UNIVERSITA DEGLI STUDI DI TRENTO Facoltà di Scienze Matematiche, Fisiche e Naturali


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1 UNIVERSITA DEGLI STUDI DI TRENTO Facoltà di Scienze Matematiche, Fisiche e Naturali Corso di Laurea Magistrale in Informatica within European Master in Informatics Final Thesis OPTIMAL RELAY NODE PLACEMENT FOR THROUGHPUT ENHANCEMENT IN WIRELESS SENSOR NETWORKS Relatore/st Reader: Prof. Alberto Montresor University of Trento Laureando/Graduant: Eduardo Feo Flushing Prof. Gianni Di Caro IDSIA ControRelatore/ 2nd Reader: Prof. James Gross RWTHAachen University Anno Accademico 292
2 Optimal Relay Node Placement for Throughput Enhancement in Wireless Sensor Networks Eduardo Feo University of Trento  RWTH Aachen A thesis submitted for the degree of European Master in Informatics December 2 Wireless Sensor Networks (WSNs) typically consist of a number of small, inexpensive, low powered devices, equipped with sensors, which are able to communicate through wireless transmissions via multihop routing. We consider a usual application, where the data generated by all sensor nodes (SNs) needs to be transmitted and gathered at a base station (BS) via multihop routing. In the recent years, the research on WSNs has proposed the deployment of relay nodes (RNs). These nodes are typically meant to receive and forward packets generated by SNs, but their deployment objective is varied. In this work we envisage their use to enhance network throughput. That is, given a restricted number of available RNs, we aim to determine the locations at which they can be positioned in order to improve network throughput and endtoend packet transmission delays. We do not make any assumptions about network connectivity. Therefore, in disconnected network scenarios, in which sensors might not be able to send their data to BSs, the deployment of relay nodes is aimed both at bringing connectivity to the BSs and optimize global network performance. We formalize the problem of RN positioning by defining a linear, mixed integer mathematical programming model. The formulation includes a number of constraints and penalty components, aimed at closely modeling the specific characteristics of the wireless environment, as well as a number of heuristics, aimed at speeding up the computations. Although our primary objective is determining the physical locations at which RNs should be placed to meet the performance improvement condition, our model also specifies the way these RNs should be used. This specification comes in the form of optimal data paths from SNs to BSs. Through a comprehensive evaluation using network simulations, we show that our approach is effective in accomplishing the network performance enhancement objective. We also compare the solutions provided by our scheme against a stateoftheart routing protocol, to assess the quality of the routes, and against a RN placement heuristic, to evaluate the positioning of the relay nodes.
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4 TO MY GRANDFATHER: GEORGE
5 Acknowledgements I would like to express my most sincere gratitude to Prof. Gianni Di Caro for his dedication, advise and interest in this work. Moreover, for accepting me as Master thesis student and his encouragement during all the project. This thesis is basically the result of long and fructiferous discussions with him. Once again, I am completely grateful for his support. To the thesis readers, Prof. James Gross and Prof. Alberto Montresor, for offering their time to evaluate this project. The research for this work was carried out at the Istituto Dalle Molle di Studi sullintelligenza Artificiale (IDSIA). I would like to thank all the staff at IDSIA. Their dedication and excellent work is admirable, and working with them has been greatly rewarding for me. Special thanks to Matteo Salani, for his help on the implementation of some parts of the system. Last, but not least, it is important for me to mention the invaluable help of my beautiful family and friends. To my parents. Although I am far from home, they have been the main source of motivation to complete this thesis. To my girlfriend for her continuous support. Without her companion and love, the last months of hard work would have been twice as difficult.
6 Contents List of Figures List of Tables ix xiii Introduction 2 Related work 5 2. Relay node placement for connectivity Relay node placement for balanced and energyefficient data gathering and maximizing network lifetime Discussion Wireless sensor networks: characteristics and modeling 3 3. Propagation and Interference Model Free space propagation model and path loss Logdistance path loss shadowing model Noise Floor Interference Model SignaltoInterference plus Noise Ratio SINR and the physical model Medium Access IEEE MAC Protocol Wireless Sensor Network simulation Linear Programming and Network flow formulation Network flow formulation Basic terminology v
7 Minimum Cost Flow Problem (MCFP) System Model Basic model Problem Formulation Limited number of relays Network bandwidth, demand and flow unit Node indegree Load balancing and preventing interflow congestion Including cost for links Number of hops vs. Link cost Candidate locations for relay nodes Effect of the number of candidate locations Convexhull pruning Adaptive grid resolution Summary Estimation of link quality MAC model and probability of MAC access Under disk model Probability of transmission Probability of accessing the channel Under logdistance path loss model Probability of Channel Busy Assessment Simulation results Diskbased model Under logdistance path loss Estimation of Packet Reception Ratio Relation between interfering nodes Destinationbusy losses Model evaluation SINR threshold Methodology Estimation vs simulation results
8 Model sensitivity to parameters Packet Success Rate Estimation P success : Integrating P MAC and Estimation of Packet Reception Ratio Model evaluation Model sensitivity to network demand Model sensitivity to packet size Results by link quality Comparison with other approaches Link distance RSSI Conclusions Model evaluation Evaluated instances Communication range Area size Node demand, flow unit and network capacity Static Node density Base station density Clustered and uniform node location generation Model solving and simulation parameters Model solving Simulation parameters Integrating P success into a link cost metric Experiment set Results Comparison of proposed solutions Experiment set Results Relay placement to improve network performance Experiment set Effect of the number of RNs available Effects of grid resolution and evaluation of proposed strategies. 3
9 Results Relay placement to provide connectivity MST heuristic Experiment set Results Conclusion Conclusion 3 References 7
10 List of Figures Unslotted CSMA algorithm Illustration of the application of MCFP to routing problems Example of wireless sensor network Using relays to achieve load balancing and prevent interflow interference Simulation analysis to valuate the efficacy of the proposed strategy Cost calculation given a flow network One hop vs. Two hop Two hop vs. Three hop Candidate locations for relay nodes Effect of parameter in number of candidate locations and size of E. Area size 3m 3m Candidate locations for relay nodes using adaptive grid resolution and convexhull pruning Disk model simulation results vs. numerical values Disk model experiment set Numerical results of p busy, under logdistance path loss model, when contender node is located at certain distance. CCA T H = 72.dBm, σ = 4., η = Logdistance model. Experiment instance example with radius = 7 m and density =. nodes/m 2 (8 contender nodes) P MAC under logdistance path loss model. Distribution of estimation error by simulation results ix
11 5.6 P MAC under logdistance path loss model. σ = 3.2, η = 3.. Simulation results vs. numerical values P MAC under logdistance path loss model. σ = 3.2, η = 4.5. Simulation results vs. numerical values P MAC under logdistance path loss model. σ = 6.4, η = 3.. Simulation results vs. numerical values Different scenarios showing two possible interactions between interfering nodes Link PRR variation over simulation time PRR evaluation. Example of topology instances PRR vs. SINR. Different models PRR estimation. Sensitivity to model parameters PRR estimation. Difference between numerical and simulation results PRR estimation. Difference error by link quality Packet success rate estimation evaluation. Example of topology instances Effect of the network demand on the link quality Difference between estimated values and simulation results by network demand Effect of the packet size on the link quality Difference between estimated values and simulation results by packet size Difference between SINR/PRR curves Difference between estimated values and simulation results grouped by link quality Relation between link distance and packet success rate for different packet sizes. Path loss exponent (η) = Relation between link distance and packet success rate for different network demands. Path loss exponent (η) = Relation between link distance and packet success rate for all instances. Path loss exponent (η) = Relation between RSSI value and packet success rate for all instances.. 84
12 6. Topology types for evaluated instances. Area size 5m 5m. Node density. nodes/m 2 = 25 SNs. Base station density.5 nodes/m 2 = 3 BSs Network delivery ratio. Comparison between values of w c Network mean delay. Comparison between values of w c Network delivery ratio difference with respect to hopcount metric. Comparison between values of w c Network delivery ratio. Comparison between values solutions provided by our model and CTP Network mean delay. Comparison between values solutions provided by our model and CTP Network average hopcount. Comparison between values solutions provided by our model and CTP Network delivery ratio. Comparison between values of K Network mean delay. Comparison between values of K Comparison of solution quality obtained by allowing different number of RNs to be deployed Comparison of solution quality obtained by different grid resolutions Comparison of solution time for each resolution considered MST heuristic algorithm MST heuristic comparison RNP to provide connectivity. Comparison between our model and MST based heuristic. Area size 5m 5m RNP to provide connectivity. Comparison between our model and MST based heuristic. Area size 75m 75m
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14 List of Tables 3. Radio model parameters Key Notations Simulation parameters for PMAC evaluation under logdistance path loss model Simulation parameters for PRR evaluation Topologies parameters for PRR evaluation Topologies parameters for Packet Success Rate evaluation Simulation parameters for Packet Success Rate evaluation Simulation parameters for model evaluation Instance parameters for link cost metric evaluation Instance parameters for RNP to enhance performance evaluation Instance parameters for RNP to provide connectivity evaluation xiii
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16 Introduction A wireless sensor network (WSN) consists of many lowcost and lowpower devices which can sense the environment and communicate with other nodes over shortdistance[]. Rapid technological advances in microelectronic and mechanical structure (MEMSs) and wireless networking have stimulated the development of multifunctional sensor nodes (SNs) devices. These devices are usually resource constrained, which means they have limited energy, limited processing and memory capability as well as limited transmission range. These sensors are usually inexpensive and can be deployed in a variety of terrains or in hostile environments to provide continuous monitoring. WSNs have been used in several applications ranging such as environmental monitoring, health care and military applications and a more recently commercial use for home appliances []. In typical applications, the data generated by all sensor nodes needs to be transmitted and gathered at a base station (BS) by means of a wireless media. Considering the transmissionenergycost model of sensor networks and the limited transmission range, the use of multihop paths for the routing has been preferred in most of the literature. One field of research on WSNs has focused on strategies to improve certain qualities of the network, after it has been deployed. A group of these works have proposed to deploy an additional number of special nodes, referred as relay nodes (RNs). These nodes can be positioned at precise locations by hand, or they can be part of a mobile robotic unit, such that they can be deployed autonomously. RNs are typically meant to receive and forward packets generated by SNs, but their deployment objective is varied. In some applications, SNs are expected to operate in an unattended manner
17 with the help of lightweight batteries, which might not able to be replaced or recharged. Therefore, a great deal of research in this field have focused on the energy conservation in WSNs, by means of RNs, so that the lifetime of the network is maximized and it can remain operational with the available energy resources for an extended period of time. In other scenarios, several factors such as hostile or dynamic environments and energy depletion may cause some parts of the network to become disconnected from BSs, causing this areas to be excluded from the monitoring process. Thus, another part of the research work in this domain has focused on the use of RNs to provide or improve connectivity. In this work we consider the problem of the deployment of RNs with the objective of general performance enhancement. That is, given a restricted number of available RNs, we aim to determine the locations at which these additional nodes can be positioned in order to improve network throughput and endtoend packet transmission delays. Energy optimization is expected to result as a byproduct of this. We do not make any assumptions about network connectivity. Therefore, in disconnected network scenarios, in which SNs might not be able to send their data to BSs, the deployment of RNs is aimed both at bringing connectivity to the BSs and optimize global network performance. We formalize the problem of RN positioning by defining a linear, mixed integer mathematical programming model. The formulation includes a number of constraints and penalty components, aimed at closely modeling the specific characteristics of the wireless environment, as well as a number of heuristics, aimed at speeding up the computations. The model is solved to optimality using a standard solver finding the best locations for the available RNs. Although our primary objective is determining the physical locations at which RNs should be placed to meet the performance improvement condition, our model also specifies the way these RNs should be used. This specification comes in the form of optimal data paths from SNs to BSs. That is, the optimal paths to route data flows. Since determining optimal routes involves the selection of wireless links which should be used for data transmissions, we are forced to take into consideration several properties of the network and the wireless media, and, in particular, to develop a link estimation metric which allows to discriminate among the possible links in the network those that have better quality. In more general terms, our approach can be seen as a set of models 2
18 and heuristics which are combined to produce the desired results for RN placement and data route definition. We perform an extended evaluation to assess individually the efficacy of the proposed schemes and of the associated solutions. We study the computational performance for solving the network flow model, as well as the effect of different heuristics and modeling choices on the resulting network performance. The effectiveness of the found RN locations is evaluated by comparing the network performance with and without the additional RNs. In turn, the network performance is extensively evaluated in simulation (using the TOSSIM simulator), considering a large set of different network topologies and traffic loads. We also compare the performance of the RN placement versus the placement defined by a minimum spanning tree heuristic in the case of initially disconnected topologies. The throughput and delay performance of the static data routes obtained by the solution of the network flow model is further evaluated by considering the throughput and delay performance obtained by the Collection Tree Protocol (CTP), which is a stateoftheart adaptive routing algorithm for WSNs delivered with TOSSIM. In all the scenarios considered, results show that our approach is clearly effective to accomplish the network performance enhancement objective. Moreover, a comparison analysis demonstrates that the proposed solutions define better quality SNBS routes, which overpass the throughput obtained by using CTP. The determined locations for RNs also prove to be adequate in pursuing their primary goal in contrast to those determined by a MST based heuristic. To the best of our knowledge, this is the first study to consider the relay node placement problem in WSNs exclusively in the context of network performance enhancement. Major contributions of this work include a MIP formulation of the problem together with a set of constraints and penalty functions aiming model several aspects of the wireless environment, in order to determine optimal RN locations and SN to BS routes to achieve better network performance in terms of delivery ratio and endtoend path latencies. Apart from determining the optimal RN locations, this study also proposes a novel link estimation model scheme, which is designed specifically to allow the selection of reliable links which support the relay node placement strategy. The rest of the work is organized as follows: Chapter 2 presents the related work. Chapter 3 introduces some basic concepts and theoretical background that precede our 3
19 work. Chapter 4 presents the system model and the problem formulation. Chapter 5 introduces a novel link quality estimation metric, which has been developed to be used in conjunction with the system model. Chapter 6 presents the evaluation and discussion of results. Finally, chapter 7 settles the conclusions and final remarks of this work. 4
20 2 Related work In the recent years, numerous studies in WSNs have proposed the deployment of RNs in a network in order to meet certain requirements and achieve different objectives. We have categorized these works based in two main aspects. First, RNs are proposed to be used in flat architectures as well as in hierarchical architectures [3]. Based on the network architecture, relay node placement (RNP) problems can be classified into either singletiered (or flat architecture) or twotiered based [3, 52], also referred as hierarchical architecture. In the singletiered RNP a SN also forwards packets received from other nodes while in twotiered RNP, a SN transmits its sensed information to an RN or to a base station (BS), but does not receive or forward packets from other nodes. The second aspect to consider while categorizing these works, is the nature of the main objective they pursue. Most of the existing work has focused on the deployment of RNs to provide connectivity [4, 6, 9, 3, 49, 5, 52, 53, 66, 7], extend the lifetime of the network [3, 68], energyefficient or balanced data gathering [2, 2, 48, 58, 76] or to provide survivability and fault tolerance [29, 3, 34, 53, 66, 69, 75]. As a large number of these works focus on providing some connectivity level, namely basic connectivity or fault tolerance, we group them together in one category. The rest of the works share similar objectives, mostly related to energyefficiency and network lifetime maximization. We place them in a second category. 5
21 2. Relay node placement for connectivity Based on the connectivity requirements, the RNP problems have been classified as Connected and Survivable (or FaultTolerant). Both aim to place the minimum number of RNs, but to achieve different goals. In the Connected problem, the goal is to make the induced network topology globally connected, assuming the nodes are disconnected. In the Survivable or FaultTolerant approach, the objective is to ensure that the network remains connected in the presence of up to k node failures. For the Connected RNP in singletiered networks, the problem can be described as deploying a minimum number of RNs in a WSN so that between every pair of SNs, there is a connecting path consisting of RNs and/or SNs and such that each hop of the path is no longer than the common transmission range of the SNs and the RNs. Lin and Xue [5] formulated the problem as the Steiner Minimum Tree with minimum number of Steiner points and bounded edge length problem (SMT MSP), proved that the problem was indeed NPhard and proposed a minimum spanning tree (MST) based 5 approximation algorithm. In [4], Chen et. al. proved that the LinXue algorithm was actually a 4approximation algorithm and also presented a 3approximation algorithm. In [6], Cheng et. al presented a faster 3approximation algorithm, and a randomized algorithm with an approximation ratio of 2.5. All of the previous mentioned works assume that the transmission range of the RNs is the same as that of the SNs. In this context, Lloyd and Xue [52] generalized the problem studied in [4, 6, 5] to the case where the RNs have transmission range R r (where r is the transmission range of SNs), and presented a 7approximation algorithm. In [9], Efrat et. al. improved the best known approximation ratio by that time [52], from 7 to 3.. In [49] Li et. al. generalized the problem to the case of heterogeneous networks with base stations (BSs). In heterogeneous wireless sensor networks (HWSNs), SNs are allowed to posses different transmission capabilities, and hence different transmission radius. In this problem the objective is to deploy a minimum number of relay to establish directed paths from each sensor to a base station. The authors presented a polynomial time 6approximation algorithm. In [53], Misra et. al. considered also WSNs with BSs, but study a constrained version of the problem, where RNs can only be 6
22 placed at a set of candidate locations and present polynomial time O() approximation algorithms. They also show that, for the particular case where there are no base stations and without restrictions on the locations of the RNs, the algorithm has an approximation ratio of 5.5. For the Connected RNP in twotiered networks, Hao et.al. [3] presented a polynomial time approximation algorithm under the assumption that RNs have a larger communication range R 4r. They aimed to deploy a minimum number of relay nodes so that () every SN is within distance r from a RN and that (2) between every pair of RNs, there is a connecting path composed only by RNs such that each hop of path is not longer than R. Tang et al. in [66] presented an 8approximation algorithm under the assumption that R 4r and the SNs are uniformly distributed. Lloyd and Xue [52] generalized the problem with the constraint that R r and without any assumptions on the node distribution, and presented a (5 + ɛ) approximation algorithm, where ɛ is any positive constant. Efrat et. al. [9] proposed a polynomial time approximation scheme for the problem. Yang et. al [7] studied a constrained version restricting the placement of the RNs to certain locations and present a O() approximation ratio algorithm. Survivable relay node placement (also known as fault tolerant relay node placement) in WSNs has been studied by many researchers [29, 3, 34, 53, 66, 69, 75]. The objective is to ensure that the network remains connected in the presence of up to k node failures. A network, in order to tolerate up to k node failures, has to be k +connected [53]. Hao et al. [3] formulated the problem where each SN must be within distance r of at least 2 RNs and the RNs (all having communication range R r) form a 2 connected network, and presented a polynomial time approximation algorithms. In [], Bredin et. al. extended the problem studied in [4, 6] in singletiered networks to the case of kconnectivity, instead of connectivity, and presented polynomial time O() approximation algorithms for any fixed k. Tang et. al. [66] gave an 4.5 approximation algorithm for twotiered networks and 2connectivity. In [34], Kashyap et al. presented a approximation algorithm ensuring 2connectivity for a single tiered network. In [75], Zhang et. al studied the RNP problems in both singletiered and twotiered networks that ensure 2connectivity under the condition R r. For the singletiered problem, presented a 4approximation algorithm and for the twotiered problem, a ( + ɛ)approximation algorithm. They then generalized the problem to the case 7
23 where there are also base stations (BSs), and presented a 6approximation algorithm for the singletiered problem with BSs and a (2 + ɛ)approximation algorithm for the twotiered problem with BSs. In [69], Wang et. al. generalized the survivable relay node placement problem to kconnectivity for both singletiered and twotiered cases and presented approximation algorithms. In [29], Han et. al addressed the problems of deploying relay nodes to provide faulttolerance with higher network connectivity in HWSN. 2.2 Relay node placement for balanced and energyefficient data gathering and maximizing network lifetime Moving apart from the deployment of RNs to achieve connectivity levels. Falck et al. [2] considered the problem briefly in the context of balanced data gathering. They formulated the problem of balanced data gathering in sensor networks taking into account the amount of data gathered from the nodes before the batteries are drained. The goal is to collect a large total amount of data, but not at the cost of ignoring some parts of the monitored area. They also presented the problem of finding an optimal routing as a linear program. Their work considered the effect of deploying RNs on the network performance, and presented and compared two simple techniques for determining good relay node locations. Hou et al. [3] considered the deployment of RNs with the objective of prolonging network lifetime. They have considered a twotiered sensor network architecture where sensors are distributed into groups; each is managed by an aggregationandforwarding (AFN) node which receives and aggregates the data from all sensors in its group. The AFNs and the BS form an upper tier network in which an AFN sends the aggregated data to the BS over a multihop path through other AFNs. Two approaches have been suggested to prolong the AFNs lifetime. The first is to provision more energy to AFNs, known as the Energy Provisioning (EP) problem The second is to deploy RNs in order to reduce the communication energy consumed by an AFN in sending the data to the basestation, known as the relay node placement (RNP) problem. Both problems are considered jointly (EPRNP) and formulated as a mixedinteger nonlinear programming (MINLP) optimization. 8
24 In [58], Patel et. al. examined the joint problem of deploying SNs, RNs and BSs on a set of feasible locations and finding bandwidthconstrained energyefficient routes with guaranteed coverage, connectivity, bandwidth and robustness. They formulated ILPs for different placement objectives: minimize cost, minimize energy consumption and maximize network utilization. Kashyap et.al. [35] studied the placement of RNs in the context of wireless optical networks with the goal of reducing the maximum link load for a given traffic imposed on the sensors. They considered a flat architecture, called backbone network. Their goal is to find appropriate positions for the relays and establish network routing paths in order to avoid network congestion and therefore, minimize transmission delay. The model restricted the placement of the relays to the lines joining the backbone nodes, so a minimum number of relays is needed to form each link. There is no notion of sink or base station, instead it considered demands between any pair of nodes in the network (called profile entries). Then formulated the routing problem as constrained knapsack problem, treating each profile entry as a commodity, which can be split over multiple paths. In [2], Ergen and Varaiya considered the problem of determining optimal locations for RNs together with the optimal energy provided to them so that the network is alive during the desired lifetime with minimum total energy. They formulate the problem as a nonlinear programming problem and proposed an approximation algorithm. In their approach, the locations where RNs are allowed to be placed are restricted to a square lattice. Wang et. al. [68] also studied the deployment of RNs to maximize network lifetime in twotiered WSNs with a base station. They called the problem as Minimum Relay Node Placement for Maximizing Network Lifetime. The network lifetime in this work is defined as the maximum number of data packets arrived at the BS before the first SN is out of energy. Since SNs do not forward packets in twotiered networks, the placement algorithm is divided in three steps. First they place RNs to connect each SN to a RN. Then place another set of RNs to connect the uppertier network with the BS. Finally a set of RNs to assure the network lifetime can achieve a maximum value. Li et. al. [48] proposed the use of Voronoi diagrams to guide the placement of relays in a twotiered network. In their scheme, the candidate locations for the RNs are determined by partitioning the field in the form of a Voronoi diagram. The authors argued that since a Voronoi vertex is placed at the same distance to each of the sensor locations that share the same vertex, the energy needed by these SNs to transmit their 9
25 data to a RN positioned at the common vertex is the same, hence a more balanced energy utilization is assured, improving the network lifetime. After the partitioning is done, a RN selection algorithm minimizes the number of RNs required to assure all SNs are covered by at least one RN. Then, an additional number of RNs might be needed to assure connectivity between the RNs and the sink. This last placement is done in a way to minimize the number of hops required for each RN to transmit its data to the sink. Zhu et. al. [76] studied the effects of increasing the number of RNs to be deployed in singletiered networks. For a WSN, they find a minimum number of RNs needed to induce connectivity using the SMT heuristic proposed by LinXue [5]. Then, onebyone an extra number of RNs are placed in order to minimize the number of hops from each node to the sink. In their placement strategy, the feasible locations for the relays are points in the straight line connecting two SNs. Authors carried out some simulations to compare the effect of increasing the number of RNs on the energy efficiency of the network. 2.3 Discussion After a brief survey on relay node placement in WSNs, we shall make some remarks about the similarities and difference to our work. Most of the studies presented in section 2. deal with approximation algorithms to the RNP problem. It has been proved that, in disconnected networks, determining the minimum number of RNs to provide connectivity is a NPhard problem [5]. For this reason there has been some interest to formulate polynomial approximations. Although our problem is similar, in the context of network connectivity, we do not focus on optimality in terms of number of RNs. The approach followed in [2] share some similarities to ours. They have also considered a LP model and established a set of possible locations for the RNs employing a grid strategy. However, they focus mostly in energyefficiency and consider a simplistic radio propagation model. Therefore, their work does not make clear the quality of the solutions in terms of network performance. The study done in [58] involves also the use of LP in the problem formulation. They have considered the objective of network performance in the form of maximize network
26 utilization. Similarly to [2], a grid is used to discretize the feasible sites where RNs can be deployed. However, they have perform poor simulations to determine the quality of the solutions and have not considered indepth the effects of interference and radio propagation in their model.
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28 3 Wireless sensor networks: characteristics and modeling 3. Propagation and Interference Model Wireless Sensor Networks, as its name suggests, rely on data transmissions through a wireless communication channel. Therefore, understanding and considering the properties and effects of radio propagation should be inherent to the design of protocols for these kind of networks. Compared to their wired analogue, wireless communications are difficult to analyze or to model due to their apparent stochastic behavior. This property is largely caused by the different phenomenas such as path loss, scattering, diffraction, shadowing, among others [6]. In this section, we introduce fundamental notions of radio propagation models and some definitions and formulas which form the basis to understand the approaches developed in the following parts of this work. Notation In the rest of the chapter, we introduce several formulas which mainly refer to received and transmitted signal power. As we try to simplify the explanation of some basic concepts, we are forced to use different measure units, depending on the definition in matter. Basically, we present some notions in mw and others in dbm. Values can be 3
29 converted easily between mw and dbm using the following formula: P [dbm] = log (P [mw ] ) (3.) P [mw ] = P [dbm]/ (3.2) For sake of convenience, we use P with subscripts in lowercase letters to denote signal power expressed in mw and with uppercase to denote signal power in dbm. 3.. Free space propagation model and path loss The Free space propagation model is relevant when the space between sender and receiver is free from obstacles, meaning a lineofsight communication. This model predicts the received signal strength as a function of the separation distance. The wellknown formula to calculate the received power (P rx ) [mw] of a transmission from a sender separated by distance d [m] is: P rx = P txg t G r c 2 (4π) 2 d 2 L (3.3) where P tx [mw] is the transmission power, G t and G r are the transmitter and receiver antenna gains, L and c are constants defined by the model. As seen, the transmitted signal suffers a distance dependent loss of power, which is called the path loss. Path loss represents the signal attenuation, and is defined as the difference (in db) between the effective radiated power and the received power [6]. This relationship is expressed in equation 3.4, where P L is the path loss, P T X [db] is the radiated power, P RX [db] is the received power, and d [m] is the separation distance. P L(d) = P T X P RX (d) (3.4) 3..2 Logdistance path loss shadowing model An accurate modeling of signal propagation is fundamental to develop protocols which aim to improve performance in wireless networks. Unfortunately, the traditional path loss radio propagation model does not reflect reality accurately enough since the received signal strength is modeled as a direct function of the distance. A more sophisticated model for radio propagation is the logdistance path loss shadowing model [6] where the signal strength perceived by a certain node does not only depend on 4
30 the distance between transmitter and receiver, but also includes some random factor. The logdistance shadowing radio propagation model accounts for the fact that antennas are not perfectly isotropic, and, even more importantly, the environment might be obstructed by, e.g., buildings or trees. In our work, we adopt this model for signal propagation. The model is given by: P L(d) = P L(d ) + η log ( d d ) + X σ (3.5) where P L(d )[db] is the reference path loss, d[m] is the transmitterreceiver distance, d [m] is the reference distance, η is the path loss exponent and X σ a zeromean normal distributed random variable (in db) with standard deviation σ, also in dbm. The received signal strength (P RX [db]) at a distance d is calculated subtracting P L(d) from the transmitting power (P T X [db]) as shown in equation 3.4. The model parameters we use in all the analysis and simulations are given in table 3.. Nonlogarithmic model The lognormal shadowing model can be also stated using a nonlogarithmic model. The received signal power P rx [mw ] is given by the following formula: P rx = P tx KYd η where P tx [mw ] is the transmitted power, K = d η P L(d ) is the reference path loss and Y is a lognormal distributed variable with variance (mσ) 2, m = ln. Probability and cumulative density functions Using the logdistance model for radio propagation allows us to capture the environment effects and therefore obtain a better approximation of the wireless communication behavior. However, this advantage comes to a cost, with an increase of complexity. As we will see in the next chapters, in order to model some properties of the network (i.e connectivity, packet reception rate) we must determine the probability that the received power, or a function of the received power, is lower (or greater) than certain threshold. Given that the received signal power is modeled as a lognormal variable, it is useful to introduce its cumulative probability distribution. Let P RX be the received 5
31 signal power, the probability that this value surpass certain threshold P T H is given by: ( ) ) P(P RX P T H ) = P (P T X P L(d ) η log X σ P T H dd (3.6) = P(X σ δ) (3.7) where: δ = P T X P L(d ) η log ( d d ) P T H (3.8) Since X σ is a normal distributed random variable, with zero mean and standard deviation σ, we obtain: P(X σ δ) = 2 ( ( )) δ + erf 2σ 2 (3.9) Distribution of the sum of lognormal distributed variables Let I be a set of nodes transmitting simultaneously. The probability that the sum of the received signal power levels surpass certain threshold P T H is defined as: ( ( P log i I P i rx ) P T H ) (3.) Let Z = P tx KYd η i where d i is the distance from node i to the receiver and P tx i I is the common transmission power. As we see, the random variable Z consists on the sum of independent lognormal distributed RV. To derive the distribution of Z we need to consider the distribution of the sum of independent lognormal RV, which is a wellstudied problem and can be approximated by another lognormal RV [6]. We consider the FentonWilkinson s (FW) approach [22] which is simple and gives more accurate results than other approximation methods [5]. Z = i I = P tx K i I P tx KYd η i (3.) Yd η i (3.2) Let Y i = Yd η i, since Y log N (, (mσ) 2),then Y i log N ( µ i, (mσ) 2) (3.3) 6
32 d P L(d ) P T. m 55 dbm dbm Table 3.: Radio model parameters ln where m = and µ i = ln(d η i ). Using the FW approach, Z can be approximated as a lognormal distributed variable, with mean µ Z and variance σz 2 : e 2µ i ( ) σz 2 = ln e m2 σ 2 i I ( ) 2 + (3.4) Finally, µ Z = ln ( i I e µ i ) i I + m2 σ 2 2 e µ i σ2 Z 2 + ln (P txk) (3.5) P ( log (Z) P T H ) = P ( log (Z) < P T H ) (3.6) = + erf P T H ( m µ ) Z (3.7) 2 2 (mσ Z ) Noise Floor Estimating the noise at the receiver, together with an appropriate model of radio propagation, is important to assess the coverage and quality of service of a given network [6]. Since all electronic components generate some noise which should not be neglected in the calculations, we introduce this element and define a suitable value for the analysis in this work. Obviating the underlying theory for thermal noise and noise power spectral density, we present the formula which allows to calculate the value of the noise floor. For radio devices, the noise power is determined by [6] : P n = ( + F )kt B (3.8) where F is the noise figure, k is the Boltzmann s constant given by J/K, T is the ambient temperature [K] and B is the device bandwidth. For wireless sensor 7
33 networks, at room temperature, operating at 2.4 Ghz, this value is approximately dbm. Since there are other sources which contribute to the amount of noise perceived in a network, such as interference generated by external devices, we rely on values determined by other works [77] and use an average noise floor of 5 dbm Interference Model One of the main factors that limits performance in wireless networks is interference, which is a direct consequence of using a shared communication medium [9]. For this reason, using an accurate modeling of interference to make data routing decisions can enhance significantly the network performance [57]. In the literature, two main interference models have been proposed [27, 33]: the protocol and the physical interference models. In the protocol model, a transmission between a pair of nodes i and j is considered successful if no other node within a specified range is transmitting simultaneously. Several works on Topology Control in wireless networks have defined metrics to estimate the possible interference along a wireless link using the protocol interference model [7, 3]. Blough et. al. defined an Interference Number for a link (u, v) as the number of nodes within u and v s range when transmitting at maximum power. The metric definitions in [7, 3] consider a node as interferer if it lies within the transmission range, however other works [4, 43] have used the concept of interference range to specify the distance within a transmission will interfere with the sending/receiving operations of two nodes attempting to communicate. Due to its simplicity, the protocol model has been used widely in the literature. On the other hand, the physical interference model, which we will refer from this point as physical model, describes the success probability of a transmission when one or more interferers are contributing to the interference at the receiver of the intended transmission. In this model, the transmission success depends on the received signal strength, the interference caused by simultaneous transmissions and the environmental noise level. A data transmission from node i to a node j is successful if the SINR (Signaltointerferenceplus noise ratio) at j is above certain threshold which depends on the physical layer design [57]. In this work we extend the physical model, coupling it with the radio propagation model and the MAC model, as we explain in detail in chapter 5. 8
34 3..5 SignaltoInterference plus Noise Ratio The Signaltointerference plus noise (SINR) ratio is defined as the ratio of the power of a received signal and the total interference perceived by the receiving node [6]. The total interference power is the sum of the noise power [mw] and the power of interfering signals [mw] emanating from simultaneous transmissions. Let P rx be the received signal power [mw], I the set of interfering nodes, P i the received interfering signal power [mw] from interferer i and P n the noise power [mw], the SINR is given by: SINR = P rx P n + i I P i (3.9) SINR and the physical model The radio reception performance can be characterized by defining the packet reception rate P RR as a function of the received SNR. As we mentioned before, the physical model gives the probability that a packet received with a specific SINR will be decoded correctly by the receiver. In order to determine the relationship between SINR and PRR, a SINR threshold should be specified. The SINR threshold (θ) is defined as the minimum SINR value which guarantees a reliable packet transmission with P RR.9 [62]. This value can be determined experimentally or examining the SNR vs P RR curve given by the manufacturer. Considering the radio model of section 3..2, we can state the probability that a SINR γ will surpass the threshold θ and therefore, the probability that a transmission will be successful. Given a transmission from node i to node j and a set of interfering nodes I, and considering the SINR equation 3.9 and the nonlogarithmic expression of the received power using the logdistance shadowing model, the SINR γ of a transmission between i and j is given by: γ = P i Kd η ij Y i P k Kd η kj Y (3.2) k + P n k I as we can see, the numerator describes the received power of the transmitted signal from i to j, where P i is the transmitted power by the sender i, d ij is the distance between sender and receiver and Y i is the lognormal random variable describing the shadowing effect of the signal emitted by node i. In the denominator, we have the 9
35 sum of the interfering signal powers, from interferer nodes set I, where P k (k I) is the transmitted power of interferer k, d kj denotes the distance between the interferer and the receiver and Y k represents the shadowing effect, as a lognormal distributed variable, of the interfering signal emanating from interferer k. As mentioned before, K is a constant given by the propagation model and P n denotes the environmental or thermal noise. Given that all nodes transmit with same power level (P i = P k = P tx ) k), we have: γ = k I d η ij Y i d η kj Y k + Λ where Λ = P n P tx K (3.2) (3.22) Distribution of γ : For computing the probability density function of γ, we perform an analysis similar to the formulation presented in [36]. In order to find an expression for the probability density function of γ (3.2) as: where Z i γ = W + Λ (3.23) Z i = d η ij Y i (3.24) W = k I d η kj Y k (3.25) Random variable Z i is lognormal distributed, hence has the following distribution: ( ( Z i log N ln d η ij ), (mσ) 2) (3.26) 2
36 W consists on the sum of Lognormal variables. Using the FW approach formulated in equations (3.4) and (3.5), we can state its distribution: W log N (µ W, σ W ) (3.27) where m = ln() ( ) σw 2 = ln e m2 σ 2 k I ( and µ W = ln ( µ k = ln ( k I d η kj e µ k ) ) k I + m2 σ 2 2 e 2µ k e µ k σ2 W 2 ) 2 + (3.28) (3.29) (3.3) (3.3) Recalling equation (3.23), γ consists on the quotient of lognormal variable Z and lognormal variable W plus the constant Λ, therefore it can be approximated as a lognormal variable. Putting all together: γ log N ( µ γ, σγ 2 ) where ( µ γ = ln d η ij (3.32) ) µ W Λ (3.33) σ 2 γ = (mσ) 2 + σ 2 W (3.34) We should note the importance of the last derivation, because with this result, it is now possible to compute the probability that γ surpass certain threshold, particularly the SINR threshold (θ). Given a transmission from a sender i to receiver j, and set I of nodes transmitting simultaneously. Under the physical model, the probability of successful reception is given by: P (γ θ) = P (γ < θ) (3.35) ( ( )) = erf ln (θ) µ γ (3.36) 2σγ 2
37 3.2 Medium Access A medium access control (MAC) is needed to manage the access of a common wireless medium. Carrier Sense Multiple Access protocols (CSMA) are commonly used in WSNs due to their simple implementation. In the rest of the paper, we assume a CSMAbased medium access protocol. In this protocol, when a node attempts to transmit, it first senses the medium. If the medium is busy by a current transmission, it defers from transmitting and schedules a new attempt based on a random exponential backoff algorithm. Typically, the wireless channel is sensed busy when the signal strength level is larger than the carrier sense threshold (β cs ) IEEE MAC Protocol IEEE [32] is a shortrange wireless technology specifically devised to support low power, low cost, low data rate Personal Area Networks (PANs). The main application of this technology is the implementation of Wireless Sensor Networks (WSNs). The standard allows two types of channel access mechanisms: beacon or nonbeacon enabled, using a slotted or unslotted Carrier Sensing Multiple Access with Collision Avoidance (CSMA/CA) protocol, respectively. The choice between these two modes depends on the application. The beacon enabled mode requires the intervention of a Wireless Personal Area Network (WPAN) coordinator which broadcasts beacons to the devices at regular intervals to maintain synchronization of the nodes. Therefore, it is not suitable for the application scenarios we consider, since we do not make assumptions on network connectivity or architecture. We consider the nonbeaconenabled mode (Fig. 3.). The algorithm uses two variables, BE which is the current backoff exponent, and NB to count the number of backoffs. Each time a node generates a packet for transmission, it waits a random number of backoff slots ranging from to 2 BE (BE is referred as backoff exponent). Initially, BE is initialized to BE min (by default 3) and its maximum value is BE max (by default 5). The variable NB is set to at the beginning. After waiting the selected amount of time, the node performs a CCA (Clear Channel Assessment) to determine whether the channel is busy or not. If the channel is idle for an amount of time, called T CCA, the procedure terminates, the channel is considered free and the node proceeds with the data transmission. When the channel is perceived as busy, BE is increased by one unit (if BE < BE max ), 22
38 Unslotted CSMA NB =, BE = macminbe Delay for random(2 BE ) backoff units of time Perform CCA Channel Idle? Yes No NB = NB +, BE = min(be +, macmaxbe) No NB > maxbackoff? Yes Failure Success Figure 3.: Unslotted CSMA algorithm the backoff counter is incremented NB = NB +, and the procedure is repeated if NB <= maxbackoff. After maxbackoff + tries, the attempt to get access to the channel fails and the packet is discarded. We conclude this section pointing out that the assumption of IEEE MAC Protocol does not limit our model and can obviously be relaxed. Numerous alternative MAC protocols have been developed by the sensor networking community [42], each of them with their specific advantages and disadvantages. Nevertheless, due to its standardized state, we consider the IEEE MAC as a natural choice. 23
39 3.3 Wireless Sensor Network simulation The evaluations in our work are based on network simulations. We have chosen TOSSIM [46] wireless network simulator for this purpose, which is a eventdriven simulation tool developed for sensor networks. TOSSIM simulates applications written for the TinyOS operating system [47]. The main reason behind the choice of TOSSIM as simulation environment for our work is because it provides accurate results due to its wireless channel model and hardware emulation. TOSSIM profits from the component based architecture of TinyOS and transparently defines a hardware abstraction layer that simulates the TinyOS network stack at the processor level. To perform simulations in TOSSIM, it is necessary to implement an application just as it would be done for a real sensor node. In this way, the same application can be ported to a sensor mote without any significant changes. One of the few drawbacks of TOSSIM is that it lacks of a node location notion. Instead, the network representation it uses consists on the set of links between the nodes, together with the gain, or attenuation power, of each of them. For an evaluation over a network topology which is represented by the positions of the nodes, it is necessary to estimate the signal attenuation power for each link and supply it to TOSSIM at the start of the simulation. There exists a software tool, distributed with TOSSIM, which allows to compute the link gains for a given topology. This tool was developed by [78] and computes the signal attenuation based on the lognormal shadowing path loss model. However, there is still an issue to be considered. Although the link gain value is passed to TOSSIM as an initial parameter, it does not change during the entire simulation. While in reality, the signal attenuation might fluctuate due to the shadowing effects, therefore to obtain more reliable results it is required to perform several simulation runs, sampling a different value for the shadowing component in the lognormal path loss model (see equation 3.5). 3.4 Linear Programming and Network flow formulation Linear programming (LP) consists on methods and techniques to find the minimum/maximum value of a linear function in the presence of linear equality and/or inequality constraints. LP has been extensively used in many areas, including computer networking and its 24
40 popularity is attributed to factors such as the ability to model large and complex problems, the possibility of obtaining solutions in a reasonable amount of time and the availability of tools and software which allow to model and solve these problems [4]. LP have been used to model many problems in communication networks, typically using the widely known network flow model Network flow formulation A basic application of a communication network is to transport or flow the data generated by its users. A common problem is to develop efficient routing decision strategies to determine the data paths or routes such that the volume of data, or traffic demand required, is delivered accomplishing certain performance goals. Network flow models are frequently used to formulate problems in communication networks, and more particularly for our interest, in wireless multihop networks. There have been many works in routing decisionmaking [33, 39, 4, 4] and node placement [2, 26, 37, 58] which have used this models. As our work is also based in a network flow formulation, we introduce some terminology and key notions, together with a basic network flow problem formulation, which are used in the following chapters Basic terminology In a network flow formulation for a given communication network, the traffic demand usually represents an amount of data per second which is required for transmission. For a pair of nodes in a network, this value represents the amount of information which needs to be carried from one node to the other. For any demand between two nodes in the network, one or more paths are possibly needed to satisfy it. The amount of data which is carried through a path is referred as flow. A network may not be able to carry all the demands between its nodes. Namely, there might be network capacity bound, which constraint the amount of flow it can circulate Minimum Cost Flow Problem (MCFP) The Minimum cost flow problem (MCFP) consists on determining flows between nodes, satisfying certain constraints, at a minimum cost. Let G = (V, E) be a network consisting on a set V =, 2,..., n vertices and a set of E of directed edges which link pairs 25
41 of them. We associate to each edge (i, j) E a flow variable f ij, an upper bound u ij of the flow, and a cost γ ij. To each vertex i V, we assign a value τ i which represents the demand for flow at that vertex. If τ i >, the vertex is referred as source, and when τ i <, it is called a sink. The formulation is the following: min γ(f) = γ ij f ij (3.37) (i, j) E subject to (3.38) f ij f ji = τ i (3.39) (i, j) E (j, i) E f ij u ij (3.4) To illustrate the applications of MCFP formulations to computer network problems, we consider the following problem: Given a network (figure 3.2a) consisting on 5 source nodes {s, s 2,..., s 5 } which send data to one sink node (b). Since we know the properties of our network, we determine the set of possible links E {(i, j) i, j {s, s 2,..., s 5, b}} between the nodes. We are interested in finding a shortest paths from each source node s i to the sink where the length of a path is defined in terms of the number of hops. As noted, this is a common problem for computer networks, especially in the context of routing. A solution to the problem consists on a subset of links which induce a routing tree, which defines direct paths from each s i to b. To proceed with the formulation as MCFP, we define the problem structures. The set of vertices corresponds to the set of nodes in our network, so V = {s, s 2,..., s 5, b}, while the set of directed edges is simply the set of all possible links in the network. Following, we determine the values u ij and γ ij for each edge. In this example problem, the values for the flow upper bound are not relevant, therefore u ij = + for all edges. As we do not make distinction between edges, and the use of a link increases the path length in one unit, we set λ ij = (i, j). The only parameter left to be defined is the value of τ i for each vertex. To end the formulation, we define τ i as: τ i = { i {s, s 2,..., s 5 } 5 for i = b (3.4) Having formulated the problem as MCFP, we can use any numerical method or solvers available. An optimal solution, with the values for f ij, is shown in figure (3.2b) 26
42 s4 s s4 f s4,s = s fs,b = 2 b b s5 s5 f s5,s2 = f s2,b = 3 s2 s2 f s3,s2 = s3 s3 (a) Network consisting on five sources and one sink (b) Routes defined by an optimal solution to MCFP, where there is a shortest path between each source and the sink Figure 3.2: Illustration of the application of MCFP to routing problems 27
43 28
44 4 System Model Let a WSN consisting of sensor nodes (SNs) and base stations (BSs) located at S and B respectively, where SNs generate data packets and also forward packets received from other SNs/RNs towards base stations. We assume that the data rate generation of each SN is known, or can be estimated. We are given a set of K relay nodes (RNs) which can be placed in the sensing field. These nodes do not generate any information and their only task is to forward data received from other SNs/RNs, therefore serving as bridge between disconnected parts of the network and/or shortening the paths the data should cover over the network. As computing the optimal locations for the RNs allowing them to be placed anywhere in the sensing field is computationally hard to solve [7].We restrict their placement to a numerable set of candidate locations R. SNs and RNs communicate to other SNs/RNs/BSs within the network communication range r using a common wireless channel. We refer as node any element of the network (SN, BS or RN), and from now on we do not make distinction between nodes and their locations, therefore when we talk about node i, we also refer to location i S B R = N at which it is located. Since we focus on improving the communication from sensors to base stations (data collection), BSs do not generate or send any data. Hence, communications between node i and node j are assumed from a sender i, either a SN or a RN (i S R) to a receiver node j which could be any of SN, RN or BS (j N). The hardware and the protocols used for wireless communications determine the channel bandwidth. Our problem consists on finding optimal locations for a maximum number K of RNs and the data paths from each SN to a BS in order to enhance the total network 29
45 Table 4.: Key Notations S Set of locations of SNs B Set of locations of BSs R Set of candidate locations for RNs N S B R K Maximum number of RNs that can be deployed r Network communication range N i N Set of locations reachable by a SN or a RN placed at location i (i, j) Denotes wireless link between node (SN or RN) located at i S R and node (SN, RN or BS) located at j N. i, j Euclidean distance between location i and j. E Set of possible wireless links between nodes. (i, j) E iff j N i Cost assigned to wireless link (i, j). γ ij throughput. The way we approach to this objective is by minimizing the endtoend delay and the percentage of packet losses. When there are no RNs available to be deployed, the problem becomes a pure routing problem, and its solution is defined by the optimal data routes. There are no assumptions about the initial deployment of SNs and BSs, which includes the case where the initial network topology is disconnected. In such scenarios, the placement of RNs should aim to connect the network and, at the same time, improve its performance. We do not require the placement strategy to deploy all the RNs available and, in some cases, it can be determined that the best performance is achieved using less than K RNs. Additionally we do not constraint the data to be transmitted over a single path, hence a data flow can be spitted over multiple paths. 4. Basic model We formalize the problem of RN deployment defining a linear, mixed integer mathematical programming (MIP) model, namely by means of a network flow formulation. This formulation includes a number of constraints and penalty components, aimed at closely modeling the specific characteristics of the wireless environment, as well as a number of heuristics, to reduce the computation time. Our problem strategy enhances data 3
46 s s7 s BASE STATIONS s6 s9 b s2 s5 s4 s s r4 s2 RELAY NODES s3 r48 s3 STATIC NODES s8 s4 Figure 4.: Example of wireless sensor network 3
47 throughput by minimizing the packet losses and network delay, and by determining the best locations of a maximum number of RNs together with optimal sensortosink data flow routes. We start from a basic network flow formulation in which we approach the relay node placement problem and minimize the SNtoBS path length. As reducing the network routes length (number of hops) is a basic approach to reduce endtoend delays and improve network throughput, we have started aiming at this objective. However, we discuss later about this common assumption, when we include link costs in our formulation. After introducing the constraints which allow to determine RN locations and model the relay node placement problem, we move forward to include more sophisticated elements which are designed to improve network performance. 4.. Problem Formulation To formulate the relay node placement, we define a variant of the Minimum Cost Flow Problem (MCFP) in which we exploit the model structure to approximate solutions for our problem. Since there are constraints and terms in the MIP model which need to be explained in more detail, we start from a simplified variant of the MCFP formulation which is increasingly enhanced and adapted to the needs of our problem. Let G = (V, E) be a connected digraph where V = N consists of the set of nodes and the set of edges E denotes the possible communications between them. Let γ : E R be a cost function. Let τ : S R be a demand function and D = τ(i) be the total i S demand of the network. For e = (i, j) E, we simplify the notation of the mapping function γ(e) as γ ij. Same as for τ(i) = τ i. We use the following decision variables in our model: Flow variable f ij denotes the amount of data flow through wireless link (i, j), in other words, data traffic from node n i to node n j, located at positions i, j N respectively. Binary variable y i indicates whether location i R is being used to circulate flow. When y i = in a solution, it indicates that a RN should be positioned at the corresponding location. Therefore, a solution to our problem is defined by this two variables. The locations of the relays to be deployed is determined by the set {i R y i = }. The sensortosink routes are defined in the routingtree induced by the set {(i, j) E f ij }. The minimum cost flow problem requires the determination of a mapping f : E R with minimal cost. 32
48 It is formulated as: (i, j) E f ij i B min (i, j) E γ ij f ij (4.) subject to (4.2) { τ i if i S, f ji = (4.3) if i R d k (4.4) (j, i) E (j, i) E f ji = k S y i = j N f ji > (4.5) 4... Limited number of relays The way this formulation computes a feasible flow network f considers that there is a RN available to be placed at each of the candidate locations. As the problem statement defines a maximum number of available relays, we must define a constraint to enforce this issue in the model. The binary decision variable y i was introduced, to indicate which RNs are involved in the flow paths. A value y i = indicates that the flow network uses the RN positioned at candidate location i R, therefore the ingoing/outgoing flow for that node has a positive value. A value y i = implies there is no flow at that location. Using the variable y i, a simple constraint is added to limit the maximum number of RNs. y i K (4.6) i R Penalty factor for relays Solving our model as it is defined now will generate solutions using a maximum number K of relays. However, there are cases with optimal solutions using less than K relays. To enforce the model to generate optimal solutions using the least amount of relays as possible, we include a penalty factor in the objective function. The reasoning is the following: since the goal of relay placement in the model is to minimize the objective function, we must justify the use of a n relays if it produces a solution with a minimum gain with respect to a solution obtained using n relays. We define the value of the minimum justified gain as the Relay penalty factor ˆR. Therefore, 33
49 we redefine the objective function introduced before as: min γ ij + ˆR y i (4.7) (i, j) E i R The value of ˆR is a parameter in our model and it can be adjusted according to the problem instance and conditions (i.e relay node availability, economic cost). In our evaluations, we fix this value to one unit Network bandwidth, demand and flow unit Flow unit and network demand Wireless channels are bandwidthlimited, meaning that there is a maximum amount of data which could be transmitted over a period of time. Thus we must consider this limiting factor to avoid solutions which saturate parts of the network and consequently produce a large amount of packet losses. To specify this condition, we should define first the previously introduced demand function τ i. As SNs are the only elements in the network which generate data at certain rate, a solution should guarantee that all the generated information arrive at any of the BSs. Having said that, we define the demand τ i for a SN i as the amount of data per second it generates. For convenience, we use the concept of Flow unit (f unit ), in bytes/sec, as a reference for the values of τ. Therefore, values for the demand function τ are specified in flow units. As an example, for SN i, the demand τ i corresponds to τ i f unit bytes/sec. Network capacity We define the network capacity N cap as the total amount of data which could be transmitted through any wireless link in the network. This amount usually corresponds to an amount of bytes/sec, and as for the network demand, we specify it using flow units. Using these concepts we are ready to include in our formulation the bandwidth limitations of the network. Therefore, we include the following constraint: + N cap i N (4.8) f ij (i, j) E (j, i) E With this constraint, the total amount of data transmitted/received by any node of the network should not exceed the network capacity. This condition avoids saturation conditions in any link or parts of the network. f ji 34
50 4...3 Node indegree For a SN s in the network, the indegree represents the number of nodes which rely their data to s to be forwarded towards a base station. Clearly, this factor influences the capacity of node s and the loadbalancing of the network. As more nodes rely data to s, higher will be the time it should spend receiving and forwarding their packets. Together with this, as more nodes rely on the same neighbor as next hop in their data path, the number of MAC collisions increases, therefore this node becomes a bottle neck in the network. Although this factor is not studied indepth in this work, it may be useful to be included in our model. For this reason we consider including a restriction for the maximum node indegree allowed (deg max ). The following constraints are included in the model formulation: b ij = f ij i, j N (4.9) b ji deg max i S (4.) (j, i) E In our evaluations, we have set this parameter to Load balancing and preventing interflow congestion To produce more balanced routing trees, we must stimulate the generation of flows which do not interfere with each other. For this reason we include a penalty component in the objective function which reflects our desire to obtain balanced trees. We define the maximum local flow ( F max ) as the maximum amount of flow we expect to circulate within any region of the network. Since it is an abstract concept and depends on the regions considered, we refine it as the maximum circulation of flow in a circular area of radio r around any SN. Figure 4.2 illustrates this strategy, and how RNs can be used to prevent interflow congestion. Using this approach we can easily include this condition in our model, using the model structure itself. Calculating the flow circulating in a circular region of radio r around any SN i accounts to sum the outgoing flow from any of its neighbors. Thus, we define the binary penalty variable p i as: p i = f jk F max (4.) (i, j) E (j, k) E 35
51 Having defined the penalty variable, we proceed to specify the penalization in the objective function. In order to do it, we must first reason about the weight the penalty should have taking into consideration the order of magnitude of the objective function value without penalties. First, for any cost γ mapping function, we can determine the maximum link cost ( γ). Secondly, we determine a lower bound of the path length for every SN. This is done considering the minimum distance between a SN i to a BS, divided by the transmission range. We argue that as the model aims to optimize the network performance, the length of the paths in the solution will not defer strongly from the minimum path length. Summing up, an estimate value of the optimal solution cost will be similar in the same order of magnitude as: ( ( ) ) i, j γ(f) = min τ i γ j B r i S The penalty factor ˆF for the exceed in maximum local flow is defined as: (4.2) ˆF = α γ(f) (4.3) The value of alpha allows to determine the weight of the penalty in the objective function. This parameter could be defined taking into consideration network characteristics such as density, clusterization coefficient, among others. However, studying this relations is subject for future work. We define the value of α =.. Figure 4.3 shows an example of solutions obtained with and without this element in the model. Red spots in the heat map correspond to links and/or nodes which suffer of high rate of packet losses. In this example, the network delivery ratio increased in 5% when this factor was considered in the solution. However, we do not perform indepth evaluation of this strategy, and we leave a complete evaluation as future work. Finally, including the penalty in the objective function defined in eq. 4.7, we have: min γ ij + ˆR y i + p i ˆF (4.4) (i, j) E i R i S 4.2 Including cost for links Since we have not yet included link costs, solving the model as it is defined now will provide solutions selecting at most K relays while minimizing the total amount of flow 36
52 s s2 s s2 r22 s7 s5 s7 s5 s s3 s s3 s8 s8 s9 s6 s9 s6 s4 s4 s s s3 s3 s s s2 r52 s2 b b s6 s6 s4 s2 s4 s2 s8 s8 s22 s22 s7 s7 s23 s23 s9 b s9 b s2 s5 s2 s34 s5 s29 s34 s3 s29 s3 r s28 s28 s38 s24 s3 s38 s24 s36 s3 s33 s26 s36 s32 s33 s26 s37 s25 s32 s37 s27 s25 s35 s39 s27 s35 s39 (a) Network with congested regions (b) Network without congested regions Figure 4.2: Using relays to achieve load balancing and prevent interflow interference 37
53 s s2 s7 s5 s s3 s8 s9 s6 s4 s s3 s s2 b s6 s4 s2 s8 s22 s7 s23 s9 b s2 s5 s34 s29 s3 s28 s38 s24 s3 s36 s33 s26 s32 s37 s25 s27 s35 s39 s s2 r22 s7 s5 s s3 s8 s9 s6 s4 s s3 s s2 r52 b s6 s4 s2 s8 s22 s7 s23 s9 b s2 s5 s34 s29 s3 r s28 s38 s24 s3 s36 s33 s26 s32 s37 s25 s27 s35 s39 (a) Network with congested regions (b) Network without congested regions Figure 4.3: Simulation analysis to valuate the efficacy of the proposed strategy f, which is equivalent to minimize the number of links used by each flow. This accounts to find the shortest routing tree, where the length of the tree is the number of hops. Some related studies [76] have focused on this objective, however several factors show that indiscriminately selecting links following the minhop rule ignores many effects of the link characteristics on the network performance. For this reason, in chapter 5 we have developed a model for estimating the link quality given certain parameters. However, the furthest decision to be made is about selecting a flow network which corresponds to selecting routes. Therefore, the total cost value that will be subject of comparison will relate to all routes. The costs to be included in the model need to be derived as function of the individual metric values assigned to each link in the route. Although several functions could be used such as multiplication, average and summation, we shall restrict ourselves to linear functions due to the limitations of Linear Programming. Recalling equation 4., we have selected summation as the function of individual link costs, consequently, the total cost of the flow network is the sum of costs of links, which form part of it, multiplied by the flow circulating through those links. Figure 4.4 shows an example for a flow network consisting on three static nodes, one relay and one base station. In this example, static nodes generate one flow unit (τ s = s). Here we can appreciate how the link costs γ and flow through links f 38
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