consisting of 𝑛 nodes numbered, 1,2,,𝑛, interconnected in an arbitrary fashion by links. 𝑁 and 𝐿, respectively, denote the sets of nodes and links. There are multiclass users in the network. 𝐶 denotes the set of user classes. Each class may have a distinct set of links available to the class. We assume that users (commodities) do not change their classes during their trips from origins and destinations. Thus, the users (commodities) of different classes can be different. We call links available to class-𝑘 users “class-𝑘 links.” 𝐿𝑘 denotes the set of “class-𝑘 links.” Then, 𝐿=𝑘𝐶𝐿𝑘. By a path for a class, say class 𝑘, connecting an ordered pair 𝜔=(𝑜,𝑑), we mean a sequence of class-𝑘 links (𝑣1,𝑣2),(𝑣2,𝑣3),,(𝑣𝑛1,𝑣𝑛) that any class-𝑘 user can pass through where 𝑣1,𝑣2,,𝑣𝑛 are distinct nodes, 𝑣1=𝑜, and 𝑣𝑛=𝑑. Then, the path is denoted by (𝑜,𝑣2,,𝑣𝑛1,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.

If 𝑣𝑖 is the same as node 𝑣𝑗 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 3. We have the following assumptions. (A1) (1) If there exists a possible series of link connections for a class between an 𝑜-𝑑 pair, there must exist a path for the class between the 𝑜-𝑑 pair. (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 rates of arrivals at each origin and of departures at each destination are given for each class.

Remark 2.1. As seen later in Sections 3 and 4, 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.
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. We see below, however, that the assumption (A2) is most general. Consider a network, named , 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 , where one class-𝑘 “final” destination is added to the network and that each of the class-𝑘 destinations in is connected to the class-𝑘 final destination via a class-𝑘zero-cost link in the network for every class 𝑘 (later in this section, we will describe zero-cost links along with the definition of link-cost functions, 𝐺𝑘𝑖𝑗). Then, the imagined network can be regarded as the one with multiple-origins and one common destination for class 𝑘 as shown in Section 4.1.1 and satisfies assumptions (A1) and (A2). The optimal solutions of the networks and 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 equivalent 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 equivalent to the network where one initial origin and one final destination are added for class-𝑘.
We therefore see that assumption (A2) is most general and covers all three kinds of networks each of which is equivalent to the corresponding one of the three networks mentioned above, respectively.

For simplicity, we assume that a node cannot be both an origin and a destination at the same time for the same class. The sets of all origin and destination nodes for class 𝑘 are denoted by 𝒪𝑘 and 𝒟𝑘, respectively. The sets of all possible paths which originate from an 𝑜𝒪𝑘 and which are destined for a 𝑑𝒟𝑘, for class 𝑘, are denoted by 𝒫𝑘𝑜 and 𝒫𝑘𝑑, respectively. 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 𝒫𝑘𝑑, that is, 𝒫𝑘=𝑜𝒪𝑘𝒫𝑘𝑜=𝑑𝒟𝑘𝒫𝑘𝑑. For every origin 𝑜𝒪𝑘 and for every destination 𝑑𝒟𝑘, respectively, let 𝑟𝑘𝑜 and 𝑟𝑘𝑑 (𝑘𝐶) be the nonnegative external class-𝑘 user traffic demands that originate at node 𝑜 for all destinations 𝑑𝒟𝑘, and that is destined for node 𝑑 from all origins 𝑜𝒪𝑘.

For a path 𝑝𝒫𝑘𝑜, 𝑦𝑘𝑝 denotes the part of 𝑟𝑘𝑜 which flows through path 𝑝. Similarly for a path 𝑝𝒫𝑘𝑑, 𝑦𝑘𝑝 is called the class-𝑘  path flow through the path 𝑝. We have the following relations:𝑝𝒫𝑘𝑜𝑦𝑘𝑝=𝑟𝑘𝑜,𝑜𝒪𝑘,(2.1)𝑝𝒫𝑘𝑑𝑦𝑘𝑝=𝑟𝑘𝑑,𝑑𝒟𝑘𝑦,(2.2)𝑘𝑝0,𝑝𝒫𝑘,𝑘𝐶.(2.3) Naturally, 𝑜𝒪𝑘𝑟𝑘𝑜=𝑑𝒟𝑘𝑟𝑘𝑑. Denote the path flow pattern by 𝐲=[𝐲𝑘], where 𝐲𝑘=[𝑦𝑘𝑝]. By a feasible path flow pattern, we mean 𝐲 which satisfies relations (2.1), (2.2), and (2.3). Denote by 𝐹𝑆𝐲 the set of feasible path flow patterns. Clearly, 𝐹𝑆𝐲 is convex, closed, and bounded.

Denote by 𝑥𝑘𝑖𝑗 the class-𝑘 user flow rate, also called the class-𝑘flow, through link (𝑖,𝑗). Let 𝐱=[𝐱𝑘] where 𝐱𝑘=[𝑥𝑘𝑖𝑗]. Furthermore, let 𝐱=[𝐱𝑖𝑗] where 𝐱𝑖𝑗=[𝑥𝑘𝑖𝑗]. We call 𝐱 the link flow pattern. Since a link appears in a path at most once, a class-𝑘 link flow is expressed by class-𝑘 path flows as follows:𝑥𝑘𝑖𝑗=𝑝𝒫𝑘𝛿𝑝𝑖𝑗𝑦𝑘𝑝,(𝑖,𝑗)𝐿𝑘,𝑘𝐶,(2.4) where 𝛿𝑝𝑖𝑗=1,iflink(𝑖,𝑗)iscontainedinpath𝑝,0,otherwise.(2.5) If link (𝑖,𝑗) is included in path 𝑝, we also express it as (𝑖,𝑗)𝑝. From (2.4), we notice that a path flow pattern 𝐲 induces a unique link flow pattern 𝐱, while it is possible that more than one path flow pattern 𝐲 induces the same link flow pattern 𝐱. Moreover, for given 𝐱, it may not be sure whether there exists a path flow pattern 𝐲 that induces 𝐱.

Let 𝐺𝑘𝑖𝑗 be the class-𝑘 link cost of sending a class-𝑘 user from node 𝑖 to node 𝑗 through link (𝑖,𝑗). 𝐺𝑘𝑖𝑗 is a function of all link flows 𝐱. We assume that, for 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 all 𝑖,(𝑗𝑖),𝑘. We also consider the possible existence of zero-cost links the flows of which do not influence other link costs, that is, for some 𝑖,𝑗,𝑘, 𝐺𝑘𝑖𝑗(𝐱)=0 for all 𝐱, and 𝑥𝑘𝑖𝑗 does not affect any other 𝐺𝑘𝑖𝑗. 𝐱𝑠 denotes the vector that consists of the elements 𝑥𝑘𝑖𝑗 such that each corresponding 𝐺𝑘𝑖𝑗(𝐱) is strictly convex in 𝑥𝑘𝑖𝑗, and 𝐱𝑠 denotes the vector that consists of the elements 𝑥𝑘𝑖𝑗 each of which is the flow through the class-𝑘 zero-cost link (𝑖,𝑗). Denote by 𝐿𝑘𝐬 and by 𝐿𝑘𝐬, respectively, the sets of class-𝑘 links with nonzero cost and with zero cost. Then we assume that 𝐺𝑘𝑖𝑗(𝐱) is strictly convex in 𝐱𝑠 for all 𝑖,(𝑗𝑖),𝑘.

𝐷𝑘𝑝(𝐱) denotes the class-𝑘 cost of a path 𝑝. Then,𝐷𝑘𝑝(𝐱)=(𝑖,𝑗)𝐿𝑘𝛿𝑝𝑖𝑗𝐺𝑘𝑖𝑗(𝐱),𝑝𝒫𝑘,𝑘𝐶.(2.6)

2.1. Overall Optimal Routing for Multiclass Networks

By using (2.4) and (2.6), the overall cost of users over all classes is expressed as 1𝐷(𝐱)=𝑅𝑘𝐶𝑝𝒫𝑘𝑦𝑘𝑝𝐷𝑘𝑝1(𝐱)=𝑅𝑘𝐶(𝑖,𝑗)𝐿𝑘𝑥𝑘𝑖𝑗𝐺𝑘𝑖𝑗(𝐱),(2.7) where 𝑅=𝑘𝐶𝑜𝒪𝑘𝑟𝑘𝑜=𝑘𝐶𝑑𝒟𝑘𝑟𝑘𝑑.

Thus, considering (2.4), the overall optimal routing problem is expressed as follows:min𝐲𝐷(𝐱(𝐲))subjectto𝐲𝐹𝑆𝐲.(2.8) We have assumed that 𝐺𝑘𝑖𝑗(𝐱) is convex in 𝐱 and, in particular, strictly convex in 𝐱𝐬, for all 𝑖,(𝑗𝑖),𝑘. Then, we can see that 𝐷(𝐱) is convex in 𝐱 and strictly convex in 𝐱𝐬 (by noting that 𝑘𝐶(𝑖,𝑗)𝐿𝑘{𝛼𝑥𝑘(1)𝑖,𝑗+(1𝛼)𝑥𝑘(2)𝑖,𝑗}𝐺𝑘𝑖,𝑗(𝛼𝐱(1)+(1𝛼)𝐱(2))=𝑘𝐶(𝑖,𝑗)𝐿𝑘𝐬{𝛼𝑥𝑘(1)𝑖,𝑗+(1𝛼)𝑥𝑘(2)𝑖,𝑗}𝐺𝑘𝑖,𝑗(𝛼𝐱(1)𝐬+(1𝛼)𝐱(2)𝐬)<𝛼𝑘𝐶(𝑖,𝑗)𝐿𝑘𝐬𝑥𝑘(1)𝑖,𝑗𝐺𝑘𝑖,𝑗(𝐱(1)𝐬)+(1𝛼)𝑘𝐶(𝑖,𝑗)𝐿𝑘𝐬𝑥𝑘(2)𝑖,𝑗𝐺𝑘𝑖,𝑗(𝐱(2)𝐬)+𝛼(1𝛼)𝑘𝐶(𝑖,𝑗)𝐿𝑘𝐬(𝑥𝑘(1)𝑖,𝑗𝑥𝑘(2)𝑖,𝑗)[𝐺𝑘𝑖,𝑗(𝐱(2)𝐬)𝐺𝑘𝑖,𝑗(𝐱(1)𝐬)]<𝛼𝑘𝐶(𝑖,𝑗)𝐿𝑘𝐬𝑥𝑘(1)𝑖,𝑗𝐺𝑘𝑖,𝑗(𝐱(1)𝐬)+(1𝛼)𝑘𝐶(𝑖,𝑗)𝐿𝑘𝐬𝑥𝑘(2)𝑖,𝑗𝐺𝑘𝑖,𝑗(𝐱(2)𝐬)=𝛼𝑘𝐶(𝑖,𝑗)𝐿𝑘𝑥𝑘(1)𝑖,𝑗𝐺𝑘𝑖,𝑗(𝐱(1))+(1𝛼)𝑘𝐶(𝑖,𝑗)𝐿𝑘𝑥𝑘(2)𝑖,𝑗𝐺𝑘𝑖,𝑗(𝐱(2)) for 0<𝛼<1 and for 𝐱(1)𝐬𝟎 or 𝐱(2)𝐬𝟎.) Then, we see that 𝐷(𝐱) is convex in 𝐲, which can be easily shown as follows: Indeed, the relation (2.4) can be regarded as a linear transformation: 𝐲𝐱 and we denote this by 𝐱=𝐱(𝐲). Then, 𝛼𝐷(𝐱(𝐲1))+(1𝛼)𝐷(𝐱(𝐲2))𝐷(𝛼𝐱(𝐲1)+(1𝛼)𝐱(𝐲2))=𝐷(𝐱(𝛼𝐲1+(1𝛼)𝐲2)), where the inequality follows from the convexity of 𝐷() and the equality follows from the transformation linearity. Thus, we see that 𝐷(𝐱(𝐲)) is convex in 𝐲.

Consider the following Lagrangian function:𝐿(𝐲,𝝓)=𝑅𝐷(𝐱)+𝑘𝐶𝑜𝒪𝑘𝜙𝑘𝑜𝑟𝑘𝑜𝑝𝒫𝑘𝑜𝑦𝑘𝑝+𝑑𝒟𝑘𝜙𝑘𝑑𝑟𝑘𝑑𝑝𝒫𝑘𝑑𝑦𝑘𝑝,(2.9) where 𝜙𝑘𝑜 and 𝜙𝑘𝑑 are Lagrange multipliers. Define 𝑔𝑖𝑗(𝐱) as follows:𝑔𝑘𝑖𝑗𝜕(𝐱)=𝑅𝜕𝑥𝑘𝑖𝑗𝐷(𝐱).(2.10) Then, the path flow pattern 𝐲 that satisfies the following relation derived from the Kuhn-Tucker condition is a solution for overall optimal routing if such a flow pattern 𝐲 exists,(𝑖,𝑗)𝑝𝑔𝑘𝑖𝑗(𝐱)=𝛽𝑘𝑜,𝑑,for𝑦𝑝>0,(𝑖,𝑗)𝑝𝑔𝑘𝑖𝑗(𝐱)𝛽𝑘𝑜,𝑑,for𝑦𝑝=0,𝑝𝒫𝑘,𝑜𝒪𝑘,𝑑𝒟𝑘,𝑘𝐶,𝐲𝐹𝑆𝐲,(2.11) where 𝛽𝑘𝑜,𝑑=𝜙𝑘𝑜+𝜙𝑘𝑑. We recall that 𝒫𝑘=𝑜𝒪𝑘𝒫𝑘𝑜=𝑑𝒟𝑘𝒫𝑘𝑑.

2.2. Individually Optimal Routing for Multiclass Networks

Informally, we define the individually optimal routing to be such that each individual user routes itself so as to minimize its own cost from the arrival at its origin node to the departure from its destination node, given the expected link cost of each link. In the equilibrium that the routing policy results in, every user of all classes may feel that its own cost is minimized and has no incentive to make a unilateral decision to change its route. In other words, the link flow pattern 𝐱 of individually optimal routing is a Wardrop equilibrium [2], or a Nash equilibrium point in the sense of noncooperative game [16]. Thus, we define the equilibrium condition of the individually optimal routing as follows.

Definition 2.2. 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: 𝐷𝑘𝑝(𝐱)=(𝑖,𝑗)𝑝𝐺𝑘𝑖𝑗(𝐱)=𝐴𝑘𝑜,𝑑,for𝑦𝑝𝐷>0,𝑘𝑝(𝐱)=(𝑖,𝑗)𝑝𝐺𝑘𝑖𝑗(𝐱)𝐴𝑘𝑜,𝑑,for𝑦𝑝=0,𝑝𝒫𝑘,𝑜𝒪𝑘,𝑑𝒟𝑘,𝑘𝐶,𝐲𝐹𝑆𝐲.(2.12)

We recall that 𝒫𝑘=𝑜𝒪𝑘𝒫𝑘𝑜=𝑑𝒟𝑘𝒫𝑘𝑑. We call the path flow pattern 𝐲 the solution of the individually optimal routing if it satisfies the above equilibrium condition. It is a Wardrop equilibrium [2].

Remark 2.3. The above definition 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 last paragraph of Remark 2.1.

2.3. Relation between Individually and Overall Optimal Routing for Multiclass Networks

We note that link-cost function 𝐺𝑘𝑖𝑗(𝐱)((𝑖,𝑗)𝐿𝑘,𝑘𝐶) is differentiable, that is, 𝜕𝐺𝑘𝑖𝑗/𝜕𝑥𝑘𝑙𝑚((𝑖,𝑗)𝐿𝑘,(𝑙,𝑚)𝐿𝑘,𝑘,𝑘𝐶) exists. In order to obtain an optimization problem that gives the same solution as the equilibrium condition (2.12) of the individually optimal routing, we consider the following function 𝐷(𝐱) as described as follows. From Patriksson [1, page 75, Theorem 3.4], the necessary and sufficient condition that we can construct a new overall cost function 𝐷(𝐱) for the same network as that of (2.8), such that𝐺𝑘𝑖𝑗𝜕(𝐱)=𝑅𝐷(𝐱)𝜕𝑥𝑘𝑖𝑗,(𝑖,𝑗)𝐿𝑘,𝑘𝐶,(2.13) is that the matrix of partial derivatives of link-cost functions, Λ(𝐱)=[𝜕𝐺𝑘𝑖𝑗/𝜕𝑥𝑘𝑙𝑚], is symmetric (i.e., 𝜕𝐺𝑘𝑖𝑗/𝜕𝑥𝑘𝑙𝑚=𝜕𝐺𝑘𝑙𝑚/𝜕𝑥𝑘𝑖𝑗 for all (𝑖,𝑗)𝐿𝑘,(𝑙,𝑚)𝐿𝑘,𝑘,𝑘𝐶).

Moreover, we consider a submatrix, Λ𝑠(𝐱), of Λ(𝐱) that contains the ((𝑖𝑗𝑘),(𝑖𝑗𝑘))th elements such that both 𝑥𝑘𝑖𝑗 and 𝑥𝑘𝑖𝑗 are in 𝐱𝑠. We note that the elements of Λ(𝐱) that are not in Λ𝑠(𝐱) are all zero. We assume that Λ𝑠(𝐱) is positive definite. Then, if the above symmetry condition holds, 𝐷(𝐱) is strictly convex in 𝐱𝑠, Λ(𝐱) is semipositive definite, and 𝐷(𝐱) is convex in 𝐱. In the traditional link-cost models, Λ(𝐱) is also symmetric and semipositive definite (see, e.g., [6, 9]). Denote by (𝐱(𝑖𝑗𝑘),𝑥𝑘𝑖𝑗) the vector with the component 𝑥𝑘𝑖𝑗 of 𝐱 replaced by 𝑥𝑘𝑖𝑗. If Λ(𝐱) is symmetric, the following satisfies (2.13):1𝐷(𝐱)=𝑅𝑘𝐶||𝐿𝑘||𝑘𝐶(𝑖,𝑗)𝐿𝑘𝑥𝑘𝑖𝑗0𝐺𝑘𝑖𝑗𝐱(𝑖𝑗𝑘),𝑥𝑘𝑖𝑗𝑑𝑥𝑘𝑖𝑗.(2.14)𝐷 corresponds to what is often called a potential in game theory [17]. We define𝐺𝑘𝑖𝑗1(𝐱)=𝑥𝑘𝑖𝑗𝑘𝐶||𝐿𝑘||𝑥𝑘𝑖𝑗0𝐺𝑘𝑖𝑗𝐱(𝑖𝑗𝑘),𝑥𝑘𝑖𝑗𝑑𝑥𝑘𝑖𝑗,(𝑖,𝑗)𝐿𝑘,𝑘𝐶.(2.15) Then, we regard 𝐺𝑘𝑖𝑗 as a new class-𝑘 link cost on link (𝑖,𝑗). Thus, 1𝐷(𝐱)=𝑅𝑘𝐶(𝑖,𝑗)𝐿𝑘𝑥𝑘𝑖𝑗𝐺𝑘𝑖𝑗(𝐱).(2.16) We recall that, for 𝑥𝑘𝑖𝑗𝐱𝑠, 𝐺𝑘𝑖𝑗=0 and 𝑥𝑘𝑖𝑗 would not influence other 𝐺𝑘𝑖𝑗(𝑖𝑖 or 𝑗𝑗). Thus, considering (2.4), as an optimization problem that gives the same solution as the equilibrium condition (2.12) of the individually optimal routing problem, we have the following overall optimization problem:min𝐲𝐷(𝐱(𝐲))subjectto(2.4)and𝐲𝐹𝑆𝐲.(2.17)

We call the overall optimization problem (2.17) an associate problem to the individually optimal routing problem. We note that it is another overall optimal routing problem. Consider the following Lagrangian function𝐿𝐷(𝐲,𝝓)=𝑅(𝐱)+𝑘𝐶𝑜𝒪𝑘𝜙𝑘𝑜𝑟𝑘𝑜𝑝𝒫𝑘𝑜𝑦𝑘𝑝+𝑑𝒟𝑘𝜙𝑘𝑑𝑟𝑘𝑑𝑝𝒫𝑘𝑑𝑦𝑘𝑝,(2.18) where 𝜙𝑘𝑜 and 𝜙𝑘𝑑 are Lagrange multipliers. Then, the path flow pattern 𝐲 that satisfies the following relation 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;(𝑖,𝑗)𝑝𝐺𝑘𝑖𝑗(𝐱)=𝐴𝑘𝑜,𝑑,for𝑦𝑝>0,(𝑖,𝑗)𝑝𝐺𝑘𝑖𝑗(𝐱)𝐴𝑘𝑜,𝑑,for𝑦𝑝=0,𝑝𝒫𝑘,𝑜𝒪𝑘,𝑑𝒟k,𝑘𝐶,𝐲𝐹𝑆𝐲,(2.19) where 𝐴𝑘𝑜,𝑑=𝜙𝑘𝑜+𝜙𝑘𝑑. We see that the above is equivalent to the condition (2.12) describing the solution of the corresponding individually optimal routing problem. Then, we have the following lemma.

Lemma 2.4. A path flow pattern 𝐲 is a solution to associate problem (2.17) if and only if it satisfies the equilibrium condition (2.12).

On the other hand, if we regard 𝑔𝑘𝑖𝑗(𝐱) as a new class-𝑘 link cost on link (𝑖,𝑗), then we have the equilibrium condition (2.11) for the individual optimization that is an associate condition to the overall optimal routing problem for multiclass networks with 𝐺𝑘𝑖𝑗(𝐱) being the class-𝑘 cost on link (𝑖,𝑗).

Corollary 2.5. A path flow pattern 𝐲 satisfies the associate condition (2.11) if and only if it is a solution to the overall optimization problem (2.8).

2.4. Existence and Uniqueness

In this section, we study the existence and uniqueness of the solutions to overall and individually optimal routing problems for multiclass networks. 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 (2.17) as long as the symmetry condition given in Section 2.3 holds, we investigate the existence and uniqueness of the solution to the associate problem (2.17) and, then, to the individually optimal routing problem (2.12).

Denote the set of feasible link flow patterns by 𝐹𝑆𝐱. That is,𝐹𝑆𝐱=𝐱Thereexists𝐲suchthat𝐱and𝐲satisfy(2.4)and𝐲𝐹𝑆𝐲.(2.20) Clearly, the set {(𝐱,𝐲)(𝐱,𝐲)satises(2.4)and𝐲𝐹S𝐲} 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., [18]), the set 𝐹𝑆𝐱 is convex in 𝐱 and a closed and bounded hyperplane (see, e.g., [18]). Note that 𝐷(𝐱) in (2.8) is continuous in 𝐱 and, thus, in 𝐲, and that the feasible set 𝐹𝑆𝐲 is closed and bounded. Then, there exists a solution of path flow patterns 𝐲 to (2.8), according to the Weierstrass theorem (e.g., [19, 20]). Since 𝐷(𝐱) is continuous and convex in 𝐱 and strictly convex in 𝐱𝑠, we have the following.

Theorem 2.6. For the overall optimal routing problem for multiclass networks (2.8), an optimal path flow pattern 𝐲 exists and, in particular, the resulting 𝐱𝑠 is unique.

The uniqueness of 𝐱𝑠 is shown by contradiction as follows. Suppose that 𝐱𝑠 is not unique and that both 𝐱1=(𝐱1𝑠,𝐱1𝑠) and 𝐱2=(𝐱2𝑠,𝐱2𝑠) give the minimum 𝐷min of 𝐷(𝐱), for 𝐱1𝑠𝐱2𝑠. Then, from the convexity of the feasible region of 𝐱, 𝛼𝐱1+(1𝛼)𝐱2, for some 𝛼 (0<𝛼<1), is also in the feasible region, and 𝐷𝛼𝐱1+(1𝛼)𝐱2=𝐷𝛼𝐱1𝑠+(1𝛼)𝐱2𝑠,𝛼𝐱1𝑠+(1𝛼)𝐱2𝑠𝐱<𝛼𝐷1𝑠,𝛼𝐱1𝑠+(1𝛼)𝐱2𝑠𝐱+(1𝛼)𝐷2𝑠,𝛼𝐱1𝑠+(1𝛼)𝐱2𝑠=𝛼𝐷min+(1𝛼)𝐷min=𝐷min,(2.21) where the inequality follows from the strict convexity of 𝐷(𝐱) in 𝐱𝑠 and the second-last equality follows from the meaning of 𝐱𝑠. 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 𝐱𝑠.

For the individually optimal routing problem, we note that 𝐷(𝐱) in (2.17) is continuous in 𝐱 and, thus, in 𝐲, and that 𝐹𝑆𝐲 is closed and bounded. Then, similarly as above, there exists a solution of 𝐲 to (2.17) according to the Weierstrass theorem (e.g., [19, 20]). With Lemma 2.4 and by noting that 𝐷(𝐱) is convex in 𝐱 and strictly convex in 𝐱𝑠, we have the existence and uniqueness of a solution to individually optimal routing as follows.

Theorem 2.7. For the individually optimal routing problem, there exists a solution 𝐲 to (2.17) and thus that satisfies (2.12), and, in particular, the resulting 𝐱𝑠 is unique.

For the overall optimal routing problem, consider the following optimization problem that involves only the link flow pattern 𝐱 and does not involve the path flow pattern 𝐲:min𝐱𝐷(𝐱)subjectto𝐱𝐹𝑆𝐱.(2.22) Similarly as above, we see that there exists a solution of 𝐱 to (2.22) according to the Weierstrass theorem (e.g., [19, 20]). The optimization problem (2.22) is a nonlinear convex optimization problem, but, clearly, (2.22) gives the solution 𝐱 that is the same as the link flow pattern 𝐱 that the solutions 𝐲 to (2.8), thus (2.11) results in.

For the individually optimal routing problem, consider the following optimization problem that involves only the link flow pattern 𝐱 and does not involve the path flow pattern 𝐲:min𝐱𝐷(𝐱)subjectto𝐱𝐹𝑆𝐱.(2.23) Similarly as above, we see that there exists a solution of 𝐱 to (2.23), according to the Weierstrass theorem (e.g., [19, 20]). The optimization problem (2.23) is another nonlinear convex optimization problem, but, clearly, (2.23) gives the solution 𝐱 that is the same as the link flow pattern 𝐱 that the solutions 𝐲 to (2.17), thus (2.19) (i.e., (2.12)) results in.

3. Link-Traffic Loop-Free Property

In this section, we show a property that holds for the overall/individually optimal routing for multiclass networks, called the link-traffic loop-free property. The link-traffic loop-free property is such that there exists no loop that consists of a sequence of links (𝑣1,𝑣2),(𝑣2,𝑣3),,(𝑣𝑛1,𝑣𝑛) where 𝑣1,𝑣2,,𝑣𝑛 are distinct nodes while 𝑣1=𝑣𝑛 such that class-𝑘 link flow 𝑥𝑘𝑣1,𝑣2>0,𝑥𝑘𝑣2,𝑣3>0,𝑥𝑘𝑣𝑛1,𝑣𝑛>0(𝑘𝐶) (Figure 1). Although it is evident that no path has a loop, it is not clear whether there exists no loop for link flows of each class. For example, if assumption (A1) does not hold, as shown by the example given later, there may exist loops for link flows in the network.

645954.fig.001
Figure 1: Link-traffic loop-free property in overall/individually optimal routing.

From relations (2.1), (2.2), and (2.4), we have the following flow-balance relation:𝑟𝑘𝑖if𝑖𝒪𝑘+𝑙𝑉𝑘𝑖𝑥𝑘𝑙𝑖=𝑟𝑘𝑖if𝑖𝒟𝑘+𝑙𝑉𝑘𝑖𝑥𝑘𝑖𝑙,𝑖=1,2,,𝑛1,𝑘𝐶,(3.1) where 𝑉𝑘𝑖 is the set of immediately neighboring nodes of node 𝑖 for class 𝑘, that is, 𝑉𝑘𝑖={𝑗(𝑖,𝑗)𝐿𝑘,or(𝑗,𝑖)𝐿𝑘}. The constraint with respect to 𝑖=𝑛 can be derived by summing up both sides of the above constraints for 𝑖=1,2,,𝑛1. Define 𝐹𝑆𝐼 as follows: 𝐹𝑆𝐼={𝐱𝐱satises(3.1)and𝐱𝟎}.(3.2) Note that the set of 𝐹𝑆𝐼 is convex, closed, and bounded. Note, furthermore, that 𝐹𝑆𝐼 includes but may not be identical to 𝐹𝑆𝐱.

We have the overall optimal routing problem (and the associate problem for the individually optimal routing) with the following new constraint (with 𝐷(𝐱) and 𝐺𝑘𝑖𝑗(𝐱), resp., to be replaced by 𝐷(𝐱) and 𝐺𝑘𝑖𝑗(𝐱) for the associate problem for the individually optimal routing):min𝐱𝐷(𝐱)subjectto𝐱𝐹𝑆𝐼.(3.3)

The necessary and sufficient condition that a solution to the above overall/individually optimal routing problem satisfies is given as follows.

Lemma 3.1. The link flow pattern 𝐱 is an optimal solution to the overall (and individually) optimal routing problem with constraint 𝐱𝐹𝑆𝐼 (3.3) if and only if 𝐱 satisfies the following set of relations (with 𝑔𝑘𝑖𝑗(𝐱) to be replaced by 𝐺𝑘𝑖𝑗(𝐱) for the individually optimal routing problem): 𝛼𝑘𝑖𝑔𝑘𝑖𝑗(𝐱)=𝛼𝑘𝑗,for𝑥𝑘𝑖𝑗>0,(𝑖,𝑗)𝐿𝑘𝛼,𝑘𝐶,𝑘𝑖𝑔𝑘𝑖𝑗(𝐱)𝛼𝑘𝑗,for𝑥𝑘𝑖𝑗=0,(𝑖,𝑗)𝐿𝑘,𝑘𝐶,(3.4) subject to 𝐱𝐹𝑆𝐼, where 𝛼𝑘𝑖(𝑖𝑁,𝑘𝐶) are Lagrange multipliers.

Proof. We show the case of overall optimization. The case of individual optimization is shown in a similar way. To obtain an optimal solution to problem (3.3), we form the Lagrangian function as follows: 𝐻(𝐱,𝜶)=𝑅𝐷(𝐱)+𝑘𝐶𝑛1𝑖=1𝛼𝑘𝑖𝑟𝑘𝑖if𝑖𝒪𝑘+𝑙𝑉𝑖𝑥𝑘𝑙𝑖𝑟𝑘𝑖if𝑖𝒟𝑘𝑙𝑉𝑖𝑥𝑘𝑖𝑙,(3.5) where 𝛼𝑘𝑖 are Lagrange multipliers.
Since function 𝐷(𝐱) is continuous and convex in 𝐱 (and strictly convex in 𝐱𝑠) and 𝐹𝑆𝐼 is convex, closed, and bounded, there exists a solution (that has a unique 𝐱𝑠) to problem (3.3) similarly as Theorem 2.6. Thus, the link flow pattern 𝐱 that satisfies the following Kuhn-Tucker condition is an optimal solution (that has a unique 𝐱𝑠) to problem (3.3) (see, e.g., [19]): 𝜕𝐻𝜕𝑥𝑘𝑖𝑗=𝑔𝑘𝑖𝑗(𝐱)+𝛼𝑘𝑗𝛼𝑘𝑖𝑥0,𝑘𝑖𝑗𝜕𝐻𝜕𝑥𝑘𝑖𝑗=𝑥𝑘𝑖𝑗𝑔𝑘𝑖𝑗(𝐱)+𝛼𝑘𝑗𝛼𝑘𝑖𝑥=0,𝑘𝑖𝑗0,(𝑖,𝑗)𝐿𝑘𝑟,𝑘𝐶,𝑘𝑖if𝑖𝒪𝑘+𝑙𝑉𝑖𝑥k𝑙𝑖=𝑟𝑘𝑖if𝑖𝒟𝑘+𝑙𝑉𝑖𝑥𝑘𝑖𝑙,𝑖=1,2,,𝑛1,𝑘𝐶.(3.6) Rearranging the above relations, we have, 𝛼𝑘𝑖𝑔𝑘𝑖𝑗(𝐱)=𝛼𝑘𝑗,for𝑥𝑘𝑖𝑗𝛼>0,𝑘𝑖𝑔𝑘𝑖𝑗(𝐱)𝛼𝑘𝑗,for𝑥𝑘𝑖𝑗=0,(𝑖,𝑗)𝐿𝑘,𝑘𝐶,𝐱𝐹𝑆𝐼.(3.7) The above set of relations is equivalent to the set of relations (3.4) and 𝐱𝐹𝑆𝐼, and it is the necessary and sufficient condition for a link flow pattern 𝐱 to be a solution to the overall optimal routing problem (3.3).

With Lemma 3.1, we proceed to have the link-traffic loop-free property in the overall/individually optimal routing for problem (3.3).

Lemma 3.2. The class-𝑘 link traffic in a solution to the overall (and individually) optimal routing problem with constraint 𝐱𝐹𝑆𝐼 (3.3) is loop-free for all 𝑘𝐶. That is, there exists no class-𝑘 link traffic such that 𝑥𝑘𝑣1𝑣2>0, 𝑥𝑘𝑣2𝑣3>0,,𝑥𝑘𝑣𝑚1𝑣𝑚>0, 𝑥𝑘𝑣𝑚𝑣1>0 (for all 𝑘𝐶), where 𝑣1,𝑣2,,𝑣𝑚 are distinct nodes in the solution to the overall (and individually) optimal routing problem and where at least one of the links involved is not a zero-cost link.

Proof. We show the case of overall optimization. It is proved by contradiction. Assume that there exists class-𝑘 link traffic in the solution to overall optimal routing problem (3.3) for multiclass users such as 𝑥𝑘𝑣1𝑣2>0, 𝑥𝑘𝑣2𝑣3>0,,𝑥𝑘𝑣𝑚1𝑣𝑚>0, 𝑥𝑘𝑣𝑚𝑣1>0  (𝑘𝐶), where 𝑣1,𝑣2,,𝑣𝑚 are distinct nodes. According to Lemma 3.1, in the solution, we have 𝛼𝑘𝑣1𝑔𝑘𝑣1𝑣2(𝐱)=𝛼𝑘𝑣2,𝛼𝑘𝑣𝑚𝑔𝑘𝑣𝑚𝑣1(𝐱)=𝛼𝑘𝑣1.(3.8) Then, we have 𝑔𝑘𝑣1𝑣2(𝐱)+𝑔𝑘𝑣2𝑣3(𝐱)++𝑔𝑘𝑣𝑚𝑣1(𝐱)=0,(3.9) which contradicts the fact that 𝑔𝑘𝑖𝑗(𝐱)>0 if 𝑥𝑘𝑖𝑗>0 for at least one of (nonzero-cost) links involved.
The case of individual optimization is shown in a similar way as above by replacing 𝑔𝑘𝑖𝑗(𝐱) by 𝐺𝑘𝑖𝑗(𝐱).

Since the constraint 𝐱𝐹𝑆𝐼 of the optimization problems may be weaker than the set of constraints 𝐱𝐹𝑆𝐱, there may be the possibility that a link flow solution 𝐱(𝐱𝐹𝑆𝐼) may not be realized by any path flow pattern. In the following, however, we confirm that there exists a path flow pattern 𝐲 that results in any loop-free link-flow pattern 𝐱 such that 𝐱𝐹𝑆𝐼.

Proposition 3.3. There exists a path flow pattern 𝐲 satisfying the constraint 𝐲FS𝐲 that results in a link-traffic loop-free flow pattern 𝐱 satisfying the constraint 𝐱FSI. That is, for networks with a link-traffic loop-free flow pattern, FSI=FS𝐱.

Proof. Consider an arbitrary loop-free link-flow pattern 𝐱 that satisfies the constraint 𝐱𝐹𝑆𝐼. We show how to make a path flow pattern 𝐲(𝐹𝑆𝐲) that results in the loop-free link-flow pattern 𝐱(𝐹𝑆𝐼).
We consider the following for each class. Consider a path (𝑜,𝑣1,𝑣2,,𝑣𝑖,𝑣𝑖+1,,𝑑) where 𝑜 is an origin node, 𝑑 is a destination node, and 𝑣1,𝑣2,,𝑣𝑖,𝑣𝑖+1, are called “intermediate nodes.” 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 say that the sequence (𝑜,𝑣1,𝑣2,,𝑣𝑖) is the intermediate path at node 𝑣𝑖 that is included in the intermediate path (𝑜,𝑣1,𝑣2,,𝑣𝑖,,𝑣𝑗) longer than it, for 𝑖<𝑗 and 𝑣𝑗𝑑.
We can assign a path flow pattern 𝐲 such that the constraint 𝐱𝐹𝑆𝐼 is satisfied, as follows. We note that the flow through each intermediate path must be the sum of the flows of the paths that go though the intermediate path for each class. The allotment of the flow to an intermediate path of node 𝑣𝑖+1 is done by splitting the flow to the intermediate paths of node 𝑣𝑖, in a way proportional to the link flow 𝑥𝑘𝑣𝑖,𝑣𝑖+1 on the link (𝑣𝑖,𝑣𝑖+1), say, for class 𝑘 (in the case where node 𝑣𝑖 is neither an origin nor a destination for the class, and other cases are also treated in a formal manner below). In that sense, we obtain a proportionally fair allotment.
More precisely, given a link-traffic loop-free flow pattern, the path flow pattern for class 𝑘, 𝑘𝐶, can be obtained as follows. Since the used links have 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 may receive class-𝑘 flow directly from node 𝑖 (Figure 2). (i)The case where node 𝑖 is a pure origin 𝑜. A node 𝑗 directly connected to a pure origin 𝑜 (𝑗𝐨𝑘𝑎) has the class-𝑘 link flow 𝑥𝑘𝑜𝑗. There must be only one class-𝑘 intermediate path from the origin at a node 𝑗𝐨𝑘𝑎, that passes through one class-𝑘 link to the node 𝑗 from the origin, and, thus, the allotment of class-𝑘 flow to the class-𝑘 intermediate path is straightforward. Clearly, this allotment is relevant to (2.4) and (2.1) but does not violate them since 𝐱𝐹𝑆𝐼 must hold. (ii)The case where node 𝑖 is neither an origin nor a destination: each of the class-𝑘 intermediate paths at node 𝑗, 𝑗𝐢𝑘𝑎, that go through node 𝑖, will be allotted the ratio 𝑥𝑘𝑖𝑗/𝑙𝐢𝑘𝑎𝑥𝑘𝑖𝑙 of the flow of the corresponding class-𝑘 intermediate path at node 𝑖. For example, if the class-𝑘 intermediate path (𝑜,,𝑖) included in a class-𝑘 intermediate path (𝑜,,𝑖,𝑗) has the flow 𝑦𝑘𝑜𝑖, the class-𝑘 intermediate path (𝑜,,𝑖,𝑗) is to be allotted the flow 𝑦𝑘𝑜𝑖𝑗=𝑦𝑘𝑜𝑖𝑥𝑘𝑖𝑗𝑙𝐢𝑘𝑎𝑥𝑘𝑖𝑙.(3.10) Clearly, this allotment is relevant to (2.4) and (2.1) but does not violate them since 𝐱𝐹𝑆𝐼 must hold. (iii)The case where the node 𝑖 is not a pure origin but an origin that has the external arrival rate 𝑟𝑘𝑖: each of the class-𝑘 intermediate paths at node 𝑗, 𝑗𝐢𝑘𝑎, that go through node 𝑖, will be allotted the ratio 𝑙𝐢𝑘𝑎𝑥𝑘𝑖𝑙𝑟𝑘𝑖𝑙𝐢𝑘𝑎𝑥𝑘𝑖𝑙𝑥𝑘𝑖𝑗𝑙𝐢𝑘𝑎𝑥𝑘𝑖𝑙(3.11) of the flow of the corresponding class-𝑘 intermediate path at node 𝑖. In addition, a new set of intermediate paths starting at node 𝑖 is added to the group of node-𝑗 intermediate paths, and each is allotted the flow 𝑟𝑘𝑖((𝑥𝑘𝑖𝑗)/(𝑙𝐢𝑘𝑎𝑥𝑘𝑖𝑙)). Clearly, this allotment is relevant to (2.4) and (2.1) but does not violate them since 𝐱𝐹𝑆𝐼 must hold. (iv)The case where the node 𝑖 is not a pure destination but a destination that has the departure rate 𝑟𝑘𝑖: each of the class-𝑘 intermediate paths at node 𝑗, 𝑗𝐢𝑘𝑎, that go through node 𝑖, will be allotted the ratio 𝑙𝐢𝑘𝑎𝑥𝑘𝑖𝑙𝑙𝐢𝑘𝑎𝑥𝑘𝑖𝑙+𝑟𝑘𝑖𝑥𝑘𝑖𝑗𝑙𝐢𝑘a𝑥𝑘𝑖𝑙(3.12) of the flow of the corresponding class-𝑘 intermediate path at node 𝑖. In addition, we have a set of complete paths ending at node 𝑖, and each is allotted the ratio 𝑟𝑘𝑖/(𝑙𝐢𝑘𝑎𝑥𝑘𝑖𝑙+𝑟𝑘𝑖) of the flow of the corresponding intermediate path of node 𝑖. Clearly, this allotment is relevant to (2.4) and (2.2) but does not violate them since 𝐱𝐹𝑆𝐼 must hold. (v)The case where node 𝑖 is a pure destination for class 𝑘. All the class-𝑘 paths that reach this node terminate at this node, and no further path-flow allotment for class 𝑘 is needed anymore. Clearly, we see that, at this node, class-𝑘 path flow allotment so far is relevant to (2.4) and (2.2) but does not violate them since 𝐱𝐹𝑆𝐼 must hold.
We therefore see that, at every step of the above five allotments in cases (i), (ii), (iii), (iv), and (v), the set of constraints (2.4) and 𝐲𝐹𝑆𝐲 is satisfied.
Therefore, for an arbitrary 𝐱 (𝐹𝑆𝐼) with the loop-free property, 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 allotting the amount of class-𝑘 flows to intermediate paths, and finally we can complete the assignment 𝐲 of class-𝑘 path flows, for all 𝑘𝐶, that result in the above-mentioned 𝐱 (𝐹𝑆𝐼).
We note that the above-obtained 𝐲 satisfies both the constraints (2.4) and 𝐲𝐹𝑆𝐲 and, thus, that the above-mentioned 𝐱 satisfies the constraint 𝐱𝐹𝑆𝐱.
Since 𝐹𝑆𝐼 includes 𝐹𝑆𝐱, then, for networks with a link-traffic loop-free flow pattern, 𝐹𝑆𝐼=𝐹𝑆𝐱.

645954.fig.002
Figure 2: Assigning flows to paths on the basis of link flows.

From the above Proposition and Lemma 3.2, we can confirm the following.

Lemma 3.4. For the solution 𝐱 with the link-traffic loop-free property for the overall/individually optimal routing with constraint 𝐱𝐹𝑆𝐼 (3.3) there exists a 𝐲 that satisfies both the set of constraints (2.4) and 𝐲𝐹𝑆𝐲, that is, 𝐹𝑆𝐼=𝐹𝑆𝐱.

We, therefore, see that the solution 𝐱 with the link-traffic loop-free property for the overall/individually optimal routing with constraint 𝐱𝐹𝑆𝐼 (3.3) is also the solution for the overall (and individually) optimal routing problem ((2.8), (2.17), (3.3)), since its solution exists as discussed in Section 2.4.

Then, we have the following theorem.

Theorem 3.5 (Link-traffic loop-free property). The class-𝑘 link traffic in a solution to the overall (and individually) optimal routing problem ((2.8), (2.17), (3.3)) for a multiclass network is loop-free for all 𝑘𝐶. That is, there exists no class-𝑘 link traffic such that 𝑥𝑘𝑣1𝑣2>0, 𝑥𝑘𝑣2𝑣3>0,,𝑥𝑘𝑣𝑚1𝑣𝑚>0, 𝑥𝑘𝑣𝑚𝑣1>0 (for all 𝑘𝐶), where 𝑣1,𝑣2,,𝑣𝑚 are distinct nodes in the solution to the overall (and individually) optimal routing problem and where at least one of the links involved is not a zero-cost link.

Now consider the case where two nodes are connected by two links. We have the following result.

Corollary 3.6 (One-way traffic property). For any optimal solution 𝐱 to an overall/individually optimal routing problem ((2.8), (2.17), (3.3)) for multiclass networks, the following relations hold true: 𝑥𝑘𝑖𝑗=0,if𝑥𝑘𝑗𝑖>0,(3.13) where either (𝑖,𝑗) or (𝑗,𝑖) is not a zero-cost link for class 𝑘, for (𝑖,𝑗),(𝑗,𝑖)𝐿𝑘,𝑘𝐶.

Proof. It is a direct result from Theorem 3.5.

The property shown in Corollary 3.6 is called the one-way traffic property for the overall/individually optimal routing in multiclass networks. The physical meaning is clear. It shows that the traffic from the node 𝑖 to node 𝑗, 𝑥𝑘𝑖𝑗, and the user flow rate from the node 𝑗 to node 𝑖, 𝑥𝑘𝑗𝑖 cannot be positive both at the same time as shown in Figure 3.

645954.fig.003
Figure 3: One-way traffic property in overall and individually optimal routing.

Recall the definitions on the networks given at the beginning of Section 2. Since we assume that the users (commodities) do not change their classes during flowing through their path, we can partition classes into disjoint subsets. Consider that each subset of classes is associated either with a decision maker (or an atomic player) or with infinitely many decision makers (nonatomic users). Consider a case where an equilibrium exists where all of atomic users and nonatomic users achieve their own cost minimization unilaterally. In such an equilibrium, we can see that each class has no link loop, by applying Theorem 3.5 to each atomic user or to the collection of nonatomic users with the behaviors of other users being given.

4. Examples

4.1. The Cases Where Assumption (A1) and (A2) Hold Naturally

As to many networks, we can naturally assume the assumption (A1). As we noted before, however, assumption (A2) looks somewhat awkward. In the following two cases, however, the assumption (A2) holds naturally.

4.1.1. Multiclass Routing in Networks with a Common Destination

Consider the overall/individually optimal routing problem for a multiclass network with one common destination and multiple origins for a class (we call it the problem with a common destination for the sake of brevity) as shown in Figure 4. Note that the problems of load balancing in distributed computer systems [21, 22] are equivalent to the routing problems in the networks with one common destination and multiple origins. Clearly, the assumption (A2) holds naturally for the networks. The two link-traffic loop-free properties, Theorem 3.5 and Corollary 3.6 shown in the above section hold for the networks under overall and individually optimal routing.

645954.fig.004
Figure 4: A network with one common destination and multiple origins.

In contrast, consider a case of noncooperative optimal routing with a finite (but plural) number of players for this model, that is, users are divided into groups each of which is controlled by a decision maker that strives to optimize unilaterally the cost for its group only. Link-traffic loops have been found in the above-mentioned load-balancing problems (shown in [14, 15] if the definition of class given in those papers is changed to be the same as the one given in this paper).

4.1.2. Multiclass Routing in Networks with a Common Origin

We proceed to consider another network where there are multiple destinations but only one common origin (we call it the network with a common origin for the sake of brevity) as shown in Figure 5. The two link-traffic loop-free properties, Theorem 3.5 and Corollary 3.6 shown in the above section, hold for the networks for overall and individually optimal routing. Clearly, the assumption (A2) holds naturally for the networks also.

645954.fig.005
Figure 5: A network with one common origin and multiple destinations.
4.2. Examples Where Assumption (A1) Does Not Hold

In this section, we examine two examples wherein either (1) or (2) in the assumption (A1) is violated whereas the assumption (A2) holds. 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 overall and individually optimal routing in multiclass networks.

4.2.1. 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 6. Nodes 2 and 3 are connected by links (2,3) and (3,2). We consider the case where, in the one-class network, there exist only two paths (1,2,3,4) and (1,3,2,4) connecting the O-D pair (1,4) of the one class, but (1,2,4) and (1,3,4) are not paths connecting the O-D pair (1,4), which violates (2) in the assumption (A1). 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 for the one class.

645954.fig.006
Figure 6: A network with one origin and one destination that satisfies assumption (A1)(1) but does not satisfy (A1)(2).

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 for the one class. Then, in optimal routing, only paths (1,2,4) and (1,3,4) are used, and the network has no link-traffic loop.

4.2.2. 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 7. We consider the case where, in the network, there exist only two distinct O-D pairs 1-3 and 4-2 (two O-D pairs for one class) and where only (1,2,3) and (4,3,2) are possible paths. On the other hand, there exists no path connecting 1 (as an origin) and 2 (as the corresponding destination) for the one class although link (1,2) exists, and there exists no path connecting 4 (as an origin) and 3 (as the corresponding destination) for the one class although link (4,3) exits. That is, O-D pairs neither of 1-2 nor 4-3 works as an origin -destination pair, which violates (1) in the assumption (A1). Thus the commodity that enters the network at the origin 1 can get out of the network only at the destination 3 but not at 2, and the commodity that enters the network at the origin 4 can get out of the network only at the destination 2 but not at 3. 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 overall/individually optimal routing problem under the set of constraints (2.4) and 𝐲𝐹𝑆𝐲. In this example, it is clear that we have a link-traffic loop for the one class, that is, 𝑥23>0 and 𝑥32>0.

645954.fig.007
Figure 7: A network with two origins and two destinations that satisfies assumption (A1)(2) but does not satisfy (A1)(1).

On the other hand, if (1) in the assumption (A1) holds, both of (1,2) and (4,3) can be paths for the one class, and the solution under the constraint 𝐱𝐹𝑆𝐱 is such that 𝑥23=𝑟13𝑟420 and 𝑥32=0 if 𝑟13𝑟42 (under assumption (A2)), which shows the freedom of link-traffic loops that holds under assumption (A1).

So far, we consider overall optimization (in which only one decision maker, or a player, is involved) and individual optimization (in which infinitely many decision makers, or infinitely many players, are involved) are involved. In this section, we mention an extension of the above-mentioned loop-free property.

5. Conclusion

In this paper, we have studied both overall and individually optimal routing problems for multiclass networks with generalized link-cost functions and network configurations. We have seen that there is an associate overall optimal routing problem to each individually optimal routing problem for multiclass networks with the same solution under some condition. We have discussed the existence and uniqueness of the solutions to overall and individually optimal routing. Furthermore, we have shown that the link-traffic loop-free property holds for the overall and individually optimal routing in a wide range of networks. While doing so, we have discussed the relation between the formulations with path and link flow patterns. We have shown the condition that characterizes the category of multiclass networks that have the link-traffic loop-free property for overall and individually optimal routing.

References

  1. M. Patriksson, The Traffic Assignment Problem—Models and Methods, VSP, Utrecht, The Netherlands, 1994.
  2. J. G. Wardrop, “Some theoretic aspects of road traffic research,” Proceedings of the Institution of Civil Engineers, vol. 1, pp. 325–378, 1952.
  3. D. Bertsekas and R. Gallager, Data Networks, Prentice-Hall, Englewood Cliffs, NJ, USA, 2nd edition, 1992.
  4. L. Fratta, M. Gerla, and L. Kleinrock, “The flow deviation method: an approach to store-and-forward communication network design,” Networks, vol. 3, pp. 97–133, 1973. View at Zentralblatt MATH
  5. R. G. Gallager, “A minimum delay routing algorithm using distributed computation,” IEEE Transactions on Communications, vol. 25, no. 1, pp. 73–85, 1977. View at Zentralblatt MATH
  6. E. Altman and H. Kameda, “Equilibria for multiclass routing problems in multi-agent networks,” in Advances in Dynamic Games: Annals of International Society of Dynamic Games, Vol. 7, A. S. Nowak and K. Szajowski, Eds., vol. 7, pp. 343–367, Birkhäuser, Boston, Mass, USA, 2005, An extended version of the paper that appeared in Proceedings of the 40th IEEE Conference on Decision and Control (CDC ’01), Orlando, Fla, USA, pp. 604–609, December 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. J. E. Cohen and C. Jeffries, “Congestion resulting from increased capacity in single-server queueing networks,” IEEE/ACM Transactions on Networking, vol. 5, no. 2, pp. 305–310, 1997.
  8. A. Orda, R. Rom, and N. Shimkin, “Competitive routing in multiuser communication networks,” IEEE/ACM Transactions on Networking, vol. 1, no. 5, pp. 614–627, 1993.
  9. T. Roughgarden, “On the severity of Braess's Paradox: designing networks for selfish users is hard,” Journal of Computer and System Sciences, vol. 72, no. 5, pp. 922–953, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. P. Gupta and P. R. Kumar, “A system and traffic dependent adaptive routing algorithm for Ad Hoc networks,” in Proceedings of the 36th IEEE Conference on Decision and Control, pp. 2375–2380, San Diego, Calif, USA, December 1997.
  11. S. C. Dafermos, “Extended traffic assignment model with applications to two-way traffic,” Transportation Science, vol. 5, no. 4, pp. 366–389, 1971.
  12. S. C. Dafermos, “The traffic assignment problem for multi-class user transportation networks,” Transportation Science, vol. 6, pp. 73–87, 1972.
  13. S. C. Dafermos and F. T. Sparrow, “The traffic assignment problem for a general network,” Journal of Research of the National Bureau of Standards, vol. 73B, no. 2, pp. 91–118, 1969. View at Zentralblatt MATH
  14. H. Kameda, E. Altman, T. Kozawa, and Y. Hosokawa, “Braess-like paradoxes in distributed computer systems,” IEEE Transactions on Automatic Control, vol. 45, no. 9, pp. 1687–1691, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. H. Kameda and O. Pourtallier, “Paradoxes in distributed decisions on optimal load balancing for networks of homogeneous computers,” Journal of the ACM, vol. 49, no. 3, pp. 407–433, 2002. View at Publisher · View at Google Scholar
  16. R. B. Myerson, Game Theory: Analysis of Conflict, Harvard University Press, Cambridge, Mass, USA, 1991.
  17. D. Monderer and L. S. Shapley, “Potential games,” Games and Economic Behavior, vol. 14, no. 1, pp. 124–143, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, NJ, USA, 1970.
  19. M. D. Intriligator, Mathematical Optimization and Economic Theory, Prentice-Hall, Englewood Cliffs, NJ, USA, 1971.
  20. M. W. Jeter, Mathematical Programming, vol. 102, Marcel Dekker, New York, NY, USA, 1986.
  21. J. Li and H. Kameda, “Load balancing problems for multiclass jobs in distributed/parallel computer systems,” IEEE Transactions on Computers, vol. 47, no. 3, pp. 322–332, 1998. View at Publisher · View at Google Scholar · View at MathSciNet
  22. A. N. Tantawi and D. Towsley, “Optimal static load balancing in distributed computer systems,” Journal of the Association for Computing Machinery, vol. 32, no. 2, pp. 445–465, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH