 1  Outline  Model  problem statement  detailed ARENA model  model technique  Output Analysis

 2  Model 5-1: An Automotive Maintenance and Repair Shop  additional maintenance and repair facility in the suburban area  customer orders (calls)  by appointments, from one to three days in advance  calls arrivals ~ Poisson process, mean 25 calls/day  distribution of calls: 55% for the next day; 30% for the days after tomorrow; 15% for two days after tomorrow  response missing a desirable day: 90% choose the following day; 10% leave

 3  An Automotive Repair and Maintenance Shop  service  Book Time, (i.e., estimated service time) ~ *BETA(2, 3) min  Book Time also for costing  promised wait time to customers  wait time = Book Time + one hour allowance  actual service time ~ GAMM(book time/1.05, 1.05) min  first priority to wait customers  customer behavior  20% wait, 80% pick up cars later  about 60% to 70% of customers arrive on time  30% to 40% arrive within 3 hours of appointment time

 4  Costs and Revenues  schedule rules  at most five wait customers per day  no more than 24 book hours scheduled per day (three bays, eight hours each)  normal cost: $45/hour/bay, 40-hour per week  overtime costs $120/hour/bay, at most 3 hours  revenue from customers: $78/ book hour  penalty cost  each incomplete on-going car at the end of a day: $35  no penalty for a car whose service not yet started

 5  System Performance  simulate the system 20 days to get  average daily profit  average daily book time  average daily actual service time  average daily overtime  average daily number of wait appointments not completed on time

 6  Relationship Between Models  Model 5-1: An Automotive Maintenance and Repair Shop  a fairly complicated model  non-queueing type  Model 5-2: Enhancing the Automotive Shop Model  two types of repair bays for different types of cars  customer not on time

 7  The Structure of the ARENA Model  Five parts  Control Logic to initialize variables and count days  Generate appointment calls, including a representative initial condition  Make appointments, considering priority of jobs  Service activities  Update performance variables

 8  Details of Model  logic control and submodels  for each day  first simulate the calls for appointments (of future days)  then simulate the work of the day  vectors  variables and expressions

 9  Steps to Prepare a Simulation Program  assumption: already formulated the problem, i.e., fully understood how the system works  for a simple problem: use the crude to detailed pseudo code approach to build the flow of the model  for a complicated problem  first play around with a simplified problem  use paper and pencil to simulate

 10  An Illustration for Model  a simplified version of Model  a week of three days  reservations made two days in advance  Book Time = 1 w.p. 3/4 and = 2 w.p. 1/4  actual Service time  = 1.2 Book Time w.p. 1/3  = 0.8 Book Time w.p. 2/3

 11  An Illustration for Model  each customer equally likely to be leave or wait  every day 4 hours, with at most 1 hour OT  at most 1 customer to leave his car  number of customers in each day  = 2 w.p. 1/3 and = 3 w.p. 2/3  simulation duration: 4 days

 12  Before Simulation  terminating or non-terminating process?  non-terminating  typically simulated for a long time and the initial condition being unimportant  how to set the initial condition if a non- terminating system is simulated for a short time?  empty: is it representative?  not empty: how to make it representative?

 13  To Generate a Representative Initial Condition  representative initial condition  day 1: appointments made in previous two days, i.e., day -1 and day 0  day 2: appointments made in day 0  idea  generate calls for day -1 and then for day 0  whenever applicable, schedule appointments on days 1 and 2  implicitly drop appointments for days -1 and 0

 14  Paper and Pencil Simulation of the Simplified System Day

 15  Very Crude Pseudo-Code  1  Generate a representative initial condition  2  Simulate the system for 4 days  assumption for the model: ignore the time of calls, assuming that all happen in the morning

 16  Refinement of the First Step of the Pseudo-Code Generate a representative initial condition Start with an empty 6-day schedule Generate Book Times and schedule them for calls in Day -1 Generate number of calls for Day -1 Generate Book Times and schedule them for calls in Day 0 Generate number of calls for Day 0

 17  Refinement of the Second Step of the Pseudo-Code Refinement of the Second Step of the Pseudo-Code

 18  To Implement in ARENA  need further refinement of the pseudo-code  need modifying the pseudo-code to suit the structure of ARENA, e.g.,  what are the entities in the ARENA model?  what are the correspondence between the steps in paper and pencil simulation and ARENA?  ….. lots of details

 19  Output Analysis  simulation: estimate  = E(X) by observing sample values from the distribution of X  output analysis  point estimator of  ?  unbiased estimator of  ?  variance of estimator?  efficient estimator of  ?  confidence on the range estimator?  # of simulation runs (replications) required?

 20  Desirable Functions of Software  interval estimation  comparing alternatives  automatic statistical tests  handy housekeeping for scenarios  automatic searching for optimal  all features available in ARENA

 21  Output Analysis  two types of estimates, point and interval  theoretical basis  point estimates: SLLN  interval estimate: CLT

 22  Strong Law of Large Numbers  i.i.d. random variables X 1, X 2, …  finite mean  and variance  2  define   = E(X)  (X 1 + … + X n )/n

 23  Additional Facts  X 1, X 2,..., X n be i.i.d.; finite mean  and variance  2 unbiased estimator of  unbiased estimator of  2

 24  Central Limit Theorem  i.i.d. random variables X 1, X 2, …  finite mean  and variance  2

 25  Central Limit Theorem - Basis to Analyze Terminating Systems  t,  2, and F: useful distributions for range estimation and hypothesis testing of normal random variables X i ’ s  CLT: statsitics approximately normal for “ large enough ” n  can use t,  2, and F for (approximate) range estimation and hypothesis testing

 26  Differences Between Terminating and Non-Terminating Processes Differences Between Terminating and Non-Terminating Processes  termination condition and run length  terminating: well-defined, i.i.d. replications  non-terminating: no well-defined length  initial condition  terminating: clear, defined by the problem  non-terminating: unclear, biased by any fixed initial value  random variables for estimation  i.i.d. random variables  stationary version of random variables

 27  Non-Terminating Processes         time

 28  Terminating Processes  standard outputs  interval estimate of mean Model 05-02Model  hypothesis testing of mean Model 05-02Model  number of runs Model 06-01Model  saving results in an output file for further processing  export from Output Analyzer to a text file Model 06-02Model  processed by first by Excel and then Input Analyzer to analyze the output data Model 06-02Model  confidence intervals by Output Analyzer Model 06-02Model  comparison by Output Analyzer Model 06-03Model  sequential determination of number of runs comparison by Output Analyzer Model 12-03Model 12-03

 29  Non-Terminating Processes  non-terminating process Model 07-02Model  Output Analyzer  replication/deletion Model replication/deletionModel  batch means  sequential batch means  auto-correlation  regenerative simulation