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Simulation of Single Server Queue and Drive-In Restaurant: Modeling for Decision Support

This lecture reviews simulation techniques using the next-event increment method to analyze a single server queue. We simulate the arrival and serving of customers, focusing on key parameters like mean inter-arrival and service times. The examples include a drive-in restaurant's operations, showcasing the uniform distribution of service times and customer arrivals. We will also discuss the system state transitions and the simulation clock functionality necessary for accurate modeling. Key takeaways include understanding the application of probability distributions in queue management.

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Simulation of Single Server Queue and Drive-In Restaurant: Modeling for Decision Support

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  1. WOOD 492 MODELLING FOR DECISION SUPPORT Lecture 27 Simulation

  2. 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

  3. 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

  4. 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

  5. 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

  6. Final comments • 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

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