Modeling and Optimization of Traffic Flow in Urban Areas - Czech ...

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48 Chapter 4 TORSCHE Scheduling Toolbox for Matlab>> T = taskset([T1 T2 T3 T4 T5 T6 T7 T8],prec)Set of 8 tasksThere are precedence constraints4. Define solved problem.>> p = problem(’P|prec|Cmax’)P|prec|Cmax5. Call List Scheduling algorithm with taskset and problem created recentlyand define number of processors and desired heuristic.>> TS = listsch(T,p,2,’SPT’)Set of 8 tasksThere are precedence constraintsThere is schedule: List SchedulingSolving time: 1.1316s6. Visualize the final schedule by standard plot function, see Fig. 4.7.>> plot(TS)P 1leg 1leg 3 seat assembly 1/2assembly 2/2P 2Fig. 4.7: The manufacture scheduling in the Gantt chart formleg 2leg 4backrest0 10 20 30 40 50 t4.3.2 SAT Based Scheduling AlgorithmThis subsection presents a satisfiability of boolean expression based algorithmfor the scheduling problem P |prec|C max , i.e. the scheduling oftasks with precedence constraints on the set of parallel identical processorswhile minimizing the schedule makespan. The main idea is to formulatethe scheduling problem in the form of CNF (conjunctive normal form)clauses [Cram 06, Memi 02].In the case of the P |prec|C max problem, each CNF clause is a functionof the Boolean variables in the form x ijk . If the task T i is started at time
4.3 Implemented Algorithm 49unit j on the processor k then x ijk = true, otherwise x ijk = false. Foreach task T i , where i = 1 . . . n, there are S × R Boolean variables, where Sdenotes the maximum number of time units and R denotes the total numberof processors.The Boolean variables are constrained by the three following rules (modestadaptation of [Memi 02]):1. For each task, exactly one of the S × R variables has to be equalto true. Therefore two clauses are generated for each task T i . The firstguarantees having at most one variable equal to true:(¯x i11 ∨ ¯x i21 ) ∧ · · · ∧ (¯x i11 ∨ ¯x iSR ) ∧ · · · ∧ (¯x i(S−∞)R ∨ ¯x iSR ).The second guarantees having at least one variable equal to true: (¯x i11 ∨¯x i21 ∨ · · · ∨ ¯x i(S−∞)R ∨ ¯x iSR ).2. If there are a precedence constraints such that T u is the predecessorof T v , then T v cannot start before the execution of T u is finished. Therefore,x ujk → (¯x v1l ∧ · · · ∧ ¯x vjl ∧ ¯x v(j+1)l ∧ · · · ∧ ¯x v(j+pu−1)l) for all possible combinationsof processors k and l, where p u denotes the processing time of taskT u .3. At any time unit, there is at most one task executed on a givenprocessor. For the couple of tasks with a precedence constrain this rule isensured already by the clauses in the rule number 2. Otherwise the set ofclauses is generated for each processor k and each time unit j for all couplesT u , T v without precedence constrains in the following form:(x ujk → ¯x vjk ) ∧ (x ujk → ¯x v(j+1)k ) ∧ · · · ∧ (x ujk → ¯x v(j+pu−1)k).The toolbox cooperates with the zChaff solver [Mosk 01] to decidewhether the set of clauses is satisfiable. If it is, the schedule within S timeunits is feasible. An optimal schedule is found in an iterative manner. First,the List Scheduling algorithm (see 4.3.1) is used to find the initial value of S.Then the algorithm iteratively decreases the value of S by one and tests thefeasibility of the solution. The iterative algorithm finishes when the solutionis not feasible.An example of the P |prec|C max problem can be taken from the digitalsignal processing area. A typical scheduling problem is to optimize thespeed of a computation loop, e.g constituting the Jaumann wave digital filter[Groo 92]. The goal is to minimize the computation time of the filterloop, shown as a directed acyclic graph in Fig. 4.8. The nodes in the graphrepresent the tasks (i.e. operations of the loop) and the edges represent the
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Czech Technical University in Pragu
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This thesis is dedicated to my wife
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Nomenclatureα|β|γ Graham and B̷
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NomenclatureviiR 0+ set of non-nega
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AbbreviationsCCPN Constant speed Co
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Goals and ObjectivesThis thesis wil
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ContentsGoals and Objectivesxi1 Int
Page 17 and 18: List of Tables2.1 Traffic data for
Page 19 and 20: Chapter 1IntroductionThe urban traf
Page 21 and 22: 1.1 Related Work 3Traffic stream mo
Page 23 and 24: Chapter 2Simple Light ControledInte
Page 25 and 26: 2.2 Extended Queue Model 72 E( k)AB
Page 27 and 28: 2.2 Extended Queue Model 9E [s]4000
Page 29 and 30: 2.4 Control of The Simple Intersect
Page 37 and 38: Chapter 3General Light ControlledIn
Page 39 and 40: 3.2 Continuous Petri Net 21take the
Page 41 and 42: 3.2 Continuous Petri Net 23v 5V 5vD
Page 43 and 44: 3.3 Light Controlled Intersection M
Page 49 and 50: 3.4 Performance Evaluation 31-1V[uv
Page 51 and 52: 3.4 Performance Evaluation 33-1V [u
Page 53 and 54: Chapter 4TORSCHE SchedulingToolbox
Page 55 and 56: 4.2 Tool Architecture and Basic Not
Page 63 and 64: 4.3 Implemented Algorithm 45tasks a
Page 65: 4.3 Implemented Algorithm 47leg 1p=
Page 69 and 70: 4.3 Implemented Algorithm 51P 2P 1
Page 71 and 72: Chapter 5Traffic Flow OptimizationT
Page 73 and 74: 5.2 Traffic Region Model 5514 1516
Page 75 and 76: 5.3 Tasks Definition for Intersecti
Page 77 and 78: 5.4 Scheduling with Communication D
Page 79: 5.5 Summary 61Intersection 6T 2,6T
Page 82 and 83: 64 Chapter 6 Conclusionsoptimal und
Page 85 and 86: Appendix ATORSCHE AlgorithmsA.1 Sch
Page 87 and 88: Bibliography[Abou 09] K. Aboudolas,
Page 89 and 90: Bibliography 71[Febb 06] A. D. Febb
Page 91 and 92: Bibliography 73[Kwak 72]Cybernetics
Page 93: Bibliography 75[Tolb 01] C. Tolba.
Page 96 and 97: 78 Indexphase, 3offset, 3, 53split,
Page 99 and 100: List of Author’s PublicationsPape
Page 101: List of Author’s Publications 83P
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