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Energy -efficient strategies for deployment and resource allocation in wireless sensor networks
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Energy -efficient strategies for deployment and resource allocation in wireless sensor networks
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E N E R G Y -E F F IC IE N T S T R A T E G IE S F O R D E P L O Y M E N T A N D R E S O U R C E A L L O C A T IO N IN W IR E L E S S S E N S O R N E T W O R K S b y M o rteza M alak i A D issertatio n P resen ted to the F A C U L T Y O F T H E G R A D U A T E S C H O O L U N IV E R S IT Y O F S O U T H E R N C A L IF O R N IA In P artial F u lfillm en t o f the R eq u irem en ts fo r th e D egree D O C T O R O F P H IL O S O P H Y (C O M P U T E R E N G IN E E R IN G ) M ay 2006 C o p y rig h t 20 0 6 M o rteza M alak i Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number: 3233853 INFORMATION TO USERS The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleed-through, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. ® UMI UMI Microform 3233853 Copyright 2006 by ProQuest Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. ProQuest Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, Ml 48106-1346 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Acknowledgements I would like to express my deep and sincere gratitude to my advisor, Professor Massoud Pedram. I am greatly indebted to him for his generous support throughout this work. His understanding, encouraging and personal guidance have provided a good basis for the present thesis. His constructive comments have been o f greatest help at all times. I warmly thank Professor Michael Neely for his valuable advice and friendly help. His extensive discussions around my work have been very helpful for this work. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table of Contents A c k n o w led g em en ts.............................................................................................................................. ii L ist o f F ig u re s .........................................................................................................................................v A b s tra c t...................................................................................................................................................vii C hapter 1 in tr o d u c tio n ........................................................................................................................1 1.1 R elated W o r k ....................................................................................................................... 5 1.2 N etw o rk S e tu p ......................................................................................................................9 1.2.1 E n erg y M o d e l.......................................................................................................... 10 1.2.2 K ey A ssu m p tio n s...................................................................................................11 1.3 M o d el o f R an d o m D e p lo y m en t...................................................................................12 1.4 C o n trib u tio n s...................................................................................................................... 13 C hapter 2 :Q o M and L ifetim e-co n strain ed R an d o m D ep lo y m en t o f S ensor N etw o rk s fo r M in im u m E n erg y C o n su m p tio n ....................................................................... 16 2.1 Q oM D e fin itio n ................................................................................................................. 16 2.2 N e tw o rk life tim e ..............................................................................................................21 2.3 Q uality and L ifetim e-co n strain ed S ensor D ep lo y m en t (Q L S D ) P roblem 21 2.3.1 M o d elin g the Obj ectiv e F u n c tio n ...................................................................22 2.4 O ptim al R o u tin g A lg o rith m s in th e C o n tin u o u s S pace M o d el o f R andom D e p lo y m e n t...................................................................................................................................... 23 2.4.1 C o n tin u o u s S pace M odel and R o u tin g A lg o rith m ...................................24 2.4.2 E n erg y D en sity and T o tal E n e rg y ...................................................................27 2.5 S im u latio n R e s u lts ...........................................................................................................31 2.6 C h ap ter S u m m ary ............................................................................................................. 36 C hapter 3 :E n erg y -E fficien t R an d o m D ep lo y m en t o f a T w o -T ier S en so r N etw o rk in a F ie ld ....................................................................................................................... 37 3.1 P ro b lem D e sc rip tio n ........................................................................................................ 38 3.2 M o d elin g th e O b jectiv e F u n c tio n ..............................................................................39 3.3 C o n tin u o u s S pace M o d el and A n aly sis o f the R o u tin g S c h e m e .................... 42 3.3.1 C o n tin u o u s S pace M odel and a R o u tin g A lg o rith m .............................. .43 3.3.2 E n erg y A n aly si s ..................................................................................................... 48 3.3.3 A v erag e P ath -len g th A n aly sis o f C S C R ...................................................... 54 3.4 S im u latio n R e s u lts ............................................................................................... 56 iii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.4.1 P aram eter S electio n ..................................................................... 57 3.4.2 E n erg y D e n sity ....................................................................................................... 57 3.4.3 T otal E n erg y E v a lu a tio n .................................................................................... 59 3 .4.4 A v erag e P ath L en g th (A P L ) E v a lu a tio n ......................................................61 3.4.5 E ffect o f D ep lo y m en t A rea S iz e .....................................................................62 3.5 C h ap ter S u m m ary ............................................................................................................63 C hapter 4: L ifetim e-aw are D ep lo y m en t o f T w o -tier W ireless S en so r N e tw o rk s 64 4.1 T h e M D E A P ro b le m ..................................................................................................... 65 4.2 T he M D E A -1 P ro b lem in a C o llin ear N e tw o rk ................................................. 65 4.3 M D E A -1 P ro b lem S ta te m e n t.....................................................................................67 4.3.1 M D E A -lc P ro b lem S ta te m e n t.........................................................................72 4.4 T he M D E A P ro b lem in a P lan ar N e tw o rk ........................................................... 77 4.4.1 M D E A -2 P ro b lem S ta te m e n t........................................................................... 79 4 .4.2 T h e F irst P ro p o sed H eu ristic S o lu tio n fo r M D E A -2 P ro b le m 79 4.4.3 T h e S econd P ro p o sed H eu ristic S olution fo r M D E A -2 P ro b lem ....80 4.5 S im u latio n R e s u lts ......................................................................................................... 83 4.6 C h ap ter S u m m ary ............................................................................................................92 C hapter 5: S u m m ary and P o ssib le E x ten sio n s......................................................................... 93 B ib lio g rap h y .......................................................................................................................................... 95 A p p en d ix A ............................................................................................................................................99 A p p en d ix B ..........................................................................................................................................104 iv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. List of Figures F igure 1 -1 ................................................................................................................................................11 F igure 2 - 1 ................................................................................................................................................20 Figure 2 - 2 ............................................................................................................................................... 25 Figure 2 - 3 ................................................................................................................................................25 F igure 2 - 4 ............................................................................................................................................... 29 Figure 2 - 5 ................................................................................................................................................33 Figure 2 - 6 ................................................................................................................................................34 Figure 2 - 7 ................................................................................................................................................34 Figure 2 - 8 ................................................................................................................................................35 Figure 2 - 9 ................................................................................................................................................36 Figure 3 - 1 ................................................................................................................................................38 Figure 3 - 2 ............................................................................................................................................... 45 Figure 3 - 3 ............................................................................................................................................... 47 Figure 3 - 4 ............................................................................................................................................... 49 Figure 3 - 5 ................................................................................................................................................50 Figure 3 - 6 ................................................................................................................................................58 Figure 3 - 7 ................................................................................................................................................60 Figure 3 - 8 ................................................................................................................................................61 Figure 3 - 9 ................................................................................................................................................62 v Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. F igure 4 - 1 ................................................................................................................................................67 Figure 4 - 2 ................................................................................................................................................81 Figure 4 - 3 ................................................................................................................................................83 F igure 4 - 4 ................................................................................................................................................85 F igure 4 - 5 ........................................................................ 86 F igure 4 - 6 ................................................................................................................................................88 F igure 4 - 7 ................................................................................................................................................88 F igure 4 - 8 ................................................................................................................................................89 Figure 4 - 9 ................................................................................................................................................90 Figure 4 - 1 0 .............................................................................................................................................91 Figure 4 - 1 1 .............................................................................................................................................92 Figure B - l ............................................................................................................................................ 105 vi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Abstract A W ireless Sensor N etwork (W SN) comprises o f a collection o f sensor nodes that perform sensing and communication tasks and possibly a set o f additional nodes (called micro servers) which receive and aggregate local sensor data and send it to a central base station. Two figures o f merit for assessing the quality and efficacy o f a deployed WSN are Quality of M onitoring (QoM ) and M onitoring Lifetime (M oL).The QoM is strongly influenced by the density o f sensors inside the monitoring region whereas the M oL is directly related to the initial energy level assigned to the nodes. C hief among design problems o f W SNs are optimal sensor node deployment and energy resource allocation. In chapter 2, considering a flat network architecture comprised o f randomly-deployed sensor nodes, we determine the energy and node densities that result in allocating the minimum total resources while meeting constraints on the QoM and M oL figures o f merit. After showing that the optimal solution is a strong function o f the routing scheme used to send a packet o f data from sensors to the base station, we propose and analyze a novel data routing scheme for such a network. In chapter 3, we investigate the same problem for a two-tier W SN comprising of randomly-deployed sensors and micro-servers at its lower and higher levels. The sensor density is preset in order to achieve a minimum level o f QoM. The micro-server and energy resource densities are, however, to be determined. In chapter 4, we investigate lifetime-aware strategies for detailed placement and energy allocation o f infrastructure in the second level o f a two-tier W SN. Various forms o f the following problem are formulated: Assign positions and initial energy levels to the micro- vii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. servers, and concurrently, assign groups o f sensors to each m icro-server so as maximize MoL o f the tw o-tier WSN subject to a total energy budget. Simulation results show that the optimal design and deployment o f a two-tier network can increase the WNS lifetime by one order o f magnitude compared to a flat WSN with the same total initial energy and quality o f monitoring. viu Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 1 Introduction Sensor nodes can be deployed to m onitor a phenomenon o f interest in an area. These sensor nodes collect a variety o f data, such as sound, motion, temperature, or vibrations and pass this data up toward a base station whose goal is to perform a high level monitoring tasks such as detecting seismic activity or securing some region. Sensor nodes tend to be severely constrained by the amount o f battery energy that is available to them. This limitation in turn greatly impacts the service lifetime and the QoM that can be achieved by each sensor node or by the sensor network as a whole. A sensor network derives its strength from the rather large number o f sensors that includes. This brings to mind a variety o f design and scale issues. C hief among these problems are those o f sensor node deployment and energy resource allocation. The first problem is that o f assigning a sensor node density, X, to the deployment region in the case o f random deployment o f sensors and assigning positions to the sensors in the case o f precise placement o f sensors whereas the second question relates to that o f allocating the am ount o f initial energy, eh to each sensor. Assuming a fixed initial energy level, eo, the total energy allocated to the network in a region 5^with area ||< 2 fl| is thus etot=X \\<S^eo- In other words, for a fixed total energy bound eto t in a fixed area % the parametric solutions (k!a, a.e0 ) for any a >0 are completely equivalent. Furthermore, a may change in different sub-regions (it can spatially vary.) Therefore, in this thesis, we will treat the sensor deploym ent problem (i.e., 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. determining node density or positions o f sensors) and the energy allocation problem (i.e., setting e, per node) as related and interacting problems. When we consider random deployment, we assume that sensor node density is controlled at a coarse granularity e.g., through aerial deployment. That is why when we refer to sensor node deployment, we avoid referring to the exact coordinates o f individual sensors in the network. Instead we assume that the coarse-grain deployment mechanisms can only guarantee a random distribution with known density function1. The problem that we address in the case o f random deployment is that o f finding (2,, e,) that minimize < K subject to constraints on QoM and network lifetime. Here index i identifies some sub- region A, in < % . The minimum size o f A, is determined by the granularity o f the deployment control. For example, if there is sensor deployment through ground vehicles Aj is in the range of tens o f meters whereas for the aerial sensor deployment is in the range o f hundreds o f meters. In our solution technique, we will assume that we have full control over node densities within some thin circular regions. N ote that the situation is different after the network is deployed and has become operational since at that point the approximate locations o f all sensors in the network can be determined and exploited by the scheduler. Given a fixed region o f coverage and initial energy levels for all sensor nodes in the network, two figures o f merit may be used to assess the effectiveness o f a deployed sensor network: 1) QoM , i.e., the accuracy or fidelity o f the gathered data in a coverage region, 1 We realize that having complete control over how the node density varies over space is in the limit equivalent to having precise control over the coordinates of individual sensors. However, we assume that there is a sufficient level of deployment control to allow for slowly varying node densities across space. 2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. and 2) the m onitoring lifetime (MoL), i.e., the duration o f time that this monitoring service is provided. Both figures o f merit are functions of the sensor density and placement, sensor sampling rates (and hence average power consumption), network connectivity, and communication cost. Sensor deployment is a key network design step, which greatly influences both o f these metrics. In particular, the optimal placem ent o f the sensors from MoL viewpoint is not the same as the one that maximizes the QoM point o f view. It follows that, with a preset total energy allocation to all sensor nodes in the network and a fixed coverage region for a WSN, one can trade QoM for longer M oL or can achieve higher QoM with shorter MoL. It is thus interesting to study the problem o f maximizing the M oL subject to a constraint on QoM. Several architectures have been proposed for designing a sensor network. A simple architecture is “flat” architecture with homogenous sensor nodes where data from sensors are pushed toward a base station with multi-hop communication. In this structure, the communication burden is entirely with the sensor nodes. Sensors have limited communication and computation power. To reduce the communication burden o f sensors, two-tier (or multi-tier) hierarchical heterogeneous network architecture will be employed. The first level o f this hieratical network is sensor nodes and the second level (or higher) is a network o f m icro-server1 nodes, which are more power full than sensors in terms of communication and computation capability. M icro-server nodes are added as data aggregator and local manager nodes to improve network performance. They build an overlay network on top o f sensor nodes. Each m icro-server forms a cluster o f sensors such that each sensor node in that cluster sends its data to the m icro-server directly or by multi hop routing through the other sensors inside the cluster. The m icro-server then forwards ' In this thesis the term micro-server is used for the nodes at the second level o f a two-tier WSN. 3 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. received data to the base-station by multi-hop routing through other micro-servers in the network. There are operational advantages to a hierarchical heterogeneous layering that cannot be achieved with a “flat”, homogeneous network o f sensors, with its inherent limitations on power and processing capabilities. For instance, the relays help preserve limited battery resources o f sensors by eliminating the need for sensors to m onitor communications from their neighbors. In data gathering networks, the micro-server layer offers the advantage of caching and forwarding compressed data to the destination. Thus for a variety of applications, it appears that a relatively small number o f higher-tier network elements with access to more power and better computing and communication capabilities could greatly improve the performance o f the overall system in terms o f throughput, reliability, longevity, and flexibility. In flat sensor networks, data can only be forwarded by the sensor nodes in a multi-hop manner toward the base station. W hile' in a hierarchical network, data may be forwarded in a single-hop or multi-hop manner toward the micro-servers (cluster heads) and then aggregated and forwarded in a single-hop or multi-hop manner to the base station. It is o f great interest to understand what performance gains can be achieved by the hierarchical networks. For a hierarchical hybrid network which consist o f N mobile nodes (size o f the first level o f the network) and m base-stations (size o f the second level o f the network), it has been shown that if m grows faster than J n , the throughput capacity increases linearly with the number o f base-stations, providing an effective improvement over a pure Ad Hoc network [24], Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1.1 Related Work Many o f the requisite features o f a sensor network such as connectivity and coverage are significantly affected by the deployment strategy. Hence, sensor deployment strategies play a significant role in determining the performance o f static sensor networks. To this end, there have been a number o f studies that address this problem as explained in the following discussion. The work in [40] studies the infrastructure tradeoff for wireless sensor networks considering both phenomenon-driven and continuous delivery models. The study focuses on meeting the application accuracy and delay requirements via sensor deployment. A number o f deployment strategies are studied, including random deployment, grid-based deployment, and intelligent placem ent (nodes placed closest to the locations where the occurrence o f a sensed phenomenon is most likely). The work relies on a linear relationship between the monitoring accuracy and the num ber o f deployed nodes. However, this does not appear to be the best possible way o f studying deployment. In particular, this reference does not consider different node distributions. N or does it consider the effect o f distributed properties like aggregation on the energy consumption. In [1] parameters such as accuracy, latency and network lifetime are treated as runtime optimization constraints with respect to node density. These parameters change since some nodes fail or die out over time, and consequently, the node density decreases. The lifetime o f the network has been defined as a network utility function, which is computed as the product o f the netw ork’s connectedness with monitoring accuracy. The value o f this utility function must be kept above a certain threshold for the network to remain alive. The aim is to come with parameters for runtime network optimization that can be tuned by a cluster management scheme, as previously suggested by the authors. 5 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. There have been a couple o f research results on deriving an upper bound on the service lifetime o f sensor networks. The service lifetime o f a sensor network has been derived in [8][3] by multi commodity flow non-linear programming formulation. It can be shown that the problem w e will state later chapters 2 and 3 which is finding minimum total energy allocated to all nodes subject to the constraint on network lifetime is dual o f finding the maximum network lifetime subject to constraint on the initial energy budget. Traffic or load balancing can optimize the utilization o f sensor network resources in two ways. First, balancing the energy consumption rate o f all nodes in the network (load balancing) enhances the network lifetime which second, since sensor networks are space and bandwidth limited, congestion may often happen in typical application o f sensor networks [31] [42], The work in [32] proposes a stochastic routing strategy for routing packets along an NxN regular grid topology. A single source is located in the bottom left of the grid topology and the base station is on the top right. Any node (except nodes that are at the boundaries o f the grid topology) along a path from the source to the base makes a random local decision to forward the packet up or to the right. Each link is assigned a probability o f forwarding such that most o f the nodes eventually have almost same amount of traffic to forward. In our proposed research, for the scenario in which all (active) nodes in grid cells are sources o f traffic generation, we will show how to assign a fraction of incoming or locally generated flow for a grid cell to the up or the right link so that the traffic load is evenly distributed across all grid cells. For the scenario in which traffic is non-deterministic in terms o f the source-sink pair or in terms o f the rate-durations (i.e., on-demand request-reply),on-demand routing schemes are proposed in [25] [26] [27] with the goal o f extending network lifetime by finding routing solutions that tend to minimize the variance o f the remaining energies o f the nodes in the 6 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. network or in other words balancing the energy depletion rates o f all nodes in the network. On demand request-reply scenario often occur in M obile Adhoc Networks (MANETs). They may also happen in query-driven sensor networks when every node can send a query and trigger some other nodes or sinks for that query to send back responses to that query [22], We believe that on-demand routing solutions proposed for M ANETs can be even quite effective for sensor networks. The abovementioned works are the most relevant works to our proposed research. The sensor placem ent (or deployment) problem for minimum total energy transmission (or maximum network lifetime) subject to coverage constrained has also been addressed in the literature. The work presented in [16] has tackled the optimization problem o f sensor placement with constraints on distortion o f the reconstructed sensed signal at the base and with the objective o f minimizing the total transmission power for a given number o f sensor nodes. This reference provides solutions for sensor placement on a line and a special case of sensor placement in a plane assuming that number o f sensors is given and the sensors can be precisely placed at their desirable locations. Our work in chapter 4 extends this work to the case o f sensor deployment in a plane where, because o f lack o f precise control in the deployment phase, the sensor locations are random although the sensor density can be controlled. In [23] a similar problem, but with the objective o f maximizing sensor network lifetime, has been addressed. References [16] and [23] provide solutions for sensors placem ent on a line and a special case o f placement in a plane. In [29], the authors calculate the optimum average energy cost for transm itting data to a base-station, where two types o f nodes are randomly deployed in a large area. This work only considers the case o f single-hop communication between nodes which greatly simplifies the calculations, but does not result in the optimum level o f energy consumption for the communication. In 7 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [37], the authors consider the problem o f purely random deploym ent o f sensor and aggregator nodes and find node densities as well as initial energy level o f aggregators and sensors such that the total amount o f allocated energy is m inimized while achieving a certain network lifetime. This work considers multi-hop routing (with a fixed radio range) among sensors and single-hop routing for aggregators. M oreover, it takes two fixed initial energy levels for sensors and aggregators. This work fails to achieve the optimum total energy for several reasons because (a) it assumes a fixed radio range for the sensors and (b) it supposes direct transm ission for the aggregators. Moreover, as we will show later, spatial distribution o f initial energy level is a super-linear function o f distance from the sink node and assigning fixed initial energy levels is not the optimum way o f allocating energy to the nodes. The Low Energy Adaptive Clustering Hierarchy (LEACH) protocol [18] is a protocol for forming clusters in a self-organized homogeneous sensor network when the base station is located far from the sensors. In LEACH some nodes are elected as cluster-heads while the other nodes communicate with the base-station through the nearest cluster-head. This protocol randomly rotates the job o f cluster-head based on the node’s remaining energy in order to uniformly distribute the energy consumption throughout the network. Geographic Adaptive Fidelity (GAF) [43], which uses intelligent node scheduling techniques to conserve the energy, is an example o f energy conservation techniques that rely on node clustering and network routing. GAF divides the coverage area where the nodes are distributed to small virtual grid cells such that at any instant, only one node in each cell is active while all other nodes in that cell are in their power saving mode (sleep or discovery). After a node remains active for a period o f time, it changes its state to power saving mode in order to give a chance to other nodes within the same cell to become active. 8 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. We will show in our deployment scheme how to apply a proper duty cycle for turning on/off the nodes distributed in a ring such that all the nodes in the ring expire at the same time. The abovementioned works are the most relevant works to our work in the thesis. There is a large body o f other works that address the connectivity [33] and coverage problems [33][4][17] in distributed sensing for generic applications (e.g., target tracking). Most o f these works use uniform deployment strategy or a deploym ent scheme that minimizes the probability o f detection along the exposure paths in the case o f target tracking. 1.2 Network Setup We assume that boundaries o f a sensor deployment region (or m onitoring region) 5^ and location o f a sink node are known. The sink node gathers information from the monitoring region and has direct connection to a base station. We wish to calculate the value o f a particular parameter, Z, related to some phenomenon o f interest in region <2(,by monitoring the region with a number o f sensor nodes appropriately deployed in that region. Each sensor node frequently senses the environment and measures a value o f Z at its present location or in its local environment (the size o f this local environment is determined based on the sensing range o f the sensor.) Next it reports its measurem ent by sending the quantized values o f Z to the designated sink node by using a (possibly) multiple-hop routing scheme. The reporting can be periodic or aperiodic. In periodic reporting, a sensor node periodically reports its measurement to the base. However, in aperiodic monitoring, the sensor node reports its measurement when the value o f Z at its location or its sensing range exceeds (and/or drops below) a pre-set threshold, which indicates occurrence of 9 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. some event in the local environment o f the sensor. In either periodic or aperiodic reporting, we work with the time average o f the data reporting rate fj, . In other words, we assume that ju is constant for all sensor nodes inside the monitoring region. The gathered information is then sent to the base, which subsequently aggregates the received sensor data. The notion o f sensing range has different definitions depending on the task for which the sensors are deployed. In network interdiction (more generally, event detection) applications, the sensing range o f a sensor is defined as the surrounding area o f a sensor where some event o f interest can be detected by the sensor. In environmental monitoring applications such as temperature reading, each sensor measures the exact value of z at the point where the sensor is located. The value o f 2 at a point in 3^ where there are no sensors is calculated from the measurements o f sensors, which are spatially correlated to that point. 1.2.1 Energy Model For a first order radio model o f a wireless sensor node [18], the transm ission energy per bit,S’ ,*,is related to the transmission distance d as shown and calculated in Figure 1-1. Furthermore, energy dissipated in receiver part for receiving a bit is taken to be constant. The path-loss exponent is denoted by /? where ft > 2 and a T 0 and a T are constants. We emphasize that, given an Additive W hite Gaussian Noise (AW GN) channel, the above equation is only valid for a fixed modulation scheme for each link(i.e., the number o f bits per symbol and the symbol period, for example, in the M -ary Quadrature Amplitude M odulation (M QAM ) scheme are fixed) and a given probability o f bit error at the 10 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. receivers. These assumptions translate to a fixed transm it bit rate for all the sensors and micro-severs. stt — < X j - a +arci sr x — cte Figure 1-1. Energy dissipate in transmitter and receiver for sending a bit of information over distance d. 1.2.2 Key Assumptions Before For formulizing and calculating total energy consumption in the network for communication we state a number o f assumptions. 1. For sake o f simplicity in calculation, when we consider random deployment we assume that all sensors have the same height and they are deployed in a 2-D circular region where the base is located at the center o f the region. This case clearly encompasses the case where region < 2 ^ is a line. 2. The M edium Access Control (M AC) layer is considered to be collision free, i.e., there is no energy loss due to the MAC layer collision. This assumption may in fact be close to reality because typically WSNs have small synchronized traffic load. The synchronization in the traffic can be broken by introducing small jitter delay to the transmission o f each node such that inter packet delay at the output flow o f each node is randomized, which in turn reduces the probability of interference. Another reason that makes this assumption realistic is that the new physical layer technologies enable us to use orthogonal channel frequencies (i.e., OFDM) and orthogonal codes (i.e., 11 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CDM A) for channel coding. In this way interference and collisions can be almost entirely avoided at the MAC layer. 3. The sensors are time-synchronized (see [12][15] and there is no energy loss due to the lack o f synchronization between sensor nodes. In other words each sensor node is scheduled to wake up only when required (i.e. when it wants to send or receive data.)At all other times, the sensor nodes are sleep, and hence, they do not dissipate any energy. 4. Sensors can find their locations after they are deployed. They can use localization algorithms such as those in [35] to obtain their spatial coordinates with respect to the sink node. This information is then used by the sensors, which must also be knowledgeable about the deployment solution, to calculate and set their wakeup probabilities. 1.3 Model of Random Deployment Sensor nodes can be deployed in several ways in a given deploym ent region. The level o f control in deployment o f sensor nodes sets the degree o f randomness in the locations of nodes. In a fine-grained deployment, there is no randomness in placem ent o f sensor nodes and sensor nodes can be exactly placed in any location within the deployment region. In practice, however, it is usually infeasible to devise a deployment strategy whereby each sensor is placed precisely at some location. Practical deployment in large sensor networks is usually random, or at best, can be controlled with coarse granularity. As a result, adopting a random deployment model with slowly varying node densities is more realistic. 12 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Consider a homogeneous Poisson point process n in ^ representing locations of sensors where « (,c R 2 is the deployment region. The density o f the process n is denoted by X . Furthermore, consider that region 3^is partitioned into n disjoint sub-regions, A/...A„. Now for each A ,- £ , let N(A) denote the number o f sensor nodes in A, and ||4|| denote the area size o f A, .We will make use o f the following two key assumptions: • N ode distribution over region Aj is a Poisson distribution with mean 2||41| • • N(A 1 ) .. .N{A„) are independent random variables. 1.4 Contributions The contribution o f chapters 2 and 3 o f this dissertation is the investigation and development o f energy-efficient strategies for random deploym ent o f wireless sensor networks (W SN) for the purpose o f monitoring some phenomenon o f interest in a given coverage region. M ore precisely, the following optimization problem is formulized and solved. The problem is to find the sensor node density and initial energy level o f each sensor node inside a given deployment region that results in allocating the minimum total energy allocated to the nodes subject to constraints on the Quality o f M onitoring (QoM) and network lifetime. The QoM is defined in terms o f the average spatial distortion in the reconstructed signal at the base station and can be upper bounded for a random deployment of sensor nodes when sensors are points o f a Poisson process in the deployment region. In chapter 2, closed form approximations which express QoM and total energy as functions o f sensor node density are derived. Next, considering a flat network architecture comprised o f homogeneous, randomly-deployed sensor nodes, we determine the sensor node density and the energy resource density that results in allocating the minimum 13 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. number o f deployed nodes and the minimum total energy while meeting constraints on the QoM and M oL figures o f merit. The problem is formulated in a circular region where the node densities are the same within any circular disc around the base station, but otherwise vary as a function o f the radius o f that disc. It is shown that the optimal solution is a strong function o f the routing scheme used to send a packet o f data from some sensor node to the base station through multi-hop routing. Therefore, a novel routing scheme for such a network, called the Continuous Space Radial Transmission is proposed and experimentally evaluated. Simulation results show that the analytical minimum total energy value is close to the actual energy required in a randomly deployed dense network. In chapter 3, we investigate the same problem for a two-tier network comprising of randomly-deployed sensors at lower level and randomly-deployed micro-servers at higher level o f the WSN. The sensors monitor their surrounding environment, while micro-servers provide connectivity between sensors and a base station. The node density for the sensors is preset in order to achieve a minimum level o f QoM. The m icro-server density is, however, to be determined. The objective is then to determine the distribution o f energy resource (for both sensors and micro-servers) inside the deploym ent region so as to minimize the total amount o f allocated energy in the network while meeting a minimum level o f MoL. The solution to this problem is also strongly dependent on the routing scheme used to send a packet o f data from some sensor node to the m icro-server that acts as the cluster head for that sensor node, and subsequently, from the micro-server to the base station. Both cases, we consider multi-hop routing and seek to determine the expected path length from sensor node at given location to the base station placed at the center o f the deployment region. Therefore, a novel routing scheme for such a network, called Continuous Space Conic Routing, is presented and experimentally evaluated. 14 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. In chapter 4, we investigate energy-efficient and lifetime-aware strategies for detailed placement and energy allocation o f infrastructure in the second level o f a two-tier WSN. In contrast to chapter 3, the sensor nodes in this problem setup have a priori and arbitrary positions. Various forms o f the following problem are thus formulated and solved: Assign positions and initial energy levels to the micro-servers, and concurrently, assign groups of sensors to each m icro-server so as maximize the monitoring lifetime o f the two-tier WSN subject to a total energy budget. The problem is solved for both collinear deployment and planar deployment situations. Solutions are obtained by transform ing the problems to the convex optimization forms or by proposing heuristic solutions. By performing extensive simulations, the operational advantages o f optimal deployment o f a two-tier WSN in terms of network lifetime compare to flat network architecture are investigated. The results show the optimal design and deployment o f such a network can increase the WNS lifetime by one order o f magnitude compared to a flat WSN with the same total initial energy and quality o f monitoring. 15 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 2 QoM and Lifetime-constrained Random Deployment of Sensor Networks for Minimum Energy Consumption In this chapter, considering a flat network architecture com prised o f homogeneous, randomly-deployed sensor nodes, we determine the sensor node density and the energy resource density that results in allocating the minimum number o f deployed nodes and the minimum total energy while meeting constraints on the Quality o f M onitoring (QoM) and M onitoring Lifetime (M oL) figures o f merit. The problem is formulated in a circular region where the node densities are the same within any circular disc around the base station, but otherwise vary as a function o f the radius o f that disc. It is shown that the optimal solution is a strong function o f the routing scheme used to send a packet o f data from some sensor node to the base station through multi-hop routing. Therefore, a novel routing scheme for such a network, called the Continuous Space Radial Transmission is proposed and experimentally evaluated. 2.1 QoM Definition The QoM is intended to represent the “similarity” between the actual values o f some parameter o f interest in 5^and the reconstructed values for the same param eter in the base. These values can be thought o f as a random field in the space and time. Therefore, QoM should be analyzed in two respects: temporal and spatial accuracy. The temporal accuracy 16 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. is defined based on the reporting period whereas the spatial accuracy is related to accuracy of spatial estimation o f the value o f interest at locations where no exact values are available at the reconstruction time. of this analysis, we assume that the sampling frequency is equal to or greater than the Nyquist frequency, which can thus fully track and/or reconstruct changes o f any phenomenon o f interest inside region % Consequently, the base station can completely reconstruct the changes in time. In other words, we will not worry about temporal distortion. The base should also be able to spatially reconstruct the value o f interest or events in 5 ?, within the required minimum QoM. In this chapter, spatial distortion is the primary concern. By reducing this distortion, “similarity” between the reconstructed value and actual value o f the parameter o f interest increases, thereby improving the QoM metric. More precisely, we define an average spatial distortion as a m ean square error metric, MSE(<R), as follows: (i.e., x=(r,0) in polar coordinates) in region , respectively. According to this metric, first the error variance is calculated at any point x given locations X o f II particles (which are the set o f all sensors.) Next, the second expectation is taken with respect to distribution of sensor nodes in 3(,, which are points o f a Poisson process. The temporal part o f QoM is represented by frequency o f sampling. For the remainder MSE(<IQ where Z(x) and Z(x) represent the actual and reconstructed values o f interests at point x 17 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. It is assumed that the network designer or application developer imposes an upper bound constraint, D m a x , on the distortion metric as follows: M SE(<rS ) < Dm m (2 . 1) We assume that Z is a stationary spatial process where the covariance between any two points xi and X j is a function o f Euclidian distance (denoted by |.|) between the two points i.e., Note that stationary property holds for a phenomenon which is observed in an isotropic area [17]. M oreover, let C0 = £ [ z 2(x)] .The auto-correlation function, ac(.J, between any two points with a relative distance o f d is also defined as follows: Function C(d) is a monotonically decreasing function o f d. M oreover, we consider that power spectral density o f auto-correlation function (i.e., the Fourier transform o f ac(d)) is a bandwidth limited signal which is translated so that there is a maximum distance do above which the auto-correlation becomes nearly zero, i.e., ac(d) - 0 for d > d 0.We now define the neighbor set o f each point x as the set of sensors, which are within a disc centered at x and with radius do i.e.,: One can easily estim ate the value o f Z at any point in 5^by linearly combining readings from its neighbor set as follows: C ov(x,,x; ) = £ '[Z (x,)Z (xy)] = C{dtj) where dtj = |x,. - x neighix) = |x ( e FI ||x - x ,. | < d0 j Z(x)= a,Z(x,) (2.2) Xj^neigh(x) 18 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where the exact values o f Z are known all x, e neigh(x) and Z(x) represents estimation of value o f Z at point x. For each point x the coefficients a, are com puted so as to minimize variance o f error at that point. The computation method is presented in appendix A. We assume that the covariance (or auto-correlation) function o f Z is given. Furthermore, < R ^ is sufficiently large. T heorem 1: When using the linear estimation (cf. equation (2.2)) to estimates the value of parameters at every point x in region < R ,, MSE(<S) is bounded as follows: D,(X)<MSE(^)<Dh(X) f d. \ AW = c0 \ - 2 t iX ^ra2 c(r)dr 0 t/j) \-2nX jra 2 c(r)e~/l' r r dr o do 'I Dh (A) = C0' (2.3) Proof: See appendix A. Notice that in deriving the upper bound, Dh we estimate the value o f each point by the value o f its nearest sensor whereas for deriving the lower bound, D t, we assume that the readings o f sensors within the neighbor set o f any point are spatially uncorrelated. O f course, tighter bounds can be derived. E xam ple: Consider that the auto-correlation function has an exponential form: ac(d) = e'fJ . As a result, , D, (Z) = C0 (1 - - ^ - ) , and Dh (2) = 5 = r Figure 2-1 depicts functions Di (X) and Dh (2) for different values o f y. For small values of X, the difference between Di and Dh is small, which means that any o f the bounds can approximate actual value o f MSE(<R) with small error. Furthermore, it can be seen that as y increases, the M ean Square Error (M SE) (which lies between the Di and Dh) also increases. 19 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. This is expected because the auto-correlation reduces, which in turn introduces more error in linear estimation technique. MSE vs lambda 0.3 UJ C O 0.25 0.2 0.15 0.45 0.05 0.35 0.4 0.5 0.1 0 1 5 1 0.5 o 0 .2 0 .4 0 .6 0 .8 1 gamma lambda Figure 2-1. The bounds (Dh and Dh) onMSE(30 at different node density when the covariance between any two points is exponential with parameter y and CV=0.5. Theorem 1 gives closed form equations by which MSE(<K) can be approximated. Hence, the constraint on MSE(<R) (cf. equation (2.1)) can be imposed on these approximations. Clearly, the actual value o f MSE{<E) will be overestim ated (or equivalently its constraint will be tightened) if MSE{<E) is approximated by Dh . M oreover, the actual value o f MSEifi) will be underestimated (or equivalently its constraint will be relaxed) if MSE(<E) is approximated by Dh In this chapter, we shall use (Di(X)+Dh (X))/2 as an approximation to MSE(^.T herefore, we replace constraint (2.1) with the following constraint: 20 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. m m m < D (2 ,4 ) 2 max ' / C orollary 1: The bounds A and Dh are monotonically increasing functions o f A ,. P roof: The proof is simple and follows from > 0 and > 0 . ■ dX dA T he co n seq u en ce o f th e above corollary is that, th ere is a Am in such that: MSEiX) < Dm m < = > 2, > 2m in (2.5) 2.2 Network lifetime The sensor network lifetime is defined as the duration o f time between when all sensor energy sources are replenished (all sensors have full charge capacity) and when the spatial distortion in the reported value o f interest in region ^exceeds Dm ax. Notice that the spatial distortion increases with time as some sensor nodes exhaust their energy source, and consequently, exit the Ad hoc sensor network. Also notice that this distortion can rise slowly or rapidly over time, depending on the distribution o f sensors inside the region coupled with how traffic is routed inside the network. 2.3 Quality and Lifetime-constrained Sensor Deployment (QLSD) Problem Given are boundaries o f region ^ and the location o f the sink (base station), an upper bound, Dm a x , on average spatial distortion as a metric that represents QoM, and a lower bound, Tm in , on the Network Lifetime. Assuming random deployment o f sensors with the model presented in chapter 1, the objective is to find node density or energy density at each point in so as to minimize the number o f deployed nodes or the total energy allocated to the nodes in the network while satisfying the QoM and network lifetime constraints. 21 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.3.1 Modeling the Objective Function Let dx denote an infinitesimal sub-region o f ^aro u n d point x. Recall that p is the average reporting rate (in bits/sec.) Assume that each sensing event results in generation o f B bits of data to be transmitted to the sink node. We can thus define energy dissipation per bit for the sensing operation itself (denoted by a s), energy needed to deliver one bit o f data from a sensor to the base (denoted by w). Note that w accounts for all o f the energy consumptions related to receiving and/or sending data on its path from a source node located at point x, (i.e., Xi=(rh0,) in polar coordinates) to the base. Therefore, w is a function o f locations of source node, xh and all other intermediate nodes along the path from x, to the sink (denoted by Xj.i, Xj.2, ...) e.g., w0 = w0(xj,x j_l ,x j_2,...) .We also define W as follows: where E{ ] denotes the expected value. L em m a 1: The mean value o f the total energy consumption in sensors deployed in region 3(,during time interval T is expressed as: Proof: By partitioning area < 2 ^ to infinitesimal disjoint parts Ax, , it is easily shown that the expected value o f the total energy consumption in sensors deployed in region (^during time interval T' £ [V o '(s0 ] > can be expressed as: where /j, is the average reporting rate o f each sensor and NfAxj denotes the number o f nodes in infinitesimal area Ax,. Now because o f the properties o f a Poisson point process: W (x ,) = E [w 0 (x ,, x,_,, x,_2,...) | x, ] (2 .6) (2.7) £[e'°'(3O ] = mT % (as +W(x,))Pr{N(AXl)> 0} 22 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. N(Ax,) = > 2 with prob. X(Ax,)2 « 0 1 with prob. XAx. ,therefore in the limiting case that /Ax, — >0 , we get 0 with prob. \-XAx, equation (2.7) and this complete the proof. Now we can write QLSD problem as a mathematical program: Mm^X/xT J ( a s +W(x))dx s.t. T > T m „ D,{X) + Dh{X) SB™ w here: (2 .8) D,(X) = C0 DhW = C0 \ - 2 k X I r a 2c{r)d r ° / ' 4 > 1 - 2 kX j r a 2c(r)e~'i’rr2dr The objective function must be minimized with respect to three variables: X , T and W(x). The objective function with respect to T is a linearly increasing function. So at the optimum solution, T=Tm in , Moreover, it has been shown (cf. equation (2.5)) that there is a X m i„ value that satisfies the spatial distortion constraint. Consequently, hereafter we shall use the minimum values o f X and T (i.e., Xm im Tm in ) and will focus on minimizing the objective with respect to W(x), which is a function o f the data routing algorithm in the sensor network. 2.4 Optimal Routing Algorithms in the Continuous Space Model of Random Deployment In this section we introduce a continuous-space model o f random deployment o f sensor networks and propose algorithms for data routing under this model such that minimization 23 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. of equation (2.8) is achieved. We define an energy density metric, ed , which captures the spatial distribution o f energy resources inside area % . As stated previously, we consider the case that < K . c R 2 is circular and the base is located at the center o f region. The area is radially symmetric, which in turn enables one to derive a relatively simple form for the objective function in equation (2.8). An example application o f sensor deployment in a circular region is the task o f guarding a restricted area around a base against any intruders/attackers trying to break into the base by deploying and m onitoring the restricted area with sensors. 2.4.1 Continuous Space Model and Routing Algorithm Equation (2.7) shows that the formulation that we described earlier in order to calculate the total energy required by the sensors relies on continuous variables, and hence, it is a continuous space model. Here, an infinitesimal area, dx, can be treated as source o f traffic. The local data flow for the sensors in dx is X\d\p ■ Furthermore, each source node must find a path to the base so as to minimize the total energy consumption for sending its data to the base. As described in cahpterl, the transmission energy per bit, elx,. is related to the transmission over distance d is as follows: £a = ^ro +ai ^ p 9 ) where /? > 2 recalls the path-loss exponent and a T o and ctr are constants. Moreover, the energy for receiving a bit is constant and denoted by aR . Consider a source node which is located at distance r from the sink and its traffic is relayed by m-1 intermediate sensors before reaching the sink. The work in [18] has proved 24 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. that in this case when each hop has the same length (set to r/m), the total transmission energy is minimized. Therefore, the minimum energy for sending and relaying a bit to the sink is calculated as W \ r ) = m ( a T0 + a T(— )p ) + (m - 1 ) a R . (2.10) m By setting the derivative o f W*(r) with respect to m to 0, we obtain m* that minimizes W*(r) as follows: m* = r / ^ ( / ? - l ) a r /(a R + a T0) (2 . 11) Note that W* has only one extremum with respect to m when m>0. Now, the optimal number o f relaying hops should be an integer, i.e., either ]„t[m*] = [ m *J 0r [m *J + l and the distance o f each relaying hop ds is constant and satisfies ds=r/Int(m*). Figure 2-2. Deployment region is divided to annuluses wherein nodes within an annulus have same hop-count to the sink. Figure 2-3. Optimal radial path for a node located in S4 . 25 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The optimal hop length o f each sensor can be calculated iteratively starting from the center o f the circular region along the radial line segment on which the sensor is placed. In other words, because the area is radially symmetric in the polar coordinates, the 2-D sensor deployment problem in a circular region is transformed to a series o f 1-D sensor deployments on different radial line segments (starting from the sink node which is at the center o f the region and extending to the boundary o f the circular region). Details are included in the Continuous Space Radial Transmission (CSRT) model provided below. In this algorithm, R denotes the length o f the 1-D deployment region. N ow consider a point x which is at distance r < R from the sink. We define r,„ as the r value which results in equalization o f the minimum energy dissipation for forwarding a packet from x to the sink by using exactly m hops and that for forwarding the packet by using exactly m+ 1 hops. Notice that for all points x that lie in the range (rm ./,rm \, the m inimum-energy hop length is r/m. From this definition and equation (2.10), we have: I'™ W '(rm +Sr) = W ‘(rm -Sr) => m(am +a/(-f)+ (m -\)a R ={m+\)(am +ar{-^! L -f)+m aR A m O m m + \ By solving the above equation rm is obtained as follows: r = jccR + o c T 0) / o c T vw M ~(m + 1)H j (2 . 12) Equation (2.12) shows how in CSRT the various rm values are calculated for m= 1, ..., M where M is called network diameter and is a function o f radius o f the deployment area. M is determined so as to satisfy the following inequalities: {aR + a T 0)/a T < R < (iaR + a T0)/ aT , VM ^ - ( M + (2.13) X M - Y j - 1 3 - M x ~ p Let the successor o f a node located at x=r be the node that is closer to the sink and who relays the data sent out from this node toward the sink. In the CSRT model, a successor 26 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. xp=(r-r/m, 0) is assigned to each point x=(r,6) along the radius toward the sink. The CSRT model (when applied to all radial lines with angle 6) divides the 2-D deploym ent region, % to concentric annuluses Sm as shown in Figure B -l where each annulus, Sm , specifies a subregion com prising o f points (r, 0) as follows: Sm = {(r,6 ) e : rm _] < r < rm and 0 < 0 < 2 k } for 1 <m<M. (2.14) Therefore, for each node x = {r,0)eSm ,its successor is x =(r- L !g)e s ■ Figure 2-3 is an p m’ ”'1 example showing how a node which is located inside S4 sends its packets toward the sink through the optimal path. All hops have equal length except for the last hop which connects to the sink. 2.4.2 Energy Density and Total Energy Now we set out to determine the optimum energy density in 3^ and the total energy allocated to the sensor nodes in the network under the condition that radial routing presented in CSRT model is usedfor data forwarding toward the base station. T heorem 2: By m odeling data routing in the sensor network deployed in ^ with CSRT, energy density, ed(r), at a point (r,0) e % can be calculated as: M + m - l { < ■ a -K) M J M - m + 1-1 M / 2m ri+as - a R m AC,, i] = a T\ — \ + a Ta+ a R V (r,0)eSm m d l < m < M (2.15) and 1 1 if r > — R M 0 otherwise 27 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Proof: To find the flow distribution, and consequently, the energy density at distance r from the sink, the deployment region is divided into infinitesimally narrow annuluses around the sink with width Ar (see Figure 2-4.) We denote such an annulus as p(r) and specify it as follows: As stated earlier for the continuous space model, each infinitesimal area may be treated as a source o f traffic with the local flow being proportional to the probability that one node is found in that infinitesimal area. Notice that from the Poisson process property, if a node is in p(r), then it will have equal probability o f residing in any location inside that p(r). Therefore, the local flow o f each is uniformly distributed in the area o f that ring. Therefore, the energy density inside the ring is uniform. Let AI{r) and AF{r) denote total input/output flow to/from p(r) and let AL(r) denote locally generated flow in p(r) which is expressed as follows: By w riting flow conservation for any p(r) where 0 < r < R, we get AF(r)=AL(r)+AI(r) and A/(r) = 0 ,fo r(r,0 )e SMN otice that for 1 < m < M-2 , if (r , 0 ) e s m then (H±Lr,e) e sm tl -Thus, we can write: p(r)={(r',9): r < r ' < r + Ar and 0 < 6 < 2jt\ (2.16) (2.17) m AF(r) = y'A L (r+—) for (r,9 )e S m and 1 < m < M to m (2.18) where K = M - m - 1 and 1 0 otherwise 28 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The lifetime o f a node that resides in p(r), denoted by x(r), can be calculated as follows: (ia T0+ a T(r/m)p )AF (r) + a R (A F(r) - AL(rJ) + a sAL(r) To fully utilize total energy allocated to all nodes in the deploym ent region and avoid energy waste, we want to ensure that all sensor nodes have the same lifetime, which is equal to the network lifetime, Tm in . Consequently, from the above equation, energy density, ed(r), is calculated as follows: ^ w = ( K ^ « rWm)-’)AFWt a ,(a F(r) - M W)t% A t (r))7 - . , By substituti„g A?(r) and AL&) 2nrt\r given by equations (2.17) and (2.18) in the above equation, we derive equation (2.15) and this complete the proof. r/m Figure 2-4. A I(r) and AF(r) are total input and output flows of an inflnitesimally narrow ring with width Ar (denoted by p(r)) and r/m is optimal hop-Iength for a sensor within the annulus Sm . As explained earlier sensor deployment problem and the energy allocation problem are related and interacting problems. Indeed there are two ways to provide energy density for a 29 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. p(r). One way is to allocate ||p(0|| ed (r) amount o f energy to a node inside p(r) .However, if the battery capacity o f each node is fixed to Eo another way to allocate the required energy is to adjust node density inside p(r) according to the following equation: (r \ M+m - 1 m V M ' J V M 7m 1+Os~aR foe\<m<M and (r,ff)eSm (2.19) 3 This adjustm ent increases the node density above than the density ^,tH n that requires for satisfaction o f QoM constraint. Therefore a policy must be em ployed to control the node density such that at any reporting cycle, density o f active nodes which are participating in sensing the environm ent and data routing reduces to Xm in and the rest o f the nodes are in the sleep mode (power saving).Therefore, not every node is awakened in each reporting cycle. More precisely, we assume that each sensor node is given a probability o f wakeup and wakes up in a given cycle with that specified probability. That probability for a sensor which is located at distance r from the sink is given as XmJX{r) where Xm in /X(r) <1. To run this wake up scheduling algorithm, each node must know its relative distance from the base. This information can be obtained by a simple flooding procedure as follows. The procedure starts by broadcasting a packet (called localization packet) by the base. Each node within the radio coverage o f the base which receives that packet can measure its distance (with some accuracy) from the base by m easuring the average radio signal strength during reception o f the packet. At the next step those nodes which received localization packet directly from the base append their distance to the received packets and rebroadcast them. Therefore, similarly nodes located two-hop away from the base measure their distances from the base. The flooding procedure continues until all sensors within 5^ receive at least one copy o f the localization packet originated by the base. 30 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The underlying assumption in CSRT and theorem 2 is that data traffic for a sensor at any point in the region is sent through an optimal radial path to the sink which consumes the least energy (i.e. Jf*(r)) according to equation (2 .10); therefore, the actual energy cost, W(r), is always greater than or equal to W (r) .Consequently, CSRT derives minimum the total energy needs to be allocated to %. Corollary 2: The minimum o f the expected total energy required in the sensor network is deployed in 2^ is: E [e,J> Tm in — A J 27tred{r)dr \ r = 0* iV l = ^ m i n Z J 2vred(r)dr m=l \ ( r * ) e S „ Assuming that rM =R, a simple formulation for deriving the expected total energy is expressed as follows: M - > (a s . - a R) R2 + (M + m){M - m + 1) m =1 p + 2 m p+ i + (anl +dx) (rm ~K m -l ^ \ 2 m J J (2 .20) It should be mentioned that equations similar to equations (2.15) and (2.20) can be derived for a collinear network (one in which sensors are placed on a line) where nodes are deployed along a line and the sink is located at origin. The difference in calculation is that in collinear network, infinitesimal area size is ||p(r) |= ^ therefore AZ(r) equal to Xm m dr instead o f 2% xXm in dr as in the circular network (cf. equation (2.17).) 2.5 Simulation Results We have performed several simulations to evaluate CSRT model and to find energy density in a given deployment region. We compare the optimum solution obtained by equation (2.20) with that obtained when employing a Minimum Total Energy (MTE) 31 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. routing policy [36], Given the node locations, the network is m odeled as a network graph where energy cost o f each edge is calculated from equation (2.9). By applying the MTE routing, an optimal path which has the minimum total energy cost is selected for every pair of source and destination nodes. Therefore, the M TE produces the minimum total energy for a network graph with given locations o f sensors and micro-servers. W hen employing the M TE policy for routing, we measure the total energy consumed in the network separately for each run, which takes Tm j„ seconds and this is repeated for many simulation runs. In each run, the initial placement o f sensors is given. In the end, the average over all runs o f the total measured energy consumed in the network is reported. Note that in each run o f the simulation for generating the initial placement o f nodes (which has Poisson point process distribution), the placement method proposed in [30] is utilized. The system and network parameters selected for the simulations are: T/Eo-1, fi-\ bit/sec, as=0, R= 1000 m (the radius o f deployment region), and the set o f parameters for energy model as o f equation (2.9) is taken from [18].Those energy model parameters are measured for a sensor node with a radio communication subsystem comprising of transm itter/receiver electronics, amplifiers, and antenna. The parameters for ffee-space communication are: /?=2, aR = a T o~50 nJ/bit,a-i=\0pJ/bit/m2 , and those for two-ray communication model are: /?=4, aR = a T o = 50 nJ/bit, a 7 =0.00013 pJIbit/m2. The energy density as a function o f distance from the sink is illustrated in the Figure 2-5. The simulation has been performed for A.=0.01 nodes/m 2 /?=4. The figure shows that as we get close to the sink, inside each annulus, the energy density has several peaks with increasing height. Also sharp density transitions occur at the boundary between annuluses. 32 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. x iq ’7 S patial distribution of optim al allocation of energy £ | Z 3 O — D S10 S11 S12 CO I 0.6 CD £ 0 .4 0.2 100 2 0 0 300 400 500 600 700 800 900 1000 distnac from the base(m ) Figure 2-5. Energy density vs. distance from the sink. In the next simulation, we compared the total energy allocation obtained from equation (2.20) (i.e. by CSRT) and the actual value o f the total energy for different node densities. Figure 2-6 depicts the total energy versus density o f sensors. The results demonstrate that CSRT can estimate the total energy with relatively small error and that the total energy as a function o f sensor density is a semi-linear function. This simulation was performed for set of parameters for energy model corresponds to P=2,4. The results o f total energy obtained by CSRT and actual total energy obtained by M TE for the collinear case are depicted in Figure 2-7.The results show that in the collinear case, CSTR can estimate total energy with very small error and the error even gets smaller as node density increases. 33 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Total energy vs. node density 45 CSRT(beta=4) • MTE(beta=4) CSRT(beta=2) x MTE(beta=2) 40 35 30 25 20 15 10 5 0 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 lambda(node/m*'m) Figure 2-6. Actual value of total energy obtained by MTE and total energy obtained by CSRT versus node density for a circular region with R=1000m. Total energy v.s. node density 10 • MTE(beta=2) - ■ CSTR(beta=2) - MTE(beta=4) < CSTR(beta=4) 9 8 7 6 5 4 3 1 — 0.04 0.05 0.06 0.07 0.08 0.09 0.1 lambda(node/m) Figure 2-7. Actual value of total energy obtained by MTE and total energy obtained by CSRT versus node density for a collinear network with size R=10000m. 34 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. As aforem entioned network diameter, M, is obtained from the inequalities expressed in (2.13).Those inequalities can be solved numerically. Figure 2-8 illustrates the results for two sets o f parameters o f energy model and for different area diameters (i.e. 2R) The results show that network diameter has a semi-linear functional relation with area diameter and the slope increases as P increases. Network diameter vs. Area diameter 30 25 beta=4 20 15 10 5*^ 1000 4000 1500 2000 2500 3000 3500 Area diameter(2*R) Figure 2-8. Network diameter versus diameter of deployment area. Figure 2-9 depicts the effects o f the area size (radius o f area 5 0 on functional relation of total energy and node density. The figure shows that by fixing node density at any value, total energy has a power law relation with the area size which means that energy grows super-linearly when area size increases. 35 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. la m b d a R Figure 2-9. Total energy versus radius of the deployment region and node density. 2.6 Chapter Summary In this chapter we showed how to calculate the sensor node density, or alternatively the energy resource density, at every point inside a given deployment region that results in allocating the minimum number o f deployed sensors, or alternatively the total energy resource allocated to the sensors while meeting constraints on the QoM and network lifetime o f a flat WSN. First, we defined QoM and then found closed form expressions that express QoM and total energy as functions o f the sensor node density. These closed form equations in turn enabled us to formulate and solve the problem o f interest as a mathematical optimization problem. 36 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 3 Energy-Efficient Random Deployment of a Two-Tier Sensor Network in a Field In this chapter, we investigate the problem o f energy-efficient deploym ent o f a two-tier network com prising o f sensors at lower level and micro-servers at higher level o f the WSN. The sensors m onitor their surrounding environment, while micro-servers provide connectivity between sensors and a base station. In one version o f this problem, node densities for the sensors are preset in order to achieve a minimum level o f QoM. The micro-server densities are however to be determined. The objective is to determine the distribution o f energy resource (for both sensors and micro-servers) inside the deployment region so as to minimize the total amount o f allocated energy in the network while meeting a minimum level o f MoL. The solution to this problem is strongly dependent on the routing scheme used to send a packet o f data from some sensor node to the m icro-server that acts as the cluster head for that sensor node, and subsequently, from the micro-server to the base station. Both cases, we consider multi-hop routing and seek to determine the expected path length from sensor node at given location to the base station placed at the center o f the deployment region. Therefore, a novel routing scheme for such a network, called Continuous Space Conic Routing, is presented and experimentally evaluated. 37 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.1 Problem Description Consider that the boundaries o f a sensor deployment region, (or coverage region) 3^, and the location o f the base-station, b, are known. Furthermore, consider that reporting rate of the sensors, p., is constant for all sensor nodes. The sensors are deployed such that a minimum required QoM inside 3^ is achieved. Moreover, assume that the deployment of sensors is uniform over with density X q. As stated previously, sensors have limited communication and computation power. In order to reduce the communication burden o f sensors, two-tier hierarchical heterogeneous network architecture will be employed. The first level o f this hieratical network is sensor nodes and the second level is a network o f m icro-server nodes, which are more power full than sensors in term s o f communication and computation capability. In other words, micro servers build an overlay network on top o f sensor nodes. BS Micro server Sensor o Figure 3-1. Voronoi tessellation of deployment region with a two-tier WSN consists of sensors, micro-servers and a base station at levels 1,2 and the origin, respectively. 38 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Each m icro-server forms a cluster of sensors such that each sensor node in that cluster sends its data to the m icro-server directly or by multi-hop routing through the other sensors inside the cluster. The micro-server then forward received data to the base-station by multi hop routing through the other micro-servers inside the network (see Figure 3-1.) Assuming random deploym ent o f sensors and micro-servers wherein distribution o f nodes can be modeled as a Poisson process in a 2-dimensional space [17], the goal is to find the energy density (or the node density) in 3^ so as to minimize total energy allocated to the nodes in the network (or the number o f deployed nodes i.e., sensors and micro-servers) while satisfying the QoM and network lifetime constraints. The constraint on QoM is simply imposed on sensor density which is set to be X q at its minimum value. Moreover, the network lifetime is set to T in the problem formulation. 3.2 Modeling the Objective Function Consider two independent homogeneous Poisson point processes n0 and ni in % representing locations o f sensors and micro-servers, respectively. The base station is considered as a m em ber o f the micro-servers group. W hen we speak o f a random process, its support points will be called particles. The densities o f the processes no and nj are denoted by k0 and 'k\, respectively (in our application, X\ « V ) Each n 0 particle x, is assigned to a unique particle yr In our model particle yj is chosen to be the closest to xt among a set o f particles comprising o f all the ni particles. Ties are broken arbitrarily but deterministically. In this way, ni particles partition 3fynto cells Q such that particle y} is the center o f Q- 39 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Let dx denote an infinitesimal subregion o f 3^ around point x. Recall that p is the average reporting rate (in bits/sec.) Assume that each sensing event results in generation of B bits o f data to be transmitted to the sink node. We can thus define energy dissipation per bit for the sensing operation itself (denoted by a s), energy needed to deliver one bit o f data from a sensor to a m icro-server (denoted by Wo), and energy to deliver one bit o f data from a m icro-server to the base (denoted by wj.) Note that wn and wj account for all of the energy consum ptions related to receiving and/or sending data on its path from a source node located at point x, (i.e., x,=(r„6() in polar coordinates) to a sink node (which may be a micro-server or the base.) Therefore, W o is a function o f locations o f source node, x„ and all other intermediate nodes along the path from x, to the sink (denoted by x,_/, x,.2, ...) e.g., w0 = w0(xi,xi^],xl_2,...) .We also define Wo as follows: where £[] denotes expected value. It is easily shown that the total energy consumption in sensors deployed in any cell C o during time interval T, e0'°'(C 0) , may be expressed as: H is the average reporting rate o f each sensor. Before calculating the expected value of C ' ( c 0) (denoted by £[e0'°'(C 0) ] , we first state an important result. W0 (x,) = E[w0 (x,, x,_,, x,_2 where 1 (3.1) 40 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. T heorem 3: Consider particles n 0 and ni have 2-D Poisson distributions as described above and letX *) be a non-negative function where / : M2 -> M . If Sf = ^ /(x,. )1{ J C eCo), then M oreover, the mean value o f the total energy needed by all sensors inside the deployment region, <£, is: where ||.|| denotes the size. The second term in R.H.S. o f equation (3.3) is added to take into account the total energy dissipation o f those sensors that have been directly assigned to the base, that is, these sensors are closer to the base than they are to any other micro server. Similarly, consider a m icro-server as a source located at yj and intermediate nodes along the path to the base located at y,./, y^ , ..., then w, = wi(y j,y l_ _ l,y l_2,.,.). If we define ^ ( y ;) = £[wl(y/,y_,,>’ | y.) , then it can be shown that the total energy needed by all micro-servers and its mean value are: E[Sj ] = A0 J /(x )e dx where |x| denotes distance o f x from origin. P roof: See [13], From theorem 3, we get: E[e0 'o,(C0)] = \ju T l(a s +W0( x ) ) e - ^ d x (3.2) E[e0,o,]= ^ I I * I I E[e0lol( C 0)] + E[e0lol( C 0)] (3.3) (3.4.a) yjS. n 41 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. E[e;o,] = ^ T / £[W (C.)] \ { a x + W,(y))-Ag(y)dy y e < R , (3.4.b) where A^yJ) is a function which represents the aggregation operation in a micro-server located at point y} , E[N(C*)] denotes the expected number o f sensors located in any cell (i.e., averaged over all cells, C,.) We shall later introduce different forms o f f u n c tio n ^ .). By setting/(x,)= 1 for all x, in theorem 3, it can easily be shown that: £ [ 7 V(C,)] = 4 l (35) 'h Therefore, equation (3.4.b) can be rewritten as follows: £[e,to '] = \ M T J {as +Wx(y))-Ag(y)dy (3 6) We will later propose a model to numerically compute equations (3.3) and (3.6) to derive the total allocated energy. N ow we can express the objective function minimization as follows: Min E [e'°']=E [er} + E[e;°'} (3.7) Mr)’" ! v ' The objective function is clearly dependent on W0(x) and Wj(x) (cf. equations (3.3) and (3.6).) Therefore, this function must be minimized with respect to variables W0(x) and W/(x), which are in turn dependent on the data routing algorithm in the sensor network. 3.3 Continuous Space Model and Analysis of the Routing Scheme In this section we introduce a continuous-space model o f random deployment o f sensor networks and propose algorithms for data routing under thus model such that minimization 42 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (3.7) is achieved. We define an energy density metric, ed, which captures the spatial distribution o f energy resources inside area % , As stated previously, we consider the case where ^ c l 2 is circular and the base is located at the center o f region (see Figure 3-1.) The area is radially symmetric, which in turn enables us to derive a relatively simple form for the objective function in equation (3.7). An example application o f sensor deployment in a circular region is the task o f guarding a restricted area around a base against any intruders/attackers trying to break into the base by deploying and m onitoring the restricted area with sensors. 3.3.1 Continuous Space Model and a Routing Algorithm Equations (3.2) thru (3.7) show the formulation that we described earlier in order to calculate the total energy required by the sensors and micro-servers relies on continuous variables, and hence, it is a continuous space model. Here, an infinitesimal area, dx, can be JlX, I I2 treated as source o f traffic. The local data flow for the sensors in dx is dx (cf. equation (3.2)) and for the micro-servers in dx is p \ A g{x)dx (cf. equation (3.6).) Furthermore, each source node must find a path to its closest sink node (micro-server or the base), in order to minimize the total energy consumption for sending its data to the sink node. Reference [28] presents a radial least-cost path routing in continuous space, which is called Continuous Space Radial Transmission (CSRT.) In CSRT, data packets o f each node are forwarded along the radial path to the sink using the optimum number o f hops. The routing strategy is based on a strong optimistic assumption which states that a node can always find its successor in any infinitesimal area that it desires. Therefore, the total energy obtained by CSRT is simply a lower bound on the actual energy needed. Thus 43 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. bound is close to actual total energy only in densely-deployed sensor networks, where there are so many sensors whereby, with probability approaching 1, another sensor exists at any desired distance from a given sensor. The model that we propose in this chapter is a continuous space model which is based on conic routing for data forwarding for both sensors and micro-servers. The conic routing has been proposed in [20], We analyze conic routing in the continuous space wherein the exact location o f nodes, and consequently, the network topology are unknown although the distribution o f nodes inside the deployment region is given. The model aims to find the total energy which must be allocated to sensors and micro-servers in a two-tier network. We will refer to the routing approach as the Continuous Space Conic Routing (CSCR.) CSCR is constructed such that each node selects as its successor a unique node and sends its data to that node. The node selects its successor from the set o f nodes that are inside a cone with solid angle co formed at the node’s fan-out (see Figure 3-2.) The selection o f a node’s successor is quite important and optimization problem (3.7) strongly depends on that selection. In CSCR, we consider a routing policy that selects the n,h closest nodes inside its solid cone. 44 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. X l L Figure 3-2. A cone1 with angle 2co (where 0 < to < n/2 ) at a fan-out of a sensor node and its nearest sensors inside the cone where it selects its successor among those nodes. Lemma 2: Assume that the distribution o f deployed nodes follows a Poisson point process with density X . Let d denote the relative distance o f a node from its n,h closest node inside its solid cone. The p d f o f d ,fd„(d),,and its moments are: f id) = (cd) ~ ~ ~ exp(-c%/2) (3'8) (« — 1 )! E[d] = c~m r ^ + 0~ 5) (3.9) n « ) E[dp] = c'm r ( ^ t Q- . 52 T{n) where T(.) is the gamma function, (3-10) c=Aa» ,and (3 > 2 Proof: See [20], Let h(r) denote the number o f hops which are traversed by data packets from intermediate nodes located at points X h(r ),Xh(r j.i,... to reach a destination (which can be a 1 The geometric shape is actually like a slice of pie. However, in this chapter, we will simply call this 2-D conic shape a cone. 45 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. micro-server or base station) starting from a node located at some distance r from the destination. In addition, let w=w{xh(r),xh(r)A ,...) denote the energy cost per bit for traversing h(r) hops. Note that w(.) is equal to wtj{.) when the energy cost is calculated for a sensor node to reach its assigned micro-server while it is equal to w/(.) when the energy cost is calculated for a m icro-server to reach the base. We have: h ( r ) W ( X h(r)-’ X h(r)~l-’ " - ’ X i ’ X i - l ’ ' " ) = & T 0 ) _ & R . 1 = 1 ( j . 11 .a) w here d t =| x( - xjA \ In other words, d, denotes the distance between node i and its successor (i.e., the next upstream node along the path from node i to its destination.) Therefore, we can write the mean value o f w{.) as follows: W (r) = W(x„(r)) = E[w(xh(r),xHryi,...,xn xi_i,...)\ = E[h(r)] (a TE [d f ] + a ro + a R ) - a R In CSCR, it can easily be shown that E[/z(r)] is bounded as follows: (3-1l.b) r <E[h(r) ] < - r E[d] E[d]Qosco Consequently, the communication cost, W(r), can also be bounded as follows: (a TE [d f ] + a T 0 + a R) - a R < W{r) < —— ^-------(a TE [d f ] + a T 0 + a R) - a R ' ' h W n r \ c r»\ ' 7 E [dV ' E[d]coso) By substituting the above bounds on W(f) in equation (3.7), similar bounds for the total energy can be obtained. Therefore to minimize the upper bound o f the objective function in (7), selection o f n must be done so that following equation holds: 46 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. By solving the above equation, the optimal «th closest node within a solid cone at the node’s fan-out is determined. This means that each node can select its energy-optimal successor to forward its data toward the destination. Again recall that if the node which forwards the data is a sensor node, then the destination is the closest micro-server to that node whereas if the node itself is a m icro-server then the destination is the base station. Let «0 p denotes the value o f n which minimizes 7(«).The plot in Figure 3-3 shows T(n) as a function o f n and for several values o f c where /i=2 and aR +aT o=10. As can be seen, the behavior o f T(n) depends on value o f c (cf. equation (3.9).) Furthermore, when c > 0.9, the minimum o f T(n) is obtained for a relatively large value o f n (nop >10). 30 x:T(rf) rf is n optimal 25 20 c=lambda*omega c=0.5_ 15 c=0.l 10 5 50 60 70 80 90 100 0 10 20 30 40 n Figure 3-3.Variation of T(n) as a function of n for different values of c=Aat. 47 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.3.2 Energy Analysis Now we set out to determine the optimum energy density in < 2 ^ and the total energy allocated to the sensor nodes in the network by employing the CSCR algorithm for data routing i.e., the routing strategy is such that each node selects its energy optimal successor to forward its data toward the destination. 3.3.2.1 Energy density and total energy computation for sensors To find flow distribution and as a result energy density at distance r from a micro-server inside any cell C o, the cell is divided into infinitesimally narrow rings around the sink with width Ar (see Figure 3-4.) We denote such a ring as p(r) and precisely specify it as follows: p{r)={{r \ 0 ) : r < r ' < r + Ar and 0 < 6 < 2 nj ||p(r) | \=2nrAr As stated earlier for the continuous space model, each infinitesimal area may be treated as a source o f traffic with the local flow being proportional to the probability that one node is found in that infinitesimal area. Notice that, from the Poisson process property, if a node is in ring p(r), then it will have equal probability o f residing in any location inside that ring. Therefore, the local flow o f each p(r) is uniformly distributed in the area o f that ring and the energy density inside the ring is uniform. 48 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 3-4. AI(r) and AF{r) are total input and output flows of an inflnitesimally narrow ring with width length Ar (p(r)). Let AI{r) and AF(r) denote total input and output flow in p(r), and AL(r) denote the locally-generated flow in p(r) (see Figure 3-4.) M {r)= ^ \ p ( r ) \ e - ^ 2 (3.13) By writing flow conservation for any p(r) where 0< r<R, we get AF(r) = AL(r)+ AI(r) and AI(r) = 0 for r > R .The input flow to p(r) can be expressed as follows: AI(r)= ^ AF(r')gn(r',r)Ar r':r<r'<R (3.14) {AL(r') + AI(r'))gn(r',r)Ar r'.r<r'<>R 49 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Function g„(r’,r) is actually the transition probability density function (pdf) of the output flow. N ote that gn (.) is a p d f because relative distance from the n h closest node is a random variable that follows the distribution in equation (3.8). M oreover, we have: r=0 jg„(r',r)dr = 1 \/r':0<r'<R r~r' S(r\r) (a) r>r'smco S(r',r) (b) r < r 's i n £0 Figure 3-5. Different forms of S(r’ ,r) when r varies from r' to 0. 50 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. T heorem 4: gn(r ’,r) is calculated as follows: gn( r ',r) = 2 « V (^ ll^ ( r ' ^ ll)" ' e- (3.15) (» -!)! where S(r \r) is the intersection area o f a pie-like slice and a circle. The pie-like region (or 2-D cone) is centered at distance r ’ from the origin while the circle is centered at origin with radius r (see Figure 3-5.) P roof: See appendix B. Now that g„(.) is derived and considering the fact that area is radially symmetric, we can rewrite the flow conservation equation in a narrow ring p(r) as: A /(r)= X (A L (0 + A /(r’ ))g„(r',r)Ar r':r<r'<R Ar r.,% iR Ar' Ar' We next define i(r) and l(r) as follows: i(r)= lim ^ l , l ( r ) = lira A r-> 0 A r-> 0 ^ y Consequently, l(r) = In ju ^ re^ and i(r)= J(i(r') + l(r'))g n(r',r)dr' (3.16) r The boundary conditions are given as [0, if r > R \lnp?^ Re /(r)= 0, if r > R l{r) \ 2 ^ R e - A 'r f , if r = i? (3.17) 51 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Let AE0{r) denotes energy which is consumed in p(r) for communication. We have: &E0(r) = T Z {{dTQ+ccT{r-r'Y)l^F{r)gn{r,r')^') + aRM{r) + asEL{r) \r':r< r'< R The energy density is defined as follows: A AE0(r) e0 (r) = lim — — — A r-> 0 I p(r) || Therefore, e0“(r) = 2nr (i(r) + l(r)) J( « 7 0 + a T( r - r Y ) g„ (r, r ')dr' r + a R i(r) + a sl(r) (3.18) Collecting all the above equations, computation o f the total energy required for the sensors deployed in 3^ and energy density at any point inside any cell C o may be summarized as follows: i(r) = J(/(r ') + l(r')) g„ (r ', r)dr' r i{r)=0, i fr > R fO, if r>R l(r) = \2 n p \ Re-^ 2, if r < R ( n - l ) ! < (/• ) = 2 nr (i(r) + l(r)) J ( a ro + a T( r - r Y ) gn (r,r ')dr' r + a R i(r) + a sl(r) E[e0 '°' Y 2 n \p T { \n R 2 +1) e0 " (r)rdr 52 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The above equations can be solved numerically. The integrals are approximated with finite summations and the functions evaluated numerically starting from the boundary. By taking step size Ar for r, we get: k i(r) = Yj (*(r + j&r) + Kr + /A r)) g„ (r + jAr, r)Ar 7 = 0 (3.19) where k ----- - A r After computing i(r), the energy density and the expected value o f total energy required for all sensors located in 3^can be computed. As explained earlier, the sensor deployment problem and the energy allocation problem are related problems. Indeed there are two ways to provide energy density for a p(r). One way is to allocate 2nrAre0 J (r) o f energy to a node inside p(r). However, if the battery capacity o f each node is fixed to En , another way to allocate the required energy is to adjust the node density inside p(r) according to the following relation: 4 > V ) = ^ (3.20) This adjustment increases the node density above density X o which is required to meet the QoM constraint. Therefore, a policy must be employed to control the node density such that at any reporting cycle, density o f active (awake) nodes that are participating in sensing the environment and data routing is no more than X( ) and that the remaining nodes are in the sleep (power saving) mode. Therefore, not every node is awakened in each reporting cycle. In particular, we assume that each sensor node is given a probability o f wakeup i.e., the sensor wakes up in a given cycle with the specified probability. This probability for a sensor located at distance r from the sink is given as (X0 d (r)l X0 ) where (X0 d (r)/ Xn ) < 1. 53 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.3.2.2 Energy density and total energy computation for micro-servers Computation o f the energy density and total energy required for the micro-servers is similar to the computation performed for the sensors. However, the aggregation operation in the micro-servers makes the flow equations somewhat different from the sensors case. As a result o f the aggregation operation, the outgoing flow is generally smaller than the incoming flow to the micro-servers. By taking a general aggregation function, Ag(.), the locally generated flow in a narrow ring p{r) can be rewritten as: In this chapter we consider an intra-cell aggregation where each m icro-server performs data aggregation for all o f the sensors assigned to it (i.e., the sensors inside its cell) such that data o f all sensors inside a cell is fused into one packet o f data. Therefore, aggregation function Ag(.) which captures the above operation has the following form: details for the sake o f brevity. 3.3.3 Average Path-length Analysis of CSCR In this section we will further analyze CSCR to determine the Average Path Length (APL) o f the network. Note that length o f a path is the number o f edges (hops) in that path. To reach the base station, each sensor takes a path o f some fixed length. The APL represents the average length o f all paths which connect all sensors inside to the base. As stated before, the path from any sensor to the base consists o f two parts: from the sensor to a micro-server and from the micro-server to the base. One motivation to calculate APL is to AL(r) = ||p(r)||((A1 S[A(C,)])ylJ ,(r )/r) = InrSr [ \ A g(r)iu) (3.21) E[N(<C.)]} V (3.22) The rest o f derivations are similar to those presented in the previous section; we skip 54 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. find an approximation o f the average end to end delay o f the network. The approximation is accurate particularly in the network without any hop-by-hop retransmission which is consistent with our M AC layer assumption, which is assumed to be contention-less. The other motivation for analyzing APL is to show that the routing algorithm in CSCR is convergent in the sense that it has a bounded APL. In the following calculation, we separately calculate APL for sensors to reach the micro-servers and the micro-servers to reach the base station. Let apl^Co) denote average path length for the sensors located in any cell C o to reach the m icro-server located in that cell. The expressions for apl0 and its mean value (which is derived by using theorem 3) are as follows: where ho{xi) denotes the number o f hops (path length) to connect a sensor located at x, eCo to the m icro-server placed in that cell. Similarly, if apli denotes the average path length for any m icro-server to reach the base, we can write: (3.23.a) E[N( C 0)] (3.23 .b) = j - | h0(x)e~*Vxfdx (3.24.a) J (3.24.b) = I ^ {y)dy I I A. || y e < x _ 55 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where N(^) is the number o f micro-servers deployed in 5^and hi(y,) denotes the number of hops for connecting m icro-server located at yj to the base. Finally, the APL for the two-tier network is calculated as follows: APL = E[apl0] +E[aplJ (3.25) = f f h0( x ) e - ^ J ho ^ e^Wldx + T^~Tl j W yyfy s < R . I I A - I I p e« Equation (3.25) expresses the APL as a function o f h0{.) and ht{.) which must in turn be calculated. Now g„(), which was discussed in the previous section, enables us to calculate /*»(.) (and in a similar way hj{.)) as follows: The above equation is an integral equation with the following boundary condition: Equation (3.26.a) can be calculated numerically starting from boundary (i.e. r=0.) By taking step size Ar, equation (3.23 .a) can be approximated as follows: In a similar way ht{.) is calculated, which completes the APL calculation. Results of numerical evaluation will be shown in the next section. 3.4 Simulation Results In this section we evaluate the CSCR model by performing several simulations. The goals o f the simulations are to evaluate the accuracy o f the model in calculation of minimum (3.26.a) (3.26.b) + (/Ar)) Sn (r > iAr)Ar (3.26.C) where k = — Ar 56 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. total energy, to investigate sensitivity o f the model to various parameters, and consequently, to select the parameters which result in the optimum solution. We compare the optimum solution obtained by solving the CSCR equations with that obtained when employing a Minimum Total Energy (MTE) routing policy [36]. Given the node locations, the network is modeled as a network graph where energy cost o f each edge is calculated from equation (1.1). By applying the M TE routing, an optimal path which has the minimum total energy cost is selected for every pair of source and destination nodes. Therefore, the M TE produces the minimum total energy for a network graph with given locations o f sensors and micro-servers. When employing the M TE policy for routing, we measure the total energy consumed in the network separately for each run, which takes T seconds and this is repeated for many simulation runs. In each run, the initial placement of sensors and micro-servers is given. In the end, the average over all runs o f the total measured energy consumed in the network is reported. 3.4.1 Parameter Selection The system and network parameters selected for those simulations are: T/E0= l, ju=1,J3=2 (which indicates that the path-loss model for communication is that o f ffee-space), aK = a T o= 10 pJlbit,a7 = \ pJ!bit/m2 ,a s=0, and 7?=100( corresponds to radius o f the deployment region.) 3.4.2 Energy Density In the first simulation, we compute the required energy density for both sensors and micro-servers. Results are depicted in Figure 3-6, which shows that as we get close to the sink (located at the center o f a cell for the lower network and at the center o f < 2 ^ for the top- tier network), the energy density increases super-linearly. The simulation has been 57 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. performed for Ao=0.3 and ^i=0.05. Furthermore, the energy density (which in fact is average per cell energy density) o f sensors is almost zero for r > 6, which indicates that for this set o f system parameters, the effective (or average ) radius o f cells is 6. E n e r g y d e s n i t y o f s e n s o r s 0.9 0.8 0.7 0.6 0.5 I 0.4 0.3 0.2 0.1 0 3 4 5 6 7 D i s t a n c e f r o m a m i c r o - s e r v e r a t o r ig in 8 9 10 (a) E n e r g y d e s n s i t y o f m i c r o - s e r v e r s 350 300 250 £ 200 5 150 100 1 0 0 D i s t a n c e f r o m o r ig in (b) Figure 3-6. Energy density vs. distance from the sink for (a) sensors with X«=0.3 (b) micro-servers with Xi=0.05. 58 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.4.3 Total Energy Evaluation In the next simulation, we compared the total energy allocation computed by the procedure explained in section 4 (i.e., the CSCR model) and the actual value of the total energy for different node densities. Moreover, we studied the effects o f second level micro server density and the data aggregation policy performed by second level nodes on the total energy consumption. (In all o f these simulation runs, we set 7 .0=0.3.) Figure 3-7(a) depicts the total energy versus density o f micro-servers when the m icro-servers aggregate data of the sensors that have been assigned to them (cf. equation (3.6).) The results demonstrate that CSCR can estimate the total energy with relatively small error and that the total energy as a function o f m icro-server density is a convex function. Consequently, there is a micro server density, which minimizes the total energy that must be allocated for a given sensor node density. Figure 3-7(b) shows total energy versus micro-server density where no aggregation is done by the second level nodes. As can be seen in this case, the total energy is a decreasing function o f the m icro-server density. In fact, a small density o f micro-servers (0.05) is helpful in significantly reducing the total energy. Further increases in the micro-server density, however, result in only a small reduction in the total energy. 59 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Total energy vs.lambda! CSCR MTE T O O h - 0.8 0.6 0.4 0.2 lambdal (a) Total energy without aggregation 1b CSCR MTE (act) 14 12 10 8 6 4 2 0 0.05 0.1 0.15 0.2 0.25 lambdal (b) Figure 3-7. Total energy vs. micro-server density a) micro-servers performs data aggregation b) no aggregation by micro-servers. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.4.4 Average Path Length (APL) Evaluation Figure 3-8 depicts the results o f APL calculation which is computed by numerical evaluation o f equations (23) thru (26).The figure depicts APL as a function o f micro-server density (we set Ao=0.3) for different values o f ©. The figure shows that APL is a super- linear increasing function o f node density; however, the function is always bounded and tends to be saturated at a node density, where © is close to jr/2. Furthermore, the figure shows that by increasing co, APL increases fast for small values o f © and increases fast when © nears 7i/2, To compare CSCR with M TE in terms o f APL value, the result o f APL measurement for M TE obtained by simulating the M TE routing strategy is depicted in the figure. As can be seen there is a © value where APL o f CSCR is greater than MTE. APL vs. Lambdal &Omega 25 CSCR * ■ ■ MTE om ega 20 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 lam bdal Figure 3-8. APL versus node density evaluated for t t /40 < © < ji/2. 61 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.4.5 Effect of Deployment Area Size Figure 3-9 depicts the effects o f the area size (radius o f area <20 on total energy. The figure shows that total energy has a power law relation with the area size which means that energy grows super-linearly when area size increases. Note that for this simulation run, we consider that micro-servers perform data aggregation. Iambda1 /lambda0=0.120.3 / m u .-"'"'lambda 1 2 1 a mb da0=0.0520.3 100 150 200 Figure 3-9. Total energy versus size of the deployment region. 62 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.5 Chapter Summary In this chapter, we considered the problem o f energy efficient random deployment o f a two-tier sensor network comprising o f sensors and micro-servers. Given a fixed sensor density which satisfies QoM, we showed how to calculate the m icro-server density as well as the initial energy levels (or energy resource density) for both sensors and micro-servers so as to use the minimum number o f deployed micro-servers and total energy resource inside the network. The optimal allocation was such that the network lifetime met a user defined constraint. 63 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 4 Lifetime-aware Deployment of Two-tier Wireless Sensor Networks In this chapter, we investigate energy-efficient and lifetime-aware strategies for detailed placement and energy allocation o f infrastructure in the second level o f a two-tier WSN. In contrast the previous two chapters, the sensor nodes in this problem setup have a priori and arbitrary positions. Various forms o f the following problem are thus formulated and solved: Assign positions and initial energy levels to the micro-servers, and concurrently, assign groups o f sensors to each micro-server so as maximize the monitoring lifetime of the two-tier W SN subject to a total energy budget. The problem is solved for both collinear deployment and planar deployment situations. Solutions are obtained by transforming the problems to the convex optimization forms or by proposing heuristic solutions. By performing extensive simulations, the operational advantages o f optimal deployment o f a two-tier WSN in terms o f network lifetime compare to flat network architecture are investigated. The results show the optimal design and deployment o f such a network can increase the WNS lifetime by one order o f magnitude compared to a flat WSN with the same total initial energy and quality o f monitoring. 64 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.1 The MDEA Problem The problem o f designing a two-tier sensor network with m aximum service lifetime is addressed. The constraints are a) the total initial energy o f the network, Eto t, and b) the number o f m icro-sever nodes, M. The problem o f interest is m icro-server deployment, sensor-to-micro-server assignment (clustering), and energy allocation to the nodes at first and second levels such that, for a given number o f micro-servers and a constraint on the total energy budget (i.e. total initial amount o f charge o f all battery capacities), the WSN lifetime is maximized. The problem is called Micro-server Deployment and Energy Allocation (M DEA.) The micro-server deployment problem without the energy allocation aspect is related to a class o f Covering Location problems, which have received wide attention in the literature [34]. An early coverage model is the Location Set Covering Problem, introduced by Toregas et al [41], which seeks to situate facilities in such a way that all o f the demand nodes are covered within a specified distance standard and the number o f facilities is minimized. A well-studied extension o f the LSCP is the Maximal Covering Location Problem, introduced by Church and ReVelle [9], which seeks to deploy a limited num ber o f facilities in such a way that demand node coverage is maximized. 4.2 The MDEA-1 Problem in a Collinear Network In this section, first a 1-D version o f the problem is studied where sensors are deployed along a ray with the base station located at closed end o f this ray. Possible applications of collinear deploym ent include sensor networks for border surveillance, highway traffic monitoring, safeguarding railway tracks, oil and natural gas pipeline protection, structural monitoring and surveillance o f bridges and long hallways [23], The key advantage o f a collinear network is that it is more amenable to rigorous mathematical analysis and 65 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. derivation o f closed form expressions, which provides insight into understanding the more complicated case o f planar sensor networks. Notice that although the analysis is initially focused on the collinear deployment o f sensor nodes, the 2-D version o f this problem is also solved. A number o f definitions for network lifetime have been proposed in the literature [25] [2]. In this chapter the network lifetime is defined as the duration o f time after which the first node (sensor or micro-server) dies out because o f energy depletion. Though routing and placement strategies for sensors and micro-servers favor a solution which balances the energy depletion rate and consequently all sensors and micro-servers die out at the same time. The assumption is that if any sensor node dies out after time T$, the QoM for the application drops below the acceptable threshold. At the same time, if any micro-server dies out after time TM , then there will be loss o f connectivity between some sensors and the sink node, and hence, again QoM drops below its threshold. Consider a collinear network (see Figure 4-1), where N sensors with initial energy o f Ei are uniformly deployed with density X along a line segment o f length L, and therefore, the distance between two neighbor sensors is d=l/X. The base station is located at the origin (one end o f the line segment.) The problem o f interest is to place M (where M <N ) micro-servers, at positions x/, ...,x M along the x-axis, and to assign initial energy levels of E2 to micro-servers and E/ to the sensors such that the total network energy, i.e., M.E2 +N.Ei, is less than a user specified budget, E,oh while the lifetime o f the network, T„et, is maximized. N ote that in collinear case the solution to the sensor clustering problem is trivial, because each sensor will consume lower energy if it connects to its closest micro server. The multi-hop routing is allowed from a sensor node to its m icro-server and then multi-hop routing from the micro-server to the base station (but only hopping thru other micro-servers is allowed at this level.) 66 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ** sensor --------------- ▲ A A ▲ _ __ ▲ ___ ▲ r T N T N P i ~ \ BS (a) -► M- " Micro-server (b) Figure 4-1. Collinear network a) flat network b) two-tier hybrid network 4.3 MDEA-1 Problem Statement Given a line segment o f length L, N sensors placed at Hk = i.N/L ( i=l,N), M micro servers, and total network energy Etoh the goal is determine the initial energy levels o f sensors and micro-servers, Ei and E2, and the locations o f M micro-servers, X/, xM so as to: Tnet = Max Min (Tm,Ts) (4.1) s.t. N ■ Ei +M ■ E2 £ Etot h = \xM - xi\ (4. l.a) To transfer the MDEA-1 problem into a mathematical program, we notice that for a given deploym ent o f micro-servers, a chain o f sensors, chj, is formed around each micro server j i.e., the monitored data o f these sensors goes to j. In addition, another chain is formed among the micro-servers to relay the gathered data to the base-station. This simple 67 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. cluster structure enables us to employ the results in [14] where the authors have obtained the following close form equation for optimal lifetime o f a chain o f sensors uniformly (with node density X ) placed along a line with length / . 7], ( / , £ , ) = — ^ PM-D 1 E x A A -1 ( A . / + !).(!- M M ) A' I (4.2) Here, /x denotes the sensor data reporting rate o f each sensor (which is assumed to be fixed), e,(\/X) denotes energy per bit for data transmission over a distance 1A, between two consecutive sensors, and Ei denotes the initial energy level o f each sensor. N ote that the optimal lifetime obtained with optimal routing which was firstly proposed and formulated in the form o f a mathematical programming in [8],The equation shows that the network lifetime is linearly proportional to the energy level o f sensors and is inversely proportional to the size o f the chain (i.e., I). In deriving this equation, similar energy model as described in chapter 1 is employed for the communication among sensors. As described in chapter 1 the energy for communication a bit o f data, is related to the communication cost over distance d is et0 -d?' +er In this equation, e, denotes energy dissipation per m eter per bit o f transmitted data, er denotes energy dissipated at the receiver for reception o f a bit, d denotes the distance between transm itter and receiver, and /?, > 2 0=1,2) denotes the exponents o f the path loss function for the first level sensor network , pi ,and the second level micro-server network, P2. 68 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The lifetime o f k h micro-server, Tm (k,E2 ), with initial energy level E2 can be calculated as follows: T m{k,E2) ^ — ^ W fk ‘ (ei0 ' + a k ' er ) **«(*) = / / r ( e,o V 2 + a k -er ) (4-2.a) where f k is the outgoing flow o f micro-server k , ak denotes aggregation factor o f m icro server k and ik =| xk - xk_x \. Based on equations (4.2) and (4.2.a), the MDEA-1 problem can be rewritten as: Max Tnet = Min (Min(Ts (A ■ I,, E{ ), Tm (i, E2))) E\ E 2,h ,-,lM+i i s.t. N - E x+ M ■ E 2 S E tot, (4.3) V / = 1 : M f t = - P s(ii) T m{i,E2) ^ ^ r ( h + 1 +li) + fi + 1 a, 1 and } /,• = L ' ;= 1:M + 1 / \ \ A (4-3.a) ^ ('1 yHe( (W/2+«rT) where the objective, T^,, is the network lifetime, P m (/) is pow er consumption o f micro server i, a, recalls the aggregation factor o f micro-server i, and f denotes the total flow which is forwarded toward the base station by m icro-server i. In addition, et and er denote the energy per meter per transmitted bit and the energy per received bit. The base station is at xo=(), m icro-server i is at position x„ andXm+i=L. 69 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. As stated earlier, we assume that energy for transmission o f a bit o f data over distance d in the m icro-server network is proportional to e,0 -dP l .Notice that (2 < /?2 < A ^ 4) • This is because different radio propagation schemes and antennas may be used by the sensors and the micro-servers. For example, the micro-servers may use a more sophisticated channel coding technique. Therefore, their path loss exponent becomes lower. O f course, this comes at the cost o f additional computation on the data encoder on the micro-server side and data decoder on the base station. As another example, it may be feasible and cost- effective to place the micro-servers at a higher elevation com pared to the sensors. The higher elevation o f micro-servers reduces the effect o f reflections, w hich in turn decreases the path loss in the links among the micro-servers. Another way to decrease the path loss for the micro-servers is to increase the antenna height and/or gain for the micro-servers, which results in higher power efficiency. In our formulation, each micro-server forwards all o f its data to its nearest micro-server in the direction o f the base station (this is called nearest-neighbor routing) i.e., the sensor data is relayed from one micro-server to next until it reaches the base station. This nearest- neighbor routing also reduces the MAC layer interference, which is caused by bypassing neighbors. In any case, MDEA-1 can be solved by any standard mathematical program solver, including M ATLAB. As an interesting variant, we consider a related problem in which, as before, the initial energy level o f all o f the sensors is Eh but, this time, the initial energy levels o f micro-servers can vary. The rationale here is that the sensor nodes are homogenous whereas the micro-servers can have different amount o f energy i.e., they have batteries with possibly different full-charge capacitates. 70 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. T heorem 5: The M ax-M in constrained optimization problem in (4.3) can be transformed to a convex epigraph form as follows: Min co (4-4) s.t. V i = 1: M Ps (li) - E r co< 0 Pm( i ) - E 2 -co<Q ] T / , - L = 0 i= \. M N .E] + M ,E 2 - Etot / Ps(h) = M-el( h m + l ) - (4.4.a) j lo g (M )' Lk , / B x ^ Y ' { l M + l i ) + f i + \ p m ( i ) = f i \ ef l r + a i * r ) and f i = - ^ ------------------------- P roof: The equivalency can easily be shown by considering that the objective function in (3) implies that T n el< Ts and Tnel < Tm therefore, — < — and — < — ! — and by taking Pn ct Pm Pnct c o = — — , the optimization problem in (4.3) is transformed to (4.4). The objective in (4.4) is Pm t linear. Furthermore, the constraints and the objective have linear or polynomial forms with respect to the variables Eh E2 and lk hence it is straight forward to prove the constraints are convex by showing that the second derivatives o f constraints with respects to the variables are positive. Am ong the constraints, the most com plex function is Ps(.) where its second derivative is 9 Ps^ = X• /, -2■ logU•/,■ ) + 3 .Let take h ( y ) = y - 2 ■ log(v) + 3 .Note that d i p lim h ( y ) = +« .Furthermore, h(y) has no root except for y -»+<»; therefore, h(y) is always o+ 71 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. positive. Consequently, the conditions for convexity o f the optimization problem are met; therefore, optimization problem in (4.4) is convex. 4.3.1 MDEA-lc Problem Statement Given a line segm ent o f length L, N sensors placed at \tX=\.N/X (i=l:N), M micro-servers, and total communication energy budget, Em, the goal is determine the initial energy levels of sensors and micro-servers, E] and E\’s and the locations o f M micro-servers, xt, xM so as to : with the additional constraints given by (4.3.a). T heorem 6: The M ax-M in constrained problem in (4.5) which is stated for M D EA -lc problem can be transferred to the following M in-Sum convex problem: S.t. (4.5) N-El + ^ 4 < E tot i = \ : M Min m ax ) + Pmih’h ..... * 1 rm a v (4.6) S.t. lo g C i - U x ) Ps ( lm ax) = ) ' ( 4 ' / max+ 1 ) ' 1 m a x (4 .6 .a) 72 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. P roof: We first state that, in the optimal solution of M D E A -lc, the lifetime o f sensor nodes in any cluster (chain) is upper-bounded by the lifetime o f the sensors in the longest chain connected to a micro-server. This is straight forward to check since the initial energy 1 44 levels o f all sensors are equal, and therefore, according to equation (4.2) when ( > - — the X function Ps(.) is an increasing function which implies that the chain with maximum number o f sensors, i.e., the longest chain(its length is denoted by lm ax), will have the minimum lifetime. N ext we point out that, in the optimal solution o f M D E A -lc, the lifetime o f the longest chain is equal to the lifetime o f the micro-server network. The reason is that if this is not the case, then one can decrease the allocated energy to micro-servers, i.e., E‘ 2, and increase the allocated energy to the sensors, and thereby, increase the lifetime o f the network. M ore precisely, we can write: T L = T * ^ > E 0 = K ^ E 2 ) w h e r e V i' : / max = k ^ l i ■ Given a placement o f the micro-servers, we can bound the lifetime o f the two-tier WSN as follows: T <-__________ ^tot *net — PsUmax) + Ps (/m ax ) = N.ju. e, ( i ) • (2 • /m ax +1) • [ 1 l0g(A' /max} (4.7) ) — 'y ’ (7) fi ' [^t{ ) ' + a i ' j=l:M i=\\M where Pm(h’h<--^M+\)= ^ p,„ (J) denote the total power consumption for the M m icro- servers. By plugging equation (4.5) into objective function o f M D E A -lc problem in (4.5), equation (4.6) is obtained. 73 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The proof o f convexity is similar to theorem 5; however, note that here constraints in (4.6.a) are linear and the second derivatives o f the objective function with respect to the variables I, and lm ax are positive which indicates that the constraints are convex. Furthermore, the function in the objective with respect to the variables is polynomial and hence it is convex besides it can easily be shown that P s(lm ax) is a convex function. Consequently, the objective and the constraints in problem (4.6) are convex and this concludes the proof. ■ Note that in procedure o f transferring problem (4.5) to problem (4.6) an extra variable, lm ax, was added to the set o f variables and E\ and E '2 ’s were eliminated. After solving problem (4.6) the eliminated variables are calculated as follows: J- _ __________ Etot__________ ^s(^max) + Pm(h’h ’"-’lM+0 E\ ~ Tv(^max) ' T n et Ei2 = f r (eto- l ^ + ar er)-Tmt Also note that the function P s(.) is uniquely known when we assign a value to lm ax- Furthermore, we know that that, — - — < lm m < L . (M + 1) m a x The optimization problem in (4.6) is a convex problem and can be solved by standard mathematical solver such as MATLAB. Next we show how to solve problem statement (4.6) by finding its dual problem. First, for simplicity, we assume a,=a, for all i. However, the approach can easily be generalized to handle different aggregation factors for the micro-servers. We thus get: 74 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Pm(h>h>->lM+0 = etO' Z f i ’lih +Tl - er- Z ' ’h i = l : M + l ju-A i = \ : M f i = 1 ’ Z v j = i : M + l (4.8) a We can write the Lagrange dual function o f the optimization problem in (4.6), 0 (.), as follows: = N./x-et (—) ■ (4 ■ /max +1) ■ l°s(4- • /m ax) X-l, + et0 Z + r l ' e r I H /= 1 :M + 1 where v is the Lagrange multiplier. The last term captures the constraints on total length of sensor chains and the maximum length o f each sensor chain. Then the dual optimization problem is expressed as follows: S.t. (4.9) v > 0 By solving the dual problem, we get / = v -i-rj-er Pi Pi-1 V / = 1: M (4.10.a) i= V M *Pi v - ( M + -er (4.10.b) 75 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where I* ’s are the optimal sensor chain lengths. Note that equations (4.10.a) and (4.10.b) result in a set o f non-linear equations in terms o f /* ’s because the amount o f data flow through each m icro-server,/, is itself a function o f I * ’s as in equation (4.10.a). Another equation that we have is the following one: Equations (4.10.a), (4.10.b), and (4.10.c) yield (M+2) relations from which /* and v can be calculated as follows. First, assuming l * M + x =6 values o f /w, I m-i, ■ ■ ■ , h are calculated from equation (4.10.a) as a function o f 0 and v. Next, from equations (4.10.b) and (4.10.c), 0 and v are calculated. Lemma 3: The optimal value o f lm a x (denoted by f max) is 1.44/A. or L/(M +1). Proof: By taking derivative o f 0 (.) w.r.t variable lm a x we get: (4.10.C) i= V M + \ ' ^m ax ) max max _ d /^ (/m a x ) _ ' ^max) ^ ‘ ^max l° g ( A • / m ax) 1 1 7 Pil I - > 7 \2 max 1 .4 4 dl max X However, as aforementioned the value o f / is bounded as --------- < /m a x < L .Therefore, (M + 1) ■ e 1.44 1.44L L if -----= ------- < --------- X N (M + 1) M < - 1 regarding that /m ax > '— > 0 hence the optimal 1.44 X o/max max value is / max 76 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Note that due to the convexity o f (4.6) Slatter’s conditions for this problem are satisfied consequently the strong duality holds for (4.6) and there is no gap between problem (4.6) and its dual problem in (4.6) [6], 4.4 The MDEA Problem in a Planar Network MDEA problem in two dimensional case (i.e., the deployment region is a land-based supply network or distribution system) is a difficult problem since in addition to sensor clustering, m icro-server placement, and energy allocation, network routing itself adds a new degree o f freedom. To simplify the presentation, in this part, we constrain ourselves to the case where each sensor has direct transmission to its m icro-server and each micro servers has direct transm ission to the base'(see Figure 4-2.) This removes the routing variables from the clustering problem. We assume that locations o f the sensors in the Euclidian space (denoted by x(s,) for sensor s,) are known and the base-station is located at the origin. Let S={.y/, s2, ..., sN } and Ms={/w/, m2, ..., mM } denote the set o f N sensors and the set o f M micro-servers (cluster heads) in the tw o-tier network, respectively. Furthermore, let X(S)={x(5/), x(s2), ..., x(s„)} and X(Ms)={x(w/), x(m2), ..., x(mm )} denote the set o f all sensor locations and micro server locations, respectively. Let mk=head(s,) be the cluster head mk for sensor s„ and Pt(sh mk ) and P,(mhb) denote the energy dissipations per bit o f transm itted data from sensor s, to mk and from micro-server mk to reach base station b, respectively. Define sset(mk) = {si eS\mcp(Sj) = mk} and Ck as cardinality o f set sset(mk ). As before, we have 1 This kind o f two-tier WSN with single-hop routing within each level is very practical and highly efficient. Many applications that utilize a sensor network actually benefit from such a simple, yet cost-effective and energy-efficient, hierarchical structure. 77 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I]{s,,mk) = er \x{s,) - x(mk)fl and P,(mk,b) = e,.\x(mk)- x(b)f2 . Pr denotes the energy dissipation per received bit. Notice that I)(sl,mk)- f (,s,) and P r • / {s,) denote the corresponding transm it and receive power dissipation levels. Furthermore, assuming constant bandwidth efficiency for all the transmitters, it is easy to see that the following relations hold: Let p tm a x denote the maximum transm it energy level per bit for all sensors. This means that the distance between any sensor st and its cluster-head head(s,) is upper bounded by some maximum distance, Dm a x . Finally, let b and x(b) denote the base station and its location, and E2 denote the energy allocated for all micro-servers mk. W ith this notation, the lifetime, , o f sensor s, and the lifetime, T mk, o f micro-server tnk can be calculated as follows: £ /(*:> sl \mk = h e a d (s l ) (4.11) a(m k) {Pt(si’head(Sj)) + Pr ) - f { s t) (4.12.a) 7 ]■ £ / w ' st esset(mk) (4.12.b) 78 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.4.1 MDEA-2 Problem Statement Given S , X(S) , Ptmax, M, and the total communication energy budget, Eto l, the objective is to find a sensor to micro-server assignment function, head(.), locations o f the micro-servers, X(MS) , and the sensor/micro-server energy allocations, Ei and E2 , in such a way that: where a(m,0>l denotes the data aggregation factor for m icro-server mk. Constraint (4.13.a) ensures that every sensor has sufficient radio power to reach its assigned micro-server. Finally, constraint (4.13 .b) ensures an upper bound on the total energy allocated to WSN. Problem (4.13) in fact is a M ixed Integer Non-Linear Programming (MINLP). Note that general M INLP is addressed as a NP-complete problem. N ext we propose two heuristic methods to obtain an approximate solution for problem (4.13). 4.4.2 The First Proposed Heuristic Solution for MDEA-2 Problem Max Min (7 [ h e a d ( , ) M iM . s),E ] ,E 2 ) mk e M s Sj e ss e t(mk) (4.13) subject to: Vs,- € S, mk = head(Sj) e M s : Pt {Sj,mk) < P\ i m ax (4.13.a) M-E2+N-Ex<Elot (4.13.b) Similar to the procedure performed in theorem 5, it can be shown that problem (4.10) is equivalent to the following optimization problem: M in co (head(.\X{mk),Ex,E2 ) (4.14) 79 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. s.t. (Pt (sn head(s,)) + Pr )- /(.?,) - £, • co < 0 pt(mk,b) . ^ — + p \ £ f ( Si) - E 2 -w < 0 «(«*) J , , (4.14.a) ' s^ssetim^) v ' Vi, e S mk = h e a d (S j) e Ms : P t(Sj,mk) < P tm a x M ■ E2 + N • E\ < Etot The new problem (4.14) has epigraph form and consists o f simultaneous finding optimal locations o f micro-servers and optimal assignment o f sensors to micro-servers. The new problem is also concurrent finding location o f micro-servers and assigning sensors to micro-servers hence it is a MINLP problem which as aforem entioned is as a NP-hard problem. However, the problem size can be reduced by taking an separating assignment phase from the optimization and taking an assignment head(.) which maps each sensor to its closest m icro-server (called Voronoi assignment.) By the Voronoi assignment approach, problem (4.14) is converted to a convex problem that can be solved by a standard mathematical programming tool. The converted convex problem is an approximation solution to problem (4.14). 4.4.3 The Second Proposed Heuristic Solution for MDEA-2 Problem To solve the MDEA-2 more competently, we propose a heuristic approach as follows. To begin with, we have a physical map o f the sensors, X(S) , in the Euclidean space, reflecting the actual layout o f the sensor network on a plane. W e construct a graph, G, whose nodes are the set o f sensors, S. The edges in G are formed between pairs o f sensor nodes whose Euclidean distance is less than or equal to 2.Dm a x . Now, we identify the Maximum Independent Set (MIS) of this graph o f size r. Each node in this MIS is the seed of a m icro-server cluster. Clearly, r should be less than or equal to M; otherwise, the 80 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. problem has no feasible solution for the given M. If r is less than M, then we will split the final clusters as will be explained in the following. In this way, we guarantee that we will always start with less than or equal to M cluster seeds. Suppose we have performed some node clustering and a partial solution, z h ..., zM- , has already been generated. The graph comprises o f these cluster nodes, z*’s, and a set of unassigned single nodes, s ’ s. Next, we identify an edge connecting some zk to some s, such that E(zk) + \^center(zk,si) - xb\^2 -\x ( z k) - x bf 2 + j ■ / T has the minimum value. Here c e n te r ( z k ,S j) and x ( z k ) are tentative locations o f the cluster heads o f U {Sj} and zk, respectively. Let z;- and si be such a pair. Then 5/ is collapsed into zy in the graph i.e., zy =zv -U{j/}. In addition, we set x(zJ) = center(zJ,sl) and E(zj) = ^x(zj)-xb\P2 +pr \z]\^ - f -t ■ This process is continued until all nodes are assigned to some cluster. Micro server Sensor BS Figure 4-2. Single-hop routing within each level of the WSN and sensor clustering. 81 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The only remaining question is how to calculate the center(.) for a set o f sensor nodes { ,v u, .s\, sw , ...} .If p 2= 2 , center(.) is in fact the center o f mass o f identical weights placed at locations 0 ,y ) 0 u ), (x,y)(sv ), (x,y)(sw ), ... i.e., centerx(su,s v,s w, . . . ) = ~ ^ x(sv) , v=\...U center}l(stl,sv,sw,...) = — • ^ } ( s v) . For other values o f p 2 , a similar center o f mass solution can be v=l..U obtained (but solved with respect to the appropriate ft 2-norm.) This choice reflects our desire to produce equal radio communication costs from the sensors to the cluster heads. If the number o f final clusters is smaller than the available number o f micro-servers, M, we will first sort the clusters based on descending order o f their diameter, i.e., the maximum distance o f sensors in each cluster from its cluster head. Next, we assign one extra micro server to each cluster starting from top o f the list until we use all available micro-servers. Then, we run the aforesaid approach to find the new clusters. The procedure o f the proposed solution is illustrated in Figure 4-3. 82 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Construct G I____ Find MIS of G r = sizeof(MIS(G)) If r <= M ■ N o - Yes Yes No • ( end ) No. of clusters M "T s there any un- .dustered sensop ^ \ l e f t ? ^ > ^ Update CenteriZk) Assign extra cluster heads starting from top Sort clusters based on their diameter Find a pair of (Sj, zk ) such that iE(Cenfe/tzk )) is minimal Assign each node in MIS(G) to a cluster zk and set them to be the cluster head Cenferfzk ) &E(Center(zk)) = || Center(zk,s,) - xh f 1 - \x(zk) - xb| ' + Pr j ■ f ■ T Figure 4-3. Flow chart which demonstrates the procedure of heuristic-1 solution for MDEA problem. 4.5 Simulation Results We performed a number o f simulations to assess the effectiveness o f proposed solutions for various forms o f M DEA problems. In particular, we com pared the lifetime of a two-tier W SN with that o f a flat network to explore the operational advantage o f two-tier structure to flat network. We have taken uniform deployment o f sensors in our simulation where sensors are uniformly placed along a line or randomly deployed in a square-shaped 83 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. plane. When sensors deployed their locations were kept unchanged throughout the simulations. The set o f parameters for energy model is taken from [18].Those energy model parameters are measured for a sensor node with a radio communication subsystem comprising o f transm itter/receiver electronics, amplifiers, and antenna. The parameters for ffee-space communication are: 2, e,=10 pJIbit/m2 , ef=50 nJ/bit. Before demonstrating and discussing the simulation results, we first define a term called Lifetime Gain Factor (LGF) which is calculated by the ratio o f network lifetime two-tier network to lifetime o f fla t network. Therefore, if the ratio is greater than one it shows the improvement factor in network lifetime which is achieved by optimal placement and energy allocation (i.e. by solving various forms o f MDEA problem ) to the second level. In the first simulation the effect o f number o f deployed micro-servers on LGF is studied. MDEA-1 problem stated in equations (4.4) and (4.6) is solved by using optimization tools o f M atlab and measured lifetimes o f the flat and two-tier WSNs for a given num ber o f micro-servers and a fixed per-node energy, E3. This means that the total communication energy o f the flat network is Etot=N.E3. We used the same total energy for the two-tier network o f N sensors and M micro-servers, i.e., in this case Etot=N.Ei+M.E 2. We reported results in Figure 4-4 for N— 50 and A=0.1 nodes/m in two different cases: the case o f hom ogenous propagation model, p i = f 2=2, and the case o f heterogeneous propagation model, /?;=4 and f}2=2 (i.e., we assumed different propagation models for the two levels o f the W SN.) 84 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 18 MDEA-1c MDEA-1 beta1=4 / b e ta 2 = 2 , ' 16 14 12 lambda=0.1 miu=1 N=50 10 8 beta1=beta2=2 6 4 2 0 5 10 20 25 30 35 45 50 0 15 40 # o f micro-servers (M) Figure 4-4. Lifetime comparison of two-tier and flat Collinear WSNs. Figure 4-4 shows the effectiveness o f adding few nodes at the second level. More precisely, it shows that having allocated the same amount energy, the two-tier network lifetime is up to 6 times higher than that o f the flat network in the case o f equal p and up to 17 times higher in the case o f non-equal. Furthermore, the figure indicates an important fact that for any given network size (i.e. number o f deployed sensors) there is a particular micro-server count for which the lifetime o f two-tier network is maximized (we next referrer to that particular count as optimum size o f the second level.). As can be seen for a network with size 50 and node density /l=0.1 nodes/m the optimal num ber o f micro-servers are M=10 and M=15 for two different cases. In this simulation, we also solved M D EA -lc problem by using the iterative Lagrange dual approach. As expected the lifetime o f WSN with fixed E\, but variable E2’s is higher than that with fixed E t and fixed E2. 85 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. x 10' 8 2-tier flat lam bda=0.05 n od es/m Etot=10*N b eta 1 = b et a2=2 2 0 2 0 3 0 4 0 5 0 6 0 70 8 0 9 0 100 Network size(N ) Figure 4-5. Network lifetime of two-tier and flat network versus network size. In the second simulations, the effect o f network scalability on lifetime o f two-tier and two-tier networks at different network sizes is studied. For the flat network, we assume optimum routing so that the maximum lifetime can be achieved as indicated in equation (4.2). For the tw o-tier network, by solving MDEA-1 formulation we find the optimum number o f micro-servers, their locations, and their initial energy levels, which yield the maximum lifetime. Figure 4-5 and Figure 4-6 show the results o f the comparison. As before, we allocated the exact same total energy Eto l to the flat and two-tier networks. We set 2=0.05 nodes/m and increased the network size o f the first level, N, from 10 to 100 and reported network lifetime for both two-tier and flat network which was depicted in flat networks is studied. In other words, the lifetime o f flat network with the lifetime o f 86 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 4-5 (the total length o f the line segment, L, is N/X.) As can be seen in this figure, the lifetime o f flat network degrades more rapidly than two-tier network as network size increases which indicates that adding few nodes at the second level can significantly improves the scalability o f the network in terms o f lifetime. The effect o f network size on LGF for the same simulation is illustrated in Figure 4-6. Furthermore, the figure shows results for different aggregation factor and also for the case that each m icro-server fuses all its incoming packets to a single packet (denoted by SPO in the figure.) Note that in this simulation we took a homogenous aggregation function for all micro-servers (i.e. a* = a for /=1 :M )T he results o f this figure show that as the network size increases, the effectiveness o f two-tier network (i.e. LGF) in extending the network lifetime increases with some factors. M ore precisely, without aggregation LGF increases but is saturated at some value of network size whereas when micro-servers perform aggregation the LGF is always monotonic increasing function o f the network lifetime. The saturation effect in the case of no aggregation can be explained by the result demonstrated in Figure 4-5. As can be seen in the figure the network lifetime for both two-tier and flat network degrade and saturate when network size increases. As part o f scalability study the optimal sizes o f the second level for various network sizes are reported and the result is shown in Figure 4-7. As can be seen the optimal value of the second level has a piece-wise linear relationship with the size o f the first level. 87 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Iambda=0.025 nodes/m beta1=beta2=2 tpha=1.02 — ntr aggregation(alpha=1) 1 0 0 Network size(N) Figure 4-6. Effectiveness of two-tier network versus network size. 16 14 12 10 8 6 lambda=0.2 nodes/m beta1=beta2=2 4 2 90 80 100 20 40 50 60 70 " 1 0 30 Network size Figure 4-7. Optimal size of the second level versus the size of first level in a two-tier network. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. In the next part o f our simulation study, we have found functional dependency o f LGF as well as optimal size o f second level in a collinear two-tier network to density o f deployed sensors at the first level. We took a fixed number o f nodes, N, but with variable node density, X. Therefore, the total length o f the line segment on which nodes are located, L - N/X, is also variable (in fact the collinear network is stretched as X increases and it spread when X decrease.)Figure 4-8 depicts LGF versus node density. As can be seen two- tier network is much more effective in sparse network than a dense network. However, as can be seen in Figure 4-9 in sparse network higher count o f m icro-server is required to achieve maxim lifetime. LGF vs. S e n so r den sity(n od es/m ) b eta1=2 b eta1=4 5 25 0 .2 5 0.2 0 .1 5 Node density(lam bda) 0 .0 5 Figure 4-8. Effectiveness of two-tier network versus node density at the first level. 89 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 20 18 16 14 12 10 8 6 0 .08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 lambda(node/m) Figure 4-9. Optimum size for the second level of a two-tier network versus node density at the first level. To evaluate our proposed methods for solving the MDEA-2 problem, we considered a 50 X 50 m2 square-shaped deployment region wherein the base is located on a vertical line which crosses the lower horizontal line o f the deployment region (see Figure 4-2.) We used a uniform random process to deploy sensors in the given region. N ext the lifetime o f the two-tier network is measured and is compared it with the lifetime o f a flat network wherein each sensor sends its data with single-hop routing to the base. In both cases, Eto t is kept the same. Figure 4-10 shows the results. The (near) optimal locations o f micro-servers and their initial energy levels are calculated by solving optimization problem in (4.14) using two heuristic methods proposed in sections 4.4.3 and 4.4.2 (those are called heuristic-1 and heuristic-2 in the figure legend, respectively.)Note that Figure 4-10 shows the results for the values o f M fo r which the optimization problem 90 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. solved by proposed heuristic methods could get feasible solutions. It can be seen that the heuristic-1 method produces results which are close to the optimal solution with much shorter computation times (the heuristic method uses a factor o f 15 lower CPU time.) Unlike to the case o f a collinear two-tier network, the effectiveness o f the second level in 2-D case is a monotonic increasing function o f the size o f the second level. Finally, we point out that the two-tier network exhibits a lifetime improvement. Figure 4-11 demonstrates the effectiveness o f the two-tier network as function o f density o f the sensors deployed in the squared-shape plane. As can be seen the behavior is similar to the collinear two-tier network. 4.5 lam bda=0.05 beta1=beta2=2 Are a:50 X 50 — Heuristic 1 - Heuristic2 3.5 I L . 2.5 # Microservers Figure 4-10. Lifetime comparison of two-tier and flat WSNs deployed inside a square shaped plane. 91 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.6 Heuristic-1 — Heuristic-2 2.4 M=20 b eta1=beta2=2 Are a: 5 0 X 50 2.2 LL o 0.6 0.0 5 lam bda(nodes/m ) Figure 4-11. LGF versus node density for a two-tier WSNs deployed inside a square shaped plane. 4.6 Chapter Summary In this chapter, we proposed optimal strategies for design and deployment o f two-tier hierarchical W SN. The design goal was to maximize network lifetime. Furthermore, we studied the operational advantages (in terms o f extending network lifetime) o f a two-tier architecture over a flat network architecture. M ore precisely, w e formulated and solved the problem o f assigning positions and initial energy levels to the micro-servers and concurrently partitioning the sensors into clusters assigned to individual micro-servers so as maximize the m onitoring lifetime o f the two-tier WSN subject to a total energy budget. This problem, called M DEA, was solved for both collinear deployment and planar deployment situations. Experimental results showed that an order o f magnitude lifetime increase for a tw o-tier WSN can be achieved compared to an energy-efficient flat WSN. 92 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 5 Summary and Possible Extensions In this thesis, we proposed optimal strategies for design and deploym ent o f flat and two- tier hierarchical W SN under both random, continuous-space position models and fixed position models. The primary consideration for us was that o f m inim izing the total energy usage in a W SN so as to meet constraints on the Quality o f M onitoring (QoM) and M onitoring Lifetime (MoL). We showed that node and energy densities as well as average hop count in WSN are strong functions o f the routing schemes used. Furthermore, we showed that with the same energy allocation and subject to the same M oL and QoM targets, a properly designed two-tier network significantly outperforms a flat network. More precisely, we showed that by optimally assigning positions and initial energy levels to the micro-servers and concurrently partitioning the sensors into clusters assigned to individual micro-servers, significant lifetime increase for a tw o-tier W SN can be achieved compared to an energy-efficient flat WSN The work in this dissertation can be extended in a number o f important directions, including consideration o f W SNs with heterogeneous sensor nodes connected by possibly multiple levels o f intermediate/aggregator nodes (micro-servers.) Another key extension is that o f optimal energy allocation, velocity and trajectory assignment in a WSN with mobile micro-servers e.g., Unmanned Arial Vehicles (UAVs) or ground vehicles following controllable or randomized tour o f the deployment region. 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Han, and M .B.Srivastava,’’Dynamic Fine Grained Localization in Ad-Hoc Sensor Networks,” Proc. o f M obile Computing and Networking (Mobicom), pp. 166-179, July 2001. [36] S.Singh, M. W oo and C.S. Raghavendra, “Power-Aware Routing in Mobile Adhoc Netw orks,” Proc. o f M obile Computing and Networking (Mobicom), pp. 181-190, July 1998. 97 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [37] J.Seung , G. de Veciana, Su, “M inimizing Energy Consumption in Large-scale Sensor Networks through Distributed Data Compression and Hierarchical Aggregation,” Proc. o f IEEE JSAC Special Issue on Fundamental performance limits o f wireless sensor networks , Vol. 22, No. 6, pp. 1130-1140, Aug. 2004. [38] A.Savvides, C.C. Han, and M.B. Srivastava, “Dynamic Fine Grained Localization in Ad-Hoc Sensor Networks,” Proc. o f M obile Computing and Networking (Mobicom), pp. 166-179, July 2001. [39] J. Seung , G. de Veciana, Su, “M inimizing Energy Consumption in Large-scale Sensor Networks through Distributed Data Compression and Hierarchical Aggregation,” Proc. o f IEEE JSAC Special Issue on Fundamental Performance Limits o f Wireless Sensor N etw orks, Vol. 22, No. 6, pp. 1130-1140, Aug. 2004. [40] S. Tilak, N. B. Abu-Ghazaleh and W. Heinzelman, “Infrastructure Tradeoffs for Sensor Networks," Proc. o f A CM In t’ l Workshop on Wireless Sensor Networks and Applications (WSNA), Sep. 2002. [41] C. Toregas, R. Swain, C. ReVelle, and L. Bergman, “The Location o f Emergency Service Facilities,” Operations Research, Vol. 19. pp. 1363-1373, 1971. [42] C-Y. Wan, S.B.Eisenman, and A.Campbell, “CODA: C ongestion Detection and Avoidance in Sensor Netw orks,” Proc. o f Int’ l Conference on Embedded Networked Sensor Systems ( SenSys), pp. 148-161, Oct. 2003. [43] Y. Xu, J. Heidemann, and D. Estrin, “Geography-informed Energy Conservation for Ad Hoc Routing,” Proc. o f M obile Computing and Networking ( Mobicom ), pp. 70-84, July 2001. [44] H. Zhang and J. Hou,”On Deriving the Upper Bound o f a-lifetim e for Large Sensor Netw orks,” Proc. A C M Int ’ l Symp. on Mobile A d Hoc Networking and Computing (M obiH oc),, M ay 2004. 98 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Appendix A Before presenting a proof for the theorem 1, we first state several lemmas as follows. L em m a 4: Given that covariance between two points x, and Xj, and assuming that £[Z(x)] = 0 and £’[ z 2(x)] = C0 then the variance o f error at any point x, cr2 E(x), for the linear estimation as equation (2.2) can be calculated follows: kx kx kx fZ o v ix ^ x f)- 2 Z aiCov{x, Xj) + C 0 /= l j = 1 /= 1 where k x = |m (g * (* )| denotes the cardinality o f neigh(x) and the weights o f linear estimation are obtained from the following matrix equation: [Cov(x(,x ,)] ^ ^ [a, \ kx)) = [CovCx.x^] ^ P roof: Proof is simple by expanding e (z(x)-Z(x))2 and by plugging equation (2.2) into the expansion. < t 2 ( x ) = E (z(x)-z(x))2 |*(n) = E \2 Z W - X ^ U ) \x { n ) = Z X a>a i Co i>(x,, Xj) - a,'Cov(x, x,) + C0 The weights a\, a2,... are assigned such that minimizes the error variance. Therefore, it can easily be shown that the optimal weights are obtained from the following matrix equation: d a 2(x) da. [Cov(Xi, Xj ) ] {^ [a, ] (M|) = [C o v (x , x . ^ 99 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The average spatial o f the variance o f error within an area MSE(%'}, is defined as follows: Clearly deploying more sensors in the region reduces the error o f estimation. Then we may need to calculate MSE (El in an area as a function o f node density in that area. According to the result o f lemma 4 to calculate the error variance at any point x according to linear estimation method the locations o f II particles (which are the sensors),X(n),m ust be given whereas in random deployment the exact locations o f sensor nodes are unknown; however, we can calculate the bounds for MSE as follows The upper bound on MSE is calculated by estimating the value Z at each point x from the closest sensor (so-called 1 -nearest linear estimation). Indeed, 1-nearest is a special case of linear estimation according to the equation (2.2) when all weights are zero except the weight o f nearest point. L em m a 5: The error variance o f the linear estimation at a point x, < j 1 ( x ) , is bounded as follows where d„ is the distance between x and its closest sensor which is located at a point x„. P roof: Let o f (x) denote the error variance o f 1-nearest estimation then from lemma 4 we have cr2 (x) < erf (x) for any point x. MSE ((E) a 2(x )< C 0( l - a 2 c(d„)) 100 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Meanwhile: kx kx a h W = 2 Z a,aj C ov(x, , xj ) - 2 S a,Cov(x, x,) + C0 / - I 7=1 where a„ = and a, =0, V; * « then: « y-~r I 5 '-'n C T „2< X > = «„2C0 - 2a„C0 cov(x,x„) + C0 =Cn Covjx, xn) r v '-'o y s2'\ Lemma 5 enables us to derive an upper bound for MSE('% ). First it can easily be shown that: The following lemma and theorem compute the RHS o f the above equation. L em m a 6: Assume that each particle in n is center o f a disc with radius rs. Let V denote the area consisting o f all points o f <^(i.e. V c %) which are not covered by any disc then: £[WI] Ml Proof.- See [17]. Now we can present the proof for the main theorem. 101 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. T heorem 1: W hen using the linear estimation (cf. equation (2.2)) to estimates the value of parameters at every point x in region < % ,, MSE(<E) can be bounded as follows: w here: Dt(A) < MSE(<RS) ^ Dh(X) ( < i „ D, (A) = C0 1 - 2nX J ra2 c (r)dr V o d a 1 - 2kX | ra2 c (r)e~X K r dr Dh(A) = C0 Proof: Let Y(r„) = < jl2(x) = C0( l - a 2c(rn)) (0 < Y(rn) < C0) where r„ is the distance between point x and its closest sensor. Note that Y is a random variable which is a function of relative distance from closest sensor to any point x. Let FY (y) denote the cumulative distribution function (cd j) o f random variable Y. By considering the fact that ac(r) is a non increasing function o f r then we can proceed as follows: Pr { Y(r„) > y} = Pr{r„ > r} = - jJLM = «r Fr(y) = 1-e -A n r2 1 C ° MSE(<H) < ^ J E [a„2{ x ) ] h = J ydF r ( y ) I I I I JceO l. 0 = J C0 (l - a 2 (r)) 12nAre"x"r jcfr = Cn 1 - 2t t A jra2 c (r)e X xr dr Considering that ac(d) = 0 for d > d0 then we g e t: d a \ Dh(A) = C0 1 - 2nA | ra2 c (r)e~X x r dr 102 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. For deriving lower bound, D b we assume that the reading o f sensors within the neighbor set o f any p o in tx are spatially uncorrelated, i.e. Cov(Xj,x-) = 0, for i * j Then: a , = ■ Cov(x, xi) C„ <r,\x) = C0 1 - I x,eneigh(x) / \ 2 Cov(x,xt) r v '-'o y = C„ Let take Sc(x)= X xt eneigh(x) i - X y Xizneigh(x) where = |x - x (| ,\2 Cov(x, x,) The expected value o f <ScC) with respect to location of particles n which reside inside a disc centered at a point x and with radius d0 .It can easily be shown th a t: w 0 £ [Sc(x)} = |2^A ra2 c(r)<ir x(n) Therefore, MS£(30 > E [a ,2 (x)] = C0 1 - 2^2 J ra2 t. (r)dr Finally we get: Cn ' d 0 1 - 2^/1 J ra2c (r)dr \ o ■ N ( d , < M S E (^ )< C 0 l-2 n X ^ra2c(r)e~lnr dr AW AW 103 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Appendix B T heorem 4: gn(r » ,c a n be obtained as follows: g,(rr) = 2 a \ r W l ^ ’.-O lir1 e-w<~» («-!)! Proof: S (r’ ,r) recalls the intersection area o f a pie-like slice and a circle. The pie-like (or cone) is centered at distance r' from the origin while the circle is centered at origin with radius r (see Figure 3-5.) Note that at the origin a m icro-server is located if we compute gn {.) for sensors and the base is located if computation is done for micro-servers. Let N (S(r’ ,r)) denote number sensors inside S and rn denotes distance o f nth closest node from the origin inside the fan-out cone o f a node at radius r ’. By using properties of Poisson point process we can compute g„(.) as follows: P„(rr) ± P r{rn >r} = Pr{N(S(r \r)) < n} n — \ <= Point process property where * k\ _ Pr{r <rn< r + Ar} _ dPn(r',r) gnK Ar dr _ o h 9 I I S (r ',r)]| -^,||y(f',f) n — / l 0 U n - \ - C or 104 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. By looking at the geometry o f dS(r',r) which is dawn in Figure B -l it can easily be shown that: lim || S (r \ r + Ar) || - 1 | S(rr) ||= IrArB Ar~>0 ^ £ M ( ^ 0 ]| = 2rt9 dr where 6 = 6 (r\r) is the solution o f the following quadratic equation: I I S(r \r) ||= r a-rr'sinB + r B —f - tg~1 co)tg2 0 - 2tg 'a.tgd + ( - r J r Therefore, we get g j r » = ( n -l)\ Figure B-l. Geometry of dS(rr) _ 105 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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Malaki, Morteza
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Energy -efficient strategies for deployment and resource allocation in wireless sensor networks
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