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WOOD 492 MODELLING FOR DECISION SUPPORTPowerPoint Presentation

WOOD 492 MODELLING FOR DECISION SUPPORT

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WOOD 492 MODELLING FOR DECISION SUPPORT

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WOOD 492 MODELLING FOR DECISION SUPPORT

Lecture 27

Simulation

- Simulated a single server queue with Next-event increment method
- 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

- Next-event increment has two steps:
- Advance time to the time of the next event
- Update N(t)

Example 16

Wood 492 - Saba Vahid

- A drive-in restaurant has one queue and two servers for bringing the food to the cars
- The cars arrive every 1 to 4 minutes according to the probabilities in the table below. CDF (same as F(x)) values are given in the last column.
- Cars wait for the first server who’s free or has been free the longest
- The servers have different times for serving cars
- Server 1: uniform distribution between 2 to 4 minutes
- Server 2: uniform distribution between 3 to 5 minutes

Wood 492 - Saba Vahid

- Uniform distribution: all values have the same probability of occurring
- For example: the probability of server 1 taking 3 minutes, or 4 minutes or 2.5 minutes to serve a car is all the same and is calculated as:
a and b are min and max values of the random variable x (e.g. 2 and 4 for server 1)

- The CDF values for this distribution is:
So the probability of a service time smaller than 3 minutes is: (3-2)/(4-2)=50%

- The inverse of CDF is calculated with this formula:
Where p is the random number you draw and t is the corresponding service time

Wood 492 - Saba Vahid

- Use Next-event increment method
- Assume at t=0 there are 2 cars in line and both servers are busy
- State of the system = N(t) = number of cars in the line
- Potential events:
- Arrival of cars (arrival)
- Car served by server 1 (exit to 1)
- Car served by server 2 (exit to 2)

- System transition formula:
- Arrival: N(t)=N(t-1)+1
- Exit to 1 or exit to 2 : N(t)=N(t-1)-1

- Simulation clock: moves to the next event time, decided by a random draw and inverse CDF transformation

Example 17

Wood 492 - Saba Vahid

- Exam on November 19th, 9:00 am, same room as usual
- Grades will be posted at my door about one week later
- Check the course website for any potential updates
- Quiz 5 answers posted online
- Some extra simulation and network problems will be uploaded next week
- Friday, 16th 10 to 12 office hours room 2026

Wood 492 - Saba Vahid