Operations Management Waiting Lines. 2 Ardavan Asef-Vaziri Dec-2010Operations Management: Waiting...

Operations Management

Waiting Lines

2Ardavan 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

Experiment # 1

Arrival time: Service time:0 910 920 930 940 950 960 970 980 990 9


A Deterministic System: Example 1

Interarrival time Throughput TimeDeparture TimeWaiting Time in Queue9.0 9.0 0.0

10.0 9.0 19.0 0.010.0 9.0 29.0 0.010.0 9.0 39.0 0.010.0 9.0 49.0 0.010.0 9.0 59.0 0.010.0 9.0 69.0 0.010.0 9.0 79.0 0.010.0 9.0 89.0 0.010.0 9.0 99.0 0.0



789

10

Graph #2: Total #in system:

0

1

2

0 20 40 60 80 100 120

Graph #1: Total time spent in System per Job

0

2

4

6

8

10

12

0 20 40 60 80 100 120 140

TIME

Job

#


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



Utilization: 90%

Variability: 0.00

Average Throughput time: 9.00 minutes

Average Wait in Queue: 0.00 minutes

Average Number in system: 0.90 jobs


What if arrivals are not exactly every 10 minutes? Let’s open the spreadsheet.

Known but Uneven Demand: Example 2



Graph #1: Total time spent in System per Job

0

2

4

6

8

10

12

0 20 40 60 80 100 120 140

TIME

Job

#



Arrival Time

Service Time

Interarrival time

Throughput time Departure

Waiting time in Queue

0 9 9 9 0

12 9 12 9 21 0

20 9 8 10 30 1

34 9 14 9 43 0

40 9 6 12 52 3

44 9 4 17 61 8

51 9 7 19 70 10

69 9 18 10 79 1

85 9 16 9 94 0

90 9 5 13 103 4



Graph #2: Total #in system:

0

1

2

3

4

0 20 40 60 80 100 120


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.


Average Interarrival time 10.000 minutes Utilization 86%

Average Service time 9.000 minutes

Average Throughput Time 11.70

minutes

Std Service time 0.000 minutesAverage Wait in Queue 2.70

minutes

Thoughput rate 0.096jobs / min

Average # in the system 1.12 jobs

Capacity (Service rate) 0.111

jobs / min


Why do we have idleness (low utilization) and at the same time orders are waiting to be processed?

Answer: Variability



How to measure variability?

Coefficient of variation:CV = Standard Deviation / Mean

Known but Uneven Demand: Example 2


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


Simulation of Uncertain Demand (Inter-arrival times): Example 3

Arrival Start Finish Waiting Idleness

5 5 14 0 0

20 20 29 0 6

25 29 38 4 0

30 38 47 8 0

35 47 56 12 0

40 56 65 16 0

55 65 74 10 0

70 74 83 4 0

75 83 92 8 0

90 92 101 2 0

105 105 114 0 4

120 120 129 0 6

135 135 144 0 6

150 150 159 0 6

165 165 174 0 6


(Recall that in Example 1, no job needed to wait.)


Average Interarrival time 10.200 minutes

Average Througput time 18.98

Average Service time 9.000 minutes

Average wait in queue 9.98

Std Service time 0.000 minutesAverage # in queue 0.98


Average in the system

1.86004


jobs / min


Suppose we change the previous example and assume: Inter-arrival time 17 0.5 probability Inter-arrival time 3 0.5 probability Average inter-arrival times as before 10 min.

Uncertain Demand (Inter-arrival times): Example 3


The effect of variability: higher variability in inter-arrival times results in higher average # in queue.


Average Interarrival time

10.200

minutes

Average Througput time 27.94

Average Service time 9.000

minutes

Average wait in queue 18.94

Std Service time 0.000minutes

Average # in queue 1.86


Average in the system

2.73812


jobs / min


Can we manage demand?

What are other sources of variability/uncertainty?

Can we reduce demand variability/uncertainty?


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)


The factors that determine the performance of the waiting lines: Variability Utilization rate Risk pooling effect

Key Concepts and Issues


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


As the utilization increases the waiting time and the number of orders in the queue increases exponentially.

Rule 2


In general, pooling the demand (customers) into one common line improves the performance of the system.

Rule 3


What is the queue size?What is the capacity utilization?

Arrival Rate at an Airport Security Check Point

Customer Number Arrival Time

Departure Time

Time in Process

1 0 5 5

2 4 10 6

3 8 15 7

4 12 20 8

5 16 25 9

6 20 30 10

7 24 35 11

8 28 40 12

9 32 45 13

10 36 50 14

0 10 20 30 40 50

Time

1

2

3

4

5

6

7

8

9

10

Cust

omer

Num

ber



Flow Times with Arrival Every 6 Secs

Customer Number Arrival Time

Departure Time

Time in Process

1 0 5 5

2 6 11 5

3 12 17 5

4 18 23 5

5 24 29 5

6 30 35 5

7 36 41 5

8 42 47 5

9 48 53 5

10 54 59 5

0 10 20 30 40 50 60

Time

1

2

3

4

5

6

7

8

9

10

Cust

omer

Num

ber




Customer Number Arrival Time Processing Time

Time in Process

1-A 0 7 7

2-B 10 1 1

3-C 20 7 7

4-D 22 2 7

5-E 32 8 8

6-F 33 7 14

7-G 36 4 15

8-H 43 8 16

9-I 52 5 12

10-J 54 1 11

0 10 20 30 40 50 60 70

Time

1

2

3

4

5

6

7

8

9

10

Cu

sto

mer

Queue Fluctuation

0

1

2

3

4

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64

Time

Nu

mb

er




Customer Number Arrival Time Processing Time

Time in Process

1-E 0 8 8

2-H 10 8 8

3-D 20 2 2

4-A 22 7 7

5-B 32 1 1

6-J 33 1 1

7-C 36 7 7

8-F 43 7 7

9-G 52 4 4

10-I 54 5 70 10 20 30 40 50 60 70

1

2

3

4

5

6

7

8

9

10

Operations Management Waiting Lines. 2 Ardavan Asef-Vaziri Dec-2010Operations Management: Waiting...

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