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WOOD 492 MODELLING FOR DECISION SUPPORT. Lecture 27 Simulation. Review. 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:

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WOOD 492 MODELLING FOR DECISION SUPPORT

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## WOOD 492 MODELLING FOR DECISION SUPPORT

Lecture 27

Simulation

### Review

• 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

### Example 17: drive-in restaurant simulation

• 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

• 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

### Simulating the drive-in system

• 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