Operations management waiting lines
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Operations Management Waiting Lines. Example: A Deterministic System. 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?. A Deterministic System: Example 1.

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Operations Management Waiting Lines

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Operations management waiting lines

Operations ManagementWaiting Lines


Example a deterministic system

Example: A Deterministic System

  • 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?


A deterministic system example 1

A Deterministic System: Example 1


A deterministic system example 11

A Deterministic System: Example 1


Utilization

Utilization

  • 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


A deterministic system example 12

A Deterministic System: Example 1


Known but uneven demand example 2

Known but Uneven Demand: Example 2

  • What if arrivals are not exactly every 10 minutes?

  • Let’s open the spreadsheet.


A deterministic system example 2

A Deterministic System: Example 2


A deterministic system example 21

A Deterministic System: Example 2


A deterministic system example 22

A Deterministic System: Example 2


A deterministic system example 23

A Deterministic System: Example 2

Observations:

  • Utilization is below 100% (machine is idle 14% of the time).

  • There are 1.12 orders (on average) waiting to be processed.


A deterministic system example 24

A Deterministic System: Example 2

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

  • Answer: Variability


Known but uneven demand example 21

Known but Uneven Demand: Example 2

  • How to measure variability?

  • Coefficient of variation:

    CV = Standard Deviation / Mean


Uncertain demand interarrival times example 3

Uncertain Demand (Interarrival times): Example 3

  • 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.


Simulation of uncertain demand inter arrival times example 3

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


Uncertain demand interarrival times example 31

Uncertain Demand (Interarrival times): Example 3

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


Uncertain demand inter arrival times example 3

Uncertain Demand (Inter-arrival times): Example 3

  • 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 interarrival times example 32

Uncertain Demand (Interarrival times): Example 3

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


Can we reduce demand variability uncertainty

Can we reduce demand variability/uncertainty?

  • Can we manage demand?

  • What are other sources of variability/uncertainty?


Uncertain demand inter arrival times

Uncertain Demand (Inter-arrival times)

  • 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?


Key concepts and issues

Key Concepts and Issues

  • The factors that determine the performance of the waiting lines:

    • Variability

    • Utilization rate

    • Risk pooling effect


Rule 1

Rule 1

  • 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 2

Rule 2

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


Rule 3

Rule 3

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


Arrival rate at an airport security check point

Arrival Rate at an Airport Security Check Point

What is the queue size?

What is the capacity utilization?


Flow times with arrival every 6 secs

Flow Times with Arrival Every 6 Secs

What is the queue size?

What is the capacity utilization?


Flow times with arrival every 6 secs1

Flow Times with Arrival Every 6 Secs

What is the queue size?

What is the capacity utilization?


Flow times with arrival every 6 secs2

Flow Times with Arrival Every 6 Secs

What is the queue size?

What is the capacity utilization?


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