Joint route selection and resource allocation in multihop wireless networks based on a game theoretic approach

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Contents lists available at SciVerse ScienceDirect

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Joint route selection and resource allocation in multihop wireless networks based on a game theoretic approach

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Jorge Ortín a,⇑, José Ramón Gállego b, María Canales b a

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b

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Telematic Engineering Department, University Carlos III, Madrid, Spain Aragón Institute of Engineering Research, University of Zaragoza, Zaragoza, Spain

a r t i c l e

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i n f o

Article history: Received 16 October 2012 Received in revised form 14 February 2013 Accepted 1 May 2013 Available online xxxx

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Keywords: Game theory Multi-hop networks End-to-end resource allocation Power control SINR model

a b s t r a c t In this work we aim to design simple, distributed self-configuring solutions for the problem of route selection and channel and power allocation in multihop autonomous wireless systems using a game theoretic perspective. We propose and compare three games with different levels of complexity: a potential flow game where players need complete network knowledge, a local flow game requiring full information of the flow and a low complexity cooperative link game which works with partial information of the flow. All these games have been designed to always assure the convergence to a stable point in order to be implemented as distributed algorithms. To evaluate their quality, we also obtain the best achievable performance in the system using mathematical optimization. The system is modeled with the physical interference model and two different definitions of the network utility are considered: the number of active flows and the aggregated capacity in bps. Results show that the proposed games approach the centralized solution, and specially, that the simpler cooperative link game provides a performance close to that of the flow games.  2013 Elsevier B.V. All rights reserved.

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1. Introduction

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Autonomous, self-configuring multihop wireless systems present a versatile solution to provide broadband services with infrastructure-less deployments based on a decentralized management. One of the main research challenges to face in this kind of networks is the proposal of efficient and distributed radio resource management solutions that accomplish the channel and power allocation for the links of each flow in the network. These solutions should be simple enough to be implemented in real systems, where overall information on the environment is not assured, and should get good results in terms of the global network performance.

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⇑ Corresponding author.

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E-mail addresses: [email protected] (J. Ortín), [email protected] (J.R. Gállego), [email protected] (M. Canales).

The use of game theory to tackle these challenges has recently received an increasing interest, since it can be used to analyze the strategic interactions among the different autonomous elements forming the network, giving an insight into its expected behavior and performance [1–6]. Additionally, in the context of self-configuring wireless systems without an available centralized controller, noncooperative games can provide efficient and distributed solutions for radio resource management [1]. The joint channel and power allocation problem for distributed wireless networks have been already studied with a game theoretical perspective [7–12], but restricted to the establishment of single hop links between pairs of nodes as isolated entities. However, in a multihop wireless system, the successful activation of a flow between two nodes requires the selection of a route between source and destination and the joint resource allocation in all the links of that route. There are very few works tackling the problem of end-to-end resource allocation from a game theoretic per-

1570-8705/$ - see front matter  2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.adhoc.2013.05.002

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spective [13,14] and, to the authors’ knowledge, none of them devoted to the joint problem of route selection and channel and power allocation for each link in the flow under the realistic physical interference or SINR model [15]. In this context, the aim of this work is to analyze the behavior of competing flows within an autonomous multihop wireless system, providing strategies that maximize the network utility, measured in terms of the aggregate capacity in bps of all the established flows and of the total number of established flows. These definitions are useful to model elastic and fixed rate applications respectively. To address this problem, we propose three different games that perform joint route selection and end-to-end channel and power allocation. In two of them the flows themselves are the players: The first one is a potential game where each flow needs global information about the remaining flows in the network, while the second is a local flow game which requires information about all the links in the flow. In the third game, the individual links of each flow are the players, so each link updates its strategy locally, with a certain degree of cooperation among the links of the flow, but with much lower complexity than the flow games. The obtained results demonstrate that this simple cooperative link game can provide a feasible solution for the addressed problem with good performance, limited computational complexity and low requirements of environmental information. Finally, to analyze the efficiency of the obtained equilibria in the proposed games, we characterize the quality of the equilibria in terms of the Price of Anarchy (PoA) and the Price of Stability (PoS). These measures are obtained formulating a Mixed Integer Non-Linear Programming (MINLP) problem that can be solved with the branch and bound algorithm [16]. The remaining of the paper is organized as follows: the related work and the main contributions are presented in Section 2. Section 3 describes the system model and the definitions of capacity that will be used. In Section 4, some basic concepts of game theory are given and the three proposed games are explained in detail. Section 5 presents the formal characterization of the equivalent centralized optimization problem and the basis of the branch and bound algorithm. Section 6 shows the simulation framework and the obtained results. Finally, some conclusions are provided in Section 7.

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2. Motivation

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2.1. Related work

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In this section we provide an overview of the contributions more related to our work, in particular those focused on channel allocation and power control for multihop wireless systems using game theory. In this sense, while both channel and power allocation for single-hop links has been extensively studied, few works deals with the establishment of end-to-end multihop flows. Specifically, channel allocation for single-hop wireless networks is studied from a game theoretical perspective in [17–19]. Nevertheless all these works use the protocol

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interference model [15] and assume that the transmission power, and therefore, the transmission and the interference ranges are fixed. This causes that aspects such as the quality or the capacity of the links cannot be taken into account. In other works [7–9,20] the interference generated by each link is introduced in the definition of the utility function of the players, proposing different cooperative and selfish strategies. In these works, the cooperative strategies based on a potential game provide significant better performance, but they require that each player has global information of the strategies of all the remaining players in the network, which is not realistic in a real deployment [21]. In [12] we proposed a game theoretic solution for joint channel and power allocation of single-hop networks under the SINR model and different utility definitions depending on the requirements of the users’ applications. In that work, we proposed a game using only local information that provides a performance similar to that of a potential game. The main distinguishing features of this local game are the inclusion of SINR and power constraints to reduce the selfishness of the players without requiring explicit coordination. As stated above, all these works deals with the establishment of single hop isolated links between pairs of nodes. However, in a multihop network, the successful establishment of an end-to-end flow between two nodes requires the joint activation of all the links of the flow. In [13] game theory is used to perform the channel allocation taking into account these end-to-end requirements. Nevertheless, all the flows are assumed to reside in a single collision domain and the simplistic protocol interference model with fixed transmission and sensing ranges is used. In [14] we analyzed the behavior of competing flows within an autonomous multihop wireless system, providing strategies that maximize the number of flows that can be established. For this purpose, we proposed three different games that perform the end-to-end joint channel and power allocation using the more realistic SINR model. Nevertheless, the existence of an equilibrium point and the efficiency of the obtained equilibria for these end-to-end flow games remained as an open issue.

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2.2. Main contributions

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This work focuses on end-to-end channel allocation and power control strategies with joint route selection in distributed multihop wireless networks using game theory. We specifically aim to propose distributed and adaptive solutions to the problem. The flexibility to develop autonomous functions with light cooperation among the nodes in the single-link scenario [12] motivate to further analyze the more realistic end-to-end scenario. In [14] we proposed an initial version of the games, but only aiming at maximizing the number of established flows with a unique route between the source and the destination nodes. In [22] we extended the games to include also the aggregate capacity as a utility definition. In this paper, we extend this game theoretic framework with the following contributions:

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 The games are redefined to include the possibility of choosing between different routes to establish the flow between the source and destination nodes, instead of performing the joint channel and power allocation over a single pre-established route.  In [14] the existence of a pure Nash Equilibrium and the convergence to that equilibrium for both the local flow game and the cooperative link game were not guaranteed. In this paper we modify these games to ensure the convergence to a NE in the local flow game and to a predefined stable point in the cooperative link game.  We propose a MINLP optimization problem characterizing the same route selection and resource allocation problem considered for the games. The solution of this optimization problem with the branch and bound algorithm provides an upper bound to the performance of the distributed game-based solution and enables to obtain the PoS and PoA of the games. Unlike other approaches [23], which optimize the resource allocation assuming that the resources are enough for all the sessions to be active, we force an admission control of competing flows defining the objective function as the maximum number of active sessions or the maximum aggregate capacity of the active sessions. In addition, instead of using the continuous Shannon capacity, we consider a more realistic discrete or binary capacity.

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3. System model

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The system considered in this work is a multihop wireless network with N nodes and C non-interfering channels. All the nodes in the system try to access these channels and all of them have the same rights in using the available ones. To model in a simplified and generic way the presence of other kind of interference sources or users, it is assumed that some of these C channels are not sensed as available for the modeled users. The set of available bands depends on the geographic location of the users to introduce space variability in the resource distribution. Without loss of generality, we assume that each node is equipped with up to C transceivers, so it can establish a link at each available channel. Within this network, there is a set F of multihop flows which want to be established, each of them with a different pair of source and destination nodes. The objective is to maximize either the number of established flows or their capacity.

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3.1. SINR model

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Given a directional link l between a pair of nodes (lTX ? lRX), the channel gain from transmitter (lTX) to receic ver (lRX) is defined as g l;l ¼ dl;l , being dl,l the distance from lTX to lRX and c the path loss index. We assume that each link l can only transmit in a single channel cl and that the transmission power is discretized into Q + 1 levels q = {0, 1, . . . , Q}, equispaced between 0 and the maximum transmission power Pmax. Thus, the transmission power of link l is:

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pl ¼ ql 

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Pmax Q

ð1Þ

where ql represents the level q at which the link l transmits. Under the physical interference model, a transmission is successful if the SINR at the receiving node is higher than a certain threshold a, i.e., if it is fulfilled:

SINRl ¼

pl  g l;l

PN þ

P Pa m 2 L; m–l pm  g m;l cm ¼ cl

with wcl the bandwidth of channel cl. However, in a real system, the transmission rate is given by a predefined set of discrete values, basically depending on the modulation and channel coding schemes that guarantee a bit error rate for the actual SINRl. A simplified way of introducing this effect in the proposed model is applying the Hartley’s law, which sets the maximum capacity for a given number of levels in the modulation, Ml:

ð4Þ

Considering this upper bound and the Shannon capacity, and assuming the number of modulation levels is a power of two, the maximum allowed Ml for a given SINRl is:

Ml ¼ 2

blog2 ð1þSINRl Þ=2c

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ð3Þ

LC l 6 2  wcl  log2 ðMl Þ

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ð2Þ

where pl and cl are the power and the channel used by the link l respectively, L corresponds to the set of links in the network and PN is the background noise power. The term gm,l represents the channel gain from the transmitter of the link m (mTX) to the receiver of the link l (lRX) and is c equal to dm;l . Each link l will have a capacity LCl which will depend on the value of SINRl. This capacity is upper-bounded by the theoretical limit obtained with the Shannon theorem, which states that the maximum available capacity for a link l in an AWGN (Additive White Gaussian Noise) channel is:

LC l ¼ wcl  log2 ð1 þ SINRl Þ

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ð5Þ

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The minimum value for Ml is Mmin = 2, corresponding to a binary modulation. Arbitrarily setting a Mmax, a fixed set of capacities for each link can be defined. Replacing (5) in (4) and considering the SINR restriction of (2), the following link capacity is obtained:

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LC l ¼



2  blog2 ð1 þ SINRl Þ=2c if SINRl > a 0

otherwise

ð6Þ

where it has been assumed for simplicity that wcl ¼ 1 for all the channels in the network. This discrete capacity is useful for elastic applications, such as those based on TCP, where the transmission rate can be adapted to the available resources. However, for inelastic applications with a fixed rate (as VoIP, for example), which need a minimum SINR to guarantee a predefined bit error rate, the only requirement is to establish the link with a SINR higher than a. In this case, the link capacity is binary and it can be directly defined as follows:

LC l ¼



1 if SINRl > a 0

otherwise

ð7Þ

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3.2. End-to-end resource allocation

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As stated before, we consider a set F of multihop flows, each of them with a different pair of source and destination nodes. Each flow f in F has an associated set Rf of available routes r, with each route formed by a sequence of kr directed links from the source node to the destination node of f. It is assumed the existence of a common control channel shared by the users. The contention to access this channel or the control protocol specification are out of the scope of this paper. Through this channel, nodes can exchange control information in order to create the aforementioned routes following a multipath routing protocol such as MP-OLSR [24]. Although multiple paths can be used as backup route or be employed simultaneously for parallel data transmission, in this work we are going to use the multiple available paths to choose one where end-to-end resources can be allocated. To define the network graph, we consider that there is an edge between two nodes if and only if a link can be established between these nodes at the maximum capacity achievable according to (6) when only the background noise is present, i.e., no other link is transmitting. From this graph, routes are generated with the multipath Dijkstra algorithm [24]. Using this algorithm, we try to obtain for each flow R completely disjoint routes from the source node to the destination node. Since in some cases it is not possible to get so many routes, the value of jRfj can be lower than R and different for each flow of the network. In this scenario a flow is feasible if and only if all the links of one of its routes can be established simultaneously. The capacity RCr,f of a specific route r in a flow f will be given by the bottleneck link in the flow:

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RC r;f ¼ min LC l 8l2Lr;f

ð8Þ

where Lr,f denotes the subset of links belonging to the route r of flow f. Since only one route of the flow can be active at the same time, the flow capacity FCf can be obtained as follows:

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FC f ¼ max RC r;f 8r2Rf

ð9Þ

With these definitions, the design goal is to maximize the Network Utility (NU), calculated as the sum of the capacities FCf of all the flows f in the network:

X NU ¼ FC f

ð10Þ

f 2F

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Depending on the considered capacity (binary or discrete) used in the system, the NU will be the number of active flows in the network or the aggregated discrete capacity of these flows.

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4. Game theoretic approach

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As stated before, game theory offers an interesting perspective to deal with distributed solutions in the context of multihop wireless networks. In this sense, we model the

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relationships among nodes as a formal game and perform the algorithmic design by correctly defining the set of players and the utility functions. Let be the game C = {P, {Si}i2P, {ui}i2P}, where P is the finite set of players, Si is the set of strategies of player i and ui : S ! R is the utility function of that player, with S = i2PSi the strategy space of the game. This utility function ui is a function of si, the strategy selected by player i, and of si, the current strategy profile of the rest of the players of the game. Players will selfishly choose the actions that improve their utility functions considering the current strategies of the other players. One general key issue when designing a game is the choice of ui so that the individual actions of the players provide a good overall performance. In addition, in our specific scenario it is interesting the existence of an equilibrium point to ensure the convergence of the proposed algorithms. In this context, it is useful the concept of Nash Equilibrium (NE), defined as a situation where no player has anything to gain by unilaterally deviating. Thus, a NE of a game C is a profile s⁄ 2 S of actions such that for every player i 2 P we have:

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    ui si ; si P ui si ; si

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ð11Þ

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for all si 2 S, where si denotes any strategy of player i and si denotes the strategies of all the players other than player i in the profile s⁄. From a radio resource allocation perspective, the convergence to a NE of the game makes it possible to reach a stable solution. In addition, nodes can react to variations in the environment as any deviation from this equilibrium forces the participants to play again to lead a new NE. Another specific key issue for the problem addressed in this work is the definition of the players of the game. An adequate network model facilitates the development of realistic and implementable procedures according to the proposed game. In this regard, we consider two different choices: in the first one, the players are the flows themselves (flow game), while in the second one the players are the set of links belonging to the flows (link game). It is worth noting that in the flow game, there must be a physical entity (e.g. the source node of the flow) which actually selects the strategy for the flow. This entity gathers the set of available channels and powers of each link in the routes of the flow and performs the specific route selection and channel and power allocation for all the links of the selected route. Considering the different alternatives of players and utility functions, we propose the following three games to analyze the system.

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4.1. Flow games

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In these games, the players are the jFj flows of the network (P = F). The set of strategies Si of a flow i with m routes is formed by the Cartesian product of the set of strategies of each route of the flow Si = B1  B2      Bm. Similarly, the set of strategies Bj of a route j with kj = jLj,fj links is formed by the Cartesian product of the set of strategies Al of each link of that route: Bj ¼ A1  A2      Akj .

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The subindexes represent the position of the link in the route, from the source to the destination. Each link strategy al = (pl, cl) is the allocation of transmission power and frequency channel for that link. 4.1.1. Potential Flow Game (PFG) An exact potential game is a game for which there exists a potential function V : S ! R such that:

  Dui ¼ ui ðsi ; si Þ  ui s0i ; si ¼ DV ¼ 0  ¼ Vðsi ; si Þ  V si ; si 8i 2 P;

forallsi ; s0i

2 Si

ð12Þ

This definition implies that each player’s individual interest is aligned with the groups’ interest, since each change in the utility function of each player is directly reflected in the same change for the potential function. If only one player acts at each time step (repeated sequential game) and that player maximizes (best response strategy) or at least improves (better response strategy) [1] its utility, given the most recent action of the other players, then the process will always converge to a NE. In addition, global maximizers of the potential function V are NE, although they may be just a subset of all NE of the game. These interesting properties of potential games (assured convergence which can maximize the potential function) suggest their utilization as a first approximation. Therefore, we define a potential flow game as a reference for the considered scenario making V the objective to maximize, in this case the Network Utility NU. This NU can be the aggregate flow capacity or the number of active flows following the capacity definitions given in Section 3. A direct option is to define the utility function ui equal to the potential function (identical interest games [25]):

X ui ðsi ; si Þ ¼ FC j

ð13Þ

j2F

In this game each player needs global information about all the flows (and consequently, all the links) in the network. To compute the capacity of all the flows for each strategy, each player requires the channel gains gl,m between any pair of transmitting and receiving nodes of the links in the network and the current selected strategies al = (pl, cl) of all these links. 4.1.2. Local Flow Game (LFG) The requirements of global network knowledge do not recommend the PFG for a practical implementation, being necessary a simplification of the utility function. In this section, a local formulation tries to model the flows as selfish players which want to maximize their own capacity, but adding certain grade of cooperation to approach the PFG objective. The utility function of flow i is directly related to its capacity FCi, which depends on the capacity RCr,i of each route r of flow i. If Ri and Lr,i denote the subset of routes belonging to flow i and the subset of links belonging to route r of flow i respectively, the utility function is defined as:

  8 jLr;i j > > RC r;i þ 1  max > > r 0 2Ri jLr 0 ;i j > < > 0 and pl ¼ 0 8l 2 Lr0 ;i 8r0 – r if RC ui ¼ r;i > > > 0 if p ¼ 0 8 l 2 Lr;i 8r 2 Ri > l > : 1 otherwise

5

ð14Þ

Therefore, the utility is the capacity of a specific route of the flow if and only if this capacity is higher than 0, i.e., Eq. (2) is satisfied for all the links of the route, and the links belonging to the remaining routes of the flow are inactive. Although only one route among the available ones is effectively selected, the multi-route capability provides the possibility to choose the best path according to a preestablished criterion increasing the flexibility of the distributed solution. As reflected in (14), since the players try to maximize its utility, the route with the highest capacity will be always selected. The term added to RCr,i ensures that when there are several routes which could achieve the same capacity, the one with fewer hops is preferred. For example, the discrete capacity defined in (6) can take even values: 2 (Ml = 2), 4 (Ml = 4), 6 (Ml = 8) and so on. Let us assume that there are two routes with 3 and 5 links respectively which can provide the same capacity (for Example 6, Ml = 8). The utility obtained with the first route is 6 + (1  3/5) = 6.4 whereas for the second one is 6 + (1  5/5) = 6, so the first route is chosen. If the second route had provided the next higher capacity (8, Ml = 16), its utility would have been 8 + (1  5/5) = 8 and then it would have been chosen despite having more links. This decision rule tries to reduce the interference on the remaining flows. The value 1 in the utility function also tries to introduce a degree of cooperation to compensate the inherent selfishness of this game: if no route can be established, i.e., there is no route in the flow with all its links satisfying Eq. (2), it is better to stop the transmission of all the links of the flow to reduce the interference on the remaining flows. Likewise, links belonging to different routes of the same flow cannot be simultaneously active (i.e., transmitting) since the utility would be 1 as well. In the particular case where all the flows are single-hop routes (i.e., single links), this game can be seen as a local link game. In this kind of games the existence and convergence to a pure NE cannot be assured [11,12]. Therefore, the same problem applies for this local flow game, which is an extension of the previous one. To ensure the existence of at least one NE and the convergence of the game to one of them, we set a threshold for the maximum number of nonconsecutive times that a player can choose a specific strategy. With this simple rule, the game has a NE at least: if all the players remove the strategies that exceed their corresponding thresholds without achieving a NE, ultimately the strategy space of the game will be formed by only one possible strategy for each player, which must be a NE of the game. Finally, it must be noted that a full knowledge of all the strategy spaces for each link and route of the flow i is required to perform the strategy selection. In addition, to ob-

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tain the SINR of each link, the node of the flow which acts as the player and selects the strategy needs to know the channel gain g l;l0 from the transmitter of each link l (lTX) 0 0  of the flow to the receivers of the rest of links l lRX in the flow and the interference levels at all these receivers.

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4.2. Cooperative Link Games (CLG)

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The main drawback of the flow games is the complexity that they pose in terms of the necessity of sharing information among the nodes of the network and the computational load required to perform the strategy selection. As it will be seen in Section 4.4, the complexity to select the strategy profile in the flow games can be unaffordable. To solve this problem, we propose a cooperative link game which does not need a central entity in each flow to perform the strategy selection. This decreases both the amount of information shared between the nodes of the network and the complexity of the selection of the strategy profile. Nevertheless a certain degree of cooperation between the links of the flow is required as described below. In this game, the players are the set of links belonging to the routes of the jFj flows of the network (P = L). The set of strategies Si of a link i is its available set of power transmission and channel frequency combinations, si = ai = (pi, ci). The links of the same flow cooperate in the selection of their strategies. At each flow f, the game is played sequentially, from the route with the lowest number of hops to the one with the highest number, and at each route r, from the source to the destination for each link i of the route. The strategy selection process is divided in the following two steps:

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Step 1: For each route r, each link i selects its strategy according to the following utility function:

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ui ¼

8   jLr;f j > > if min LC l > 0 min LC þ 1  > l < max 0 jL 0 j 8l6i

> 0 > > : 1

r 2Rf

r ;f

8l6i

if pl ¼ 0 otherwise

567

ð15Þ

568

where it has been assumed that the links are ordered following the flow of information from the source node to the destination node: that is, the first link of a flow has the lowest index l and this index l is increased as the flow progresses to the destination. This expression reflects that each link tries to choose a strategy that improves the route capacity as if it were the final link in the route. The same correcting term used in (14) to provide higher utility to routes with fewer hops is also included here. To compute the terms LCl in this step, each link assumes that the rest of the links belonging to other routes r0 of its flow f and the links of its same route r but with index l0 higher than i (i.e., links closer to the destination node) are inactive. If the link cannot be activated without disrupting the previous links of its route (SINRl > a "l 6 i), it is better for itself and the following links of the route to stop transmitting and get a utility ui = 0.

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Step 2: Once all the links of the flow have performed Step 1, the following pseudo flow capacity is obtained: 

FC ¼ max uk 8r2Rf

ð16Þ

with uk the utility of the last link of each route r. This expression is similar to (9), but with a correcting term to include the number of links of each route. In this sense, when several routes have the same capacity, the one with fewer links is preferred. Likewise, the route r⁄ which provides the highest utility is: 

r ¼ arg max uk 8r2Rf

ð17Þ

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After that, the strategies of all the links of the flow are finally updated with the following utility function:

8 if i 2 Lr ;f and FC  > 0 > <1 ui ¼ 0 if pl ¼ 0 > : 1 otherwise

586

601 602

603

ð18Þ 605

Thus, if a link belongs to the route with the highest utility, it does not change the strategy selected in Step 1. In any other case, they will stop transmitting to obtain a utility ui = 0. With these definitions of the utility functions, only the route with the highest capacity and fewer hops will be active at any time. Furthermore, if no route can be established, all the links of the flow will remain without transmitting to reduce the interference on the remaining flows. As stated previously, the information required by each agent in this game to perform the strategy selection is lower than in the flow games. To determine it, the process followed to select a specific strategy must be carefully analyzed. Since in this game a link i must verify if it disrupts any of the previous links l in the route, it must compute their SINR with the strategy to be selected. These computations require the following information for each of the previous links l:

606

Transmitting channel in link l. Received power at l. Interference level at l. Channel gain from the transmitter of link i to the receiver of l. This information does not have to be exchanged, since i could estimate it.

624

   

607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623

625 626 627 628 629 630

Therefore, each link needs three parameters of each of the previous links of the flow (selected channel, received power and interference level). Since links are established consecutively from the source node to the destination, each link can add its own parameters to a list containing the set of parameters of the previous links and propagate it to the following one.

631

4.3. Timing and decision rules

638

A repeated sequential game with a round robin scheduling and a better response strategy is considered for all the proposed games. In all of them, whenever a flow or link plays, it looks first for strategies with the lowest power

639

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profile for the same utility to reduce the interference over the remaining players. For the flow games (both local and potential), the game is played until a pure NE is found. This playing rule requires searching through the entire strategy space. The cooperative link game reduces this complexity at the expense of not evaluating all the possible strategy profiles. Therefore, the profiles corresponding to the NEs may not be reached and consequently a different less restrictive rule is defined to stop the game: if no link has improved its utility (and hence no flow has improved its capacity) in a round robin cycle, the game is stopped. It is worth mentioning that this stopping rule does not imply that some flow could actually improve its capacity since the complete strategy space is not evaluated. Finally, the same rule regarding the maximum number of times that a specific strategy can be selected has been applied in the CLG to ensure its convergence.

660

4.4. Complexity of the games

661

With the better response strategy described above, each player tries to improve its current utility at each step regardless of the past history. Therefore, the complexity of the games is directly related to three factors:

643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658

662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698

 The number of possible strategies to be chosen at each step.  The computational complexity of calculating the utility of each of these possible strategies.  The number of steps required to reach an stable point. With regard to the first factor, each step of the flow games (both local and potential) may require in the worst case PjRf j kj exploring different strategies. On the other j¼1 ðQ  CÞ hand, the worst case for the CLG only requires exploring Q  C strategies per step. Nevertheless, in order to compare these measures we must take into account that each step PjRf j of the flow game comprises j¼1 kj steps of the link game. With this consideration, the number of strategies that must be evaluated in the worst case of the link game to perform the resource allocation of a complete flow is PjRf j j¼1 ðkj  Q  CÞ. As for the computational complexity of calculating the utility of each strategy, we use as a reference to compare the three games the number of link capacities (terms LC) required to calculate the game utility. This link capacity is directly related to the SINR of the link, as shown in Section 3. In the LFG the evaluation of each strategy requires the calculation of kj terms LC, corresponding to the kj links of the route j. In the PFG at most jLj terms LC are computed, since in this case the capacities of all the links in the network have to be computed for each strategy. Finally, in the CLG the number of terms LC that must be obtained for each strategy depends on the position of the link in the route: according to (15), in the first link of a route, only the capacity of this first link has to be evaluated, in the second one, the capacity of first and second link and so forth. Therefore, for the link in the kj position, kj (kj + 1)/2 terms LC must be computed.

7

Taken into account these two factors, an upper bound for the computational complexity per flow game step (or its equivalent in the CLG) in terms of LC calculations is given by: PjRf j  Local flow game: j¼1 kj  ðQ  CÞkj . PjRf j  Potential flow game: jLj j¼1 ðQ  CÞkj . PjRf j  Cooperative link game: j¼1 ðQ  C  kj  ðkj þ 1Þ=2Þ.

699 700 701 702 703 704 705 706

As it can be seen, the computational load of each step is much lower in the CLG. Regarding the last factor (the number of steps required to reach an stable point), it will be analyzed in Section 6. With those results and the analysis performed in this section the computational complexity of the three games will be completely compared.

707

5. Efficiency of the game theoretic framework

713

The efficiency of a game-based system degrades due to the selfish behavior of its agents. No equilibrium will provide better performance than the one that could be obtained using a centralized approach that maximizes the network utility without equilibrium constraints. One of the most common approaches to analyze the difference between equilibria and this centralized optimum is the use of the concepts of Price of Stability (PoS) [26] and Price of Anarchy (PoA) [27]. These terms are respectively defined as the ratio between the best and worst equilibrium and the centralized optimum solution. Although the PoS and the PoA are generally defined for games with a NE, we have also extended their definition for the CLG using as a reference the stable point defined in Section 4. Therefore, three parameters must be calculated in the proposed scenario to compute the PoS and the PoA: the maximum network utility and the maximum and minimum network utility for an equilibrium. With this purpose, a formal definition of the corresponding optimization problems is proposed in this section. Since the power levels are discretized and the computation of the SINR and the capacity of each link requires nonlinear operations, the optimization must be performed with MINLP algorithms, such as the branch and bound algorithm [16]. Depending on the NU considered in the system (number of active flows or aggregated capacity of the active flows), we propose the following optimization problems:

714

5.1. Number of active flows (binary capacity)

742

Let Cl be the subset of C containing the channels where the link l can transmit. For each link l, we define jClj binary variables, xl,c, indicating whether the link l is transmitting in channel c or not. Since each link can only transmit at one specific channel, the following restrictions must be satisfied:

743

X xl;c 6 1 8l 2 L

749

708 709 710 711 712

715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741

744 745 746 747 748

ð19Þ

c2C l

751

with L the set of links in the network. The sum in (19) will be 1 if the link l is active and 0 if it is inactive.

752

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We also force that if a link l is not transmitting, the rest of the links l0 in its route must be inactive as well to decrease the overall interference in the network. This can be expressed as follows:

t m;l;c P qLm;c  sl;c þ sLl;c  qm;c  qLm;c  sLl;c

X X xl;c ¼ xl0 ;c

t m;l;c 6 qUm;c  sl;c þ sLl;c  qm;c  qUm;c  sLl;c

811

where the superscript U or L represents the upper or lower bound of a variable. Since the bounds of the variables ql,c can vary in each step of the branch and bound algorithm, the bounds for the variables sl,c and tm,l,c, which depend on them, must be updated as well. Additionally, the SINR restriction can be expressed with the following inequalities:

812

8l; l0 2 Lr;f ; 8r 2 Rf ; 8f 2 F

ð20Þ

760

c2C l

761

Therefore, a link will be active i.e., transmitting, if and only if the rest of the links of its route are also active. This condition may seem unnecessary since the optimization algorithm will choose by itself the transmitting links that maximize the network utility, (i.e., the links that can establish complete flows), stopping those that do not increase the global capacity of the network. Nevertheless, this restriction will simplify the definition of the objective function as it will be seen later. As seen in the proposed game approach, another requirement of the proposed system is that only a route per flow can be active at most, which is equivalent to:

762 763 764 765 766 767 768 769 770 771 772

773

775 776 777 778 779 780 781 782 783 784 785 786 787

788 790

791 792 793 794 795

796

c2C l0

0

X 1 XX @ xl;c A 6 1 8f 2 F nr;f r2R l2L c2C f

r;f

801

802

805 806 807 808

809

8l 2 L;

8c 2 C l

ð22Þ

We also define the variables sl,c, which correspond to the SINR at the receiver of the link l and the channel c. These variables are function of the transmission power of the link l and the rest of interfering links using the same channel c:

PN þ

P

ql;c 

P max Q

 g l;l

m 2 L qm;c  m–l

P max Q

 g m;l

8l 2 L; 8c 2 C l

ð23Þ

This expression can be manipulated to transform the division into multiplications, which are more adequate for mathematical programming:

X

g 0m;l  qm;c  sl;c  g 0l;l  ql;c ¼ 0 8l m2L m–l 2 L; 8c 2 C l

PN  sl;c þ

804

l

xl;c 6 ql;c 6 Q  xl;c

798

800

ð21Þ

where nr,f is the number of links in the route r of the flow f. If no route is active in a flow, the summations in (21) will be equal to 0. On the other hand, if a route is active, the two inner summations will add to nr,f and the left-hand side of (21) will be 1. As stated in Section 3, the transmission power is discretized into Q + 1 levels q = {0, 1, . . . , Q}. Let us define the variables ql,c indicating the level q at which the link l transmits in channel c. According to the definition of xl,c, ql,c must be 0 if xl,c = 0 and it must be comprised between 1 and Q if xl,c = 1. These restrictions can be expressed mathematically as:

sl;c ¼

799

1

ð24Þ

with g 0m;l ¼ g m;l  Pmax =Q . To deal with the non-linear crossproducts qm,c  sl,c, we define the auxiliary variables tm,l,c = qm,c  sl,c and apply the well-known linearization inequalities proposed by McCormick [28]:

t m;l;c P qUm;c  sl;c þ sUl;c  qm;c  qUm;c  sUl;c t m;l;c 6 qLm;c  sl;c þ sUl;c  qm;c  qLm;c  sUl;c

sl;c P a  xl;c

8l 2 L;

8c 2 C l

ð25Þ

ð26Þ

That is, if the link l is transmitting in channel c (xl,c = 1), then the SINR for this channel, sl,c, must be higher than a. On the contrary, if xl,c is 0, ql,c and sl,c will also be 0 and the inequality will hold as well. This restriction implicitly involves a rate constraint for the flows, since every link in the flow will have at least the rate given by the SINR threshold a. To find the centralized optimum solution, the function to maximize in this case is the number of active flows in the network, which is equivalent to the binary capacity defined in expression (7). Thanks to the restriction that only the links corresponding to active flows can be transmitting (20), the objective function to maximize can be defined as:

XX 1  xl;c nr;f l2L c2C

813 814 815 816 817 818

819 821 822 823 824 825 826 827 828 829 830 831 832 833 834

835

ð27Þ 837

l

Additionally, to obtain the PoS of the game we also need to compute the best equilibrium of the game (namely, NE for the flow games or the predefined stable point for the CLG). In this case, the function to maximize is expression (27) as well, but with the restriction that the obtained solution must fulfill the equilibrium condition of the game. Nevertheless, this restriction is not included as an additional constraint in the mathematical formulation of the optimization problem but as a specific restriction in the heuristic function of the branch and bound algorithm as it will be seen later in Section 5.3. For the PoA the same restriction arises, and in this case the objective function is not the maximization of expression (27) but its minimization. Fig. 1 summarizes the problem when the network utility is measured in terms of the number of active flows, being the variables xl,c binary, ql,c integer in the range {0, . . . , Q} and sl,c and tm,l,c real numbers.

838

5.2. Aggregated capacity

856

When the network utility uses the definition of capacity given in Eq. (6), it is required the addition of new variables and restrictions to the previous model and a new objective function. Using the same definition of RCr,f given in Section 3 and assuming that only a route per flow is active at most, the objective function can be expressed as:

857

XX RC r;f

840 841 842 843 844 845 846 847 848 849 850 851 852 853 854 855

858 859 860 861 862

863

ð28Þ

f 2F r2Rf

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If we also define LCl,c as the capacity which can be obtained in link l and channel c, the value of RCr,f will be limited by the following inequalities:

RC r;f 871 872 873 874

875 877

878 879 880 881 882 883 884 885

886

890 891 892

893

8l 2 Lr;f ; 8r 2 Rf ; 8f 2 F

LC l;c

ðiÞ

LC l;c 6 aðiÞ  sl;c þ b aðiÞ ¼

ð29Þ b

That is, the capacity of a route is the minimum capacity of the links that form it. Similarly to (6), the variables LCl,c will be determined by:

  log2 ð1 þ sl;c Þ ¼2 8l 2 L; 8c 2 C l 2

ð30Þ

It must be noted that the condition regarding the SINR threshold a which appears in (6) is not needed here since it is already included in restriction (26) of the problem formulation. To obtain a linearization of (30), we propose to use two different sets of inequalities: first, a lower and an upper bound for LCl,c imposed by the lower and upper bounds of sl,c:

LC l;c 889

X 6 LC l;c c2C l

LC l;c

888

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J. Ortín et al. / Ad Hoc Networks xxx (2013) xxx–xxx



7 6 6log 1 þ sL 7 6 2 l;c 7 5 8l 2 L; 8c 2 C l P24 2

7 6 6log 1 þ sU 7 6 2 l;c 7 5 8l 2 L; 8c 2 C l 624 2

ð31Þ

The second set of inequalities is based on the lines formed pairs of consecutive points n

by the

o ðiÞ ðiÞ ðiþ1Þ ðiþ1Þ at the discontinuities of the sl;c ; LC l;c ; sl;c ; LC l;c function expressed in (30):

ðiÞ

¼

8l 2 L; 8c 2 C l

ðiþ1Þ LC l;c ðiþ1Þ sl;c



ðiþ1Þ sl;c

ðiÞ ðiÞ LC l;c  sl;c  ðiþ1Þ ðiÞ sl;c  sl;c





ðiÞ LC l;c ðiÞ sl;c

ð32Þ ðiþ1Þ LC l;c

895

with 0 6 i 6 M  1 and M the number of allowed modulations. The linearization for M = 3 is shown in Fig. 2. The nonlinear function for LCl,c used in expression (30) is represented in black bold line. The shadow area corresponds to the zone where all the linearization inequalities are satisfied. On the left, this area is represented for the values of sUl;c and sLl;c at the beginning of the branch and bound algorithm. As the algorithm evolves and the upper and lower bounds of sl,c get closer, the area shrinks due to changes in the right hand side of inequalities (31) (Fig. 2 right). Fig. 3 summarizes the problem for the capacity of the network, being the variables xl,c binary, ql,c integer in the range {0, . . . , Q} and sl,c, RCr,f, LCl,c and tm,l,c real numbers.

896

5.3. Application of the branch and bound algorithm

910

The branch and bound algorithm is based on the application of a divide and conquer strategy on a linear relaxation of the original problem. In our case, this relaxation treats the binary variables xl,c and the integer variables ql,c as real variables. The relaxed problem is located at the root node of a tree that the branch and bound algorithm generates dynamically to solve the original problem. Each node of the tree will be composed of this initial relaxed problem with appended restrictions that generates

911

Fig. 1. Problem formulation (binary capacity).

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4. If the lower bound LB is higher than the highest lower bound Lmax obtained so far, update the value of Lmax B B with LB. 5. Set the node as checked. If U B > Lmax select an integer B variable yi (xl,c or ql,c) whose optimal value in P is not integer and create two new problems (child nodes): i c to P and the one appending the restriction yi 6 by i c þ 1, with yi other appending the restriction yi P by i the value of the same the original integer variable and y variable obtained when P is solved in step 2. Update also the coefficients and parameters of the restrictions which may vary with the new appended restriction: qUl;c ; qLl;c ; sUl;c ; sLl;c and restrictions (25) and (31).

940 941 942 943 944 945 946 947 948 949 950 951 952 953

Fig. 2. Linear approximation for the link capacity function.

920 921 922 923 924 925 926 927 928 929 930 931 932 933 934 935 936 937 938 939

partitions of its solution space. The original problem is solved repeating the following steps: 1. Select a node of the tree. In our case, a depth-first search has been implemented to keep the size of the tree as small as possible. 2. Obtain an upper bound on the optimal solution of the relaxed problem associated to the selected node, P in the sequel, using traditional linear programming algorithms. This bound UB is obtained computing the objective value at an optimal solution of P. 3. Compute the heuristic function of P to obtain a lower bound LB of the optimal objective value of the original problem. In our case, the heuristic function simply rounds the values of xl,c and ql,c obtained in the optimal solution of P and computes the network utility using Eqs. (6)–(10). Additionally, if the best or worst equilibria of the games are calculated to obtain the PoS or the PoA, it is also verified that the solution provided by the heuristic function is a equilibrium of the game. If not, the values obtained in this step are discarded and step 4 is skipped.

Note that the previous algorithm is intended for the maximization of an objective function. If we want to use it to minimize a function (as it is the case of finding the worst equilibrium of a game), we have to change the sign of the objective function.

954

6. Results

959

The analysis and evaluation of a game model should cover two different aspects: first, the existence of some equilibrium points and second, their quality, which can be measured as the ratio between the NU obtained in the equilibrium and the maximum achievable NU by a central entity. Concerning the first issue, the convergence to an equilibrium of the different proposed games has been analyzed in Section 4. Given the well-known properties of the potential games, the existence and convergence to a NE is guaranteed for the considered potential game [29]. Additionally, the threshold for the maximum number of nonconsecutive times that a player can choose a specific strategy ensures the convergence to a NE of the proposed local flow game and to a stable point of the cooperative link game. As for the quality of the equilibria, the proposed games have been evaluated by simulations and compared to the centralized optimum solution described in Section 5. Additionally, the PoS and the PoA have been also calculated for all of them. Both the centralized optimum and the PoS and PoA have been solved with CPLEX [30]. Several scenarios have been studied varying the values of the main simulation parameters (topology size, number of nodes, noise power, available frequency channels, power levels in the transmitter). The relative differences among the game strategies hold in any case, making the proposals scalable and applicable to diverse situations. The computational complexity of the centralized optimization problem makes it difficult to obtain results as the network size grows. For this reason, this paper presents results only in the scenario described below, being their specific results a sample of the correctness of the proposed games. Since the relative performance among the games is extensible to different settings, the obtained conclusions are maintained in all the cases. The network consists of 100 nodes deployed in a square area of 700  700 m2 with different random topologies. Several numbers of flows jFj (2, 4 and 6) are generated between random source and destination nodes of the topol-

960

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Fig. 3. Problem formulation (discrete capacity).

999 1000 1001 1002 1003 1004 1005 1006 1007 1008 1009 1010 1011 1012 1013 1014 1015 1016 1017 1018 1019 1020 1021

ogy. For each flow, R = 2 disjoint routes are tried to be obtained. The maximum number of hops of the primary route is limited to 5, while the secondary route may have up to 2 more hops than the primary one. All the results for the games and the centralized optimum solution are averaged with 100 random instances of the scenario. The number of non-interfering channels is set to C = 5, nevertheless, only a subset of these channels is available to each node to model the presence of other interference sources. To do so, the scenario is divided into regions of 100  100 m2, being assigned only a random subset (between 2 and 4) of these C channels to the nodes in the area. Pmax is set to 20 dBm, the transmission power is quantized with Q = 4 levels different to 0 and four modulations, Ml = {2, 4, 8, 16}, are available in the transmitter. For simplicity, the bandwidth of each channel, wcl , is the same and normalized to 1 unit. The path loss index is c = 4, the SINR threshold a is set to 10 dB and the noise power PN to 85 dBm. Although with these parameters the maximum transmission range is 237 m, we set the maximum distance between nodes of a link to 100 m to guarantee that in the absence of interference, the SINR required to transmit with Ml = 16 levels can be obtained, therefore, achieving the maximum link capacity.

In the figures, we denote BC (Binary Capacity) to the NU definition of number of established flows (flow capacity is either 0 or 1) and DC (Discrete Capacity) to the NU definition of aggregate capacity (flow capacity can take discrete values according to the modulation level).

6

active flows

998

BC-LFG-1R BC-PFG-1R BC-CLG-1R BC-LFG-2R BC-PFG-2R BC-CLG-2R Opt-1R Opt-2R

4

2 2

4

6

number of flows Fig. 4. Active flows for the analyzed games with BC utility definition vs. optimum solution with 1 and 2 routes. Mean value and standard deviation (bars).

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6

BC-LFG-1R BC-PFG-1R BC-CLG-1R DC-LFG-1R DC-PFG-1R DC-CLG-1R Opt-1R

active flows

aggregate capacity

30

20

DC-LFG-1R DC-PFG-1R DC-CLG-1R

10

2

DC-LFG-2R DC-PFG-2R DC-CLG-2R 4

Opt-1R Opt-2R

4

2

6

2

number of flows Fig. 5. Aggregate capacity for the analyzed games with DC utility definition vs. optimum solution with 1 and 2 routes. Mean value and standard deviation (bars).

1027 1028 1029 1030 1031 1032 1033 1034 1035 1036 1037 1038 1039 1040 1041 1042 1043 1044 1045 1046 1047 1048 1049 1050 1051 1052 1053 1054 1055 1056 1057

4

6

number of flows Fig. 6. Active flows for the analyzed games with BC and DC utility definitions vs. optimum solution with 1 route. Mean value and standard deviation (bars).

Fig. 4 shows the number of active flows obtained with the BC utility definition for the three proposed games at the equilibrium point (NE for the flow games and the criterion defined in Section 4.3 for the cooperative link game) with 1 (1R) and 2 (2R) disjoint routes per flow vs. the respective optimum solutions with 1 and 2 routes. It can be seen that the cooperative link game provides a performance close to that of the flow games (local and potential), with a much lower complexity. In addition, the improvement obtained with a second route is not very important (around 5% for the three games and the optimum). As can be seen, the performance of the three games is close to the optimum. For example, for the 1 route scenario, the degradation ranges from 5% (2 flows) to 9% (6 flows) for the local flow game or from 10% to 14% in the simpler cooperative link game. Fig. 5 shows the aggregate capacity of the active flows when the DC utility definition is considered in the same scenarios of Fig. 4. Again, the cooperative link game provides a performance similar to that of the flow games. In this case, the degradation with regard to the optimum is higher than in the previous case: for the 1 route scenario, it ranges from 4% (2 flows) to 16% (6 flows) for the local flow game or from 12% to 22% in the cooperative link game. This may be due to the higher number of available combinations involved with this utility definition, which hinders a distributed resource allocation. Finally, the improvement obtained with a second route is again around 5% for both the games and the centralized optimum. Table 1 shows the PoS and PoA for the proposed games with one route. For the CLG, the expressions used to obtain

the PoS and the PoA are slightly modified to adapt them to the definition of equilibrium point of this game. For this reason, the numerator of the PoS and the PoA correspond for the CLG to the best and the worst stable points of the game as defined in Section 4. Several remarks can be extracted from these results. First of all, the PoS is always 1 for the potential flow game, since by definition, the global maximizers of the potential function (in our case the objective function to maximize) are NE of the game. Although the same condition cannot be guaranteed for the local flow game and the cooperative link game, the obtained results are very close to 1 in both cases. Regarding the PoA, the best results are obtained with the local flow game. This result is logical since the LFG is the game which obtains the best overall performance as shown previously in Figs. 4 and 5. On the other hand, the PoA values are lower for the cooperative link game. This can be due to the less strict condition used to reach a stable point in this game. Nevertheless, the happening of these stable points with low network utility are very infrequent, since the mean value of the network performance with the CLG is only slightly lower than that of the LFG. Finally, both the PoS and the PoA are higher in all games with the binary definition of capacity than with the discrete one, which confirms that the degradation with regard to the optimum is higher with the DC as pointed out previously. Figs. 6 and 7 compare the performance of the BC and DC utility definitions. On the one hand, it can be seen that the number of active flows is very similar for both utility definitions with the three games. On the other hand, the DC

Table 1 PoS and PoA for the proposed games with 1 route. Flows

CLG

LFG

BC

2 4 6

DC

PFG

BC

DC

BC

DC

PoS

PoA

PoS

PoA

PoS

PoA

PoS

PoA

PoS

PoA

PoS

PoA

1 1 1

0.66 0.585 0.515

0.998 0.997 0.996

0.562 0.525 0.479

1 1 1

0.9 0.768 0.732

1 0.995 0.993

0.847 0.751 0.712

1 1 1

0.85 0.659 0.572

1 1 1

0.537 0.523 0.488

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aggregate capacity

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20

BC-LFG-1R BC-PFG-1R BC-CLG-1R DC-LFG-1R DC-PFG-1R DC-CLG-1R Opt-1R

10

2

4

6

number of flows Fig. 7. Aggregate capacity for the analyzed games with BC and DC utility definitions vs. optimum solution with 1 route. Mean value and standard deviation (bars).

3.5

active flows

BC-LFG-1R BC-PFG-1R BC-CLG-1R

BC-LFG-2R BC-PFG-2R BC-CLG-2R

Opt-1R Opt-2R

3.25

3

2.75 2

4

6

number of flows Fig. 8. Mean number of links per active flow for the analyzed games with BC utility definition vs. optimum solution.

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definition obtains a much higher aggregate capacity than the BC definition for all the proposed games, thus providing a better overall performance. Fig. 8 shows the mean number of links of the established flows for the BC utility definition (results for DC are analogous). This number decreases as the congestion in the network (i.e., the number of competing flows) grows. In this situation, the flows with fewer links tend to be established since the requirement that all the links in the flow are active is more easily achieved with short flows. As can be seen, the number of links per flow is very similar in the games and in the centralized optimum

solution, so the higher capacity obtained with the optimum solution is not obtained at the expense of allocating more but shorter routes. As for the computational complexity of the games, we have already seen in Section 4.4 that it depends on two variables: the number of steps required to achieve the equilibrium and the computational load of each step. The analysis of the second one has been already carried out in Section 4.4 and now the results regarding the mean number of steps performed until the equilibrium is reached are also presented. Table 2 shows the mean number of normalized flow steps for the three games. For the flow games, the flow steps are directly the game steps, whereas for the link game, the flow steps are equivalent to the game steps divided by the mean number of links per flow. Since the number of flow steps is within the same order of magnitude for the flow games and for the CLG, we can state that the complexity of the cooperative link game is much lower than that of the flow games. It can also be seen that the use of two routes does not significantly raise the number of steps, although it increases their complexity. Finally, the DC utility definition barely pushes up the number of steps in all cases, which confirms the better overall performance of this utility definition.

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7. Conclusions

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In this work we have analyzed under a game theoretic framework the route selection and channel and power allocation in distributed multihop wireless networks. To perform this analysis, we have proposed three different games: a potential flow game, a local flow game and a cooperative link game, each of them requiring different grades of environmental knowledge, cooperation and complexity. Additionally, a mathematical characterization of the equivalent optimization problem has been suggested to evaluate the efficiency of the games. Some linear approximations for the nonlinear terms of the equivalent MINLP problem have been proposed to compute the optimization with the branch and bound algorithm. Simulation results under the SINR model have shown that the proposed games obtain a performance close to the centralized optimum solution. It is worth noting that the cooperative link game can provide stable configurations with a global performance similar to much more complex flow games, which suggests its potentiality as a distributed algorithm to be implemented in a protocol. Be-

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Table 2 Mean values of the normalized number of flow steps performed until convergence to a stable point for the proposed games. Flows

CLG

LFG

BC

2 4 6

DC

PFG

BC

DC

BC

DC

1R

2R

1R

2R

1R

2R

1R

2R

1R

2R

1R

2R

4.7 11.3 23.0

4.8 12.3 25.6

5.1 15.0 23.3

5.2 14.3 25.3

4.0 11.7 22.7

3.7 12.9 26.3

4.9 19.2 46.4

5.6 24.2 54.8

3.4 9.7 16.6

3.3 9.8 16.8

5.0 12.0 19.3

5.0 12.4 19.6

Q1 Please cite this article in press as: J. Ortín et al., Joint route selection and resource allocation in multihop wireless networks based on a game theoretic approach, Ad Hoc Netw. (2013), http://dx.doi.org/10.1016/j.adhoc.2013.05.002

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sides, it has been shown that a utility definition based on the channel capacity provides better overall performance than one only based on whether the flows are active or not.

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Acknowledgments

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This work has been supported by the Spanish Government through the Grant TEC2011-23037 from the Ministerio de Ciencia e Innovación (MICINN), DGA-FSE and the European STREP Project iJOIN (FP7-317941).

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References

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Jorge Ortı´n was born in Zaragoza, Spain, in 1981. He received the Engineer of Telecommunications and Ph.D. degrees from the Universidad de Zaragoza in 2005 and 2011 respectively. In 2008 he joined Aragón Institute of Engineering Research (I3A) of Universidad de Zaragoza, where he has participated in different projects funded by public administrations and by major industrial and mobile companies. In 2012 he joined the Universidad Carlos III of Madrid as a postdoc research fellow. Research interests include wireless communications systems.

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José Ramón Gállego was born in Zaragoza (Spain) on 1978. He received the Engineer of Telecommunications MS and Ph.D. degrees from the Universidad de Zaragoza, Spain, in 2001 and 2007, respectively. In 2002, he joined the Centro Politécnico Superior, Universidad de Zaragoza, where he is currently an Associate Professor. He is member of the Aragón Institute of Engineering Research (I3A). His professional research activity lies in the field of wireless communications, with emphasis on radio resource management and mobility support, in 3G/4G, ad hoc and cognitive networks.

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Marı´a Canales was born in Zaragoza (Spain) on 1978. She received the Engineer of Telecommunications MS and Ph.D. degrees from the Universidad de Zaragoza, Spain, in 2001 and 2007, respectively. In 2002, she joined the Centro Politécnico Superior, Universidad de Zaragoza, where she is currently an Associate Professor. She is member of the Aragón Institute of Engineering Research (I3A). Her professional research activity lies in the field of wireless communications, with emphasis on radio resource management and mobility support, in 3G/4G, ad hoc and cognitive networks.

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Q1 Please cite this article in press as: J. Ortín et al., Joint route selection and resource allocation in multihop wireless networks based on a game theoretic approach, Ad Hoc Netw. (2013), http://dx.doi.org/10.1016/j.adhoc.2013.05.002