Transcript Slide 1
Queueing Model for an Assemble-to-Order
Manufacturing System- A Matrix Geometric
Solution Approach
Sachin Jayaswal
Department of Management Sicences
University of Waterloo
Beth Jewkes
Department of Management Sciences
University of Waterloo
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Outline
Motivation
Model Description
Literature Review
Analysis
Future Directions
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Motivation
Get a better understanding of Assemble-to-Order
(ATO) production systems
Develop a queuing model for a two stage ATO
production system and evaluate the following measures
of performance:
Distribution of semi-finished goods inventory
Distribution of order fulfillment time
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Model Description
External Supplier
1
Q1
Infinite Store
Semi finished
goods
BO
B1
Finished
goods
2
Q2
N2
N1
Stage 1
Stage 2
λ
Demand Arrival
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Model Description
Notations:
λ : demand rate (Poisson arrivals)
μj : service rate at stage j, j=1, 2 (exponential service times)
B1: base stock level at stage 1 (parameter)
N1: queue occupancy at stage 1
N2: queue occupancy at stage 2
I1 : semi-finished goods inventory after stage 1. I1 = [B1 – N1]+
BO :Number of units backordered at stage 1. BO = [N1 – B1]+
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Model Description
If B1 = 0, the system is MTO and operates like an
ordinary tandem queue:
The process describing the departure of units from
each stage is Poisson with rate λ
Individual queues behave as if they are operated
independently. In equilibrium, N1 and N2 are
independent
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Model Description
For the ATO with B1 > 0:
Arrival process to stage 2 is no longer poisson.
There is a positive dependence between the arrival
of input units from stage 1 to stage 2. Times
between successive arrivals to stage 2 are correlated.
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Related Literature
Buzacott et al. (1992) observe that C.V. of inter-arrival
times at stage 2 is between 0.8 and 1 and, therefore,
recommend using an M/M/1 approximation for stage2 queue. Lee and Zipkin (1992) also assume M/M/1
approximation for stage 2. (BPS-LZ approximation)
Buzacott et al. (1992) further improve upon this
approximation by modeling the congestion at stage 2 as
GI/M/1 queue. (BPS approximation)
Gupta and Benjaafar (2004) use BPS-LZ approximation
to compare alternative MTS and MTO systems
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Solution – Matrix Geometric Method
State space representation 1
Consider a finite queue before stage 2 with size k
State description:
{N = (N1, N2) : N1 ≥ 0; 0 ≤ N2 ≤ k+1}
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Infinitesimal Generator
Q=
B0
A
2
A0
A1
A2
A0
A1
A0
A2
A1
A2
A0
A1
A2
A0
A
A2
A0
A1
A0
This is a special case of a level dependent QBD
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Solution…
State space representation 2
Consider
State
a finite queue before stage 1 with size k
description:
{N = (N2, N1) : N2 ≥ 0; 0 ≤ N1 ≤ k+1}
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Infinitesimal Generator
B0
A
Q= 2
A0
A1
A0
A2
A1
A0
Q is a level independent QBD process and hence can be
solved using standard Matrix-Geometric Method
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An Exact Solution
The
above methods are not truly exact as one of
the queues is truncated
We next present an exact solution for the doubly
infinite problem, using censoring (Grassmann &
Standford (2000); Standford, Horn & Latouche
(2005))
State
description: {N = (N2, N1) : N1 ≥ 0, N2 ≥ 0}
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Censoring
Infinitesimal Generator
B0'
'
A2
Q=
'
A0
'
A1
'
A2
'
A0
'
'
A1 A0
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Censoring
Transition Matrix: P 1Q I ; maxqii
P
B0
A
2
=
A0
A1
A2
A0
A1
A0
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Censoring
Censoring all states above level 1 gives the
following transition matrix:
B0 A0
P(1) =
A2 U
Censoring level 1 gives:
P0 B0 A0 I U 1 A2
B0 RA2
where R A0 I U 1
P(0) infinite only in one dimension
However, P(0) may no longer be QBD
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Censoring
R matrix: R A0 RA1 R A2
2
R
R00
R
10
= R20
Rk 0
R01
R02 R0 k
R11
R12
R21
R22 R2 k
R1k
Rk 1
Rk 2 Rkk
limi Ri,ik Rk i
R matrix possesses asymptotically block Toeplitz form
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Censoring
P0 B0 RA2
P(0) =
P01
P02
P00
P
P11
P12
10
P20
P21
P22
Pk 0
Pk1
Pk 2
P
Pk 11 Pk 12
k 10
Pk 20 Pk 21 Pk 22
P0 k
P1k
P2 k
P0
P1
P2
P0 k 1 P0 k 2 P0 k 3
P1k 1 P1k 2 P1k 3
P2 k 1 P2 k 2 P2 k 3
P1 P2
P0
P1
P1 P0
P(0) is also asymptotically of block Toeplitz form
Hence, one can use GI/G/1 type Markov chains to study P(0)
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Censoring
GI/G/1 type Markov chain is of the form:
B0
B
1
P=
B
2
B
3
C1
C2
C3
Q0
Q1
Q2
Q1
Q0
Q1
Q2
Q1
Q0
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Censoring
B0
B
1
B
2
B
3
C1
C2
C3
Q0
Q1
Q2
To make P(0) conform to GI/G/1 type Markov chain, we
Q1 Q0 Q1
choose B0 to be sufficiently large to contain those
elements
Q2 Q1 Q0
not within a suitable tolerance of their asymptotic
forms
P(0) =
P01
P02
P00
P
P11
P12
10
P20
P21
P22
Pk 0
Pk1
Pk 2
P
Pk 11 Pk 12
k 10
Pk 20 Pk 21 Pk 22
P0 k
P1k
P2 k
Q0
Q1
Q 2
P0 k 1 P0 k 2 P0 k 3
P1k 1 P1k 2 P1k 3
P2 k 1 P2 k 2 P2 k 3
Q1 Q2
Q0 Q1
Q1 Q0
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Censoring
Transition matrix with all states beyond level n censored
(Grassmann & Standford, 2000)
n 1
Q
Q
Q
0
1
2
Q0
Q1n1
Pn 1 Q1
n 1
n 1
n 1
Q
Q
Q
2
1
0
Qn3 2 Qn2 2 Qn1 2
Q3n 2
Q2n 2
Q1n 2
Q0n 2
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Censoring
Qi*
Qi*
Ci*
Qi
Qi* j
j 1
Qi
Q*j
j 1
I
i0
i0
*
* 1 *
Ci j I Q0 Q j
j 1
1
Q*j I Q0* Bi* j
j 1
Ci
B0
Ci*
j 1
i0
* 1 *
Q0 Qi j
Bi* Bi
B0*
I
* 1 *
Q0 Q j
I
i0
* 1 *
Q0 Bi
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Solution to level-0 probabilities
Non-normalized probabilities αj for censored process
0 0 B0* ; 00 1
n1
n
n1Qi*
i 1
n 1
i 1
I
* 1
Q0
n1Vi 0Cn*
*
I
0Cn*
I
1
Q0*
* 1
Q0
;
Vi
*
Qi*
I
* 1
Q0
Normalized probabilities for censored process
0 x0 0, x1 0, x2 0,...
x j 0
j
t
t determined using generating function
(Grassmann & Standford (2000))
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Solution
Stationary vector at positive levels
k k 1R
Performance measures
EK2: Expected no. of units stage 2 still needs to
produce to meet the pending demands. EK2 = E(N2+BO)
EI: Expected no. of work-in-process units.
EI = E(I1+N2)
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Initial Results
B1
EK2
EI
1
4.1693
1.1693
3
3.0006
2.0006
5
2.2709
3.2709
7
1.8100
4.8100
9
1.5171
6.5171
λ=1; μ1 =1.25; μ2 =2
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Future Directions
To construct an optimization model using the
performance measures obtained
To compare the results obtained with the
approximations suggested in the literature
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