WOOD 492 MODELLING FOR DECISION SUPPORT

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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

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### 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
• 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
• 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)

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
• 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
• 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
• 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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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
• 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