___________________________________________________________________________

1. Waiting Line Models ___________________________________________________________________________ Operations Research  Jan Fábry

2. Service Arrival Process Exit {Server} Waiting Line Models Waiting Line System Source Waiting Area {Queue} {Customers} {Potential Customers} ___________________________________________________________________________ Operations Research  Jan Fábry

3. Waiting Line Models Examples of Waiting Line Systems ___________________________________________________________________________ Operations Research  Jan Fábry

4. Service Arrival Process Exit {Server} Waiting Line Models Waiting Line System Source Waiting Area {Queue} {Customers} {Potential Customers} ___________________________________________________________________________ Operations Research  Jan Fábry

5. Waiting Line Models Arrival Process Source Infinite – tourists Finite – machines in factory {Potential Customers} ___________________________________________________________________________ Operations Research  Jan Fábry

6. Waiting Line Models Arrival Process In batches – BUS of tourists Arrivals Individually – patients Scheduled – trams, trains Arrivals Unscheduled – patients ___________________________________________________________________________ Operations Research  Jan Fábry

7. Arrival Arrival Arrival Time Waiting Line Models Arrival Process Arrival rate – number of arrivalsper time unit (POISSON distribution) Average arrival rate = –average number of arrivalsper time unit (mean of POISSON distribution) ___________________________________________________________________________ Operations Research  Jan Fábry

8. Arrival Arrival Arrival Time Waiting Line Models Arrival Process Interarrival time – time period between two arrivals (EXPONENTIAL distribution) Average interarrival time = 1/–average time period beetween arrivals (mean of EXPONENTIAL distribution) ___________________________________________________________________________ Operations Research  Jan Fábry

9. Service {Server} Waiting Line Models Service Process Service rate – number of customers served per time unit (POISSON distribution) Average service rate = –average number of customers served per time unit (mean of POISSON distribution) ___________________________________________________________________________ Operations Research  Jan Fábry

10. Service {Server} Waiting Line Models Service Process Service time – time customer spends at service facility (EXPONENTIAL distribution) Average service time = 1/–average timecustomers spend at service facility (mean of EXPONENTIAL distribution) ___________________________________________________________________________ Operations Research  Jan Fábry

11. Arrival Exit Queue Server Waiting Line Models Service Process • Service configurations (type, number and arrangement of service facilities) 1. Single facility ___________________________________________________________________________ Operations Research  Jan Fábry

12. Exit Arrival Queue Servers Waiting Line Models Service Process 2. Multiple, parallel, identical facilities (SINGLE queue) ___________________________________________________________________________ Operations Research  Jan Fábry

13. Exit Arrival Queues Servers Waiting Line Models Service Process 2. Multiple, parallel, identical facilities (MULTIPLE queue) ___________________________________________________________________________ Operations Research  Jan Fábry

14. Exit Arrival Queues Servers Waiting Line Models Service Process 3. Multiple, parallel, but not identical facilities ___________________________________________________________________________ Operations Research  Jan Fábry

15. Arrival Exit Queue Queue Queue Server Server Server Waiting Line Models Service Process 4. Serial facilities 5. Combination of facilities ___________________________________________________________________________ Operations Research  Jan Fábry

16. Service {Server} Waiting Line Models Waiting Line • Discipline of the queue FCFS (First-Come, First-Served) ___________________________________________________________________________ Operations Research  Jan Fábry

17. Service {Server} Waiting Line Models Waiting Line • Discipline of the queue LCFS (Last-Come, First-Served) ___________________________________________________________________________ Operations Research  Jan Fábry

18. Service {Server} Waiting Line Models Waiting Line • Discipline of the queue PRI (PRIority system) ___________________________________________________________________________ Operations Research  Jan Fábry

19. Service {Server} Waiting Line Models Waiting Line • Discipline of the queue SIRO (Selection In Random Order) ___________________________________________________________________________ Operations Research  Jan Fábry

20. Waiting Line Models Analysis of Waiting Line Models Cost • Waiting cost • Service cost (facility cost) - cost of construction - cost of operation - cost of maintenance and repair - other costs (insurance, rental) ___________________________________________________________________________ Operations Research  Jan Fábry

21. Waiting Line Models Analysis of Waiting Line Models Time characteristics • Average waiting time in the queue • Average waiting time in the system ___________________________________________________________________________ Operations Research  Jan Fábry

22. Waiting Line Models Analysis of Waiting Line Models Number of customers • Average number of customers in the queue • Average number of customers in the system ___________________________________________________________________________ Operations Research  Jan Fábry

23. Waiting Line Models Analysis of Waiting Line Models Probability characteristics • Probability of empty service facility • Probability of the service facility being busy • Probability of finding N customers in the system • Probability that N> n • Probability of being in the system longer than time t ___________________________________________________________________________ Operations Research  Jan Fábry

24. Size of customer’s source Maximum length of queue Queue discipline Number of parallel servers Probability distribution of service time Probability distribution of interarrival time Waiting Line Models Classification of Waiting Line Models Kendall’s notation A/B/C/D/E/F ___________________________________________________________________________ Operations Research  Jan Fábry

25. Waiting Line Models Standard Single-Server Exponential Model ___________________________________________________________________________ Operations Research  Jan Fábry

26. Arrival Exit Queue Server Waiting Line Models Standard Single-Server Exponential Model ( M / M / 1 / FCFS/∞/∞ ) ___________________________________________________________________________ Operations Research  Jan Fábry

27. Waiting Line Models Standard Single-Server Exponential Model Assumptions • Single server • Interarrival times - exponential probability distribution with the mean = 1/λ • Service times - exponential probability distribution with the mean = 1/μ ___________________________________________________________________________ Operations Research  Jan Fábry

28. Waiting Line Models Standard Single-Server Exponential Model Assumptions • Infinite source • Unlimited length of queue • Queue discipline is FCFS ___________________________________________________________________________ Operations Research  Jan Fábry

29. Waiting Line Models Standard Single-Server Exponential Model Example – Grocery • One shop assistant – serves 25 customers per hour (on the average) • From 8 a.m. to 6 p.m. – 18 customers per hour arrive (on the average) ___________________________________________________________________________ Operations Research  Jan Fábry

30. Waiting Line Models Standard Single-Server Exponential Model Example – Grocery • Average arrival rate λ = 18 customers per hour • Average service rate μ = 25 customers per hour ___________________________________________________________________________ Operations Research  Jan Fábry

31. Waiting Line Models Standard Single-Server Exponential Model Example – Grocery • Utilization of the system – probability that the server is busy – probability that there is at least 1 customer in the system • Probability of an empty facility (server is idle) ___________________________________________________________________________ Operations Research  Jan Fábry

32. Waiting Line Models Standard Single-Server Exponential Model Example – Grocery • Average waiting time in the system • Average waiting time in the queue ___________________________________________________________________________ Operations Research  Jan Fábry

33. Waiting Line Models Standard Single-Server Exponential Model Example – Grocery • Average number of customers in the system • Average number of customers in the queue ___________________________________________________________________________ Operations Research  Jan Fábry

34. Waiting Line Models Standard Single-Server Exponential Model Example – Grocery • Probability of finding exactly N customers in the system ___________________________________________________________________________ Operations Research  Jan Fábry

35. Waiting Line Models Standard Single-Server Exponential Model Example – Grocery • Probability that N> n ___________________________________________________________________________ Operations Research  Jan Fábry

36. Waiting Line Models Standard Single-Server Exponential Model Example – Grocery • Probability of being in the system longer than time t ___________________________________________________________________________ Operations Research  Jan Fábry

37. Computer Simulation ___________________________________________________________________________ Operations Research  Jan Fábry

38. Computer Simulation Analytical tools Solution Computer simulation Computer simulationis a special method using computer experiments with the model of a real system ___________________________________________________________________________ Operations Research  Jan Fábry

39. Computer Simulation • Entity- object that goes through the model • Resource- agent required by the entity • Event- significant change of the system • Activity- process between two events • Generating of random values • Simulation time • Computer simulation language • Animation ___________________________________________________________________________ Operations Research  Jan Fábry