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/ department of mathematics and computer scienceJJ J N I II 1/21JJ J N I II 1/21

Performance Analysis of Assembly Systems

Marcel van Vuuren

Joint work with Ivo Adan

June 12, 2006

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Presentation Outline

• Assembly system

• Literature

• The analysis

– Decomposition

– Subsystem analysis

– Iterative algorithm

• Numerical results and conclusions

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An assembly station

Model:

• General arrival processes

• Finite buffers

• Blocking after service

• General service process

• All components have to be available before service starts

• A component can wait in the server for other components

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Model description

S: Service time of assemblyserver

Ai: Arrival process at buffer i

bi: Buffer size of buffer i

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Literature

• Fork-join queue in an open network(Hemachandra and Eedupuganti)

• Fork-join queue in a closed network(Rao and Suri, and Krishnamurti et al.)

• An exact analysis of system with two parts(Gold)

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Analysis approach

• Decomposition

• Subsystem Analysis

• Iterative Algorithm

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Decomposition in subsystems

WAi: Wait to assembly atbuffer i

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Wait to assembly at buffer i

The wait to assembly time consists of the waiting time time for theother components.So,WAi = maxj 6=i RAj

Note:

• The RAj’s can be obtained from the subsystems

• Both the WA’s and RA’s have mass in zero

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Wait to assembly at buffer i

max1≤j≤k

RAj = max{RAk, max1≤j≤k−1

RAj}

P ( max1≤j≤k−1

RAj = 0) =∏

1≤j≤k−1

(1− pe,j)

pne,i =∏j 6=i

(1− pe,j)

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Maximum of two random variables (1)

E1 and E2 Erlang distributed with parameters ki and µi (i = 1, 2)

First a number op phases with parameter µ1 + µ2 then either µ1 or µ2

qk,j: k wins and the other finished j phases

q1,j =

(k1 − 1 + j

k1 − 1

) (µ2

µ1 + µ2

)j (µ1

µ1 + µ2

)k1

, 0 ≤ j ≤ k2 − 1

q2,i =

(k2 − 1 + i

k2 − 1

) (µ1

µ1 + µ2

)i (µ2

µ1 + µ2

)k2

, 0 ≤ i ≤ k1 − 1

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Maximum of two random variables (2)

EM1,j =k1 + j

µ1 + µ2+

k2 − j

µ2

EM 21,j =

(k1 + j)(k1 + j + 1)

(µ1 + µ2)2+

(k1 + j)(k2 − j)

(µ1 + µ2)µ2+

(k2 − j)(k2 − j + 1)

µ22

E(max{E1, E2}) =

k2−1∑j=0

q1,jEM1,j +

k1−1∑i=0

q2,iEM2,i

E(max{E1, E2}2) =

k2−1∑j=0

q1,jEM 21,j +

k1−1∑i=0

q2,iEM 22,i

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Subsystem analysis

• Fit aEk−1,k orC2 distribution on the first twomoments ofAi, S andWAi

• Construct MAP ’s of the arrival and departure processes

• Construct a QBD of the subsystem

• Analyse the QBD by using matrix analytic methods

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Constructing the QBD (1)

Q =

B00 B01

B10 A1 A0

A2. . . . . .. . . . . . A0

A2 A1 C10

C01 C00

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Constructing the QBD (2)

Construct a MAP of the arrival process:AR0 and AR1

Construct a MAP of the departure process (WA and S):DE0 and DE1

Construct a MAP of the departure process in level 0 (WA):D̃E0

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Constructing the QBD (3)

A0 = AR1 ⊗ Inwac+nsa

A1 = AR0 ⊗ Inwac+nsa+ Ina ⊗DE0

A2 = Ina ⊗DE1

B01 = AR1 ⊗ Inwac+nsa

B00 = AR0 ⊗ Inwac+nsa+ Ina ⊗ D̃E0

B10 = Ina ⊗DE1

C01 = AR1 ⊗ Inwac+nsa

C00 = Ina ⊗DE0

C10 = Ina ⊗DE1

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Analyzing the QBD

πi = x1Ri−1 + xbR̂

b−i, i = 1, . . . , b (1)

0 = A0 + RA1 + R2A2

0 = A2 + R̂A1 + R̂2A0

0 = π0B00 + π1B10

0 = π0B01 + π1A1 + π2A2

0 = πb−1A0 + πbA1 + πb+1C01

0 = πbC10 + πb+1C00

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Characteristics of RA

pe: The probability that the queue is empty on departure

pe =π1B10e

T,

α: The distribution of the phase of the inter-arrival time A just after adeparture to level 0

α =π1B10

π1B10e,

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Iterative algorithm

1. Choose initial characteristics for the wait to assembly times at eachbuffer

2. For each subsystem (from subsystem 1 to n):

• Determine the wait to assembly time at buffer i

• Determine the queue-length probabilities of the subsystem

• Determine a new RAi

Repeat step 2 until the characteristics of the WA’s of the subsystems haveconverged.

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Numerical results

The following parameters are varied:

• number of parts: 2, 4, 8

• buffersize: 0, 2, 4, 8

• SCV of the arrivals: 0.2, 0.5, 1, 2

• SCV of the service process: 0.5, 1

• Occupation rate: 0.75, 1

• Imbalance in the arrival rates

• Imbalance in the SCV’s of the arrivals

A total of 768 test cases.

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Numerical results (2)

Perf. char. Avg. 0-5 % 5-10 % > 10 %Throughput 1.5 % 97.4 % 2.6 % 0.0 %

Mean sojourn time 2.8 % 84.9 % 13.4 % 1.7 %

Most sensitive for:

• different buffer sizes

• different number of parts

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Conclusions

Conclusions:

• Very good results

• Fast computation

• A good method for analyzing assembly systems

Future research:

• Incorporate the method in a network setting