Study of Throughput Improvement for
an Unreliable Work Center in terms of
Alternating Renewal Process
Chin-Tai Chen1, John Yuan2 and Chih-Hung Tsai3
1,3
Department of Industrial Engineering and Management, Ta-Hwa Institute of
Technology, Hsinchu, 307, Taiwan, R.O.C.
2
Department of Industrial Engineering and Engineering Management, National
Tsing-Hwa University, Hsinchu 30047, Taiwan, ROC.
1.
Introduction
Abstract
This paper presents both numerical
simulation and closed form methods to
calculate
performance
measures
in
production line, such as expected the transient
throughput, and the probability that measures
the delivery in time, for an unreliable work
center without intermediate buffers. Such
approaches are based on the assumptions that
(1) work center alternates between Normal
and Failed; (2) up times and down times are
i.i.d./independent (but with exponential
distributions required in closed form method);
(3) work center has the fixed production rate
. Numerical experiments that compare
values of such performance measures by
different methods are presented in terms of an
unreliable work center cited from literature.
The sensitivity of such measures with respect
to either mean up time or mean down time is
also presented to investigate how and how
much improvement can be achieved.
Keywords: probability, transient throughput,
machine failure, alternating renewal process.
Many studies try to investigate the
characteristic of an unreliable work center by
calculating the average of the system
throughput (i.e. the production rate in the long
run) [1~12, 14, 16, 17], etc. However, an
increasing need to calculate the variability of
the output besides the average performance
arises in order to measure delivery on time
(due time T) and etc. Tan [14] proposed a
method to calculate both average E() and
variance Var() of the steady state throughput
 (i.e.  = lim  T where  T  UTT   ) for an
T 
unreliable work center and a series system of
unreliable machines with the same production
rate  and without buffer between machines
based on that each machine has i.i.d.
exponential up and down times so that the
system at steady state is further modeled in an
irreducible Markov chain. He also proposed
to use lim Pr{U T    D} to measure
T 
delivery on time T for the demand D in
sufficiently large T case by observing that that
UT will be very close to the normal
distribution with mean E()T and variance
Var()T for any large enough T by applying
the Central Limit Theorem.
Without further requiring that the system
is in an irreducible Markov chain as did in
International Journal of The Computer, The Internet and Management, Vol. 10, No3, 2002, pp. 23 - 39
23
Tan [14], this article is: (1) to develop
failures and under no maintenance.
simulation methods to estimate Pr{UT  D}
(4) The repair is taken immediately for a
and E( T ) for any T based on that up times
work center upon failure.
and down times follow any distributions F
(5) The operation (or normal) time and
and G respectively, (2) to develop closed form
repair time of the ith machine are
formulas to calculate them also for any T in
exponential or Gamma with rate i
case F and G are exponential. Then the values
and i for each i = 1, 2,…, N
of each system performance measure
respectively.
obtained by such three methods (including
(6) Machine has the fixed production rate
Tan’s [14]) are compared numerically and the
.
sensitivity analysis of each measure with
From assumptions (3), (4) and (5), the
respect to either mean up time or mean down
time in the system to see how and how much unreliable work center can be modeled in an
improvement can be achieved is presented.
alternating renewal (AR) process { X i , Yi } of
type (F, G) with F = k
Organization of the remaining part of this
paper is as follows. In Section 2, we list the
model assumptions, definition of the
measures, and the notations. In section 3, we
develop closed forms of Pr{UT  D} and
E( T ) for any T. In section 4, basic formulas
on which simulation algorithms are
developed are presented. In section 5, we
investigate numerically the difference of each
performance measure among such three
methods (including Tan’s [14]) and the
sensitivity of each measure with respect to
either mean up time or mean down time to see
what improvement can be made. In section 6,
some concluding remarks and future possible
study will be presented.
1
, , and
G = k 2 ,  . That
is,
(1) X i , is i.i.d. exponential or Gamma
with rate ; Yi , is i.i.d. exponential or
Gamma with rate .
(2) X i and Yi are independent.
Therefore, the unreliable work center can
be modeled in an AR process. Let N(T)
denote the total number of renewals of the
renewal
process { X i  Yi } during [0, T],
n
S n   ( X i  Yi ) and U T be the total up time
1
of thei unreliable
work center during [0, T].
Then
 N (T )1 
(T-SN(T)-1)XN(T)
UT 
(1)
where
 N (T )1  X 1  X 2  X 3  ....  X N (T )1 ;
2. Basic Assumptions and Notations on an
Unreliable Work Center
S N (T )  {[ X 1  Y1 ]  [ X 2  Y2 ]  ....  [ X N (T )  YN (T ) ]}
.
For convenience, we will let T (s)
=Pr{ U T s} and  T ( s)  1   T ( s) . In
particular,  T ( D /  ) = Pr{ U T  D} for
most continuous type F and G.
Throughout this article, the unreliable
work center without buffers must further
satisfy following assumptions:
(1) The system produces only one type
of products without defective.
(2) There is infinite material supply and
sufficient final storage to an
unreliable work center.
(3) Machine is subject to independent
3.
24
Closed
form
formulas
for
 T ( D /  ) , E( T ) and Var ( T )
(Exponential type)
n
T
m i
/( m  i)! +
 Bi  e  t
i 0
First of all, we know that
E( T ) =
E (UT )
T
Lemma A: Suppose that the AR process
for a system is of general type (F, G). Then
E (UT )
2
and E (U T )
=
k j 1
j 1
i 0
  jT
t
k j i
/( k j  i)!
where
The closed form formulas to calculate
 T ( D /  ) and E( T ) respectively are
described in Theorem C. Its proof needs to
verify the following two Lemmas first.
(A1)
N
  C ji  e
T
= 0
[1   T (s)]ds
20T  x[1   T ( x)]dx
Ai 
1
( k i )
( k i )
( k i )
 ( m i )  1 1  2 2     N N (1) i 
(m  1)!
i

l 0
i
 (i  l  n)! l 
l 

s1  s 2 ...  s N  l
(k  s  1)! (k 2  s 2  1)!  (k N  s N  1)!
l!
 1 1

s1 ! s 2 !  s N !
(k1  1)! (k 2  1)!  (k N  1)!
1i s   2 i s     N i s
1
2
N
for i= 0, 1,…, N-1
(A2) Suppose that min{ F (T ), G (T )}  1 .
Then
T(s) =

F ( s )   [ F ( n1 ) ( s )  F ( n ) ( s )] G ( n ) ( T  s )
n 1
for 0  s  T
Bi 
1
( )  (i  N ) (  1   )  ( k1  i ) (  2   )  ( k 2  i )    (  N   )  ( k N  i ) (1) i 
( N  1)!
i
i 
 (i  l  N  1)! ( ) l 

l 0  l 
(k  s  1)! (k 2  s 2  1)!  (k N  s N  1)!
l!
 1 1


s
!
s
!



s
!
(k1  1)! (k 2  1)!  (k N  1)!
s1  s 2  ...  s N  l 1
2
N
(  1   ) i  s1 (  2   ) i  s2    (  N   ) i  s N
1-T(s) =

1- F ( s )   [ F ( n ) ( s )  F ( n 1) ( s )] G ( n ) (T  s )
for i= 0, 1,…, m-1
C ji 
n 1
N
1
(  j ) (i  N ) (   j )  ( n 1 i )  (  i   j ) ( ki  i ) (1) i 
( N  1)!
i  j , i 1
for 0  s  T and
i
 (i  l  N  1)! (  j ) l 
l 0  l 
N
(n  s j )! (k i  s i  1)!
l!
 

n! (k i  1)!
s1  s 2  ...  s N  l s1 ! s 2 !  s N ! i  j , i 1
i

1-T(T-) = Pr{U T  T } = 1-F(T)
Proof: See Appendix A for its proof.
(  1   ) i  s1 (  2   ) i  s2    (  N   ) i  s N
Lemma B: Let
for j= 1,…, N and i= 0, 1,…, k j  1
N
H m ;;k11,,k22,...,,...,kN N ( )  (  1 ) m  N  (  1 j )
j 1
kj
and
In case m = 0, Bi  0 for each i.
hm;;k11,,k22,...,,...,kN N (t ) be its Laplace inverse. Then
 ; 1 ,  2 ,...,  N
Proof: See Appendix B for its proof.
N 1
hm;k1 ,k2 ,..., k N (t ) =  Ai  t N i /( N  i)!+
Theorem C: For a reliable work center,
i 0
International Journal of The Computer, The Internet and Management, Vol. 10, No3, 2002, pp. 23 - 39
25

 E{[ 

(C1)  T ( D /  ) =Pr{ U T 

D
}= e

N
i D
i 1
N


n 1
T

  i 
N
n!

 ( N i ) ki e i 1
k1  k 2 ...  k N  n k1 ! k 2 !  k N ! i 1   j
D

j 1
 , 1 ,  2 ,...,  N
  ... h0,k ,k
k1
1
k2
2
kN
N
1
2 ,..., k N
for 0 D <T
N

Pr{ U T =T}= e
i 1
(T 
D

N
n 1
i 1
D
 (T  S n )  X n 1 ] N (T )  n} Pr{ N (T )  n}
n
n 1
=
)

 Pr{[ 
n 1
 iT
n
 (T  S n )  X n 1 ]  D /  N (T )  n} Pr{ N (T )  n}
(3)
1

[e
N

i 1

n
(2)
( i  )  (D /  ) =
T

 i 1 n! 
 Pr{U T  D /  N (T )  n} Pr{N (T )  n}
N
Their procedure is described in a flow
chart, as shown in Figure 1. Algorithm is to
generate pairs of (Xi, Yi)’s under the
restriction that the AR process being of type
(F, G).
(C2)
E(U T ) 
n 1
N
  iT
i 1
 1] 
i
k
 ;  ,  ,..., 
k
k
n
 (  i ) 1 1  2 2     NN  hn1;1k1 ,k22 ,..., kNN (T )
5. Results and Discussion
Proof: See Appendix C for its proof.
4.
Simulation
to
estimate
 T ( D /  ) , E( T ) and Var ( T )
The unreliable work center used for
numerical comparison here has no buffers and
its machines has  = 0.1,  = 2 and  = 1. This
example and its values of E ( lim  T ) are
the
T 
quoted from Tan [14]. The proposed closed
form
and simulation methods
are
For a reliable work center whose AR
implemented in Fortran software to calculate
process is of Gamma type (F, G), simulation
approaches to estimate  T ( D /  ) and E( T )  T ( D /  ) and E( T ) for the unreliable
are proposed. Our simulation is based on the work center where D denotes a given demand.
following results obtained by Total Our numerical experiments on  T ( D /  ) and
Probability Theorem (Ross [13], Tijms [15]):
E( T ) are summarized into Figure 2(a)~ (d),
E (UT )
E( T ) =
Figure 3(a)~(d) and Figure 4(a)~(d).
T
Numerical evidences indicate first that (1)
=

most results of our two methods are

consistent with difference 10-2; (2) the
T  E{U T N (T )  n} Pr{ N (T )  n}
n 1
results on E( T ) , approach Tan’s when T is
=
getting larger, and then that (3)


N (T )  n}
T  E{[  N (T )  (T  S N (T ) )  X N (T ) 1 ] N (T )  n}Pr{
T ( D /  ) increases in T given D; (4)
n 1
 T ( D /  ) decreases in D given T; (5) both
 T ( D /  ) and E( T ) decreases in  given T
=
=1, 10, and 100; (6) both  T ( D /  ) and
26
E( T ) increases in  given T =1, 10, and 100;
(7) E( T ) increases in k1 given T =10, and
100.
3.For future/further study, preventive
maintenance and defective rate may be
taken into consideration.
Now we would like to investigate how
and how much system performance
improvement can be achieved by changing
either mean up time rate  or mean down time
rate . To apply such results, it should be
noted first that given fixed T, D and , system
performance should be improved in the
direction to either increase  T ( D /  )
and E( T ) simultaneously. As Figure 5
shown, the larger  T ( D /  ) and E( T ) can
be achieved by decreasing  (i.e. increasing
mean up time). It is noted that both  T ( D /  )
and E( T ) are increasing function in . It is
suggested
that
system
performance
improvement can be achieved by decreasing
mean up time rate  or increasing mean down
time rate .
6. Conclusions
1. In this article, a numerical simulation
for an unreliable work center with the
fixed production rate and without buffers
is modeled by an AR process of Gamma
type (Fi, Gi) is proposed to
measure  T ( D /  ) and where D is a given
demand. Closed form expressions for
such measures in case Fi and Gi of
exponential type are also derived.
2. For simplicity, the numerical experiments
that compare the values of such measures
by different methods (including Tan’s
[14]) in this article are presented in terms
of an unreliable work center has the fixed
(, , ). The sensitivity analysis of each
measure with respect to mean up time or
mean down time is investigated also in
terms of such an unreliable work center to
see how and how much improvement can
be achieved.
International Journal of The Computer, The Internet and Management, Vol. 10, No3, 2002, pp. 23 - 39
27
Acknowledgments
multistation unreliable production lines,
European Journal of Operational
Research, 68 (1993) 69-89.
This article comes out from the research
project supported under the National Science
Council, Taiwan (Grant No. NSC
89-2213-E-007-111).
[8] Y. Hong, C. R. Glassey, and D. Seong, The
analysis of a production line with
unreliable machines and random
processing Times, IIE Transactions, 24
(1992) 77-83.
_____
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29
Appendix A
T
Proof of (A1): E (U T )  
xd Pr{U T  x}  T  Pr{U T  T }
0
=
T

0
T
xd T ( x)  T  (1  F (T )) =  
0
T
xd(1   T )( x)  T  (1  F (T )) = 
[1   T (s)]ds
0
T
E (U T2 )   x 2 d Pr{U T  x}  T 2  Pr{U T  T }
0
= 0T  x 2 d T ( x)  T 2  (1  F (T )) =
= 20T  x[1   T ( x)]dx
 0T  x 2 d (1   T )( x)  T 2  (1  F (T ))
Proof of (A2): First of all, we have to use another expression for U T to prove Lemma
1:
UT 

X 1 U T  X 1 Y1
X1
if X 1 Y1 T
if X 1 T , X 1 Y1 T
(A-1)
Lemma 1: For 0  s  T ,
s
T s
0
0
T(s) = F ( s)[1  G (T  s)] +  [
 T  x  y 1 (s  x1 )dG( y1 )] dF ( x1 )
(A-2)
1
where 0<T(s) - F ( s )[1  G (T  s )]  G (T  s ) F ( s ) ]  1
(A-3)
Proof: T(s) = Pr{U T  s; X 1  T , X 1  Y1  T}+ Pr{U T  s; X 1  Y1  T }
= Pr{ X 1  s, Y1  T  X 1  0} + Pr{U T  X1 Y1  s  X 1 ; X 1  Y1  T }
where
s
 Pr{Y  T  X X  x}dF ( x)
=  Pr{Y  T  x X  x}dF ( x)
=  Pr{Y  T  x}dF ( x) ( X & Y
=  [1  G(T  x)]dF ( x)
Pr{ X 1  s, Y1  T  X 1} =
1
0
s
1
1
1
0
s
1
1
0
s
1
1
are independent )
0
and
s
Pr{U T  X1 Y1  s  X 1 ; X 1  Y1  T } =  Pr{U T  xY1  s  x; Y1  T  x X 1  x}dF ( x)
0
s
 s  x; Y  T  x}dF ( x)
 Pr{U
=  [ Pr{U
 s  x}dG( y)] dF ( x)
=  [ 
(s  x)dG( y)] dF ( x)   
=
T  x Y1
0
s
T x
0
s
0
T s
0
0
1
T  x y
s T x
T  x y
0 T s
1 dG( y) dF ( x)
(Note that T  x  y  s  x if T  s  y  T  x and so Pr{U T  x  y  s  x  T }  1 )
=
s
T s
0
0
 [
s
 T  x y (s  x)dG( y)] dF ( x)   [G(T  x)  G(T  s)]dF ( x)
0
30
=

s
0
T s
[
 T  x y (s  x)dG( y)] dF ( x)dx
0
s
+ [1  G(T  s)]F (s)   [1  G(T  x)]dF ( x)
0
s
T s
0
0
Hence, T(s) = F ( s )[1  G (T  s )] +  [
s
T s
0
0
 [
Also,
 T  x y (s  x)dG( y)] dF ( x)
 T  x y (s  x)dG( y)] dF ( x)  
s T s

0 0
(A-4)
dG( y) dF ( x)  G (T  s) F ( s)  1
q.e.d.
(A3) can be obtained by applying Lemma 1 recursively. First of all, we note that
 T  x1  y 1 (s  x1 )  F (s  x1 )[1  G(T  s  y1 )]
+
s  x1
0
T  s  y1
[
 T  x  y  x  y 2 (s  x1  x2 )dG( y2 )] dF ( x2 )
1
0
1
2
and so
s
T s
T(s) = F ( s)[1  G (T  s)] +  [
0
= F (s)  [1  G(T  s)] F
s T s
s  x1
T  s  y1
0 0
0
0
+
  
0
( 2)
 T  x y (s  x)dG( y)] dF ( x)
(A-5)
(s)  [G(T  s)  G ( 2) (T  s)]
 T  x  y  x  y (s  x1  x2 )dG( y2 )dF ( x2 )dG( y1 )dF ( x1 )
1
1
2
2
where
0<T(s) –( F (s)  [1  G(T  s)] F ( 2) (s)  [G(T  s)  G ( 2) (T  s)] )
s T s
s  x1
T  s  y1
0 0
0
0

s
T s
0
0
  [
  
dG( y 2 )dF ( x2 )dG( y1 )dF ( x1 )
G(T  s  y1 ) F (s  x1 )dG( y1 )]dF ( x1 )  F ( s) 2 G(T  s) 2 < F (s) G(T  s)  1 .
Hence, it will be routine job to prove that
n
0<T(s) -  F ( k 1) ( s ) [G ( k ) (T  s )  G ( k 1) (T  s )]  F ( s) n1 G(T  s) n1
k 0
In particular,
T(s) =

F
( n 1)

n
n 0
k 1
( s ) [G ( n ) (T  s )  G ( n 1) (T  s )]
n 0
However,
T(s) =  {[  ( F ( k 1) ( s )  F ( k ) ( s ))]  F ( s )} [G ( n ) (T  s )  G ( n 1) (T  s )]

= F ( s ) [G ( n ) (T  s )  G ( n 1) (T  s )]
n 0

 [ F ( 2) ( s )  F (1) ( s )] [G ( n ) (T  s )  G ( n 1) (T  s)]  ...
n 1
International Journal of The Computer, The Internet and Management, Vol. 10, No3, 2002, pp. 23 - 39
31

 [ F ( k 1) ( s)  F ( k ) ( s)] [G ( n ) (T  s )  G ( n 1) (T  s)]  ...
nk
= F (s )  [ F
( 2)
(s)  F (s)]  G(T  s)  ...  [ F ( k 1) (s)  F ( k ) (s)]  G ( k ) (T  s)  ...
(1)

= F (s )   [ F ( k 1) ( s )  F ( k ) ( s )]  G ( k ) (T  s )
k 1
32
Appendix B
Proof of Lemma (B): First of all, we note that
N
H m ;;k11,,k22,...,,...,kN N ( )  (  1 ) m  N  (  1 j )
j 1
Bm 1 /( v   ) +
N

j 1
kj
=
A0 / v N
+…+
AN 1 / v
+
B0 /( v   ) m
+…+
{C j 0 /(v   j ) j  ...  C j ,k j 1 /(v   j )}
k
1 di N
for each i= 0, 1, 2,…, N-1
[v H (v)] v 0
i! dv i
1 di
Bi 
[(v   ) m H (v)] v  for each i = 0, 1, 2,…, m and m = 0,1, 2,…, n+1
i
i! dv
1 di
k
and C ji 
[(v   j ) j H (v)] v   j for each i = 0, 1, 2,…, kj-1; j = 1,…,N
i
i! dv
where Ai 
The formula to calculate such Ai , Bi & C ji can be found in [3]. The rest then follows from
the fact that the inverses of 1 / v k , 1 /( v   ) k and 1 /( v   ) k are T k 1 /( k  1)! ,
e T T k 1 /( k  1)! and e  T T k 1 /( k  1)! respectively.
Appendix C
To prove Theorem C, we need to verify Lemmas I, II and III first.
Lemma I: 1-T(s)

=e
N

i s

 e
i 1

N

i s (
N
 i s )n

i 1
i 1
n!
n 1
n! N i ki
( N ) k1 ,1 * k2 , 2 * ... * k N , N (T  s)

i 1
k1  k 2 ...  k N  n k1!k 2 !  k N !   j
j 1
Proof: Note that the N-series system is denoted by the AR of type (F, G) with G =
N

i 1
i
N
 j
j 1
1,  . Hence for each n = 1, 2, 3,…,
i
G(n)(T-s)=
N
n!
(k )
(k )
(k )

( N i ) ki 1, 1 1 * 1,  2 2 * ... * 1,  N N (T  s )

i 1
k1  k 2 ...  k N  n k1 ! k 2 !  k N !
 j
j 1
(Note that 1, 
=
(k )
i (T  s ) 
i
k , 
i
(T  s ) )
G(n)(T-s)
i
n!
 ( N i ) ki k1 , 1 * k2 ,  2 * ... * k N ,  N (T  s )
i 1
k
!
k
!



k
!
k1  k 2 ...  k N  n 1
 j
2
N
N

j 1
Hence, we conclude this Lemma by (A3) in Lemma A.
2 ,...,  N
Lemma II: k1 , 1 * k2 ,  2 * ... * k N ,  N ( s ) = 1k1  2k2 ... Nk N h0;k;1 ,1k,2,...,
k N ( s)
Lemma III:

T

[e
0
N
N
i s )n
 i s (
i 1
i 1
n!
 k1 , 1 * k2 ,  2 * ... * k N ,  N (T  s)]ds
N
= (  i ) n  1k1  2k 2     Nk N  hn;1;1k,1,k22,...,,...,kNN (T )
i 1
N
T
i 1
0
Proof: let     i then £(  e s
(s ) n
n!
k1 , 1 * k2 ,  2 * ... * k N ,  N (T  s )ds )()
=£ ( f n * k1 , 1 * k2 ,  2 * ... * k N ,  N (T ))( ) = [£ ( f n ) £ (k1 , 1 ) £ (k2 ,  2 ) …£ (k N ,  N )]( )
N
=

N

n 1

[  N  (   j j ) j ] = 1   N (   ) n 1  (   j j )
 (   )
1
k
j 1
N
= (  i ) n  1k1  2k 2     Nk N  H n;1;1k,1,k22,...,,...,kNN ( )
i 1
j 1
kj
T

0

[e
N
N
i s )n
 i s (
i 1
i 1
n!
 k1 , 1 * k2 ,  2 * ... * k N ,  N (T  s)]ds
N
= (  i ) n  1k1  2k 2     Nk N  hn;1;1k,1,k22,...,,...,kNN (T )
i 1
Proof of (C1): This follows by Lemma I and Lemma II.
Proof of (C2): By (A1) in Lemma A and Lemma I, we obtain that
E(U T ) =
T

0
[1   T (s)]ds
T
=  {e

N
i s
i 1
0
N


n 1
N
  i s (   i s )
N
n!
i
ki
i 1

(
)
e
 i 1 n! 
N

i 1
k
!
k
!



k
!

k1  k 2 ...  k N  n 1
 j
2
N
n
j 1
k1 , 1 * k2 ,  2 * ... * k N ,  N (T  s)}ds
=
1

[e
N

i 1
N
  iT
i 1

N
n 1
i 1
 1]   (  i ) n 1k1  2k2     Nk N  hn;1;1k,1,k22,...,,...,kNN (T )
i
where the last equation follows by Lemma III.
International Journal of The Computer, The Internet and Management, Vol. 10, No3, 2002, pp. 23 - 39
35
Figure 1: The flow chart of the simulation method to estimate  T ( D /  ) and E( T )
36
Figure 2:  T ( D /  ) vs. time period T, , and  for an unreliable work center
International Journal of The Computer, The Internet and Management, Vol. 10, No3, 2002, pp. 23 - 39
37
Figure 3: E( T ) vs. time period, k1, and k2 for an unreliable work center (= 0.1 and = 2)
38
Figure 4: The effect of  on the E( T ) for single unreliable work center ( = 2)
International Journal of The Computer, The Internet and Management, Vol. 10, No3, 2002, pp. 23 - 39
39
Figure 5: The effect of  on the E( T ) for single unreliable work center ( = 0.1)
40