Operations Management Waiting Lines

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

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

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

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

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

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 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 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 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 * * 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 Ardavan Asef-Vaziri Dec-2010Operations Management: Waiting Lines1 Simulation of Uncertain Demand (Inter-arrival times): Example 3 ArrivalStartFinishWaitingIdleness

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 system Capacity (Service rate)0.111 jobs / min

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 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 Capacity (Service rate)0.111 jobs / min

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 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 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 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 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 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

25 Ardavan Asef-Vaziri Dec-2010Operations Management: Waiting Lines1 What is the queue size? What is the capacity utilization? Arrival Rate at an Airport Security Check Point Customer NumberArrival Time Departure Time Time in Process

26 Ardavan Asef-Vaziri Dec-2010Operations Management: Waiting Lines1 What is the queue size? What is the capacity utilization? Flow Times with Arrival Every 6 Secs Customer NumberArrival Time Departure Time Time in Process

27 Ardavan Asef-Vaziri Dec-2010Operations Management: Waiting Lines1 What is the queue size? What is the capacity utilization? Flow Times with Arrival Every 6 Secs Customer NumberArrival TimeProcessing Time Time in Process 1-A077 2-B C D E F G H I J54111

28 Ardavan Asef-Vaziri Dec-2010Operations Management: Waiting Lines1 What is the queue size? What is the capacity utilization? Flow Times with Arrival Every 6 Secs Customer NumberArrival TimeProcessing Time Time in Process 1-E088 2-H D A B J C F G I5457