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Modeling and analysis of cooperative search systems
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ABSTRACT: The analysis of performance gains arising from cueing in cooperative search systems with autonomous vehicles has been studied using Continuous Time Markov Chains; where the search time distributions are assumed to follow the exponential distributions. This work proposes the use of Petri Nets to model and analyze these systems. The Petri Net model considers two types of autonomous vehicles: a searchonly vehicle and n searchengage vehicles. Specific performance metrics are defined to measure system's performance. Through simulation, it is shown that the search time with stationary targets and cues that provide exact target location follows a triangular distribution. A methodology for approximating general distributions and incorporating them into the Petri Net model for performance analysis is presented.
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Advisor: Ali Yalcin, Ph.D.
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Petri Net
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Target detection
UAVs
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Modeling and Analysis of Cooperative Search Systems by Carlos A. Portilla A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Industrial Engineering Department of Industrial and Management Systems Eng ineering College of Engineering University of South Florida Major Professor: Ali Yalcin, Ph.D. David Jeffcoat, Ph.D. Bo Zeng, Ph.D. Date of Approval July 8, 2010 Keywords: Petri Net, cueing, simulation, target det ection, UAVs Copyright 2010, Carlos A. Portilla
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Dedication This thesis is dedicated to my loving parents Jorge and Dorys for instilling the importance of hard work and higher education; they are an example to follow. Last but not least, it is dedicated to my brother Jorge for always supporting me in every possible way.
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Acknowledgements This thesis would not be possible without the inval uable help of many people. First and foremost, I would like to thank my thesis supervisor Dr. Ali Yalcin for his guidance and support during all these years. I am very grateful for the help and advice from my thesis committee Dr. Jeffcoat and Dr. Zeng. Finally, I want to thank my family th at has been the major motivation in my life and my friends for all their support.
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i Table of Contents List of Tables ................................... ................................................... .......................... iii List of Figures .................................. ................................................... ......................... iv Abstract ........................................ ................................................... ............................ vi Chapter 1: Introduction .......................... ................................................... .....................1 1.1 Search Theory ................................ ................................................... ................1 1.2 UAV Applications and Cooperative Search Systems .......................................1 1.3 Cueing ....................................... ................................................... .....................2 1.4 Motivation ................................... ................................................... ...................3 1.5 Research Objective ............................ ................................................... .............4 1.6 Proposal Organization ........................ ................................................... ............4 Chapter 2: Literature Review ...................... ................................................... .................5 2.1 Search Theory ................................. ................................................... ................5 2.2 Petri Nets ................................... ................................................... .....................8 2.2.1 Motivation for the Use of Petri Nets to Model and Analyze UAV Systems ....................................... ..............................................8 2.2.2 Formal Definition and Basic Terminology of Pe tri Nets ..................9 2.2.2.1 Reachability Set an d Reachability Graph ..........................11 2.2.2.2 Stochastic Timed Pe tri Nets (STPN) ................................11 2.2.2.3 Generalized Stochas tic Petri Nets (GSPN) .......................11 Chapter 3: Problem Description .................... ................................................... .............13 3.1 SystemÂ’s Description .......................... ................................................... ..........13 3.2 Petri Net Model .............................. ................................................... ..............15 3.3 Analytical Solution ........................... ................................................... ............18 3.3.1 Eliminating the Vanishing Marking s ................................................1 8 3.4 SystemÂ’s Performance Measures ................. ................................................... .21
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ii 3.4.1 First Passage Times in CTMCs .... ................................................... .21 3.4.2 Transient Analysis: Uniformizatio n ................................................ .25 Chapter 4: General Distributions ................. ................................................... ..............30 4.1 Simulation Description ........................ ................................................... .........30 4.1.1 Assumptions for the Model .................. ............................................32 4.2 Simulation Results ............................ ................................................... ............32 Chapter 5: General Distributions Analysis in the Pe tri Net Model ..............................36 5.1 Analysis of General Distributions ............ ................................................... ....36 5.2 Incorporating General Distributions into the Pr oposed Model .......................39 Chapter 6: Stochastic Petri Net Package (SPNP) ... ................................................... ...45 6.1 SPNP Description .............................. ................................................... ...........45 6.2 SPNP and Petri Net Model Validation ........... .................................................46 Chapter 7: Contributions and Future Research Direct ions ...........................................51 7.1 Contributions ................................. ................................................... ...............51 7.2 Future Research Directions ................... ................................................... .......54 References ........................................ ................................................... ..........................56
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iii List of Tables Table 1 Â– Transition rates/weights of Figure 6 ... ................................................... .......18 Table 2 Â– Detection rates and cueing weight ...... ................................................... ......24 Table 3 Â– Expected time to engage n targets with one searchonly vehicle and two searchengage vehicles ........ ................................................... ................24 Table 4 Â– Histograms of the distributions of the ti me associated with each search process .................... ................................................... .........................33 Table 5 Â– P values of the chi square test for the time of the cued search process ........34 Table 6 Â– Goodness of fit tests ................... ................................................... ................42 Table 7 Â– Transition rates/weights of Figure 19 .. ................................................... ......46
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iv List of Figures Figure 1 Â– Petri net example ..................... ................................................... .................10 Figure 2 Â– Searchonly vehicle states ............ ................................................... ............14 Figure 3 Â– Searchengage vehicle states .......... ................................................... ..........15 Figure 4 Â– Petri net searchonly vehicle .......... ................................................... ...........16 Figure 5 Â– Petri net searchengage vehicle ....... ................................................... .........16 Figure 6 Â– One searchonly vehicle and two searche ngage vehicles ...........................17 Figure 7 Â– Reachability graph of the GSPN of Figure 6 ..............................................1 9 Figure 8 Â– CTMC rate diagram of the GSPN of Figure 6 ............................................20 Figure 9 Â– Transition rate matrix of the CTMC for o ne searchonly vehicle and two searchengage vehicles ... ................................................... .............22 Figure 10 Â– Transition rate matrix of the CTMC for one searchonly vehicle and two searchengage vehicles ( rates Table 2) .................................... .....26 Figure 11 Â– P matrix for one searchonly vehicle an d two searchengage vehicles (rates Table 2) ...... ................................................... .....................27 Figure 12 Â– P matrix for one searchonly vehicle an d two searchengage vehicles at t = 0.5 (rates Table 2) ............................................... ................28 Figure 13 Â– Expected number of targets engaged with two searchengage vehicles ...................... ................................................... .............................29
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v Figure 14 Â– Expected number of targets engaged with two improved searchengage vehicles ............... ................................................... ........................29 Figure 15 Â– Simulation environment ............... ................................................... ..........31 Figure 16 Â– Erlangk approximation of the triangula r distribution (2, 10, 18) .............41 Figure 17 Â– Petri net model for a searchengage veh icle with Erlangk approximation ................. ................................................... ........................43 Figure 18 Â– One searchonly vehicle and two searchengage vehicles with Erlangk approximation ........ ................................................... ..................44 Figure 19 Â– Petri net model with one searchonly ve hicle and five searchengage vehicles in SPNP ....... ................................................... .................47 Figure 20 Â– Probability of detection of all searchengage vehicles varying l ..............48 Figure 21 Â– Probability of detection of all searchengage vehicles varying k .............48 Figure 22 Â– Marking with targets engaged by all sea rchengage vehicles ...................49 Figure 23 Â– Probability of detection of all searchengage vehicles varying l (Petri net) ................... ................................................... .............................50 Figure 24 Â– Probability of detection of all searchengage vehicles varying k (Petri net) ................... ................................................... .............................50 Figure 25 Â– Petri net model with cueing and without cueing .......................................52 Figure 26 Â– Petri net model with 2 actives searche ngage vehicles ..............................53
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vi Modeling and Analysis of Cooperative Search Systems Carlos A Portilla Abstract The analysis of performance gains arising from cuei ng in cooperative search systems with autonomous vehicles has been studied u sing Continuous Time Markov Chains; where the search time distributions are ass umed to follow the exponential distributions. This work proposes the use of Petri Nets to model and analyze these systems. The Petri Net model considers two types of autonomous vehicles: a searchonly vehicle and n searchengage vehicles. Specific perf ormance metrics are defined to measure systemÂ’s performance. Through simulation, i t is shown that the search time with stationary targets and cues that provide exact targ et location follows a triangular distribution. A methodology for approximating gener al distributions and incorporating them into the Petri Net model for performance analy sis is presented.
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1 Chapter 1 Introduction 1.1 Search Theory Search theory is one of the oldest areas of operati ons research and encompasses all the models and algorithms that refer to the pro blem of finding a hidden target (L. D. Stone 1992). It uses the principles and methods of operations research to resolve search problems. The search scenarios that have been stud ied in search theory are: single searcher, cooperative search and coordinated search In single searcher, there is basically one entity performing the search or the exploration Cooperative and Coordinated search involve more than one entity working toward a goal in which there is a common interest or reward (Cao, Fukunaga and Kahng 2004). Coordinat ed search implies that there is collaboration between the entities. The scenario studied in this research involves cooperative search. The entities work together to cover the search area faster but there is no coordination among them to search and engage tar gets. 1.2 UAV Applications and Cooperative Search Systems The entities that will be considered throughout thi s research are unmanned autonomous vehicles (UAVs). UAVs are robots which c an perform tasks without continuous human guidance. UAVs are becoming increa singly prevalent; their use has increased exponentially over the last decade (Oracl e Corporation 2007). UAVs have
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2 applications on land, in sea, and in the air. They are used for a broad range of applications, including police observation of civil disturbances, for work and measurement in radioactive environments, and reconn aissance support in natural disasters. Some missions are better performed by a team of UAV s instead of only one single agent. For instance a dangerous and/or extensive mi ssion, where it is unlikely that a single UAV survives to complete the task, a team is more suitable to perform it. In addition, research has shown that searching a parti cular area can be completed more quickly using multiple UAVs (Cole, et al. 2009). In all cases, cooperation among the UAVs is required for efficient and/or successful co mpletion of the mission. 1.3 Cueing The type of cooperation that will be studied in thi s research is cueing. In cooperative search applications, cueing is defined as any information that provides focus to a search; such as limiting the search area or pr oviding a search heading (D. Jeffcoat 2004). Research has shown that cueing can significa ntly improve the probability of locating targets in cooperative search applications over a fixed period of time (Jeffcoat, Krokhmal and Zhupanska 2007). In addition, experien ce in Kosovo showed that cueing enhances battle space awareness by making UAVs much more efficient and survivable. The information transmitted in the cue let a UAV kn ow where to look and thus decrease wasted surveillance time. In addition, it reduces t he exposure to point air defenses of the UAVs, making them more survivable (Bingham 2001).
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3 1.4 Motivation There are many situations in which search processes might be facilitated via cueing. For example: a person, choosing from the m enu in a restaurant, can receive additional information from the waiter to find the desired food. Or a student, looking up an article, can be helped by a librarian who points out the correct database. Or a team of UAVs, looking for survivors after a hurricane, migh t receive data about the possible location of the targets from another vehicle with s uperior capabilities. All these search processes have something in common. First there is an entity looking for something, and then it either finds it or receives additional info rmation that will expedite the search process (cueing). The motivation of this work comes from the need to easily characterize these search processes and to measure systemÂ’s perf ormance. The existing research in quantifying the performanc e gains arising from cueing utilizes continuous time Markov chains (CTMC) to mo del and analyze the system under study (Alexander and Jeffcoat 2007). CTMCs are a st ate orientated modeling formalism which requires the modeler to determine the state s pace of the complete system and assign transition probabilities between each of the se states as a part of the modeling effort. This is viable for small cooperative search systems (Jeffcoat, Krokhmal and Zhupanska 2007); however, it is not practical for l arge systems due to the difficulty in visualizing apriori the interaction among all the c omponents and determining its state transition probabilities. In addition, the sojourn time in CTMC is restricted to the exponential distribution. This work proposes the use of Petri Nets to model a nd analyze search processes. Petri Net formalism allows us to visualize the stru cture of the rulesbased system, making
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4 the model easier to understand, and to express the behavior of the system in mathematical forms (Lundell, Tang and Nygard 2005). Finally, Pet ri Nets and its extensions (general stochastic PNs (GSPN), extended SPNs (ESPNs) and de terministic SPNs (DSPN)) offer an activityoriented formalism that facilitate the use of discrete event system analysis tools, including simulation and numerical analysis, to study the performance of systems. 1.5 Research Objective In this thesis, we: Develop a Petri Net model to quantify the performan ce gains from cueing in cooperative search systems. Define and analyze systemÂ’s performance measures fo r the proposed model 1.6 Proposal Organization The rest of the proposal is organized as follows: C hapter 2 reviews the previous work in literature concerning search theory and per formance gains due to cueing in cooperative search systems. Additionally, it introd uces the theoretical foundations of Petri Nets Chapter 3 describes the specific problem to be addr essed and defines the performance measures to be used. Chapter 4 argues t he relevance of general distributions in the problem addressed, and Chapter 5 shows how t o analyze them in the proposed Petri Net model. Chapter 6 introduces software to analyze Petri Net models. Finally, Chapter 7 summarizes the contributions and outlines the futur e research directions.
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5 Chapter 2 Literature Review This chapter is divided into two sections. Section 2.1 defines search theory and presents several problems that have been addressed in this area throughout the years. In addition, it presents the research done concerning the performance gains due to cueing in cooperative search systems. Section 2.2 introduces the theoretical foundations of Petri Nets and its extensions. 2.1 Search Theory Research in search theory was initiated during the Second World War by Bernard Koopman; who derived the probability of detection a s a function of time, and studied the optimal allocation of search effort to detect a sta tionary target (Verkama 1996). Many applications of search theory have been orientated towards research with autonomous vehicles (Cole, et al. 2009), (Schultz, Parker and Schneide 2002), (Chandler and Pachter 2002). Depending on the level of human interaction, there are three types of autonomous vehicles (Committee on Autonomous Vehicles in Suppo rt of Naval Operations 2005): Scripted autonomous systems: use a preplanned scrip t to accomplish the mission objective. These systems do not have human interact ion after they are deployed. As an example consider guided rockets.
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6 Supervised autonomous systems: do most of the funct ions of planning, sensing, and networking to carry out activities. Human opera tors, via communication link, make decisions based on the data sensed by the vehi cle. Intelligent autonomous systems: use intelligent tec hnology to embed attributes of human intelligence in the software of autonomous ve hicles and their controlling elements. This research focuses on intelligent autonomous sys tems. Throughout the thesis, they are referred to as unmanned autonomous vehicle s (UAVs). Some advantages of UAVs over manned aircraft systems include: no casua lties, easier to store and ship, less expensive per aircraft and can fly longer missions. Three different scenarios have been studied in sear ch problems: single searcher, cooperation, and coordination. Uryasev et al. formu lated the single searcher problem as a stochastic program. Their objective function was to minimize the expected search time before a target is found (Uryasev and Pardalos 2001 ). The total search area was divided in subregions and determined the average time that a searcher would require spending in a specific subregion (assuming x targets within the search region). The work was extended into a cooperative search concept in which the search was concurrently performed by more than one vehicle. It was found th at cooperative searching is not only dividing search effort among each agent; particular ly when the target is able to detect and evade searchers. The cooperative search problem was approached with two opposing objectives: maximize the effectiveness of a single searcher and maximize the effectiveness of the group with multiple searchers.
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7 The second scenario studied in search problems is c ooperative search. Some missions are better performed by a team of UAVs ins tead of only one single agent. For instance a dangerous and/or extensive mission, wher e it is unlikely that a single UAV survives to complete the task, a team is more suita ble to perform it (Cole, et al. 2009). Several authors concentrated on cooperative search. Polycarpou et al. developed and evaluated the performance of strategies for coopera tive search with autonomous vehicles that seek to gain information about the environment (Marios, Yang and Liu Yang 2003). The vehicles share the information that they have t o enable cooperation. No vehicle tells another what to do nor are there any negotiations a mong them. Each seeks to enhance a global goal, not only its own goal. Chandler and Pachter looked at cooperative rendezvo us and cooperative target classification and attack in a hierarchical distrib uted control system (Chandler and Pachter 2002). The vehicle doing path planning and trajectory generation is at Decision Level 1. At Decision Level 2 is the subteam that c oordinates the activities of classification and attack. When more than one vehi cle is used to search and attack, the decision whether to continue the search or go attac k previously found targets has to be made. This decision making process leads to the wor k done in optimal stopping The theory of optimal stopping studies the problem of c hoosing a time to take a particular action, in order to maximize an expected reward or minimize an expected cost. In this work, optimal stopping is not considered. However, a good direction for future research on the problem addressed in this thesis might come from this area. The third and last scenario studied in search probl ems is coordinated search. While cooperation entails more than one entity work ing toward common goal,
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8 coordination implies a coupling between entities th at is designed to achieve the common goal (Hsieh, et al. 2007). An example of coordinate d task execution is provided in (Schultz, Parker and Schneide 2002). A robot should not start analyzing a rock until two others have moved into place to provide assistance. Thus, a distributed executive facilitates one robot monitoring the execution of a nother robot and helps it recover from faults. The search problem scenario studied in this researc h is cooperation. The effects of cueing in cooperative search system have been studi ed in (Alexander and Jeffcoat 2007). It is demonstrated that cueing increases significan tly the probability of detection over a fixed period of time and that its effect on systemÂ’ s effectiveness is bounded. Continuous time Markov chain is used to model the cooperative search system and Kolmogorov equations are solved to determine the effects of cu eing on the systemÂ’s effectiveness. This is viable for small cooperative search systems ; however, it is not practical for large systems due to the difficulty in visualizing aprior i the interaction among all the components and determining its state transition pro babilities. Hence, this work proposes the use of Petri Nets to model and analyze search p rocesses. 2.2 Petri Nets 2.2.1 Motivation for the Use of Petri Nets to Model and Analyze UAV Systems Petri Nets have proven to be very useful in the mod eling, analysis, simulation, and control of UAV systems (Cao, Fukunaga and Kahng 200 4), (Palamara, et al. 2009). They provide very useful models for the following reason s:
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9 Petri Nets capture the precedence relations and str uctural interactions of stochastic, concurrent, and asynchronous events. In addition, the graphical interface helps to visualize such complex systems ( Desrochers and Al'Jaar 1995). Petri Net models represent a hierarchical modeling tool with a welldeveloped mathematical and practical foundation. Petri Nets and its extensions (general stochastic P Ns (GSPN), extended SPNs (ESPNs) and deterministic SPNs (DSPN)) allow for bo th qualitative and quantitative analysis of performance measures (Ajmo ne, et al. 1994). The analysis of timed Petri Nets can be automated a nd several software tools such as SPNP and TimeNET are available for this purpose. Finally, Petri Net models can also be used to imple ment realtime control systems for UAVs (Cao, Fukunaga and Kahng 2004). 2.2.2 Formal Definition and Basic Terminology of Pe tri Nets A Petri Net is graphically represented by a directe d graph with two kinds of nodes: places and transitions Place nodes model states or conditions, while tra nsition nodes model events of functions of the system (Ajmo ne, et al. 1994). Petri Nets (PNs) are intended to visualize the dynamics of a system. The state of a Petri Net is called marking and is defined by the number of tokens in each pl ace. Each place may be considered as a local state of the system; it descr ibes the condition of a resource. Places and transitions are connected by arcs. According to certain rules, the transition can move the tokens from one place to another, and thus chan ge the state of the system. Formally, a Petri Net can be defined as follows:
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10 PN = (P, T, I, O, Mo) ; where P = {p1, p2,Â…, pm} is a finite set of places, T = {t1, t2,Â…, tn} is a finite set of transitions, P U T places, and P T , I: (P x T) N is an input function that defines the arcs from pla ces to transitions, where N is a set of nonnegative integers, O: (P x T) N is an output function which defines directed arcs f rom transitions to places, and Mo: P N is the initial marking. It gives the numbers of ind istinguishable tokens which are initially in each place. In the graphical representation places are drawn as circles, transitions are drawn as rectangles, and arcs have an arrowhead at their destinations. Tokens are drawn as black dots; larger number of tokens in a place is represe nted by their number. A simple example of a Petri Net is shown in Figure 1. Figure 1 Petri net example The occurrence of events or execution of operations in a Petri Net model changes the distribution of tokens in places. Thus, one can study dynamic behavior of the modeled system.
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11 The following rules are used to govern the flow of tokens: A transition t is said to be enabled in marking M, if at least one token is in all input places. An enabled transition t can fire by removing a toke n from each input place and putting one token in each output place. 2.2.2.1 Reachability Set and Reachability Graph The firing rule defines the dynamics of Petri Net models. From initial marking is possible to determine the set of all markings reach able from it and all the paths that the system may follow to move from marking to marking. The initial state must be completely specified for the computation of the set of reachable markings. The representation of all reachable markings (state spa ce of the net) is called reachability graph. 2.2.2.2 Stochastic Timed Petri Nets (STPN) STPN are Petri Nets in which stochastic firing tim es are associated with transitions. The transitions times are allowed to b e random variables. 2.2.2.3 Generalized Stochastic Petri Nets (GSPN) A GSPN is an extension of an SPN. The Petri Net co ntains two types of transitions: immediate transitions and timed transi tions. Timed transitions are associated with random, expon entially distributed firing delays.
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12 Immediate transitions fire in zero time with firing probabilities. In the graphical representation, timed transitions are drawn as thick bars and immediate transitions as thin bars. When a new mark ing is reached, it can be classified into two types. A marking that enables only timed t ransitions is called tangible, whereas a marking that enables at least one immediate transit ion is called vanishings. Markings of the latter type have zero sojourn time. An example of a GSPN is discussed in Section 3.2.
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13 Chapter 3 Problem Description This chapter is divided into 4 sections. Section 3. 1 discusses the basis of the work in this thesis. The problem addressed in this secti on was first introduced in (Jeffcoat, Krokhmal and Zhupanska 2007). Section 3.2 presents the Petri Net model proposed to model the system studied. Section 3.3 shows how to analyze the system using the Petri Net model. Finally, Section 3.4 introduces the perf ormance indices used to evaluate the systemÂ’s performance. 3.1 SystemÂ’s Description The cooperative search mission that is considered t hroughout this thesis presents a search and engage scenario. It includes two types o f UAVs: (i) a dedicated searchonly vehicle and (ii) nsearchengage vehicles. The job of the searchonly vehicle is to provide cues to all searchengage vehicles. It is assumed t hat the searchonly vehicle has better search capabilities than the searchengage vehicles ; thus, it has a higher detection rate. The searchengage vehicles can engage one target on ly and it is assumed that searchengage vehicles do not cue each other. The mission is completed when the nsearchengage vehicles have engaged n targets. Therefore, it is assumed that there are at least n targets within the search area.
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The searchengage vehicle search it is cued. If it is cued, it orientates its search toward will find a target at the location provided by and 3 illustrate all the possible states of each type of UAV. As show searchonly vehicle has two cueing searchengage vehicles. detection function to model and analyze search proc esses which the searchonly vehicle detects and cues search cues a searchengage vehicle, it starts over to look for new targ ets. As shown on Figure 3 search uncued, (ii) search uncued a nd then it either engages a target with a vehicle and starts searching qu of the detection function comes from an exponential distribution. The rate in which the searchengage v ehicle goes from that there are n search vehicles and cues are 14 engage vehicle search es (uncued) for a target until either it it orientates its search toward s the specified location. Eventually, it the location provided by cue from the searchonly vehicle. the possible states of each type of UAV. As show n has two possible states: (i) it can be either searching for targets or engage vehicles. Traditionally the exponential distribution is used as detection function to model and analyze search proc esses (L. D. Stone 1983) only vehicle detects and cues search engage vehicles is engage vehicle, it starts over to look for new targ ets. Figure 2 Searchonly vehicle states As shown on Figure 3 the searchengage vehicle ha s three possible states: (i) (ii) search cued or (iii) detect and engage a target. Initially, it is searching nd then it either engages a target with a rate qu or it is cued by the search starts searching based on this cue. Similar to the searchonly vehicle, the rate of the detection function comes from an exponential distribution. The rate in which the ehicle goes from searching uncued to searching cued is l vehicles and cues are equally distributed). From the search cued for a target until either it finds it, or the specified location. Eventually, it only vehicle. Figures 2 n in Figure 2, the it can be either searching for targets or (ii) the exponential distribution is used as (L. D. Stone 1983) The rate at engage vehicles is l. As soon as it s three possible states: (i) Initially, it is searching is cued by the search only only vehicle, the rate of the detection function comes from an exponential distribution. The rate in which the l /n (It is assumed distributed). From the search cued
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state, the vehicle engages a target with a rate the targetÂ’s l ocation in the search cued state, the 3.2 Petri Net Model To develop a Petri components of the search team are modeled individua lly. The Petri searcho nly vehicle is shown in Figure 4 searchonly vehicle. The time transition T0 represents the search target. Finally, the immediate transitions t0 correspond only vehicle returns to the search state after it c ues a search Similarly, the Petri thi s model, immediate transition t1 transitions T1 and T2 represents detection of a tar get Note the similarity in the structur rep resentation of the vehicles in F 15 state, the vehicle engages a target with a rate qc Since there is detailed information about ocation in the search cued state, the rate qc > qu. Figure 3 Searchengage vehicle states To develop a Petri Net model for the system described in S ection 3.1 components of the search team are modeled individua lly. The Petri Net nly vehicle is shown in Figure 4 Place P0 represents the search state of the only vehicle. The time transition T0 represents the search only vehic immediate transitions t0 correspond s to the event where the search only vehicle returns to the search state after it c ues a search engage vehicle. Similarly, the Petri Net model for a searchengage vehicle is shown i s model, immediate transition t1 represents the cueing of this vehicle, and the time transitions T1 and T2 represents detection of a tar get before and after cueing respectively. the structur e of the two Petri Net models with resentation of the vehicles in F igures 2 and 3. Since there is detailed information about ection 3.1 the two Net model for the Place P0 represents the search state of the only vehic le detecting a to the event where the search engage vehicle. engage vehicle is shown i n Figure 5. In represents the cueing of this vehicle, and the time and after cueing respectively. with the conceptual
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16 Figure 4 Petri net searchonly vehicle Figure 5 Petri net searchengage vehicle Transition t0 in Figure 4 and transition t1 in Figu re 5 represent the same event, namely cueing of a searchengage vehicle. Through t his common event (transition), the component models are merged. In addition, a second searchengage vehicle can be integrated by adding an identical model to the one shown in Figure 5. Following the same approach, the Petri Net model of a system with seve ral searchengage vehicles can be readily created. Figure 6 shows a Petri Net model f or a system with two searchengage vehicles.
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17 Figure 6 One searchonly vehicle and two searche ngage vehicles The complete model allows obtaining performance mea sures for a system of n searchengage vehicles and one searchonly vehicle. An advantage of the proposed Petri Net is that the same model can be used to obtain sy stemÂ’s performance measures with cueing or without cueing. If the token on P0 is rem oved, there is no cueing in the system. Thus, the same model structure can be used to quant ify the effect of cueing in the performance of the system. The next sections explai n the numerical analysis of the proposed model and introduce the performance indice s of interest.
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18 3.3 Analytical Solution A GSPN describes an underlying stochastic process, captured by the reachability graph (RG). The analysis of a GSPN is, in principle the analysis of its underlying process, which has been shown to be reducible to a CTMC. To make the reachability graph isomorphic, with a transition rate diagram of a CTMC, the vanishing markings have to be eliminated (Ajmone, et al. 1994). 3.3.1 Eliminating the Vanishing Markings The following procedure is based on a system compos ed by one searchonly vehicle and two searchengage vehicles (Figure 4). Table 1 shows the specification firing rates of the transitions in the GSPN of Figure 6. Table 1 Â– Transition rates/weights of Figure 6 Transition Rate/Weight T0 l T1 = T4 qu T2 = T3 qc t0 a t1 1a The RG (Figure 7) contains 18 markings. The label o n the arcs connecting two markings represents the time distribution to go fro m one marking to another one. There are only two types of distributions in the RG: expo nential with rate mi mi = l, qu, or qc. and constant with k0= 0. In addition, the label in square brackets corr esponds to the probability that the arc is traversed. The markings represented with a dashed line are vanishing markings.
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19 Figure 7 Â– Reachability graph of the GSPN of Figure 6 From the 18 marking of the reachability graph, ther e are 13 tangible markings and 5 vanishing markings. The vanishing marking can be eliminated by determining the equivalent rates of moving between two tangible mar kings with intermediate vanishing markings. The rate of moving from the marking (1011 0000) to the vanishing marking (01110000) is The probability of leaving the vanishing marking t o the marking (10011000) is Hence, the equivalent rate of moving from the (10 110000) to (10011000) is: (1)
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20 Using the same procedure for each pair of tangible markings with intermediate vanishing markings, the reachability graph of the G SPN can be converted to a transition rate diagram of a CTMC (Figure 8). Figure 8 Â– CTMC rate diagram of the GSPN of Figure 6 The CTMCs allow obtaining systemÂ’s performance meas ures such as mean time to complete a mission or probability of engaging a target by time t The procedure of going from the reachability graph to the CTMC can b e automated and is computationally acceptable as long as the number of vanishing marki ng is small compared to the number of tangible markings (Ajmone, et al. 1994). In addi tion, other procedures that reduce computational complexity have been studied (Miner 2 001), (Allmaier, Kowarschik and Horton 1997).
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21 3.4 SystemÂ’s Performance Measures There are many situations that require performing a task or completing an objective in a certain amount of time. As an exampl e, consider a boat that sank in cold water with 3 fishermen. The targets of the mission are the three possible survivors and they can die from hypothermia in a few hours. Thus, it is imperative to determine the probability that a specific team of UAVs can find t he fishermen by time t The performance measures will allow us to obtain the pr obability that a team of one searchonly vehicle and nsearchengage vehicles engage m targets by time t ( m n ). The performance indices that are going to be defin ed to be able to measure the systemÂ’s effectiveness with one searchonly vehicle and n searchengage vehicles are: Expected time to engage n targets with n searcheng age vehicles; one target for each vehicle. Expected numbers of targets engaged by the system a s a function of time. In both cases, it is assumed that there is at least n number of targets and that each searchengage vehicle can engage one target only. 3.4.1 First Passage Times in CTMCs The first passage times in CTMCs is used to calcula te the expected time to engage n targets with n searchengage vehicles (Kulkarni 1 999). Let { X (t), t 0} be a CTMC with state space S={1,Â…,N} and rate matrix R. The first passage time into state N is defined to be: T = min { t 0: X ( t ) = N }
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22 let, mi = E( T  X0 = i ) ; mn = 0 The next theorem gives a method of computing mi, 1 i N Â– 1. Theorem 21: (First Passage Times) { mi, 1 i N Â– 1} satisfy the following: nnrnr (2) Theorem 2 can also be extended to the expected time to reach a set of states. The transition rate matrix of the CTMC for one searcho nly vehicle and two searchengage vehicles is shown in Figure 9. S 0 S 1 S 2 S 3 S 4 S 5 S 6 S 7 S 8 S 9 S 10 S 11 S 12 S0 q u l (1a ) l ( a ) q u S1 l q u S2 q c q u l S3 q c l q u S4 l q u S5 l q c S6 q c S7 q c q c l S8 l q c S9 q c S10 q c q c S11 l S 12 Figure 9 Â– Transition rate matrix of the CTMC for o ne searchonly vehicle and two searchengage vehicles 1 For a proof of Theorem 1 see (Kulkarni 1999).
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23 All the searchengage vehicles engage a target as s oon as the CTMC visits the set of states {11, 12}; which is the markings [10000011 ] and [01000011], respectively. Then, Theorem 2 needs to be extended to the case of reaching a set. The first passage time to reach a set of states, A is: T = min { n 0: X ( t ) A } Let, mi (A) be the expected time to reach the set A starting from state i A. Then, nn n (3) From the transition rate matrix (Figure 9) and Theo rem 2 extended to the case of reaching a set of states (3), the following can be obtained: ! "r # $%(4) r &(5) #'%& ((6) $'r ()(7) % )(8) & *(9) (10) '' ('&') r"(11) ) +(12) + (13) '' r"'*'+(14)
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24 Table 2 summarizes the cueing and detection rates f or the system; the same rates were used in (Jeffcoat, Krokhmal and Zhupanska 2007 ). Table 2 Â– Detection rates and cueing weight Rate Value l 1.5 q u 0.10 q c 0.19 a 0.5 When the set of equations 4 Â– 14 are solved, the fo llowing values for the expected time to absorption from state mi are obtained: m0 = 8.353, m1 = 5.559, m2 = 8.058, m3 = 8.058, m4 = 5.559, m5 = 5.263, m6 = 5.263, m7 = 7.895, m8 = 5.263, m9 = 5.263, m10 = 7.895 Thus, on average it takes 8.353 time units to engag e two targets with one searchonly vehicle and two searchengage vehicles. Table 3 summarize the average time for different values of l, qu, and qc. The value of a was held constant at 0.5; which means that each of the two searchengage vehicles is equa lly likely to be cued. Table 3 Â– Expected time to engage n targets with one searchonly vehicle and two searc hengage vehicles q u =0.10 qc = 0.19 E[T] l = 1.5 qu =0.10 E[T] l = 1.5 qc = 0.19 E[T] l = 1.5 8.353 q c = 0.19 8.353 q u =0.10, 8.353 l = 2.5 8.174 q c = 0.29 5.814 q u =0.12, 8.246 l = 3.5 8.095 q c = 0.39 4.583 q u =0.14, 8.142 l = 4.5 8.051 q c = 0.49 3.859 q u =0.16, 8.041 l = 5.5 8.023 q c = 0.59 3.383 q u =0.18, 7.943
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25 Results on Table 3 imply that increments on the rat e qc have a greater impact on the systemÂ’s effectiveness than improving the cuein g rate of the searchonly vehicle (l) or the individual uncued detection rates (qu). In other words, if resources were to be allocated toward decreasing the mean time to engage n targets, it is better to improve (increase) the rate qc; at least for the scenarios defined in the table. The rate qc can be increased by: Improving the quality of the information provided i n the cue. For example, if the searchonly vehicle provides the exact location of the target to the searchengage vehicles. Cueing the closer searchengage vehicle to the loca tion of the target. In the proposed model, the searchonly vehicle chooses ran domly among the searchengage vehicle to be cued. Choosing the vehicle tha t is closer to the target will increase the rate c because it will decrease the time to engage the t arget after cueing. 3.4.2 Transient Analysis: Uniformization The uniformization analysis in CTMCs is used to calculate the expected number of targets engaged by time t. Let { X (t), t 0} be a CTMC with state space S = {1,,N} and let R = {ri.j} be its rate matrix. A CTMC spends an Exp( ri) amount of time in state i (ri = ,n nr), and if ri > 0, jumps to state j with probability pi.j= ri.j / ri. Now, let r be any finite number satisfying r r./01234.
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26 Define a matrix P = [ pi.j] as follows: pi.j =5 232 67 23,82 697 : Finally, the transition probability matrix P(t)= [pi.j (t)] is given by: P(t)= ;2< 2< => =?==" (15) Using the detection rates in Table 3, we can obtain the following transition rate matrix of the CTMC for one searchonly vehicle and two searchengage vehicles (Figure 10). S0 S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12 S0 0.10 0.75 0.75 0.10 S1 1.50 0.10 S2 0.19 0.10 1.50 S3 0.19 1.50 0.10 S4 1.50 0.10 S5 1.50 0.19 S6 0.19 S7 0.19 0.19 1.50 S8 1.50 0.19 S9 0.19 S10 0.19 0.19 S11 1.5 S12 Figure 10 Â– Transition rate matrix of the CTMC for one searchonly vehicle and two searchengage vehicles (rates Table 2) Then r1=1.7, r2=1.6, r3=1.79, r4=1.79, r5=1.6, r6=1.69, r7=0.19, r8=1.88, r9=1.69, r10=0.19, r11=0.38, r12=1.5, r13=0 hence, r = 1.88
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27 Then, the P matrix is: S0 S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12 S0 0.1 0.05 0.40 0.40 0.05 0 0 0 0 0 0 0 0 S1 0 0.15 0 0 0 0.80 0 0 0 0 0 0.05 0 S2 0 0 0.05 0 0.10 0.05 0 0.80 0 0 0 0 0 S3 0 0.10 0 0.05 0 0 0 0.80 0.05 0 0 0 0 S4 0 0 0 0 0.15 0 0 0 0.80 0 0 0.05 0 S5 0 0 0 0 0 0.10 0.8 0 0 0 0 0.10 0 S6 0 0 0 0 0 0 0.9 0 0 0 0 0 0.1 S7 0 0 0 0 0 0.10 0 0 0.10 0 0.8 0 0 S8 0 0 0 0 0 0 0 0 0.10 0.8 0 0.10 0 S9 0 0 0 0 0 0 0 0 0 0.9 0 0 0.1 S10 0 0 0 0 0 0 0.1 0 0 0.1 0.8 0 0 S11 0 0 0 0 0 0 0 0 0 0 0 0.20 0.8 S12 0 0 0 0 0 0 0 0 0 0 0 0 1 Figure 11 Â– P matrix for one searchonly vehicle an d two searchengage vehicles (rates Table 2) Finally, P(t) can be computed by: @ A ;r,))< r,))< B=> C="?= (16) In numerical computations, P(t) is approximated by using the first M terms of the infinite series. We compute P(t) by using the rule to choose the value of M propose d in (Kulkarni 1999): M DEF1AGH I A J4
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28 P(0.5) = M = 20 Figure 12 Â– P matrix for one searchonly vehicle an d two searchengage vehicles at t =0.5 (rates Table 2) From the first row of the P matrix in Figure 12, it can be seen that after 0.5 time units and starting at S0 which is marking [10110000], there is a probability of 0.43 that the system is still on the same state. In addition, there is a 0.03 probability that the system has transitioned to state 1; which means that one o f the searchengage vehicles has engaged a target by itself (no cue received). Using the probabilities provided by the matrix and the numbers of targets engaged in each s tate, we proceed to calculate the expected numbers of targets engaged by a specific t ime. For example, the expected number of targets engaged at 0.5 time units is: Let X (t) be the number of targets engaged at time t, the n: E( X (0.5) = 0.43 0 + 0.03 1 + 0.16 0 + 0.16 0 + 0.03 1 + 0.02 1 + 0.01 1 + 0.11 0 + 0.02 1 + 0.01 1 + 0.03 0 + 0.0 2 + 0.0 2 = 0.1129 Following the same approach, we obtain a graph with the expected number of targets engaged as a function of time (Figure 13). S0 S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12 S0 0.43 0.03 0.16 0.16 0.03 0.02 0.01 0.11 0.02 0.01 0 .03 0.00 0.00 S1 0.00 0.45 0.00 0.00 0.00 0.33 0.16 0.00 0.00 0.00 0 .00 0.04 0.02 S2 0.00 0.00 0.41 0.00 0.04 0.04 0.02 0.30 0.03 0.01 0 .15 0.00 0.00 S3 0.00 0.04 0.00 0.41 0.00 0.03 0.01 0.30 0.04 0.02 0 .15 0.00 0.00 S4 0.00 0.00 0.00 0.00 0.45 0.00 0.00 0.00 0.33 0.16 0 .00 0.04 0.02 S5 0.00 0.00 0.00 0.00 0.00 0.43 0.48 0.00 0.00 0.00 0 .00 0.04 0.05 S6 0.00 0.00 0.00 0.00 0.00 0.00 0.91 0.00 0.00 0.00 0 .00 0.00 0.09 S7 0.00 0.00 0.00 0.00 0.00 0.04 0.04 0.39 0.04 0.04 0 .44 0.00 0.00 S8 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.43 0.48 0 .00 0.04 0.05 S9 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.91 0 .00 0.00 0.09 S10 0.00 0.00 0.00 0.00 0.00 0.00 0.08 0.00 0.00 0.08 0 .83 0.00 0.01 S11 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0 .00 0.47 0.53 S12 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0 .00 0.00 1.00
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29 Figure 13 Â– Expected number of targets engaged with two searchengage vehicles Figure 14 shows how the curve of the expected numbe r of targets shifts to the left if one of the original rates is increased. The orig inal rates were changed one at a time with an increment of 50%. The 50% is an assumption, and it can represent an improvement on the search capabilities, the speed o f the vehicle, or any other factor that affect the rate at which targets are engaged. Figure 14 Â– Expected number of targets engaged with two improved searchengage vehicles The results from Figure 14 agree with the findings of the previous section. Increments on the rate qc have a greater impact on the systemÂ’s effectivenes s than improving the cueing (l) or the individual detection rates (qu). nr nr l q qn a = 0.5 l q q n a = 0.5 l q q n a = 0.5 l q q n a = 0.5 l q qn a = 0.5
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30 Chapter 4 General Distributions This chapter is divided into 3 sections. Section 4. 1 describes the simulation developed with one searchonly vehicle and one sear chengage vehicle. Section 4.2 presents the results of the simulation and argues t he relevance of general distributions in the problem addressed. Finally, Section 4.3 present s how to incorporate general distributions into the Petri Net model proposed. 4.1 Simulation Description In Section 3.1 all the time distributions in the mo del follow an exponential distribution. A simulation was developed to determi ne whether this assumption is valid for systems with stationary targets and cues that p rovide exact target location. Since there is precise information about a targetÂ’s locat ion, better fits may come from bounded distributions. A cue with the precise location of a stationary target eliminates the need for any additional search by the searchengage vehi cle and simplifies the process to traveling from one location to another. Arguably, t his no longer is a search process; however, to preserve the association with the gener al cooperative search model introduced in Chapter 3, this process is still refe rred to as cued search in the rest of the thesis.
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31 The environment in which the vehicles are searching is shown in Figure 15. The region simulated is a grid of m by n cells; the values of m and n are inputs. There are two vehicles in the simulation: one searchonly vehicle shown as a Â“2Â”; and one searchengage vehicle, shown as a Â“1Â”. There is only one t arget, and it is represented as a Â“1Â”. It is assumed that the searchonly vehicle covers m ore area than the searchengage vehicle; the area of coverage is represented by the shaded region around the vehicle and it is an input of the simulation. The searchengage vehicle has to be in the same cell with the target to find it. The initial positions of the vehicles and the target are randomly selected, between each replication, with a uniform distribution over the region of the search environment. Once the simulation starts, bot h vehicles look for targets until either the searchengage vehicle finds it, or the searcho nly vehicle detects it. If the searchonly vehicle detects the target, it transmits the target Â’s location to the searchengage vehicle. Then, the searchengage vehicle moves to specified position. The simulation ends when the searchengage vehicle finds the target. Figure 15 Â– Simulation environment
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32 The vehicles do not follow any pattern nor have mem ory of the places they have visited. They are free to move in any direction wit hin the limits of the search environment. However, once the searchengage vehicl e receives the information about the targetÂ’s location, it moves directly to the spe cified location via the fastest way to reach that position. 4.1.1 Assumptions for the Model The searchonly vehicle provides the exact targetÂ’s location to the searchengage vehicle. The time to move between cells is the same regardle ss of the direction. For example: moving to the north direction takes the sa me time as moving to the northeast direction. Both vehicles move at the same speed. Constant spee d Target is stationary. Both vehicles can be in the same cell at the same t ime. 4.2 Simulation Results The simulation was constructed to analyze the time distribution of the three search processes. Table 4 shows the histograms of the time distribution associated with each search process and vehicle. The results are based o n 5,000 replications. From Table 4, it is seen that the assumption of the exponential dist ribution is valid for the time of the first two search processes. However, the histogram of the third search process (cued search) indicates that the exponential distribution is not a good fit. The exponential distribution is a good fit for the first two search processes becau se there is no information about the
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33 targetÂ’s location. In the third search process, onc e the targetÂ’s location is known, the time to engage a target is a function of the distance be tween the target and the searchengage vehicle, and its speed. Table 4 Â– Histograms of the distributions of the ti me associated with each search process Vehicle Search Process Histogram 1 SearchOnly Search and Cue 2 SearchEngage Uncued Search 3 SearchEngage Cued Search Table 5 shows the p values of the chisquare test for the time of the cued search process. Corresponding p values less than 0.05 indicate that the distributi on is not a very good fit; larger p values indicate better fits. It can be seen from T able 5 that the exponential distribution is not a good fit. In cont rast, the triangular, the weibull and the normal distributions are better suited to model the underlying process time. Since the
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34 search time is a positive and bounded value, the tr iangular distribution is the most appropriate distribution to model the time of the c ued search process. Table 5 Â– P values of the chi square test for the time of the cued search process Functions p value Triangular > 0.75 Normal 0.473 Weibull 0.454 Erlang 0.0092 Gamma 0.0869 Lognormal < 0.005 Uniform < 0.005 Exponential < 0.005 The simulation results indicate that the time of th e cued search process is better represented with a triangular distribution. The tim e to engage a target once the vehicle receives a cue is the distance traveled to the loca tion provided times the speed (the cue transmitted gives the exact location). Thus, the ti me distribution of the cued search process is basically the distribution of the distan ce between two random points times a constant (the speed). It can be proven, using Manha ttan metrics, that the distance between two uniformly distributed random points within a re ctangle follows a triangular distribution. The proof is shown in (Gaboune, et a l. 1993) and it is summarized below. Denote ( Xi, Yi) a point in a rectangle. Consider two random point s and define X =  X1Â–X2, Y =  Y1Â–Y2. Also let L denote the average distance between two uniformly randomly distributed points in the rectangle. For 0 x a, the distribution function of X is given by: KL F @MNrN#M F O @ N#PNrF @ N#PNrF Q
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35 RSSTU LVWL UL "FrF#XF#XFrSSTLVL U LFrF#XF#XFrY where f (x1, x2), the joint probability function of X1 and X2, is defined by: T FrF# Z r U[ TFrEE\XFrE ]A^;_`; : since X1 and X2 are independent. Therefore KL F a b c TF FE #E# TFE TFP : hence, the density function of X is triangular over [0, a]: TL F d J E e F E fTFE ghij6ki : The next chapter discusses how to incorporate gener al distributions such as the triangular distribution to the proposed Petri Net m odel and calculate performance measures.
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36 Chapter 5 General Distributions Analysis in the Petri Net Mod el It has been demonstrated that cueing increases the performance of a cooperative search system and that the proposed Petri Net model captures the interactions among the vehicles in the system. Performance indices are def ined and computed to measure systemÂ’s performance. In addition, these indices ca n be used to decide how to best allocate resources to improve the systemÂ’s performa nce. Finally, the cued search process time is shown to be accurately represented by a tri angular distribution. The feasible techniques to obtain performance measu res in Petri Nets with general distributions are simulation and approximation (Van der Aalst, Van Hee and Reijers 2000). Simulation will not be addressed in this the sis. This chapter discusses how to approximate general distributions. Section 5.1 deri ves a general expression for the coefficient of variation for the general distributi on using the search area dimensions to determine the type of approximation. Section 5.2 di scusses how the triangular distribution can be incorporated into the proposed Petri Net model. 5.1 Analysis of General Distributions Agner Erlang conceived the notion of decomposing ge neral distributions into phasetype distributions (Yee and Ventura 2000). He showed that a distribution with a coefficient of variation (CV) less than one can be represented by a series of k 2
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37 exponential stages; this is known as an Erlangk di stribution. On the other hand, a distribution with a CV greater than one can be repr esented by k 2 parallel exponential stages; this is known as the hyperexponential dist ribution (Chen, Bruell and Balbo 1989). This procedure will be used to approximate a triangular distribution to an Erlangk or hyperexponential distribution (depending on the CV) and incorporate it into the proposed Petri Net model. The density function of the distance between the ta rget and the searchengage vehicle was derived in Section 4.2 as function of t he size of the search environment. The density function is: TL F d J E e F E fTFE ghij6ki : Then, the expected value E(X) and the variance V(X) can be computed to obtain a general expression for the coefficient of variation l O N Q SFTFXFSFm J E e F E fnXFU "S JF E XFS JF#E# XFU U C C : JF#E o" U : JF$pE# o" UE JE p p E E [X] refers to the expected distance traveled along the x axis. The same way can be computed for the y axis. l O q Q p r
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38 Hence, l s l Nq l N l q E p rp Er p t O N Q lOF#Qu lONQ # l O F#Q SF#TFXFSF#m J E e F E fnXFU "S JF#E XFS JF$E# XFU U C C : JF$pE o" U : F%JE# o" U JE#p E#J v E# l O F Q#e E p f# E#w t O N Q E#v E#w E#x V [X] refers to the variance on the distance traveled along the x axis. The same way can be computed for the y axis. t O q Q r#v r#w r#x t O s Q t O Nq Q t O N Q t O q Q Jy]z F{ X and Y are independent; thus, cov (x + y) = 0 t O s Q E#x r#x E#r#x
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39 Hence, the coefficient of variation is: }O s Q tOsQlOsQ E#r#x Er p E#r#vEr The general expression of the coefficient of variat ion allows for associating the type of approximation needed for the general distri bution of the cued search process to the search boundaries. The next section discusses h ow to approximate general distributions and to incorporate the approximation into the proposed model. 5.2 Incorporating General Distributions into the Pr oposed Model Any type of general distribution with support on [0 ) can be approximated by a phasetype distribution (Asmussen, Nerman and Olsso n 1996). Phasetype distributions have been successfully used for modeling nonexpone ntial activities due to their versatility and relative ease of numerical implemen tation (Shaked and Shanthikumar 2006). Several methods have been utilized for approximatin g general distributions. A general statistical approach called the EM algorith m is presented in (Asmussen, Nerman and Olsson 1996). EM algorithm can be used to appro ximate incomplete data and continuous distributions with support on [0, ). Approximating a continuous distribution by a phasetype distribution is similar to fitting a phasetype distribution to a sample. In parametric estimation, the methods that minimize th e divergence between the assumed model density and the true density underlying the d ata, include maximum likelihood, chi squared methods based on families of chisquared di stances and Hellinger distance,
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40 among others (Basu, et al. 1998). A benchmark for p hasetype estimation algorithms is presented in (Bobbio and Telek 1994). To illustrate how a triangular distribution is appr oximated by a phasetype distribution and incorporated into the proposed Pet ri Net model, let us assume the following parameters for a triangular distribution (2, 10, 18). First, the coefficient of variation is estimated: The standard deviation and the mean of a triangular distribution are defined as: ~ m E#r#y#ErEyry x n p,JÂ€ Â Ery p Hence, the coefficient of variation is: t p,JÂ€ ,pJÂ€Â‚ The CV is less than 1; thus, the triangular distrib ution can be approximated by a series of k 2 exponential stages (Erlangk distribution). The n, we proceed to estimate the number of stages ( k ) and the mean time of each one ( u ). EasyFit software (Technologies, 2004) is used as a tool for estimati ng the parameters and performing the goodness of fit tests. The following describes how the tool is used.
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41 Initially, a sample set of 5,000 was simulated from the triangular distribution (2, 10, 18). This data was used to estimate the paramet ers of the Erlang distribution. The larger the sample size the more power2 the statistical test has (Montgomery and Runger 2002). Thus, with a large sample size the test is m ore likely to reject the null hypothesis that the Erlang distribution is the true distributi on of the data. A second sample of size 200 was simulated to compare the results of the goo dness of fit tests. Figure 16 shows the histograms of the simulated dat a and the probability density functions of the fitted Erlang distribution for the two samples. Figure 16 Â– Erlangk approximation of the triangula r distribution (2, 10, 18) The parameters estimated for the Erlang distributio n are k = 9 and u = 1.0093 and k = 9 and u = 1.0681 with the sample size of 200 and 5,000, re spectively. Table 6 summarizes the results for the KolmogorovSmirnov, AndersonDarling, and ChiSquared tests. For the sample set of size 5,000, al l the statistical tests reject the null 2 The power of a statistical test is the probability of rejecting the null hypothesis Ho when the alternative hypothesis is true.
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42 hypothesis that the data follow an Erlang distribut ion. However, for the set of size 200, the ChiSquared test fails to reject the null hypot hesis ( a 0.05). Table 6 Â– Goodness of fit tests n = 200 n = 5,000 KolmogorovSmirnov Sample Size 200 5000 Statistic 0.13325 0.09453 PValue 0.00148 0 a 0.05 0.02 0.01 0.05 0.02 0.01 Critical Value 0.09603 0.10734 0.11519 0.0192 0.02147 0.02304 Reject? Yes Yes Yes Yes Yes Yes AndersonDarling Sample Size 200 5000 Statistic 5.8211 90.929 a 0.05 0.02 0.01 0.05 0.02 0.01 Critical Value 2.5018 3.2892 3.9074 2.5018 3.2892 3.9074 Reject? Yes Yes Yes Yes Yes Yes ChiSquared Deg. of freedom 7 12 Statistic 13.694 298.38 PValue 0.05691 0 a 0.05 0.02 0.01 0.05 0.02 0.01 Critical Value 14.067 16.622 18.475 21.026 24.054 26.217 Reject? No No No Yes Yes Yes Even though the results from the goodness of fit te sts may indicate that the Erlang distribution is not consistent with the data, the a pproximation is widely used to do numerical analysis in Petri Nets with general distr ibutions (Ajmone, et al. 1994), (Yee and Ventura 2000). The proposed Petri Net model is adjusted to incorpo rate the approximation of the triangular distribution with a phasetype distribut ion. The Erlang distribution (9, 1.0093) is used for depicting what the proposed model looks like with a phasetype distribution.
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43 The series of exponential distributions can be inco rporated into the model by adding a series of places and time transitions. Figure 17 sh ows the Petri Net model for a searchengage vehicle with the Erlang (9, 1.0093). The 9 t ransient states (the phases) are represented in the model with transitions {T2, T3, T4, T5, T6, T7, T8, T9, T10}. The average time of firing each transition is 1.0093. The Petri Net model for a searchengage vehicle with the Erlangk approximation can be inco rporated into the system model to obtain systemÂ’s performance measures (Figure 18). T he next chapter introduces software that allows analyzing more complex systems such as the one depicted in Figure 18. Figure 17 Petri net model for a searchengage veh icle with Erlangk approximation
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44 Figure 18 One searchonly vehicle and two searchengage vehic les with Erlangk approximation
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45 Chapter 6 Stochastic Petri Net Package (SPNP) It has been demonstrated that cued search processes with stationary targets and cues that provide the exact targetÂ’s location are b etter represented with triangular distributions, and it was shown how to incorporate them into the proposed model. This chapter introduces software that allows rapid devel opment of stochastic reward nets (including GSPN) to evaluate performance measures. The name of the software is Stochastic Petri Net Package (SPNP) 3. 6.1 SPNP Description SPNP is a modeling tool for performance analysis of complex systems. The model type used for input is a stochastic reward ne t (SRN) and they are specified using CSPL (C based SRN Language) which is an extension o f the C programming language with additional constructs for describing the SRN m odels. The SRN models are automatically converted into a M arkov reward model which is then solved to compute a variety of transient, s teadystate, cumulative, and sensitivity measures. For SRNs with absorbing markings, the mea n time to absorption and the expected accumulated reward until absorption can be computed. 3 A full description of the software and its capabil ities can be found at: http://people.ee.duke.edu/~chirel/MANUAL/manual.pdf
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46 6.2 SPNP and Petri Net Model Validation The same cooperative search system studied in (Jeff coat, Krokhmal and Zhupanska 2007) was used to replicate its results a nd consequently validate the proposed Petri Net Model and the output of the software; bef ore using the software to obtain performance measures. The cooperative search system consists of one searchonly vehicle and five identical searchengage vehicles. Table 7 summarizes the transitions rates/weights. Table 7 Transition rates/weights of Figure 19 Transition Rate/Weight T0 l T1 = T3 = T6 = T8 = T10 qu T2 = T4 = T5 = T7 = T9 qc t0=t1=t2=t3=t4 a The transition T0 represents the event where the se archonly vehicle detects a target and cues one of the searchengage vehicles. The rate at which this event occurs is l. The transitions T1,T3,T6,T8,T10 represent the even t where a searchengage vehicle searches and engages a target without receiving any information from the searchonly vehicle (no cue transmitted). The rate at which thi s event happens is q. The transitions T2,T4,T5,T7,T9 represent the event where a searchengage vehicle detects and engages a target with information about its location; the rat e is q This event only occurs if the searchengage vehicle receives a cue from the searc honly vehicle. A parameter k (cueing effectiveness) was defined in (Jeffcoat, Krokhmal a nd Zhupanska 2007) to associate the increase in the detection rate due to the informati on transmitted in the cue; hence, q = k x q. The cues are distributed equally; thus, the proba bility of firing each immediate
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47 transition (t0,t1,t2,t3,t4) is the same. The Petri Net model for one searchonly vehicle and five searchengage vehicles in SPNP is depicted in Figure 19. Figure 19 Petri net model with one searchonly ve hicle and five searchengage vehicles in SPNP Jeffcoat et al. measured the systemÂ’s effectiveness by the probability that all searchengage vehicles have engaged targets by time t. They analyzed two different scenarios and presented their results in two graphs Both scenarios have the initial detection rate q of the searchengage vehicles equal to 0.1, but th e search rate l of the searchonly vehicle varies in the first scenario an d the cueing effectiveness k is varied in the second scenario. Figures 20 and 21 show the res ults for the two scenarios studied, respectively.
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48 Figure 20 Â– Probability of detection of all searchengage vehicles varying l Figure 21 Â– Probability of detection of all searchengage vehicles varying k In the proposed Petri Net model, all the searcheng age vehicles have engaged targets by time t when the places P12, P13, P14, P1 5, and P16 have a token. The marking of interest is shown on Figure 22.
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49 Figure 22 Â– Marking with targets engaged by all sea rchengage vehicles SPNP allows obtaining the probability of reaching t his marking as a function of time. The initial detection rate q of the searchengage vehicles is held equal to 0.1 but the search rate l of the searchonly vehicle is varied according to t he results presented in (Jeffcoat, Krokhmal and Zhupanska 2007). Figures 23 and 24 show the results obtained using SPNP and the Petri Net model for the two scen arios studied.
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50 Figure 23 Probability of detection of all searchengage vehicles varying l (Petri net ) Figure 24 Probability of detection of all searchengage vehicles varying k (Petri net) From the comparison of the respective figures, it i s clear that the Petri Net model and the results from SPNP are equal to the results in Figures 20 and 21 from (Jeffcoat, Krokhmal and Zhupanska 2007) verifying the correctn ess of the proposed modeling methodology. The next chapter summarizes the contri butions of this thesis and outlines the future work that can be done in this research.
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51 Chapter 7 Contributions and Future Research Directions 7.1 Contributions This thesis presents a Petri Net based modeling app roach to model the interaction among autonomous search vehicles in a cooperative s earch system. The cooperation among the vehicles involves cueing. Both in previou s studies and in this thesis, it was demonstrated that cueing increases system performan ce. However, the concept of cueing has not been explored in detail and there is a lack of system models and modeling approaches that involve cueing. The proposed modeling approach based on Petri Nets brings with it the well documented advantages associated with using Petri N et models, such as modularity, hierarchical modeling, well developed mathematical foundation, and a wide range of software available for model development and analys is. In addition, the approach allows the analysis of si milar systems using the same Petri Net structure greatly decreasing model develo pment and verification effort. For example, in Figure 25 by removing the token from pl ace P0, the searchonly vehicle becomes inactive in the model (transition T0 is not enabled) eliminating the cueing capability of the system. The transition T0 will no t fire and the searchengage vehicles will never be cued. This is an advantage that the P etri Net model has over CTMC because the same model can be easily modified to quantify t he performance gains from cueing.
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52 Figure 25 Â– Petri net model with cueing and without cueing In the same manner, a subset of searchengage vehic les can be deactivated by removing their corresponding tokens from the Petri Net model. Figure 26 shows a Petri Net model with four searchengage vehicles but only two of them are active corresponding to marked places (P2, P11). Such a m odification allows the system modeler to evaluate alternative scenarios with vary ing number of searchengage vehicles and analyze system's performance measures without c onstructing a new model for each scenario. The number of vehicles can also change du ring a mission due to vehicle breakdowns or the nature of the mission which may necess itate reevaluation of the expected system performance.
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53 Figure 26 Â– Petri net model with 2 actives searche ngage vehicles In this thesis, it was demonstrated both through si mulation and analytically that the time distribution of the cued search process fo llows a triangular distribution when the target is stationary and the cues provide the exact target location. Methods to approximate general distributions such as the triangular distri bution with phasetype distributions are discussed and Petri Net models incorporating phasetype distributions are developed. Finally, a cooperative search system example from ( Jeffcoat, Krokhmal and Zhupanska 2007) is modeled and analyzed to verify the propose d modeling methodology. The contributions of this thesis can be summarized as follows: A novel Petri Net based modeling methodology for mo deling cooperative search systems involving cueing is introduced. The cued search process for stationary targets is s hown to follow a triangular distribution when the cue provides the exact target location. This process is
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54 similar to traveling from one random location to an other, namely from the location of the cued searchengage vehicle and the location of the stationary target. 7.2 Future Research Directions Intelligent cueing is an immediate and natural futu re research direction to incorporate intelligent target assignment into the proposed Petri net model. The proposed Petri net model assumes that cues are assigned rand omly among the vehicles available. However, the decision of what vehicle to cue could be based on several factors such as proximity to the target or elapsed uncued search ti me. In the case of a system with heterogeneous searchengage vehicles, the decision of what vehicle to cue would also depend on vehicle capabilities. Considering these f actors may decrease the time to engage a target after receiving the cue, or may inc rease system effectiveness by selecting the vehicle(s) with appropriate capabilities for a particular mission. Controlled Petri Nets are an extension of standard Petri nets in which binary control inputs can be applied as external condition s for enabling transitions in the net. The markings of the external input places can be us ed to restrict the firing policy on the Petri Net. In the proposed Petri Net model, the sta tus of a searchengage vehicle can enable a transition to make it eligible to be cued. The theory of fuzzy logic (Carlsson and Fullr 2002 ) resembles human reasoning in its use of imprecise information to generate dec isions. Fuzzy logic does not need exact equations and precise numeric values, it allows exp ressing the states of the system with subjective concepts which are mapped into exact num eric ranges. Thus, fuzzy logic can
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55 be used to classify the status of the searchengage vehicles and consequently determine if eligible to be cued. Another extension of this research involves modelin g cooperative search systems with moving targets and imprecise cues. This extens ion would not impact the structure of the proposed Petri Net model, however it is anticip ated that the cued search time would not follow the triangular distribution since there is still a search process that has to take place once the searchengage vehicle reaches the cu ed location since the target may have moved and/or an imprecise cue requires the searche ngage vehicle to search for the exact location of the target.
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