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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

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?




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



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 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.



Uncertain demand interarrival times example 31
Uncertain Demand ( Example 3Interarrival 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 Example 3

  • 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 interarrival times example 32
Uncertain Demand ( Example 3Interarrival 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/ Example 3uncertainty?

  • Can we manage demand?

  • What are other sources of variability/uncertainty?


Uncertain demand inter arrival times
Uncertain Demand (Inter-arrival times) Example 3

  • 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 Example 3

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

    • Variability

    • Utilization rate

    • Risk pooling effect


Rule 1
Rule 1 Example 3

  • 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 Example 3

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


Rule 3
Rule 3 Example 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 Example 3

What is the queue size?

What is the capacity utilization?


Flow times with arrival every 6 secs
Flow Times with Arrival Every 6 Secs Example 3

What is the queue size?

What is the capacity utilization?


Flow times with arrival every 6 secs1
Flow Times with Arrival Every 6 Secs Example 3

What is the queue size?

What is the capacity utilization?


Flow times with arrival every 6 secs2
Flow Times with Arrival Every 6 Secs Example 3

What is the queue size?

What is the capacity utilization?


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