Wood 492 modelling for decision support
This presentation is the property of its rightful owner.
Sponsored Links
1 / 9

WOOD 492 MODELLING FOR DECISION SUPPORT PowerPoint PPT Presentation


  • 99 Views
  • Uploaded on
  • Presentation posted in: General

WOOD 492 MODELLING FOR DECISION SUPPORT. Lecture 25 Simulation. Review. Simulation: used to imitate the real system using computer software, helpful when system is too complex or has many stochastic elements

Download Presentation

WOOD 492 MODELLING FOR DECISION SUPPORT

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


Wood 492 modelling for decision support

WOOD 492 MODELLING FOR DECISION SUPPORT

Lecture 25

Simulation


Review

Review

  • Simulation: used to imitate the real system using computer software, helpful when system is too complex or has many stochastic elements

  • Discrete event simulation: if the state of the system changes at random points in time as a result of various events

  • Different probability distributions are used for different purposes

Wood 492 - Saba Vahid


Distributions

Distributions

  • Various probability distributions are used for different random events

  • Poisson : distribution of number of arrivals per unit of time

  • Exponential : distribution of time between successive events (arrivals, serving customers,…)

  • Uniform: for random number generation

  • Normal : for some physical phenomenon's, normally used to represent the distributions of the means of observations from other distributions

  • Binomial: coin flip

Wood 492 - Saba Vahid


Cumulative distribution function cdf

Cumulative Distribution Function (CDF)

Highlighted area: P(x<=t)

  • CDF is calculated using the area under the probability density graph (PDF):

  • Assume x is a random variable and t is a possible value for x. If we show the PDF of x with f(x) and the CDF with F(x):

    =(the area under f(x) up to point t)

f(x)

t

x

F(x)

1.0

P(x<=t)

t

x

Wood 492 - Saba Vahid


Example 16 a discrete event simulation

Example 16 – A discrete event simulation

  • Simulate a queuing system :

    • One server

    • Customers arrive according to a Poisson distribution (mean arrival rate λ = 3 per hour)

    • Service rate changes according to a Poisson distribution (mean service rate μ = 5 customers per hour)

Wood 492 - Saba Vahid


Probability reminder

Probability reminder

  • When the arrival rate α(number of arrivals per unit of time t) follows a Poissondistribution with the mean of αt, it means that inter-arrival times (the time between each consecutive pair of arrival) follow an exponential distribution with the mean of 1/α

  • If x belongs to an Exponential distribution with the mean 1/α:

  • Therefore, if customers arrive with the mean rate of 3 per hour, the inter-arrival time has an exponential distribution with the mean of 1/3 hour (on average one arrival happens every 1/3 hour)

    So, for example, the probability of an arrival happening in the first hour (time of event, x, is less than or equal to 1 hour, t)

  • Wood 492 - Saba Vahid


    Example 16 queuing system

    Example 16 – Queuing system

    • State of the system at each time t

      • N(t) = number of customers in the queue at time t

    • Random events in the simulation:

      • Arrival of customers (mean inter arrival times are 1/3 hour)

      • Serving the customers (mean service times are 1/5 hour)

    • System transition formula:

      • Arrival: reset N(t) to N(t)+1

      • Serve customer: reset N(t) to N(t)-1

    • How to change the simulation clock (2 ways):

      • Fixed-time increment

      • Next-event increment

    Wood 492 - Saba Vahid


    Fixed time increment for example 16

    Fixed-time increment for Example 16

    • Two steps:

      • Advance the clock by a small fixed amount (e.g: 0.1 hour)

      • Update N(t) based on the events that have occurred (arrivals and serving customers)

        Example: let’s move the clock from t=0 to t=0.1 hr

        N(0)=0

        Probability of an arrival happening in the first 0.1 hr is:

        Probability of a departure happening in the first 0.1 hr is:

        How to use these probabilities?

    Wood 492 - Saba Vahid


    Using random numbers to generate events

    Using random numbers to generate events

    • To see if the events should occur or not, we use a random number generator to generate a uniform random number between [0,1] (e.g. in Excel there is a RAND() function that does this)

    • If the random number is less than the calculated probability (in previous slide) we accept the event, if not we reject it.

    • let’s assume we’ve generated a random number for the arrival of customers with Rand() function, random_A=0.1351

      Random_A < 0.259 so we accept the arrival

      We must generate a new random number for each case, so let’s assume random_D=0.5622

      Random_D >= 0.393 so we reject the departure

      N(1) = N(0)+ 1 (arrival) – 0 (departure) = 0+1=1

    Example 16

    Wood 492 - Saba Vahid


  • Login