**1. Introduction**

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In this work we face the Routing problem defined as an optimal control problem, with control variables representing the percentages of each flow routed along the available paths, and with a cost function which accounts for the distribution of traffic flows across the network resources (multipath routing). In particular, the scenario includes the load balancing problem already dealt with in a previous work (Bruni et al., 2010) as well as the bottleneck minimax control problem. The proposed approaches are then compared by evaluating the performances of a sample network.

In a given network, the resource management problem consists in taking decisions about handling the traffic amount which is carried by the network, while respecting a set of Quality of Service (QoS) constraints.

As stated in Bruni et al., 2009a, b, the resource management problem is hardly tackled by a single procedure. Rather, it is currently decomposed in a number of subproblems (Connection Admission Control (CAC), traffic policing, routing, dynamic capacity assignment, congestion control, scheduling), each one coping with a specific aspect of such problem. In this respect, the present work is embedded within the general approach already proposed by the authors in Bruni et al., 2009a, b, according to which each of the various subproblems is given a separate formulation and solution procedure, which strives to make the other sub-problems easier to be solved. More specifically, the above mentioned approach consists in charging the CAC with the task of deciding, on the basis of the network congestion state, new connection admission/blocking and possible forced dropping of the in-progress connections with the aim of maximizing the number of accepted connections, whilst satisfying the QoS requirements.

According to the proposed approach, the role of the other resource management procedures is the one of keeping the network as far as possible far from the congestion state. Indeed, the more the network is kept far from congestion, the higher is the number of new connection set-up attempts that can be accepted by the CAC without infringing the QoS constraints,

Optimal Control Strategies for Multipath Routing:

variables.

itself. Therefore we have:

Holma and Toskala, 2002).

From Load Balancing to Bottleneck Link Management 407

In this work, we consider the possibility/opportunity of splitting the given network into sub-networks as detailed in Bruni et al., 2010 each one controlled by a separate subset of

This work is organized as follows. In Section 2, a definition for a reference communication network and its decomposition is given, which is useful for the routing problem; in Sections 3, we in depth study the optimal routing control problem with reference to a number of different cost functions; Section 4 shows some results in order to evaluate the performance and to compare the found optimal solutions for traffic balancing and bottleneck link

**2. Reference telecommunication network definition and decomposition** 

At any fixed time, the telecommunication network can be defined in terms of its topological description as well as in terms of its traffic pattern. As far as network topology is concerned, we consider the network nodes *n* ∈ Ν = {*n*1,*n*2,…,*nN*} and the network links defined as ordered pairs of nodes *l* ∈ Λ = {*l*1,*l*2,…,*lL*}. To describe the network traffic request we first define a path *v* ∈ Ω = {*v*1,*v*2,…,*vV*} as a collection of consecutive links, denoted by Λ*v*, from an ingoing node *i* to an outgoing node *j* (where *i*,*j* ∈ **N**). Moreover a certain set of different Service Classes *k* ∈ Κ = {*k*1,*k*2,…,*kK*}, is defined, each one characterized by a set of Quality of Service (QoS) parameters. According to the most recent trends, the QoS control is performed on a per flow basis, where a flow *f* ∈ Φ = {*f*1,*f*2,…,*fF*} is defined as the triple *f* = (*ni,nj,kp*), with *ni* denoting the ingoing node, *nj* denoting the outgoing node and *kp* denoting the service class. The traffic associated with a given flow *f* may possibly be routed on a set Ω*f* of one or more paths. We further

1, if (, ) 0, otherwise

For each link *l* ∈ Λ, at the given time, we may consider its occupancy level *c*(*l*) defined as the sum of all contributions to the occupancy due to the flows routed on the link itself. Each contribution of this type will be quantified by the bit rate *R*(*l*,*f*) which, in turn, is the sum of bit rates of all in-progress connections going through the link *l* and relevant to the flow *f*, possibly weighted by a coefficient α(*l*,*f*) which accounts for the specific need of the flow

*alv* <sup>∈</sup> <sup>=</sup> 

*f*

∈Φ

*cl l f Rl f* α

where α(*l*,*f*) are positive known coefficients which take into account the fact that some technologies differentiate the classes of service by varying modulation, coding, and so on. For each link *l*, we consider the so-called nominal capacity *cNOM*(*l*), that is the value of the occupancy level suggested for a proper behaviour of the link (typically in terms of QoS)1.

<sup>1</sup> *c*(*l*) and *cNOM*(*l*) can be interpreted as generalizations of "load factor" and "Noise Rise" in UMTS (see

*l v*

(1)

<sup>=</sup> (2)

management; finally, concluding remarks in Section 5 end the work.

introduce the set of indices {*a*(*l*,*v*), *l* ∈ Λ*, v* ∈ Ω}, defined as follows:

() (, ) (, )

and hence the traffic carried by the network increases. By so doing, the CAC and the other resource management procedures can work in a consistent way, while being kept independent.

This work deals with the multipath routing problem. Multipath routing is a widespread topic in the literature. For example, Cidon et al., 1999, and Banner and Orda, 2007, demonstrate the advantages of multipath routing with respect to single-path routing in terms of network performances; Chen et al., 2004, considers the multipath routing problem under bandwidth and delay constraints; Lin and Shroff, 2006, formulate the multipath routing problem as a utility maximization problem with bandwidth constraints; Guven et al., 2008, extend the multipath routing to multicast flows; Jaffe, 1981, Tsai et al., 2006, Tsai and Kim, 1999 deal with the multipath routing as a minimax optimization problem.

In this work we face the multipath routing problem formulated as an optimal control problem, with control variables representing the percentages of each flow routed along the available paths. As a matter of fact, in the most advanced networks each flow can be simultaneously routed over more than one path: the routing procedure has to decide the percentages of the traffic belonging to the considered flow which have to be routed over the paths associated to the flow in question. According to the above mentioned vision, we assume that other resource management control units (specifically the CAC) already dealt with and decided about issues such as how many, which ones, when and for how long connections have to be admitted in the network, with specific QoS constraints (related to losses and delays) to be satisfied. Therefore, the routing control unit has to deal with an already defined offered traffic. Thus, the admissible set for the routing control variables turns out to be closed, bounded and non-empty, and the existence of (at least) an optimal solution of the routing problem is guaranteed.

The goal of an optimal routing policy aims the routing problem solution towards a network traffic pattern which should make QoS requirements and consequently the CAC task (implicitly) easier to be satisfied. The quality of the routing solution will be evaluated by different performance indices, which take a nominal capacity for each link into account.

As far as the dynamical aspects of a routing problem, we first note that explicitly accounting for them would call for a reliable and sufficiently general dynamical model for the offered traffic. However it is widely acknowledged that such a model is not available and hard to design, due to unpredictable features of Internet traffic. And, in any case, the requested dynamical characters are committed to the CAC procedures, where the more reliable connection dynamics model along with the feedback structure may properly handle the issue.

In addition, a non-dynamical set up for the routing problem makes it much easier to be dealt with. Moreover, this approach could be justified by assuming that the time scale for changes in the routing policy is surely slower than the bit rate fluctuations in the in-progress connections, but it is reasonably faster than the evolution of traffic statistical features. Thus, the routing policy has to be periodically computed to fit the most likely traffic pattern at each given period of time.

and hence the traffic carried by the network increases. By so doing, the CAC and the other resource management procedures can work in a consistent way, while being kept

This work deals with the multipath routing problem. Multipath routing is a widespread topic in the literature. For example, Cidon et al., 1999, and Banner and Orda, 2007, demonstrate the advantages of multipath routing with respect to single-path routing in terms of network performances; Chen et al., 2004, considers the multipath routing problem under bandwidth and delay constraints; Lin and Shroff, 2006, formulate the multipath routing problem as a utility maximization problem with bandwidth constraints; Guven et al., 2008, extend the multipath routing to multicast flows; Jaffe, 1981, Tsai et al., 2006, Tsai and Kim, 1999 deal with the multipath routing as a minimax optimization

In this work we face the multipath routing problem formulated as an optimal control problem, with control variables representing the percentages of each flow routed along the available paths. As a matter of fact, in the most advanced networks each flow can be simultaneously routed over more than one path: the routing procedure has to decide the percentages of the traffic belonging to the considered flow which have to be routed over the paths associated to the flow in question. According to the above mentioned vision, we assume that other resource management control units (specifically the CAC) already dealt with and decided about issues such as how many, which ones, when and for how long connections have to be admitted in the network, with specific QoS constraints (related to losses and delays) to be satisfied. Therefore, the routing control unit has to deal with an already defined offered traffic. Thus, the admissible set for the routing control variables turns out to be closed, bounded and non-empty, and the existence of (at least) an optimal

The goal of an optimal routing policy aims the routing problem solution towards a network traffic pattern which should make QoS requirements and consequently the CAC task (implicitly) easier to be satisfied. The quality of the routing solution will be evaluated by different performance indices, which take a nominal capacity for each link into account.

As far as the dynamical aspects of a routing problem, we first note that explicitly accounting for them would call for a reliable and sufficiently general dynamical model for the offered traffic. However it is widely acknowledged that such a model is not available and hard to design, due to unpredictable features of Internet traffic. And, in any case, the requested dynamical characters are committed to the CAC procedures, where the more reliable connection dynamics model along with the feedback structure may properly handle the

In addition, a non-dynamical set up for the routing problem makes it much easier to be dealt with. Moreover, this approach could be justified by assuming that the time scale for changes in the routing policy is surely slower than the bit rate fluctuations in the in-progress connections, but it is reasonably faster than the evolution of traffic statistical features. Thus, the routing policy has to be periodically computed to fit the most likely traffic pattern at

independent.

problem.

issue.

each given period of time.

solution of the routing problem is guaranteed.

In this work, we consider the possibility/opportunity of splitting the given network into sub-networks as detailed in Bruni et al., 2010 each one controlled by a separate subset of variables.

This work is organized as follows. In Section 2, a definition for a reference communication network and its decomposition is given, which is useful for the routing problem; in Sections 3, we in depth study the optimal routing control problem with reference to a number of different cost functions; Section 4 shows some results in order to evaluate the performance and to compare the found optimal solutions for traffic balancing and bottleneck link management; finally, concluding remarks in Section 5 end the work.
