1. 4/11: Queuing Models Collect homework, roll call Queuing Theory, Situations Single-Channel Waiting Line System –Distribution of arrivals –Distribution of service times –Queue discipline: FCFS –Steady-state operation –Operating characteristics Multiple-Channel Waiting Line System

2. Queuing Theory, Situations Waiting line for a roller coaster Waiting line at a restaurant Need to find an acceptable balance between few workers and few lost customers. Image courtesy of ohiomathworks.orgohiomathworks.org

3. Structure Single-channel waiting line Server Customers in line

4. Distribution of Arrivals When customers arrive Assume random & independent arrivals. A Poisson probability distribution: Server Customers in line x : # of arrivals in period : mean # of arrivals per period e : 2.71828

5. Distribution of Arrivals: Example Assume that it has been calculated that 30 customers arrive per hour. Calculate the likelihood that no customers, 1 customer, and 2 customers will arrive in the next two minutes. Server Customers in line

6. Distribution of Arrivals: Example = 30 cust./60 min. = 0.50 cust./ 1 min. No customers in next minute: Server Customers in line

7. Distribution of Arrivals: Example = 30 cust./60 min. = 0.50 cust./ 1 min. One customer in next minute: Server Customers in line

8. Distribution of Arrivals: Example = 30 cust./60 min. = 0.50 cust./ 1 min. Two customers in next minute: Server Customers in line

9. Distribution of Arrivals: Example = 30 cust./ hour  ? Two customers in next TWO minutes: Server Customers in line = 1 customer / 2 min.

10. Distribution of Service Times Service times follow an exponential probability distribution.  = average # of units that can be served per time period e = 2.71828

11. Prob. of Service Time: Example Our server can take care of, on average, 45 customers per hour. What is the probability of an order taking less than one minute?

12. Prob. of Service Time: Example Our server : 45 customers per hour. What is the probability of an order taking less than one minute?

13. Prob. of Service Time: Example Our server : 45 customers per hour. What is the probability of an order taking less than two minutes?

14. Queue Discipline We must define how the units are arranged for service. We will assume that our lines are FCFS, or First-Come, First-Served. Balking, reneging, jockeying

15. Steady-State Operation There is a transient period when a line starts, when things cannot be predicted. The system must be stable, that is, able to keep up with the customers. Our models apply only to the normal operation, or the steady-state operation.

16. Operating Characteristics Probability of no units in system Average number of units in waiting line Average number of units in system Average time unit is in waiting line Average time a unit is in the system Probability that an arriving unit will have to wait for service Probability of n units in system

17. Probability of no units in system = mean arrival rate (mean # arriv./time)  = mean service rate (mean # serv./time) The system includes the customer being serviced.

18. Average # of units in waiting line = mean arrival rate (mean # arriv./time)  = mean service rate (mean # serv./time) The waiting line does NOT include the customer being serviced.

19. Average # of units in system = mean arrival rate (mean # arriv./time)  = mean service rate (mean # serv./time)

20. Avg. time a unit is in waiting line = mean arrival rate (mean # arriv./time)  = mean service rate (mean # serv./time)

21. Average time a unit is in system = mean arrival rate (mean # arriv./time)  = mean service rate (mean # serv./time)

22. Probability that an arriving unit will have to wait for service = mean arrival rate (mean # arriv./time)  = mean service rate (mean # serv./time)

23. Probability of n units in system = mean arrival rate (mean # arriv./time)  = mean service rate (mean # serv./time)

24. Group Exercise A bank has a drive-up teller window. Arrivals are Poisson-distributed, with a mean rate of 24 customers / hour. Service times are exponential-probability distributed, with a mean rate of 36 customers / hour. Calculate the operating characteristics of the system.

25. Multiple-Channel Waiting Lines Server Customers in line Server

26. Multiple-Channel Waiting Lines Assumptions: Arrivals follow Poisson distribution. Service times follow exponential prob. dist. The mean service rate is the same for each channel (server). Arrivals wait in single line, then move to first open channel.

27. Multiple-Channel Waiting Lines = mean arrival rate  = mean service rate for each channel k = number of channels

28. Operating Characteristics Probability of no units in system Average number of units in waiting line Average number of units in system Average time unit is in waiting line Average time a unit is in the system Probability that an arriving unit will have to wait for service Probability of n units in system

29. Probability of no units in system = mean arrival rate (mean # arriv./time)  = mean service rate (mean # serv./time) k = # of channels The system includes the customer being serviced.

30. Average # of units in waiting line = mean arrival rate (mean # arriv./time)  = mean service rate (mean # serv./time) k = # of channels The waiting line does NOT include the customer being serviced.

31. Average # of units in system = mean arrival rate (mean # arriv./time)  = mean service rate (mean # serv./time)

32. Avg. time a unit is in waiting line = mean arrival rate (mean # arriv./time)  = mean service rate (mean # serv./time)

33. Average time a unit is in system = mean arrival rate (mean # arriv./time)  = mean service rate (mean # serv./time)

34. Probability that an arriving unit will have to wait for service = mean arrival rate (mean # arriv./time)  = mean service rate (mean # serv./time) k = # of channels

35. Probability of n units in system = mean arrival rate,  = mean service rate, k = # of channels

36. Homework due 4/18 Ch. 12 #5 (answer all questions) Ch. 12 #6 (answer all questions) Ch. 12 #7 (answer all questions) Ch. 12 #12 (answer all questions) Ch. 12 #13 (answer all questions) Turn in ON PAPER (you can do it by hand or using Excel).