Variability
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Variability. Example. A single machine performs an operation for a unit of product. The mean operation time is 30 seconds. Units arrive at the station with an average time between arrivals of 40 seconds. There is room for three waiting units. Arrival and service processes are constant.

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Variability l.jpg

Variability

Paul A. Jensen

Operations Research Models and Methods

Copyright 2004 - All rights reserved


Example l.jpg
Example

  • A single machine performs an operation for a unit of product.

  • The mean operation time is 30 seconds.

  • Units arrive at the station with an average time between arrivals of 40 seconds.

  • There is room for three waiting units.


Arrival and service processes are constant l.jpg
Arrival and service processes are constant

  • Arrival rate: 1.5/minute

  • Service rate: 2/minute

  • Percent utilization

  • Average delay

  • Average WIP


Arrival process is random service process is constant l.jpg
Arrival process is randomService process is constant

  • Arrival rate: 1.5/minute

  • Average Service rate: 2/minute

  • Percent utilization

  • Average delay

  • Average WIP


Arrival and service processes are random l.jpg
Arrival and service processes are random

  • Average Arrival rate: 1.5/minute

  • Average Service rate: 2/minute

  • Percent utilization

  • Average delay

  • Average WIP



Analytical determination of system characteristics l.jpg

u

Unit flow

t

operation time

U

=

ut

Unit time

V

Production volume

Analytical Determination of System Characteristics

s

Number of machines


For analytical purposes l.jpg
For analytical purposes

  • Assume service and interarrival times have exponential distributions


We want to compute l.jpg
We want to compute:

  • State Probabilities

  • Average Number and Time in the Queue

  • Average Number and Time in the System

  • Percent Utilization


The state probabilities l.jpg
The State Probabilities

  • The probability that the system is empty

  • The probability of n in the system for n ≤ s

  • The probability of n in the system for n > s


Average number and time l.jpg
Average Number and Time

  • In the queue

  • In the system

  • Utilization



Equivalence property l.jpg
Equivalence Property

  • Assume: All stations have exponential service times and unlimited queues and all inputs to the system are Poisson processes.

  • Then: Each station can be analyzed independently with queuing analysis.

  • Then: System characteristics can be determined by summing station characteristics.








Individual vs lot production neglecting setup time l.jpg
Individual vs. lot production (neglecting setup time)

  • The minimum number of stations is the same

  • The traffic intensity is the same

  • The average number in the queue is the same

  • But: For individual production Lq is in units

    • For lot production Lq is in lots

    • For lot production, WIP is Q times greater

    • For lot production, W and Wq is Q times greater


Effects of setup time l.jpg
Effects of setup time

  • Setup time increases the minimum number of machines

  • Setup time increases the traffic intensity

  • The effects are reduced by increasing the lot size

  • But, increasing the lot size increases WIP and throughput time by a factor of Q


Example22 l.jpg

Oper.

1

2

3

4

5

6

7

8

9

10

Time A

5

4

18

5

min.

Time B

10

12

10

15

min.

Unit

1.272

1.272

1.272

1.272

1.272

1.111

1.111

1.111

1.111

1

Flow

WIP A

2.65

2.12

8.332

2.315

WIP B

10.6

12.72

9.258

13.89

Example

  • Production of A is 0.417 per minute, and production of B is 0.8333 per minute.





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