IE Advanced Simulation Experiment Design and Analysis. Hong Wan Purdue University

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1 IE Advanced Simulation Experiment Design and Analysis Hong Wan Purdue University Spring, 2007, based on Prof. Barry L. Nelson s notes for IEMS465, Northwestern University 1

2 SIMULATION OVERVIEW One view of dynamic, stochastic systems: INPUTS + LOGIC = SAMPLE PATHS The inputs represent the uncertainty in a system at the most basic level. The logic describes how the system reacts to realizations of the inputs. The sample path is a time-ordered collection of data representing what happened to the system in one instance. Simulation produces new sample paths by generating inputs and applying the logic. Discrete-event, stochastic simulation is used when the important changes in the system occur in jumps at a countable set of event times. 2

3 DISCRETE-EVENT SIMULATION EXAMPLE Two components work as an active and spare, so the system fails if both are failed. The lifetime of a component is equally likely 1, 2, 3, 4, 5 or 6 days. Repair takes exactly 2.5 days (only one at a time). What can we say about the time to failure (TTF) for this system? The state of the system is the number of functional components. The events are the failure of a component and the completion of a repair. 3

4 System Future Event List Clock State Next Failure Next Repair 0 2 4

5 FEATURES Simulated time (the simulation clock) jumps from event time to event time; this is called next-event time advance. The current state of the system, and the list of future events, are all we need to advance the system to the next state change. In this sense the system is Markovian. The simulation ends when a particular system state occurs. In other simulations termination may occur at a fixed time or event count. The system state over time is a sample path (output) from which we may extract performance measures. It is one realization of a stochastic process. 5

6 OUTPUTS We distinguish between within-replication and across-replication output data. The time of system failure Y and the number of functional components {S(t); t 0} are within-replication outputs. The times of system failure Y 1, Y 2,..., Y n, and the average number of functional components, S 1, S 2,..., S n, from n replications are across-replication data. Notice that S is a time-average because S(t) is a continuous-time output variable. S = 1 Y Y 0 S(t) dt = 1 e N e 0 N i=1 S(e i 1 ) (e i e i 1 ) where e 0, e 1,..., e N are the event times in a replication. 6

7 SIMULATION PROGRAMMING To code a discrete-event simulation (generically) we need the following: State variables: Everything needed to describe the system at any point in time. Clock: Keeps current simulation time. Event list: Data structure to keep track of future events in chronological order. Event routines: Subprograms that describe how the system state changes for each type of event. Timing routine: Pulls next event from event list, advances time and executes the event. 7

8 Dim Clock As Double Dim NextFailure As Double Dim NextRepair As Double Dim S As Double Dim Tlast As Double Dim Area As Double simulation clock time of next failure event time of next repair event system state time of previous event area under S curve VBA Example Public Sub MainProgram() Program to generate a sample path for the reliability example Dim NextEvent As String End Sub S = 2 Clock = 0 Tlast = 0 Area = 0 NextFailure = WorksheetFunction.Floor(6 * Rnd(), 1) + 1 NextRepair = Do Until S = 0 NextEvent = Timer Select Case NextEvent Case "Failure" Call Failure Case "Repair" Call Repair End Select Loop MsgBox ("System failure at time " _ & Clock & " with average # components " & Area / Clock) 8

9 Public Function Timer() As String Determine the next event and advance time If NextFailure < NextRepair Then Timer = "Failure" Clock = NextFailure NextFailure = Else Timer = "Repair" Clock = NextRepair NextRepair = End If End Function Public Sub Failure() Failure event Area = Area + (Clock - Tlast) * S Tlast = Clock S = S - 1 If S = 1 Then NextFailure = Clock + WorksheetFunction.Floor(6 * Rnd(), 1) + 1 NextRepair = Clock End If End Sub 9

10 Public Sub Repair() Repair event Area = Area + (Clock - Tlast) * S Tlast = Clock S = S + 1 If S = 1 Then NextRepair = Clock NextFailure = Clock + WorksheetFunction.Floor(6 * Rnd(), 1) + 1 End If End Sub 10

11 GENERALIZING To make this type of simulation easier, we need... More powerful data structure to handle many classes and multiple instances of events, and store additional information with them. Support routines to generate random observations from various distributions and to collect system performance data within replications. A structure for executing multiple replications and collecting performance data across replications. A report generator to produce statistical summary reports. 11

12 PROCESS WORLD VIEW You may be familiar with process- or network- oriented simulation languages like Arena, ProModel, AutoMod, GPSS, etc. These languages represent a simulation by describing how entities (people, parts, information, etc.) flow through a network of processes. ARRIVE; QUEUE TO SEIZE SERVER; DELAY; RELEASE SERVER; EXIT; Process-oriented languages exectute as discrete-event simulations; the network of processes imply the events, rather than explicitly defining them. Process languages provide for easier modeling, but less flexibility. 12

13 COURSE OVERVIEW 1. Quick essentials of random-number and random-variate generation 2. Estimation and output analysis 3. Comparing alternatives (including common random numbers) 4. Optimization via simulation 5. Variance reduction 6. Some topics in input modeling 7. Mathematics for simulation research 13

14 8. Design of Experiments for Discrete Event Simulation