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Queueing Model for an Assemble-to-Order Manufacturing System- A

Matrix Geometric Solution Approach

Sachin JayaswalDepartment of Management Sicences

University of Waterloo

Beth JewkesDepartment 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

1 1B

1N

BO

2N

2

External

InfiniteStore

Supplier

1Stage 2Stage

Finished goods

Semi -finished goods

1Q 2Q

λ

Demand Arrival

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Model Description

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]+

Notations:

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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 DescriptionFor 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 stage-2 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 MethodState 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 =

This is a special case of a level dependent QBD

012

02

0

AAA

AAA

A12

012

012

012

00

AA

AAA

AAA

AAA

AB

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State space representation 2

Consider a finite queue before stage 1 with size k

State description:

{N = (N2, N1) : N2 ≥ 0; 0 ≤ N1 ≤ k+1}

Solution…

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Q is a level independent QBD process and hence can be solved using standard Matrix-Geometric Method

012

012

00

AAA

AAA

AB

Infinitesimal Generator

Q =

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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}

An Exact Solution

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Censoring

Q =

'0

'1

'2

'0

'1

'2

'0

'0

AAA

AAA

AB

Infinitesimal Generator

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Censoring

012

012

00

AAA

AAA

AB

Transition Matrix:

P =

IQP 1 iiqmax;

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Censoring all states above level 1 gives the following transition matrix:

Censoring

UA

AB

2

00P(1) =

Censoring level 1 gives: 2

1000 AUIABP

20 RAB 10 UIARwhere

P(0) infinite only in one dimension

However, P(0) may no longer be QBD

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CensoringR matrix: 2

210 ARRAAR

kkkkk

k

k

k

RRRR

RRRR

RRRR

RRRR

210

2222120

1121110

0020100

R =

iRR kkiii ,lim

R matrix possesses asymptotically block Toeplitz form

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Censoring 200 RABP

P(0) is also asymptotically of block Toeplitz form

Hence, one can use GI/G/1 type Markov chains to study P(0)

32

31

30

0

1

2

22

21

20

1

0

2

1

22

12

21

11

20

10

10210

122222120

111121110

100020100

k

k

k

k

k

k

k

k

k

k

k

k

kkk

kk

kk

kk

P

P

P

P

PP

P

P

P

P

P

P

P

P

P

P

P

P

PPPPPP

PPPPP

PPPPP

PPPPP

P(0) =

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CensoringGI/G/1 type Markov chain is of the form:

0123

1012

2101

3210

QQQB

QQQB

QQQB

CCCB

P =

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CensoringTo make P(0) conform to GI/G/1 type Markov chain, wechoose B0 to be sufficiently large to contain those elementsnot within a suitable tolerance of their asymptotic forms

32

31

30

0

1

2

22

21

20

1

0

2

1

22

12

21

11

20

10

10210

122222120

111121110

100020100

k

k

k

k

k

k

k

k

k

k

k

k

kkk

kk

kk

kk

P

P

P

Q

QQ

P

P

P

Q

Q

Q

Q

P

P

P

P

P

PQQPPP

PPPPP

PPPPP

PPPPP

P(0) =

0123

1012

2101

3210

QQQB

QQQB

QQQB

CCCB

21

20

21

22

23

21

10

11

12

22

1101

23

1210

nnnn

nnnn

nn

nn

QQQQ

QQQQ

QQQQ

QQQQ

Censoring

1nP

Transition matrix with all states beyond level n censored (Grassmann & Standford, 2000)

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1

*1*0

**

jjjiii QQIQQQ 0i

1

*1*0

**

jjijii QQIQQQ 0i

1

*1*0

**

jjjiii QQICCC 0i

1

*1*0

**

jjijii BQIQBB 0i

1

*1*0

*0

*0

jii BQICBB

Censoring

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Solution to level-0 probabilities

Non-normalized probabilities αj for censored process*000 B 100 ;

1*0

*0

1

1

1*0

*1

QICQIQ n

n

iinn

1*0

*0

1

1

*1

QICV n

n

iin ; 1*

0**

QIQV ii

,...0,0,00 210 xxxNormalized probabilities for censored process

tx jj

0 t determined using generating function

(Grassmann & Standford (2000))

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Solution

Rkk 1

Stationary vector at positive levels

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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To construct an optimization model using the performance measures obtained

To compare the results obtained with the approximations suggested in the literature

Future Directions

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