1. SIMULATION EXAMPLES

2. SELECTED SIMULATION EXAMPLES 4 Queuing systems (Dynamic System) 4 Inventory systems (Dynamic and Static) 4 Monte-Carlo simulation (Static)

3. Example: A Doctor Office

4. Elements of the system 4 Entities: –patients 4 Characteristics of the system : –Arrival process (Random Interarrivals 1-6) –Examination or service process (Random Service Times 1-4) –Infinite line length –FCFS queue discipline

5. Elements of the system 4 Events: –Arrival of patients –Completion of service (examination) 4 State variables: –Number of patients in the system –Number of patients in the waiting line –Status of doctor 4 Performance measures: –Time in System –Waiting time in the line –Utilization

6. Arrival Event

7. Service Completion Event

8. Interarrival and Clock Times CustomerInterarrival Time Arrival Time on Clock 1-0 222 346 417 529 6615

9. Service Times CustomerService Time 12 21 33 42 51 64

10. Simulation Table 1-002202 22221301 34663903 Customer Arrival time Service Begins Service time Service Ends Time in queue Time in system Idle Time 417 TBA 0 0 3 417 921124 529 11223 6615 41904 0 0 3 6

11. Chronological Ordering of Events Event TypeCustomer NumberClock Time Arrival10 Departure12 Arrival22 Departure23 Arrival36 47 Departure39 Arrival59 Departure411 Departure512 Arrival615 Departure619

12. Number of Customers in the System

13. Statistics 3Average waiting time= Total waiting time / number of patients = 4/6=0.66 3Average time in system= 17/6=2.83 3Average service time =13/6=2.17 (2.5) 3Average interarrival time =15/5=3 (3.5) 3Doctor utilization= Total busy time/ total time = (19-6)/19= 68% 3Average number of patients in queue = Total time in queue/ total time = 4/19=0.21

14. Further questions 4 Can we simulate the system 10,000 patients? 4 How about more complex systems?

15. (M,N) Inventory Policy 4 N is fixed & Q varies IP(t) t M QkQk Q k+1 NN N

16. (M,N) Policy Example 4 Review period N=5 days, Order-up-to level M=11 units (The order is given at the end of the review day and arrives at the beginning of the day after the lead-time elapses) 4 The shortages are backordered and instantaneously satisfied the moment the replenishment order arrives 4 Beginning inventory = 3 units; 8 units scheduled to arrive in two days 4 Holding cost h = $1 per unit per day 4 Shortage cost s = $2 per unit per day 4 Ordering cost K = $10 per order t 4 Question: based on 5 cycles of simulation, calculate –Average number of on-hand inventory at the end of the day –Average number of shortage per day –Average cost per day

17. INPUT DATA 1. Demand Distribution 2. Lead Time Distribution

18. c

19. Newsboy Problem (Static) 4 One period problem that involves a single procurement 4 He buys the papers 33 cents each and sells them 50 cents each 4 Papers not sold are scrapped at 5 cents each 4 The optimal number of papers that the newsboy should purchase each day? 4 Profit = Sales Revenue – Cost of Papers + Salvage Revenue

20. Distribution of Newspapers Demanded DemandProbability Cumulative Probability Random-Digit Assignment 400.03 01-03 500.050.0804-08 600.150.2309-23 700.200.4324-43 800.350.7844-78 900.150.9379-93 1000.071.0094-00

21. Simulation Table in Excel

22. MONTE-CARLO SIMULATION 4 Use random numbers and random sampling to approximate the outcome –stochastic and static simulation –consists of a series random events with each event unaffected by the prior events –(the passage of time is not a part of simulation)

23. Example 4 Estimating the area of an amorphous shape Y X 50 100 Area

24. Procedure: . Choose a pair of coordinates randomly (using a uniform random variable for each dimension) . Count success if it is inside the area m=m+1 . Repeat the process n times . Estimate the area Area  5000*m/n as n 

25. Estimating the Value of  0 1 1  X, Y ~ uniform (0,1) Estimate  value by simulation

26. Convergence to 

27. Example: Approximating integrals 4 One of the earliest applications

28. Summary 4 Applications can be found in many areas 4 The ad hoc methodology applied for obtaining the simulation tables is not suitable for more complex models of dynamic systems 4 A more systematic methodology: event- scheduling