Numerical approach estimate

Transcription

1 Simulation

2 Nature of simulation Numericalapproachfor investigating models of systems. Data are gathered to estimatethe true characteristics of the model. Garbage in garbage out! One of the techniques of operations research possibly the most widely used.

3 Nature of simulation -2/2 Typical applications: Designing and analyzing manufacturing systems. Evaluating military weapons systems. Evaluating logistics of weapons supply systems. Determining hardware and software requirements for a computer system. Designing and operating transportation systems. Evaluating designs for service organizations. Re-engineering business processes. Analyzing financial systems (?)

4 Designing and analyzing manufacturing systems.

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11 Systems, Models and Simulation System: collection of entities, acting and interacting towards some logical end. The definition of a system depends on the goal of the analysis. The state of the system is the collection of variablesnecessary to describe the system at a particular time. The definitional of the state of a system is conditional on the goals of the study.

12 Systems, Models and Simulation Types of systems: Discrete: state variables change instantaneously at discrete points in time. Number of cars in a parking lot. Continuous: State variables change continuously in time. Speed of a car.

13 Systems, Models and Simulation Ways to study a system: System Experiment with the system Experiment with a model of the system Physical Model Mathematical Model Analytical Solution Simulation

14 Systems, Models and Simulation Experiment with the system: Physical Model Analytical solutions vs. Simulation

15 Systems, Models and Simulation Types of simulations: Static vs. Dynamic simulations Monte Carlo Deterministic vs. Stochastic Output is determined given the inputs. Output changes given the same inputs. Continuous time vs. Discrete Simulation Similar to continuous and discrete models.

16 Discrete event simulation The system can change at only a countable number of points in time. Events: cause changes to the simulation. Types of events: To change the state of the system. To change the simulation itself. Usually done using computers.

17 Time advance Mechanism All dynamic simulations need a system clock a counter to mimic real world time advances. This is known as the simulation clock. Fixed increment time advance: Computer Games Next event time advance: Most commonly used system.

18 Next event time advance Single server queuing system:

19 Components and organization of simulations System state: collection of state variables necessary to describe the system at a particular time. Simulation clock: a variable giving the current value of simulated time. Event list: A list containing the next time when each type of event will occur. Statistical counters: performance measuring variables. Initialization routine: Used at start of simulation. Timing routine: Next event selection routine. Event routine: Next event handling routine. Library routines: used to generate random variables. Report Generators: Performance reporting routines. Main program: Visual interface control routines:

20 Fowchartfor next-event time-advance Start Initialization routine: Set clock = 0 Initialize state variables and statistical counters Initialize event list. Main program: Invoke initialization routine Invoke timing routine invoke event routine Timing routines: Determine next event type. Advance simulation clock. Event routine: Update system state Update statistical counters Generate future events. Library routines: Generate random variables Simulation over? Report generator: Compute estimates of interest. Write report Stop

21 Simulation approaches Event scheduling approach. Process approach.

22 Simulation of single server queuing system Inter-arrival times are independent and identically distributed random variables. A customer who arrives and finds the server idle starts service immediately. Service times of successive customers are IID random variables. If server is busy, customer joins the waiting queue. Server selects customers from queue in a FIFO manner. Other issues?

23 Simulation of single server queuing system What do we want to study? Customer delays? Average number of customers in the queue:

24 Single server system Where, p i is the fraction of time there are i customers in the queue. This can also be written as:

25 Single server queuing model Fraction of time the server is idle?

26 Initialization Routine

27 Simulation

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29 Homework I How would you simulate a 4 way traffic crossing? Use excel to simulate the single server queuing system for 100 arrivals. What is the mean server utilization? Identify a data structure you would use for handling the event list.