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WOOD 492 MODELLING FOR DECISION SUPPORT

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WOOD 492 MODELLING FOR DECISION SUPPORT

Lecture 25

Simulation

- 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

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- 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
- …

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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

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- 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)

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- 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:

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)

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- 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

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- 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?

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- 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