O THE OPTIMAL SIZE OF DISTRIBUTED COOPERATIVE SYSTEMS KAM MU LOO G ATIO AL U IVERSITY OF SI GAPORE 2009 O THE OPTIMAL SIZE OF DISTRIBUTED COOPERATIVE SYSTEMS KAM MU LOO G A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY OF E GI EERI G DEPARTME T OF MECHA ICAL E GI EERI G ATIO AL U IVERSITY OF SI GAPORE 2009 Acknowledgements I am very thankful and deeply indebted to my supervisor, A.P. Gerard Leng for all the hints, suggestions and critics he has been providing throughout this research. Every time I felt like I was lost and not sure how to proceed, a short discussion with Gerard would bring my focus back. I am also grateful to have got to know a bunch of friends in the CosyLab: Tze Liang, Wei Lit, Han Yong and Azfar. The fun we had together made up for some of the frustrating moments. Last but no least, I would like to thank my family and Shiau Chien for keeping faith in me. The moral support and love you have been showering me with is indeed more than what words can describe. Summary The thesis aims to answer the question “What is the optimal size of a distributed system?”. A distributed system is broadly defined as a group of identical units executing a common task in a given operating area. Examples include swarm systems, multi-robot systems and multi agent systems. While the optimal system size is dependent on the task, operating area and chosen performance metrics, there are fundamental properties which are important for distributed systems. In the thesis, the optimal system size is studied from the perspective of four properties i.e.1) characteristics of the operating area, 2) connectivity of the units, 3) mutual interference effects between the units and 4) robustness of the system against noisy information. The first contribution of this research is the study on the effect of a concave operating environment on the connectivity of a distributed system. A novel concavity measure, termed as the blockage, is proposed and used to quantify the relative complexity of 2D concave shapes. A computer algorithm has been developed to evaluate this measure for complicated shapes. The deficiencies of a few other existing concavity measures will be contrasted and compared with the blockage measure. A general relationship relating the number of units required to ensure a certain connectivity probability will be given. The second major study proposes the existence of a power-law relationship between the communication range and the number of units from the connectivity perspective. Based on extensive simulation results, this power-law is shown to hold for distributed systems under a wide variety of mobility models and, most importantly, for small system size. In the third part of this research, the optimal swarm size is studied under the interference effect and robustness requirement. The discussion will start with the proposal of the use of the failure probability to measure the effect of hazards on a swarm. Then, another measure for the degradation in the performance as a function of swarm size is proposed. A team of imperfect swarming robots carrying out surveillance task was simulated and used to demonstrate the key ideas of this study. The fourth study aims at studying the advantage of larger system size in reducing the noise effect on the information being sensed. A stochastic model for a swarm using Master Equation is derived. With the key results of this study, the system size required to achieve a desired level of noise suppression can be predicted. Lastly, a short case study will be shown to demonstrate the applications of the results from the four major studies in this research. Insight to potential new research directions will be drawn as well. Table of Contents 1. Introduction 2. 3. 1.1 Overview 1.2 The Research 1.3 General Claims 1.4 Hypotheses 1.5 Structure of the Thesis 1.6 Contributions 11 Research Review 2.1 Artificial Intelligence 14 2.2 Distributed Intelligence 15 2.3 Swarm Intelligence and Swarm Robotics 16 2.4 Connectivity 18 2.5 Swarm Robustness 21 2.6 Stochastic Modelling 23 Connectivity of Distributed Systems in Concave Operating Environment 4. 3.1 Introduction 24 3.2 Blockage - a New Measure of Concavity 29 3.3 Analytical Evaluation of Blockage 31 3.4 Computer Algorithm to Compute Blockage 34 3.5 Blockage and Connectivity Probability 40 On the Power Law Relationship of the Critical Transmitting Range and the Number of Nodes of Ad Hoc Networks 4.1 Introduction 51 5. 6. 7. 4.2 Related Works 53 4.3 Rationale behind the Power Law 55 4.4 Empirical Evidence of the Power Law 59 4.5 Conclusion 73 On the Optimal Size of a Robust Swarm 5.1 Introduction 75 5.2 The Contradicting Effects of Large Swarm Size 77 5.3 Robust Swarm Size as an Optimization Problem 80 5.4 Theoretical vs Simulation Results 91 5.5 Conclusion 99 Stochastic Swarm Modelling 6.1 Experiment, Simulation and Mathematical Modelling 101 6.2 Swarm Modelling with Stochastic Master Equation 102 6.3 The Advantage of a Swarm in a System with Noisy Global Information 104 6.4 Relationship between System Size and Noise Magnitude 115 6.5 Simulation Results 126 Conclusion – General Framework for the Optimal Size 139 References 147 Appendix 154 CHAPTER Introduction 1.1 Overview Robotics research has gone through several phases since the first industrial robot being introduced by Unimate in the early 1960s. The study of the kinematics, dynamics and control of a robotic arm was hotly pursued at that time due to its industrial applications. The main focus was being placed at designing algorithms to achieve precise control such that tasks could be completed in a structured environment and under the supervision of an operator. However, it was soon realized that the ability to carry out a task in an unsupervised manner was the real potential of a robotic system. Autonomy was the answer to the need of unsupervised operation of robots. In order to make robots more versatile and adaptive to complicated tasks, the Artificial Intelligence and Robotics research fields have worked closely in search of true autonomy. While Robotics research focuses on the relationship between sensory inputs and actions, Artificial Intelligence research aims at understanding the ways to make computers exhibit intelligence [Ota, 2006]. Humanoid robots, unmanned ground/air vehicles (UAV, UGV) are some of the most exciting research fields which could potentially change the way humans live and work in the near future. There is a paradigm shift in the field of robotics. Instead of focusing on the development of the intelligence of a single robot, the pros and cons of multi-agent robot systems are being explored. The basic characteristics of a multi-agent system or equivalently, distributed autonomous system, are: 1) inability to solve the problem by each agent alone, 2) fully decentralized control, 3) decentralized data and 4) asynchronous computation. A multi-agent robot system has been studied for a wide range of tasks. These tasks can be classified into several categories according to the objectives and nature of the tasks, see [Parker, 2008] for a review of the common classifications of the different mission types. Searching for an object, covering a given area, transporting an object cooperatively from one location to another are some of the more common tasks being studied. In their research work of Simultaneous Localization and Mapping (SLAM) [Durrant-Whyte and Bailey, 2006] using multi-robot system, Durrant-Whyte has demonstrated that it is possible to explore an unknown environment and build a map at the same time by using an EKF (Extended Kalman Filter). The traditional multi-robot system research focuses on the motion-planning algorithms and exploration algorithms. These studies tend to increase the complexity of each robot in order to acquire more accurate sensory inputs from the environment for the more sophisticated algorithms. On the other hand, swarm intelligence, a relatively new research direction, is getting more attention of late. Inspired by the biological swarms found in the nature, e.g. bees and ants, swarm intelligence aims to reproduce the collective intelligence exhibited by the biological swarms. A robotic swarm system is similar to the multi-agent robot system, with the key difference being the emphasis on the system size and unit simplicity. Some researchers treat swarm size (number of units) as the distinguishing factor between multi-agent system (small scale) and a swarm system (large scale). The other key defining characteristic of a swarm is the simplicity of each robotic unit. “Ants aren’t smart. Ant colonies are.” This simple quote from Deborah M. Gordon, a biologist at Stanford University, highlights the interesting property that the swarm intelligence is an emergent characteristic, resulting from the collective actions of a group of simple agents. Although the multi-agent system is promising, there are significant challenges to be overcome. The major challenges of multi-agent systems are design related. How does one decide on the number of agents/units to be deployed for a particular task? How the different sizes of a multi-agent system affect the performance of the team as a whole? Should a multi-agent system be homogenous or heterogenous? Some of these questions have been answered in parts. In [Mei et al, 2004], Mei et al looked at this problem from the perspectives of energy constraints, i.e. they examined the relationships between the optimal number of nodes required to serve random requests and the given energy constraints. Hayes [Hayes 2002] defined a cost function which relates the number of robots (sensors), time taken to complete the search task and the moving speed of each robot. After that, he optimized the number of robots required for a search task by determining the minimum point of the cost function. The interference effect among the agents has also been studied using foraging, searching mission setup [Lerman and Galystyan, 2002]. As expected, agents will compete for space within a confined region, which increases the time spent by the agents in avoiding each other, rather than doing useful work. As emphasized above, a multi-agent system acquires its “intelligence” from the collective actions of all the agents. This emergent characteristic is elegant, but from the design perspective, it is hard to “reverse-engineer” a particular desired behaviour of the performance requirements for a multi-robot system which information needs to be passed. In Chapter 5, it was shown that a power law model can capture the relationship between communication range (r), system size ( ) and connectivity probability (P). Given communication range r (where r is normalized against the operating area radius), the following power law model can be used for a multi-robot system searching in a random manner within a circular operating area: r= a1 P + a2 − b1 P + b2 (7.4) The first constraint for a multi-robot system can be written as: P( ) > Pi ,min ⇒ P( ) − Pi ,min > , where P( ) is found by solving equation 7.4. 7.2.2 Failure Probability Different reasons can be contributing to the failure of an individual member of a swarm and ultimately leads to the breakdown of the swarm system. Among many other reasons, an individual robot of a swarm team could fail due to motor failure, sensors failure, and control system failure. In addition, each robot of a multi-robot system being deployed to carry out hazardous mission has a chance to be eliminated due to the nature of the mission (e.g. de-mining robots, surveillance in a hostile zone). This requirement about the critical number of active units can be translated to a constraint in the probability space. In addition, the missions require at least cr units to be functional so that the team can operate and carry out the mission. In other words, if the number of active units at time t < Tm (Tm is the total mission time) is less than cr, the 142 mission is considered a failure. The constraint is constructed by looking at the probability of mission failure: Fmax < Fallow k= cr −1 ∑ Ck [1 − Pmax ] [ Pmax ] k −k < Fallow , (7.5) k =0 where Fmax is a binomial expression with parameters ( , Pmax) and it sums up the probabilities of the swarm having to cr-1 active units. Pmax is the failure probability of one of the robots. Fallow is the maximum allowable failure probability that a given mission can tolerate. 7.2.3 Interference – Collision Probability The most obvious of reasons behind the negative impact is the interference among swarm units. In a series of works by Ijspeert, Martinoli, Lerman, Galstyan [4, 9-11], a probabilistic modelling methodology was applied to a variety of multiple robots problems (searching, foraging, puck-picking) and the interference effect was studied. It was found that increasing the swarm size beyond a certain number would “eventually lead to a diminution of the collaboration rate due to overcrowding and excessive interference”. Consider a swarm in which the controller is designed such that each unit chooses a new random direction to move in every time step and with a constant speed (this will be termed an ideal random controller from now on). Define Ar as the detection area of each unit and AT as the mission area. The probability that an active unit will detect (run into) another unit is equal to the detection ratio rd = Ar (neglecting the border effect). The AT probability of a swarm unit running into at least one other unit can be written as: 143 C = − P[not colliding with any other unit] = − [1 − rd ] −1 The constraint on the maximum swarm size such that the performance of the swarm will not be too negatively affected by interference can now be written as: C < Callow − [1 − rd ] −1 < Callow (7.6) 7.3 The Optimal System Size of Multi-robot System for a Search Mission In this section, the optimal system size of a multi-robot system will be determined by considering the three constraints listed in 7.2. Consider a multi-robot system consists of identical units being deployed in a circular operating area of radius R. Each robot is equipped with communication module of communication range rcom. The robots will carry out collision avoidance manoeuvre when they find the distance to the nearest neighbour is shorter than rcol. Each robot has a probability of Pmax to fail due to hardware and/or mission nature. To guarantee the successful completion of a search mission, the system size needs to be optimized against the following constraints: 1. Type constraint: Connectivity probability has to be greater than 0.6. P( ) > Pi ,min = 0.6 2. Type constraint: Mission failure probability has to be smaller than 0.01. F ( ) < Fmax = 0.01 144 3. Type constraint: Collision probability has to be smaller than 0.6. C ( ) < Cmax = 0.6 The fitness function for this multi-robot system can be written as: ( ) ψ ( ) = ∏ U ϕ Ci ,cr , Ci ( )  ∗ ϕ Ci ,cr , Ci ( )  i = [Cmax − C ( )][ P( ) − Pmin ][ Fmax − F ( )] (7.7) × U [ P( ) − Pmin ]U [ Fmax − F ( ) ]U [Cmax − C ( )] where P( ), C( ) and F( ) are found using the following equations: rcom = a1 P + a2 − b1P + b2 C ( ) = − [1 − rcol ] k= F( ) = cr −1 ∑ −1 Ck [1 − Pmax ] [ Pmax ] k −k k =0 Figure 7.1 shows the plot of the fitness function evaluated with the following parameter values: rcol = 0.02, rcom = 0.5, Pmax = 0.1, cr = 5. The optimal swarm size is the value which maximizes the fitness function. 145 0.00020 0.00015 0.00010 0.00005 20 30 40 50 60 Figure 7.1: Fitness function (equation 7.7) plotted against swarm size with : rcol = 0.02, rcom = 0.5, Pmax = 0.1, cr = 5. 146 References: Ackermann, J., Robust Control: The Parameter Space Approach Series: Communications and Control Engineering , 2nd ed., XIII, 483 p. 321 illus., Hardcover ISBN: 978-185233-514-4. 2002. Beni, G., From Swarm Intelligence to Swarm Robotics. Swarm Robotics WS 2004, LNCS 3342, pp. 1–9, 2005. Bettstetter C. and Zangl J., How to Achieve a Connected Ad Hoc Network with Homogenous Range Assignement: An Analytical Study with Consideration of Border Effects, Mobile and Wireless Communications Network, 4th International Workshop. 2002. Bettstetter C., On the Connectivity of Wireless Multihop Networks with Homogeneous and Inhomogeneous Range Assignment, Vehicular Technology Conference, 2002. Proceedings. VTC 2002-Fall, IEEE 56th. 2002 b. Bettstetter C., On the Minimum Node Degree and Connectivity of a Wireless Multihop Network, MOBIHOC, EPF,Switzerland. 2002 c. Camp T., Boleng J., Davies V., Mobility Models for Ad Hoc Network Simulations, Wireless Comm. & Mobile Computing (WCMC), Special Issue on Mobile Ad Hoc Networking, 2002. 147 Costa L. F., Cesar Jr. R. M. , Shape Analysis and Classification: Theory and Practice, CRC Press LLC. 2000. Dousse O., Baccelli F. and Thiran P., Impact of Interferences on Connectivity in Ad Hoc Networks, IEEE/ACM Transactions on Networking, Vol. 13, No. 2. 2005. Durfee, E. H., Lesser, V. R., Corkill, D. D. Trends in cooperative distributed problem solving. IEEE Trans. Knowl. Data Eng. KOE-11(1), 63-83. 1989. Durrant-Whyte, H., Bailey, T. Simultaneous Localisation and Mapping (SLAM): Part I The Essential Algorithms, Robotics and Automation Magazine 13, pp. 99–110, 2006. Feller, W. "Law of Large Numbers for Identically Distributed Variables." §7.7 in An Introduction to Probability Theory and Its Applications, Vol. 2, 3rd ed. New York: Wiley, pp. 231-234. 1971. Gasser, L. Social conceptions of knowledge and action: DAI foundations and open systems semantics. Artificial Intelligence. 47, 107-138. 1991. Gerkey B. P., Vaughan R. T., Howard A., The Player/Stage Project: Tools for Multi-Robot and Distributed Sensor Systems, Proceedings of the International Conference on Advanced Robotics (ICAR 2003), pg 317-323, 2003. 148 Gupta P. and Kumar P. R., “Critical Power for Asymptotic Connectivity in Wireless etworks”, pp. 547-566, in Stochastic Analysis, Control, Optimization and Applications: A Volume in Honor of W.H. Fleming. 1998. Hayes A. T., How Many Robots? Group Size and Efficiency in Collective Search Task, Proceedings of the 6th International Symposium on Distributed Autonomous Sensoric Systems, pp.289. 2002 Ijspeert A. J. et al, Collaboration Through the Exploitation of Local Interactions in Autonomous Collective Robotics: The Stick Pulling Experiment Autonomous Robots 11, 149–171. 2001 Jain N. and Agrawal D. P., Current Trends in Wireless Sensor Network Design, International Journal of Distributed Sensor Networks, 1: 101–122. 2005. Ji L., L. L. H. Andrew, H. F. Chuan, M. Zukerman, M. F. Neuts, Meeting Connectivity Requirements in a Wireless Multihop Network, IEEE Communications Letters, Vol. 10, No. 1, Jan 2006. Krishnamachari B.y, Wicker S. B. and Bejar R., Phase Transition Phenomena in Wireless Ad Hoc Networks, Global Telecommunications Conference, GLOBECOM '01, IEEE. 2001. 149 Lerman K. and Galstyan A., A General Methodology for Mathematical Analysis of Multi-Agent Systems, USC Information Sciences Technical Report ISI-TR-529. 2001. K. Lerman, A. Galstyan, A. Martinoli, A. Ijspeert, “A Macroscopic Analytical Model of Collaboration in Distributed Robotic Systems”, Artificial Life 7, pp. 375-393, 2001. Lerman K. and Galystyan A. Mathematical Model of Foraging in a Group of Robots: Effect of Interference, Autonomous Robots 13, pp. 127–141. 2002. Madsen T. K., Fitzek F. H. P. and Prasad R., Connectivity Probability of Wireless Ad Hoc Networks: Definition, Evaluation, Comparison. Wireless Personal Communications, Volume 35, Numbers 1-2, pp. 135-151(17). 2005 Martinoli A. et al, A Macroscopic Analytical Model of Collaboration in Distributed Robotic Systems, Artificial Life 7: 375–393. 2001. Mei Y.G., Lu Y.H., Hu C., and Lee C.S., Determining the Fleet Size of Mobile Robots with Energy Limitations, in Proceedings of Intelligent Sensors & Systems, pp. 1420. 2004. Ni J. and Chandler S., Connectivity properties of a random radio network, Proceedings of the IEE – Communications 141, 289–296. 1994 150 Ota, J. Multi-agent robot systems as distributed autonomous systems, Advanced Engineering Informatics, 20, pp 59–70. 2006. Parker, L. E. Multiple Mobile Robot Systems, in Springer Handbook of Robotics, pp. 921-941. Springer Berlin Heidelberg, 2008 Penrose M. D., The longest edge of the random minimal spanning tree, Ann. Appl. Probab. Volume 7, Number 2, 340-361. 1997. Press W. H., Flannery B. P., Teukolsky S. A. & Vetterling W. T., Numerical Recipes in C, Cambridge University Press. 1992. Rőmer K., Mattern F., The Design Space of Wireless Sensor Networks, IEEE Wireless Communications. 2004. Royer E. and Toh C. K., A review of current routing protocols for ad hoc mobile wireless networks, IEEE Personal Communications, 6(2), pp. 46-55. 1999. Sahin, E. Swarm Robotics: From Sources of Inspiration to Domains of Application. Swarm Robotics WS 2004, LNCS 3342, pp. 10–20. 2005. 151 Santi P., Blough D. M., The Critical Transmitting Range for Connectivity in Sparse Wireless Ad Hoc Networks, IEEE Transactions on Mobile Computing, Vol. 2, No 1, Jan 2003. Santi P., The Critical Transmitting Range for Connectivity in Mobile Ad Hoc Networks, IEEE Transactions on Mobile Computing, Vol. 4, No 3, May 2005a. Santi P., Topology Control in Wireless Ad Hoc and Sensor Networks, John Wiley & Sons, Ltd, 2005b. Trianni V., Evolutionary Swarm Robotics, Studies in Computational Intelligence (SCI) 108, 23–46. 2008 Weisstein, Eric W. "Circle-Circle Intersection." From MathWorld -- A Wolfram Web Resource. http://mathworld.wolfram.com/Circle-CircleIntersection.html Winfield, A.F.T. and Nembrini J., Safety in Numbers: Fault Tolerance in Robot Swarms, Int. J. Modelling Identification and Control, Vol. 1, No. 1, pp 30-37. 2006. Xue F. and Kumar P.R., The number of Neighbors Needed for Connectivity of Wireless Networks, Wireless Networks 10, 169-181, 2004. 152 Zhu J. and Papavassiliou S., On the Connectivity Modeling and the Tradeoffs between Reliability and Energy Efficiency in Large Scale Wireless Sensor Networks, Wireless Communications and Networking, 2003. WCNC 2003 IEEE. 2003. 153 Appendix 1: Chain codes of concave shapes studied The details about chain codes can be found in [Costa and Cesar Jr, 2000]. The chain codes of all the concave shapes are listed in Table A. The x-y coordinates of the vertices for the 3-room flat are included in Table A as well. Table A: Chain codes of the various concave shapes. Shape Absolute Chain Codes L1 0, 0, 2, 4, 6, L2 0, 0, 0, 2, 4, 2, 4, 2, 4, 6, 6, Z1 0, 0, 2, 4, 2, 2, 4, 4, 6, 0, 6, Z2 0, 0, 0, 2, 4, 4, 2, 4, 4, 6, 0, F1 0, 2, 2, 0, 2, 4, 2, 0, 2, 4, 4, 6, 6, 4, 4, 6, 0, 0, 6, F2 0, 0, 0, 2, 4, 2, 0, 2, 4, 4, 6, 6, 4, T 0, 2, 0, 2, 4, 4, 4, 6, 0, U 0, 0, 0, 2, 2, 4, 6, 4, 6, 4, 6, + 0, 2, 0, 2, 4, 2, 4, 6, 4, 6, 0, C 0, 0, 0, 0, 0, 2, 2, 4, 6, 4, 4, 4, 2, 2, 2, 0, 0, 0, 6, 0, 2, 2, 4, 4, 4, 4, 4, 6, 6, 6, 6, X1 0, 2, 0, 2, 4, 2, 4, 4, 6, 0, 6, X2 0, 2, 4, 1, 3, 4, 6, 0, 5, X3 0, 2, 2, 0, 2, 4, 4, 4, 2, 2, 4, 6, 6, 4, 6, 0, 0, 0, 6, X4 0, 0, 0, 2, 2, 0, 2, 4, 4, 6, 6, 4, 2, 4, 6, E 0, 0, 0, 2, 4, 4, 2, 0, 0, 2, 4, 4, 2, 0, 0, 2, 4, 4, 4, 6, 6, 6, 6, G 0, 0, 0, 0, 0, 2, 2, 2, 4, 4, 6, 0, 6, 4, 4, 4, 2, 2, 2, 0, 0, 0, 0, 2, 4, 4, 4, 4, 4, 6, 6, 6, 154 6, Vert. x y Vert. x y Vert. x y 12 9.1 17.1 23 7.2 11.8 13 9.1 17.3 24 11.8 6.4 14 11.6 17.3 25 19.6 7.2 6.4 15 11.6 17.1 26 19.6 3- 7.2 16 11.1 17.1 27 9.8 Room 13.6 17 11.1 12.1 28 7.2 9.8 13.6 11.9 18 13.6 12.1 29 7.2 8.2 9.1 11.9 19 13.6 19.6 30 8.2 9.1 12.1 20 12.7 19.6 31 9.6 10 10.7 12.1 21 12.7 21.6 32 9.6 11 10.7 17.1 22 7.2 21.6 Appendix 1.2: Evaluating discrete curvature. To evaluate discrete curvature, first define discrete directions. An example of a set of discrete directions can be defined as: * 155 In the figure above, “*” represents a vertex. If the next vertex is in the horizontal direction and to the right, the discrete direction is for this vertex. If the next vertex is to the top of current vertex, the discrete direction is 2. Thus, as an example, for a L shape, the discrete direction can be written as: D(i) = {0, 0, 2, 4, 2, 4, 6, 6, 0} With D(i), one can draw out a L-shape by following the discrete direction as defined in the above figure. Now, the discrete curvature, k(i), is defined as: k(i) = D(i+1) – D(i) 156 Appendix 2: Simulating constant individual failure rate Consider a system simulated in discrete time, i.e. time k =0, 1, 2…, each robot has a fixed probability of failing (becomes inactive), f(k) = C = constant, < C < 1. If robots are active at time k=0, the amount of active robots as a function of time k is: (k ) = (1 − C )k The ratio of total active robots for the whole system at time k is thus: Q (k ) = (k ) = (1 − C ) k , while the overall failure rate of the system, P(k), is P(k ) = − Q ( k ) = − (1 − C ) k One can set the failure rate C by looking at the overall failure rate at a particular time TM. If it is required that the failure model to give P(Tm) = p, C can be solved: P(TM ) = p = − (1 − C )TM C = − (1 − p ) TM In the simulations done (as discussed in Section 5.4), it was intended to simulate the constant failure rate of swarm units. The simulations were done in continuous time with the failure of individual swarm unit checked two times every second (2Hz). For a total simulation time of 50 seconds, it corresponded to Tm = 50x2 = 100 loops. The overall failure at the end of the simulation time is set to be 50%, i.e. P(Tm) = 0.5. With these settings, the value of C is found to be: C = − (1 − p ) simulations. TM = − (1 − 0.5) 100 = 6.9075 × 10−3 , which is the value used in the 157 [...]... incorporating the information from nearby swarm units, the overall noise effect can be suppressed The relationship of the swarm size and the degree of dispersion of the swarm units around a target cell is shown too 23 Chapter 3 Connectivity of Multi Agent Systems in Concave Operating Environment The connectivity of a distributed network within buildings depends critically on the shape of the operating environment... information flow among the units is vital In order to facilitate the exchange of information, the distributed system needs to be in connected status most of the time, i.e high connectivity probability is desired There are a lot of factors affecting the connectivity property of a network The number of nodes, the communication range and the mobility are all important factors which ultimately determine the connectivity... concavity of operating environment (Chapter 3), connectivity of swarm under different mobility models (Chapter 4) and robustness of the swarm (Chapter 5) 8 Minor Hypothesis: A probabilistic model based on the Master Equation can be used to study the relationship between the system size and the reduction of noise effect (Chapter 6) 1.5 Structure of the Thesis There are a total of 7 chapters in this thesis,... environment A 2D shape is convex if a line connecting any two points on the boundary of the shape does not cut the boundary at a third point On the other hand, a 2D shape is concave if there are certain points on the boundary when connected with a straight line, will cut the boundary at least one more time The rationale of studying a concave environment is that any realistic operating environment (e.g a housing... facilitate the computation of the blockage of a concave polygon Results relating the blockage values and number of sensors required for high connectivity probability for various concave polygons will be shown Comparisons with other shape measures will also be made to show the relationship of blockage measure with the connectivity probability of sensors inside concave areas 3.2 Blockage – a new measure of concavity... arriving at the optimal size would ease the design and implementation of swarm or distributed system significantly 17 2.4 Connectivity A multi-agent system or distributed network is said to be connected if there exists at least one communication path between all the units or nodes The connectivity of a distributed network is measured by the connectivity probability This is defined as a measure of the total... developed in percolation theory [Dousse et al 2005] and geometric random graph theory [Santi 2005a] to describe this phenomenon The research in the connectivity property problem is characterised by the heavy reliance on the simulation results This is mainly due to the complexity of the interactions among the nodes in a distributed network The complexity of the problem is compounded 20 by the randomness introduced... individual sensors of the network The rationale is that the network 25 can maximize the utilization of the information gathered by every node through information sharing This optimization problem is central to the ad-hoc networks [Royer and Toh, 1999] research community In the ad-hoc network connectivity problem domain, the interest is in finding out how easily connections can be established among the individual... miniaturization poses constraints on the complexities and capabilities of each swarm unit In this research work, the primary problem being tackled is the optimal size of a swarm system Optimization is the process of finding the “right values” given performance requirements and/or constraints Thus, studying the optimal swarm size would also help providing insights to the ways of quantify and characterize the. .. than the currently accepted log-law relationship (Chapter 4.4) v The parameters α and β of the power law r = α −β can be used to quantify and study the effects of the different settings of a distributed network, e.g pause time (Chapter 4.5) vi The size of a distributed network can be optimized with respect to the connectivity property, having known the empirical relationship as found from Contributions . between the units and 4) robustness of the system against noisy information. The first contribution of this research is the study on the effect of a concave operating environment on the connectivity. O THE OPTIMAL SIZE OF DISTRIBUTED COOPERATIVE SYSTEMS KAM MU LOOG ATIOAL UIVERSITY OF SIGAPORE 2009 O THE OPTIMAL SIZE OF DISTRIBUTED COOPERATIVE SYSTEMS. 4.3 Rationale behind the Power Law 55 4.4 Empirical Evidence of the Power Law 59 4.5 Conclusion 73 5. On the Optimal Size of a Robust Swarm 5.1 Introduction 75 5.2 The Contradicting