1. Operations Management Waiting Lines

2. 2 Ardavan Asef-Vaziri Dec-2010Operations Management: Waiting Lines1  Questions: Can we process the orders? How many orders will wait in the queue? How long will orders wait in the queue? What is the utilization rate of the facility? Example: A Deterministic System

3. 3 Ardavan Asef-Vaziri Dec-2010Operations Management: Waiting Lines1 A Deterministic System: Example 1

4. 4 Ardavan Asef-Vaziri Dec-2010Operations Management: Waiting Lines1 A Deterministic System: Example 1

5. 5 Ardavan Asef-Vaziri Dec-2010Operations Management: Waiting Lines1  Arrival rate = 1/10 per minutes  Processing rate = time 1/9 per minute  Utilization – AR/PR = (1/10)/(1/9) = 0.9 or 90%  On average 0.9 person is in the system Utilization

6. 6 Ardavan Asef-Vaziri Dec-2010Operations Management: Waiting Lines1 A Deterministic System: Example 1 Utilization:90% Variability:0.00 Average Throughput time:9.00minutes Average Wait in Queue:0.00minutes Average Number in system:0.90jobs

7. 7 Ardavan Asef-Vaziri Dec-2010Operations Management: Waiting Lines1  What if arrivals are not exactly every 10 minutes?  Let’s open the spreadsheet. Known but Uneven Demand: Example 2

8. 8 Ardavan Asef-Vaziri Dec-2010Operations Management: Waiting Lines1 A Deterministic System: Example 2

9. 9 Ardavan Asef-Vaziri Dec-2010Operations Management: Waiting Lines1 A Deterministic System: Example 2 Arrival TimeService Time Interarrival time Throughput timeDeparture Waiting time in Queue 09 990 129 9210 209810301 349149430 409612523 449417618 5197197010 6991810791 859169940 9095131034

10. 10 Ardavan Asef-Vaziri Dec-2010Operations Management: Waiting Lines1 A Deterministic System: Example 2

11. 11 Ardavan Asef-Vaziri Dec-2010Operations Management: Waiting Lines1 Observations: 1. Utilization is below 100% (machine is idle 14% of the time). 2. There are 1.12 orders (on average) waiting to be processed. A Deterministic System: Example 2 Average Interarrival time10.000minutesUtilization86% Average Service time9.000minutes Average Throughput Time11.70minutes Std Service time0.000minutes Average Wait in Queue2.70minutes Thoughput rate0.096 jobs / min Average # in the system1.12jobs Capacity (Service rate)0.111 jobs / min

12. 12 Ardavan Asef-Vaziri Dec-2010Operations Management: Waiting Lines1  Why do we have idleness (low utilization) and at the same time orders are waiting to be processed?  Answer: Variability A Deterministic System: Example 2

13. 13 Ardavan Asef-Vaziri Dec-2010Operations Management: Waiting Lines1  How to measure variability?  Coefficient of variation: CV = Standard Deviation / Mean Known but Uneven Demand: Example 2

14. 14 Ardavan Asef-Vaziri Dec-2010Operations Management: Waiting Lines1  The interarrival time is either 5 periods with probability 0.5 or 15 periods with probability 0.5 Notice that the mean interarrival time is 10. (mean interarrival = 0.5 * 15 + 0.5 * 5 = 10)  The service time is 9 periods (with certainty).  The only difference between example 3 and 1 is that the interarrival times are random. Uncertain Demand (Interarrival times): Example 3

15. 15 Ardavan Asef-Vaziri Dec-2010Operations Management: Waiting Lines1 Simulation of Uncertain Demand (Inter-arrival times): Example 3 ArrivalStartFinishWaitingIdleness 551400 20 2906 25293840 30384780 354756120 405665160 556574100 70748340 75839280 909210120 105 11404 120 12906 135 14406 150 15906 165 17406

16. 16 Ardavan Asef-Vaziri Dec-2010Operations Management: Waiting Lines1 (Recall that in Example 1, no job needed to wait.) Uncertain Demand (Interarrival times): Example 3 Average Interarrival time10.200minutes Average Througput time18.98 Average Service time9.000minutes Average wait in queue9.98 Std Service time0.000minutesAverage # in queue0.98 Thoughput rate0.100 jobs / min Average in the system1.86004 Capacity (Service rate)0.111 jobs / min

17. 17 Ardavan Asef-Vaziri Dec-2010Operations Management: Waiting Lines1  Suppose we change the previous example and assume: Inter-arrival time170.5 probability Inter-arrival time 30.5 probability Average inter-arrival times as before 10 min. Uncertain Demand (Inter-arrival times): Example 3

18. 18 Ardavan Asef-Vaziri Dec-2010Operations Management: Waiting Lines1 The effect of variability: higher variability in inter-arrival times results in higher average # in queue. Uncertain Demand (Interarrival times): Example 3 Average Interarrival time10.200minutes Average Througput time27.94 Average Service time9.000minutes Average wait in queue18.94 Std Service time0.000minutesAverage # in queue1.86 Thoughput rate0.100 jobs / min Average in the system 2.7381 2 Capacity (Service rate)0.111 jobs / min

19. 19 Ardavan Asef-Vaziri Dec-2010Operations Management: Waiting Lines1  Can we manage demand?  What are other sources of variability/uncertainty? Can we reduce demand variability/ uncertainty?

20. 20 Ardavan Asef-Vaziri Dec-2010Operations Management: Waiting Lines1  Up to now, our service time is exactly 9 minutes.  What will happen to waiting-line and waiting-time if we have a short service time (i.e., we have a lower utilization rate)?  What will happen if our service time is longer than 10 minutes? Uncertain Demand (Inter-arrival times)

21. 21 Ardavan Asef-Vaziri Dec-2010Operations Management: Waiting Lines1  The factors that determine the performance of the waiting lines: Variability Utilization rate Risk pooling effect Key Concepts and Issues

22. 22 Ardavan Asef-Vaziri Dec-2010Operations Management: Waiting Lines1  In general, if the variability, or the uncertainty, of the demand (arrival) or service process is large, the queue length and the waiting time are also large. Rule 1

23. 23 Ardavan Asef-Vaziri Dec-2010Operations Management: Waiting Lines1  As the utilization increases the waiting time and the number of orders in the queue increases exponentially. Rule 2

24. 24 Ardavan Asef-Vaziri Dec-2010Operations Management: Waiting Lines1  In general, pooling the demand (customers) into one common line improves the performance of the system. Rule 3