Approximate Analysis of Transfer Line with PH-service Time and Parts

Assemble

Yang Woo Shin

1 a

, Gyeong Min Baek

2 b

, Dong Ok Kim

2 c

and Dug Hee Moon

3 d

Department of Statistics, Changwon National University, Changwon, Gyeongnam 51140, Korea

Department of Eco-friendly Marine Plant FEED Engineering, Changwon National University,

Changwon, Gyeongnam 51140, Korea

School of Industrial Engineering and Naval Architecture, Changwon National University,

Changwon, Gyeongnam 51140, Korea

Keywords:

Tandem Queue, Phase Type Distribution, Decomposition, Blocking, Assembly.

Abstract:

We consider a transfer line in which workstations are linked in series. Each station consists of single machine

and two buffers of ﬁnite capacity, one for product and the other for part to be assembled. The product is

supplied to the ﬁrst workstation of the system and they are processed along the line, and a part is supplied to

each station according to independent Poisson processes. The processing time distribution of each machine is

of phase type (PH). Blocking-After-Service (BAS) rule is adopted. In this paper, we present an approximate

analysis for the system based on the decomposition method. Some numerical examples are presented for

accuracy of approximation.

1 INTRODUCTION

As a motivating example, we consider a railway ve-

hicle manufacturing process. All railway vehicles are

manufactured to the customer’s speciﬁc order. In the

railway vehicle assembly shop, various parts are as-

sembled according to the customer’s requirements.

The layout of the assembly shop is divided into two

types: ﬁxed position production and ﬂow production.

Here, we consider the ﬂow production layout. The

assembly shop has limited space to store work in pro-

cess due to space constraints. Due to the nature of

the assembly shop that performs various processes,

automation is difﬁcult, and most of the processes are

performed manually. Various parts are supplied to the

assembly station as a kit to carry out the process ac-

cording to the customer’s order.

For modeling the ﬂow production layout we con-

sider a ﬂow line in which workstations are linked

in series and there is a buffer of ﬁnite capacity be-

tween two consecutive like this ﬁgure. Workstations

are assumed to be reliable. The processing time is

https://orcid.org/0000-0002-3107-4569

https://orcid.org/0000-0001-8421-3430

https://orcid.org/0000-0002-7239-2002

https://orcid.org/0000-0001-7660-4976

assumed to be of phase type (PH) for modelling the

manual operation. Parts are supplied to each worksta-

tion M

according to a Poisson process with rate λ

The buffer for parts is ﬁnite and the parts are lost if

the part buffer is full upon an arrival of a part. Each

workstation does not work if it has no parts to be as-

sembled. Transportation times of customers through

buffers and workstations are assumed to be negligible

comparing to processing time. By setting the arrival

rate of the parts to be inﬁnite, our model reduces to

a transfer line without part assemble which has been

widely used for performance modeling of computer

systems and production systems. e.g. see the mono-

graphs Gershwin (1994), Buzzacott and Shanthiku-

mar (1993), the survey papers Dallery and Gershwin

(1992), Li et al. (2009), Papadopoulos and Heavey

(1996) and the references therein. In this paper, we

present an outline of approximate analysis for tandem

queues with a buffer of ﬁnite capacity for products be-

tween two workstations. Each workstation has a sin-

gle reliable server and a supplementary buffer of ﬁnite

for part to be assembled. Our approach is based on

well-known decomposition method, e.g. see Gersh-

win (1978, 1994), Shin and Moon (2018), Moon and

Shin (2019).

The paper is organized as follows. The model is

described in Section 2. In Section 3, an approxima-

212

Shin, Y., Baek, G., Kim, D. and Moon, D.

Approximate Analysis of Transfer Line with PH-service Time and Parts Assemble.

DOI: 10.5220/0009092802120217

In Proceedings of the 9th International Conference on Operations Research and Enterprise Systems (ICORES 2020), pages 212-217

ISBN: 978-989-758-396-4; ISSN: 2184-4372

Figure 1: Tandem queue with parts assembly.

tion method is described brieﬂy. Numerical examples

are given in Section 4. Finally, concluding remarks

are given in Section 5.

2 THE MODEL

We consider a transfer line L in which

N = N + 1

workstations W

, i = 0, 1, ··· , N are linked in series

and there is a buffer B

for customers (products) of

capacity 0 ≤ c

< ∞ between W

i−1

and W

as de-

picted in Figure 1. The station W

, 1 ≤ i ≤ N − 1

has a server (machine) M

and a buffer D

of capacity

0 ≤ d

− 1 < ∞ for parts to be assembled and denote

the station i by the pair W

= (M

, D

The customers are processed along the line and

leave the system immediately after service comple-

tion at W

. When a server completes its service at a

stage, if the buffer for the customers of next station is

full at that time, then the server is forced to stop its

service and the customer is held at the station where

it just completed its service until the destination can

accommodate it. A server M

is said to be starved if

there are no customers to be served on the server M

and the station is said to be lacked if there are no parts

in W

. The station W

, 1 ≤ i ≤ N − 1 does not work if

it is in starved, lacked and blocked.

The initial station W

and the last station W

are

for preparation and investigation of ﬁnal product, re-

spectively, and assume that they consist of a server M

and M

without buffers for parts, respectively, and

they process their work without parts. We assume

that the initial server M

is never starved and never

lacked and it starts new service immediately after a

service completion unless the server is blocked. The

last server M

is never blocked and never lacked, and

the customer at M

leaves the system immediately af-

ter completing its service.

Parts are supplied to each station W

according to

independent Poisson processes with rate λ

and arriv-

ing parts enter the server M

if there is an available

space for the part in M

. Note that the maximum num-

ber of parts that can be stored in the station W

is d

1 ≤ i ≤ N − 1. If the buffer D

is full, that is, if there

are d

parts in the station W

, the arrivals of the parts

to D

is forced to stop and begin again at the epoch

when there is an available space in W

Service time distribution of M

is of phase

type with representation PH(α

, T

), where α

(α

(1), · ·· , α

)) is a probability distribution and T

is a nonsingular matrix of size h

with negative diago-

nal elements and nonnegative off-diagonal elements.

Let T

= −T

e = (t

(1), · ·· ,t

))

, where e is the

column vector of appropriate size whose components

are all 1. See Neuts (1981) for phase type distri-

bution. Transportation times of customers through

buffers and servers are assumed to be negligible com-

paring to service time.

Stochastic processes. Let D

(t) be the number

of parts in W

at time t. The state space of D

(t) is

{0, 1, ··· , d

}. Deﬁne the state M

(t) of M

at time t

(t) =







s, M

is starved

j, M

is working with service phase j

b, M

is blocked.

The state space of W

(t) = (M

(t), D

(t)) of the

station W

= (M

, D

), 1 ≤ i ≤ N − 1 is W

{(0, 0),s

s,w

w,b

b}, where (0, 0) is the state that X

(t) ≥

1 and D

(t) = 0, s

s = {(s, k), k = 0, 1, · ·· , d

w = {( j, k), k = 1, 2, ··· , d

, j = 1, 2, · ·· , h

}, b

b =

{(b, k), k = 1, 2, ··· , d

} and let w

∗

= {(0, 0),w

w}.

Since W

and W

consist of only one server M

and

and have no buffers for part, the state space of

(t) = M

(t) and W

(t) = M

(t) are given by W

{b} ∪ { j, j = 1, 2, ·· · , h

} and W

= {s} ∪ { j, j =

1, 2, ··· , h

Let X

(t) be the total number of customers wait-

ing in the buffer B

, the customers that are being

served, lacked or blocked at M

, and the customers

blocked at M

i−1

at time t. Then X

(t) takes values on

{0, 1, ··· , K

}, where K

= c

+ 2. Note that X

(t) = K

is equivalent to M

i−1

(t) = b.

Approximation method. Approximation is based

on decomposition approach. The ﬁrst step is to de-

compose the N + 1 station system into a set of sub-

systems L

, i = 1, 2, ·· · , N. Each subsystem L

is con-

sists of upstream station W

i−1

, downstream station W

and a buffer B

between them. Model the subsys-

tem L

with a Markov chain with generator, say Q

and the unknowns in Q

are calculated by an itera-

tion method. Finally, performance measure such as

throughput is calculated with the stationary distribu-

tion of L

. Since W

is a downstream station in L

and upstream server in L

i+1

, denote the downstream

server in L

by W

and the upstream server in L

i+1

, if necessary to distinguish them.

Approximate Analysis of Transfer Line with PH-service Time and Parts Assemble

213

3 APPROXIMATION

3.1 Subsystems

Deﬁne the state W

(t) of W

at time t depending on

the state X

(t) by for 1 ≤ i ≤ N − 1, 1 ≤ j ≤ h

, 1 ≤

k ≤ d

(t) =



( j, k), W

(t) = ( j, k), X

(t) = 1,

( j, k), W

(t) = ( j, k), X

(t) ≥ 2

and the states w

(0, 0) and (b

, k), l = 1, 2 are deﬁned

similarly. Note that W

(t) = W

(t). Deﬁne the state

(t) of W

in the subsystem L

by W

(t) = W

(t),

1 ≤ i ≤ N.

We model the stochastic process Z

= {Z

(t),t ≥

0} with Z

(t) = (X

(t),W

i−1

(t),W

(t)) by a Markov

chain with generator of the form







(0)

(1)

−1)

)







where B

(n)

is the square matrix with negative diago-

nal elements and the components [A

(n)]

(x,y),(x

)

(n) and [C

(n)]

(x,y),(x

)

of C

(n) are the transition

rates of Z

from the state (n, x, y) to (n + 1, x

, y

) and

(n − 1, x

, y

), respectively.

3.2 Transition Rates

The explanation of the matrices A

(n)

, B

(n)

and C

(n)

are

presented in this subsection and detailed derivation of

the formulae is omitted. The matrices A

(n)

, B

(n)

and

(n)

are as follows, for 1 ≤ i ≤ N,

(n)

= A

i−1

(n) ⊗ A

(n), (1)

(n)

= C

i−1

(n) ⊗C

(n), (2)

(n)

= B

i−1

(n) ⊕ B

(n) − ∆

(n), (3)

where A ⊗ B denotes the Kronecker product of the

matrices A and B and A ⊕ B = A ⊗ I + I ⊗ B denotes

the Kronecker sum of the matrices A and B, I is the

identity matrix, and ∆

(n) is the diagonal matrix that

makes Q

e = 0. The matrix A

i−1

(n) corresponds to

the rates of service completion at W

i−1

and C

(n) cor-

responds to the rates of departures from W

given

(t) = n, respectively. The matrix A

(n) corresponds

to transition probabilities of W

due to an arrival from

i−1

and C

i−1

(n) corresponds to the transition proba-

bilities of M

i−1

induced by a departure from M

. The

matrices B

i−1

(n) and B

(n) are for the transition rates

of M

i−1

and M

, respectively, without changing the

state of X

(t).

The matrices A

(n). Since M

is never starved, it

can be seen that 0 ≤ n ≤ K

− 2,

(n) = T

, A

− 1) = T

Note that the state transition from w

to w

(t) is occurred by a service completion from M

with X

(t) = 2 and the transition from w

to w

oc-

curs after service completion when X

(t) ≥ 3. Let for

j = 1, 2, ·· · , h

, k = 1, 2, · ·· , d

( j, k), w

) = P(X

(t) = 2|W

(t) = w

( j, k))t

( j),

and δ

( j, k), w

) = t

( j) − δ

( j, k), w

). De-

note the matrix corresponding to the transition rates

from the states in w

∗

to the states in w

∗

by A

∗

The matrices A

∗

), A

∗

s) and A

∗

k, l = 1, 2 are deﬁned similarly. It can be expressed

the components of the matrices A

x,y

y) in terms of

known parameters T

and unknowns δ

( j, k), w

k, l = 1, 2. For example, A

) = T

⊗ I, l = 1, 2

and

) = α

⊗ D

, w

where D

, w

) be the matrix of size h

×d

whose

jth block is the d

× d

matrix D

( j), w

), j =

1, 2, ··· , h

with the (k, k

)-element

( j), w

)]

= δ

( j, k), w

)1(k

= k − 1),

where 1(A) is 0 if A is true and 0, otherwise. Other

blocks can be given similarly.

The matrices C

(n). Note that when X

i+1

(t) ≤

i+1

− 1, the states of M

are not affected by a de-

parture from M

i+1

. Thus C

) = α

and C

(n) = I,

1 ≤ n ≤ K

i+1

− 1, 0 ≤ i ≤ N − 1. When M

= b

and

a departure occurs from M

i+1

, if X

(t) = 2, then the

state of M

(t) changes from b

to w

( j) with proba-

bility α

( j) and if X

(t) ≥ 3, then a transition occurs

from b

to w

( j). Let for k = 1, 2, · ·· , d

((b

, k), w

) = P(X

(t) = 2|W

(t) = (b

, k)),

((b

, k), w

) = 1 − p

((b

, k), w

The transition probabilities in the elements of the

matrices C

∗

) can be expressed in terms of

((b

, k), w

), l = 1, 2 and α

The matrices B

(n). Let T

∗

be the square matrix

whose diagonal elements are all zero and off diagonal

elements are the same as those of T

Since M

is never starved, it can be easily seen

that

) = 0, B

(n) = T

∗

, 0 ≤ n ≤ K

− 1.

ICORES 2020 - 9th International Conference on Operations Research and Enterprise Systems

214

Note that an arrival of a customer to M

is occurred

by a service completion of M

i−1

and its conditional

rate given W

(t) = x is

(x) =

i−1

∑

j=1

i−1

∑

k=1

P(W

i−1

(t) = w( j, k)|W

(t) = x )t

i−1

( j).

The components of the matrix B

x,y

y) corresponding

to the transitions from the state in x

x and the state in y

can be given in terms of a

(x) and T

∗

. For example,

s,w

∗

) =



(s, 0) O

O α

⊗ ∆[a

s)]



where a

s) be the vectors of size d

whose kth com-

ponents are a

(s, k) and ∆[x

x] is the diagonal matrix

whose diagonal vector is x

x = (x

, · ·· , x

The matrices A

(n). Once a service completion

occurs at M

i−1

, if X

(t) = 0, then the state of M

(t)

changes from s to j with probability α

( j) and if

(t) ≥ 1, then there are no state transitions of M

(t).

Thus A

(n) = I, 1 ≤ n ≤ K

− 1, A

(0) = α

and

(0) =



w(0, 0) w

w b

(s, 0) 1 O O

(s, ·) O α

⊗ I O



The matrices C

(n). Note that when M

(t) =

w( j), if X

i+1

(t) = K

i+1

− 1, then a service comple-

tion at M

results in blocking of the server M

and if

i+1

(t) ≤ K

i+1

− 2, then a departure from M

occurs.

It follows from the observations that the blocking rate

(w( j, k), 0) and the departure rate δ

(w( j, k), 1) of

given W

(t) = w( j, k) are

(w( j, k), 0)

= P(X

i+1

(t) = K

i+1

− 1|W

(t) = w( j, k))t

( j),

(w( j, k), 1) = t

( j) − δ

(w( j, k), 0).

When M

(t) = b, a departure from M

is occurred by

a departure from M

i+1

and the rate is

(b, k)

∑

y∈{w

w,b

P(W

i+1

(t) = y|W

(t) = (b, k))δ

i+1

(y)

with δ

( j) = t

( j), where

i−1

(t) = (b, k)} = {W

i−1

(t) ∈ {(b

, k), (b

, k)}}.

The components of C

(n) are given in terms of

(w( j, k), l), l = 0, 1 and δ

(b, k).

The matrices B

(n). The components of B

(n)

which are the transition rates of M

without the

changes of X

(t) are given in terms of known parame-

ters of service time α

, T

∗

, arrival rate λ

of parts and

the unknowns δ

(w( j, k), 0), and details are omitted.

3.3 Approximation of Transition Rates

Now we assume that the system is in stationary state

and let π

be the stationary distribution of Q

. We

express the parameters a

(x), δ

(x, x

), p

(x, x

) for

(n), B

, C

(n) of the subsystem L

i+1

and δ

i−1

(y) in

i−1

, B

i−1

of the subsystem L

i−1

in terms of π

. For

example, the transition rates δ

( j, k), w

) and the

probability p

((b

, k), w

) for W

are given by

( j, k), w

) =

P(X

(t) = 2,W

(t) = ( j, k))

P(X

(t) ≥ 2,W

(t) = ( j, k))

( j),

((b

, k), w

) =

P(X

(t) = 2,W

(t) = (b, k))

P(X

(t) ≥ 2,W

(t) = (b, k))

Similarly, the parameters for W

i−1

in the subsystem

i−1

are given as follows, for example,

i−1

(w( j, k), 0)

P(X

(t) = K

− 1,W

i−1

(t) = w( j, k))

P(W

i−1

(t) = w( j, k))

i−1

( j).

Throughput. Once the stationary distribution π

is obtained, the throughput can be obtained by

Θ =

∑

( j,k)∈w

P(W

(t) = ( j, k))t

( j).

3.4 Algorithm

In this subsection, an iterative algorithm for solving

the proposed decomposition equations for the com-

ponents of Q

is presented.

Boundary Conditions. Since M

is never starved

while the iteration, the formulae for the matrices

(n), B

(n) and C

(n) for M

are not changed dur-

ing iteration. Similarly, the matrices A

(n), B

(n)

and C

(n) corresponding to M

are not changed dur-

ing iteration.

Initial Step. Initially the upstream servers are as-

sumed to be never starved. Then the matrices for

upstream servers do not contain unknown param-

eters. Compute the stationary distribution π

Step 1. Backward iteration. For i = N − 1, N −

2, · ·· , 1,

(1) Update A

(n), B

(n) and C

(n). In this step

of the ﬁrst iteration, use the matrices A

(n), C

(n)

and B

(n) under the assumptions of no starvation

and from the second iteration, use the matrices of

general formulae whose components can be cal-

culated by using the formulae for transition rates

in subsection 3.3.

(2) Compute the stationary distribution π

of Q

If i = 1, GO TO next step.

Approximate Analysis of Transfer Line with PH-service Time and Parts Assemble

215

Step 2. Forward iteration. For i = 1, 2, 3, · ·· , N − 1,

(1) Update A

(n), B

(n) and C

(n) using the for-

mulae derived in subsection3.3

(2) Compute π

i+1

of Q

i+1

If i = N − 1, then GO TO next step.

Step 3. Calculate throughput and check the stopping

criterion

TOL = |Θ

(m)

− Θ

(m−1)

| < ε, (4)

where Θ

(m)

is the throughput obtained in the mth

iteration and ε > 0 is the tolerance predetermined.

If the stopping criterion is not satisﬁed, GO TO

Step 1 and repeat the backward and forward itera-

tion until the stopping criterion is satisﬁed.

4 NUMERICAL RESULTS

The accuracy of the method is investigated numeri-

cally by comparing approximations (App) with the

simulations (Sim). We consider the system with 6

workstations where the service times are identical

with common means 1.0 and the buffer size for cus-

tomers between workstations are identical. The buffer

size c

between workstations are chosen as c

= 0, 3, 5.

We use the Erlang distribution of order k (E

) for

the squared coefﬁcient of variation of service time

≤ 1 and hyperexponential distribution of or-

der 2 (H

) with balanced mean and C

= 2.0. Tol-

erance ε = 10

−5

is used for stopping criterion (4).

Simulation models for the systems in the tables are

developed with ARENA. Simulation run time is set

to 100,000 unit times including 30,000 unit times of

warm-up period. Ten replications are conducted for

each case and 95% conﬁdence intervals (c.i.) are cal-

culated. The intervals are omitted in the following ta-

bles, but we have observed that all the approximation

results are contained in the intervals. The deviation

(D) between approximation and simulation is calcu-

lated by D(%) =

App−Sim

Sim

× 100. The throughput for

the system is presented in Tables 1-2. Numerical re-

sults show that the approximation performs well.

5 CONCLUSIONS

In this paper, an approximate analysis for tandem

queues with blocking and parts assembly has been

presented. The approximation is based on the decom-

position method. To reﬂect the dependence between

consecutive stages, the states of the servers in sub-

systems are indicated by the state of the number of

Table 1: Throughput for the system with E

-service time.

0 3 5

Sim Sim Sim

App(D(%)) App(D(%)) App(D(%))

0.5 1 0.316 0.371 0.386

0.315(-0.4) 0.371(-0.1) 0.386(-0.0)

3 0.388 0.424 0.435

0.391(+0.8) 0.425(+0.3) 0.437(+0.4)

5 0.420 0.443 0.451

0.425(+1.3) 0.447(+0.8) 0.454(+0.7)

1.0 1 0.479 0.605 0.628

0.468(-2.4) 0.603(-0.3) 0.627(-0.1)

3 0.546 0.725 0.752

0.529(-3.0) 0.722(-0.4) 0.752(-0.0)

5 0.554 0.771 0.801

0.537(-3.2) 0.767(-0.6) 0.801(-0.0)

Table 2: Throughput for the system with H

-service time.

0 3 5

Sim Sim Sim

App(D(%)) App(D(%)) App(D(%))

0.5 1 0.279 0.345 0.361

0.275(-1.3) 0.342(-0.9) 0.360(-0.5)

3 0.336 0.401 0.414

0.331(-1.7) 0.396(-1.1) 0.413(-0.4)

5 0.362 0.426 0.438

0.355(-1.8) 0.423(-1.0) 0.437(-0.2)

1.0 1 0.363 0.500 0.536

0.362(-0.4) 0.494(-1.2) 0.533(-0.6)

3 0.391 0.571 0.619

0.391(0.1) 0.559(-2.0) 0.611(-1.2)

5 0.393 0.594 0.649

0.395(0.5) 0.581(-2.2) 0.641(-1.2)

customers in upstream subsystem as well as the states

of the server (blocking, starvation, working, lack of

parts), and the transitions among the states are con-

sidered. Numerical experiments indicated that the

method works reasonably.

The approach can be applied to the more complex

systems such as the system with unreliable servers

and the system with more general service time by the

versatility of PH-distribution, see e.g. Altiok (1985),

Osogami and Harchol-Balter (2006) and references

therein. Furthermore, our approach can be applied to

the optimization problem for control of parts buffers

and arrival rates of parts.

ACKNOWLEDGEMENTS

This paper was supported by Basic Research Program

through the National Research Foundation of Korea

(NRF) funded by the Ministry of Education, Grant

Numbers NRF-2018R1D1A1A09083352 and NRF-

2019R1F1A1057692.

ICORES 2020 - 9th International Conference on Operations Research and Enterprise Systems

216

REFERENCES

Altiok, T. (1985). On the phase-type approximations of gen-

eral distributions. IIE Transactions 17(2), 110-116.

Buzzacott, J. A., Shanthikumar, J. G. (1993). Stochastic

Models of Manufacturing Systems, Prentice-Hall.

Dallery, Y., Gershwin, B. (1992) . Manufacturing ﬂow line

systems: a review of models and analytical results.

Queueing Systems 12, 3-94.

Gershwin, S. B. (1987). An effﬁcient decomposition al-

gorithm for the approximate evaluation of tandem

queues with ﬁnite sotrage space and blocking. Opera-

tions Research 35, 291-305.

Gershwin, S. B. (1994). Manufacturing systems engineer-

ing. Prentice-Hall, Englewood Cliffs.

Throughput analysis of production systems: recent ad-

vances and future topics. International Journal of Pro-

duction Research 47(14), 3823-3851.

Moon, D. H., Shin, Y. W. (2019). Approximation of tan-

dem queues with blocking. In Proceedings of the 8th

International Conference on Operations Research and

Enterprise Systems (ICORES 2019), Prague, Febru-

ary, 2019, pp. 422-428

Neuts, M. F., (1981). Matrix Geometric Solutions in

Stochastic Models. Dover Publishing Co., New York.

Osogami, T., Harchol-Balter, M. (2006). Closed form so-

lutions for mapping general distributions to quasi-

minimal PH distributions. Performance Evaluation 62,

524-552.

Papadopoulos, H. T., Heavey, C. (1996). Queueing theory

in manufacturing systems analysis and design: a clas-

siﬁcation of models for production and transfer lines.

European Journal of Operational Research 92, 1-27.

Shin, Y. W., Moon, D. H. (2018). Approximation of dis-

crete time tandem queueing networks with unreliable

servers and blocking. Performance Evaluation 120,

49-74.

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