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A stochastic model to evaluate performance of testing and taping processes on an LSI production system
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Mathematical
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.v
Computer
COMPUTER MODELLING
DIRECT*
ecleNce
Modelling
38 (2003)
1347-1355 www.elsevier.com/locate/mcm
A Stochastic Model to Evaluate Performance of Testing and Taping Processes on an LSI Production System KATSUHIRO Department Tamagawa
of Information and University, Machida-shi,
Department
of Electronics Machida-shi,
TADAAKI
NAKADA Communication Engineering Tokyo
194-8610,
Japan
YONEYAMA Engineering, Tarnagawa Tokyo 194-8610, Japan
University
Abstract-h
automated testing and taping process on a LSI production system is modelled as a variant of M/G/l finite capacity queue with multiple transactions, where each package receives either a normal service or normal service plus postservice. The system has N spaces for waiting. We successfully define states of system to regard as those of a Markov renewal process even though some time points are not regeneration points. Using the model, we derive properties of stochastic behaviour of the system, in particular, throughput, process of overflows, busy period, and idle period as cost-effective tools for system evaluation. @ 2003 Elsevier Ltd. All rights reserved.
Keywords-Evaluation, system.
Markov
renewal
process,
First-passage
time,
Reduction,
LSI
production
1. INTRODUCTION An automated testing and taping process on an LSI production system is modeled. If we wish to represent faithfully the actual operation of the system, the resulting Markovian model would involve a large number of states. Although the process keeps moving all the states on the state space of the model, we may wish to study a certain subset of states and observe the states in the subset for the purpose of obtaining key properties. Useful results can be achieved by using a reduction method of Markov renewal process. Liu and Hammond [l] use state reduction method for mean first-passage time analysis in a Markov chain. Courtois and Semal [2] have proposed a bounded aggregation to compute lower and upper bounds on the conditional equilibrium probabilities in a subset of the states of a large Markov process. A reduction method of Markov renewal process (MRP) [3] and its application [4] have also been proposed. In this paper, a key work in the modelling process is the choice of the level of observations to describe the structure and behaviour of the given system. The merit of the proposed method is in the use of an aggregated model with different levels of what could be observed. At level 1, we construct an MRP and reduce it. The underlying structure of the reduced MRP is used to obtain mass functions of an MRP constructed at level 2, where we deal with the union of the number of packages and observed events, and decide on an appropriate state space of the model. 08957177/03/t - see front doi: 10.1016/S0895-7177(03)00348-O
matter
@ 2003
Elsevier
Ltd.
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reserved.
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by d/MS-T@
K.
1348
NAKADA
AND T. YONEYAMA
The reduced MRP at level 2 is more powerful for deriving indices of performance. deal with the overflow problem.
2. MODELLING
A TESTING
AND TAPING
At level 3 we
PROCESS
It is important in the design and process management of production systems such as packaging of large scale integrated circuit (LSI), to provide cost-effective tools for system evaluation. Packaging, or assembling die, significantly affects the overall cost, performance, and reliability of the packaged die. Overall integrated automation of assembly process; not only die-bonding, wire-bonding, and molding steps but also testing and taping process, is required. In this section, we set up an abstract model to represent behaviors of a testing and taping unit of LSI packages. As shown in Figure 1, the testing unit considered here normally tests whether every package satisfies its specification or not. When a package satisfies given specifications by the test, then the result of the test is “good”-the package is transferred from testing unit to taping unit, where the package is packed on tape. While the result of the test is “no-good”, the package is removed from the testing unit by the action of removing the unit. A retest, after the primary test, is made when the primary test ends in failure. The retest is made only once.
Overflow
Figure
1. Schematic
diagram
of testing
and
taping
unit.
Whenever a package supplied from the package handler finds the testing unit busy, it is stored in the buffer. Let N denote the buffer capacity. A control device in the testing unit that finds all N spaces in the buffer full refuses to store the package. The package must go away. The package is then said to have overflowed. Packages arrive according to a Poisson process with rate X. Only one package may be tested at a time. Packages in the buffer are tested in the order of arrival, so that the testing discipline is FIFO. Packages are judged as “no-good” with probability p and the primary test ends in failure with probability CL. Normal service times are independently and identically distributed random variables according to a general distribution function Bs(t) with mean E(s(O)), and so are post-time Is (I = 1,2) to a general distribution function Bl(t) with mean E(s(‘)), respectively. The operation discipline is as follows: whenever the buffer is empty, the production unit stops and warns the operator. He supplies packages to the package handler and stores “a” pieces of packages on the buffer. The operator starts the production units again. Suppose that these elapsed times to restart are independently and identically distributed random variables according to a general distribution function G(t) with mean g.
A Stochastic Model
1349
3. CONSTRUCTION OF MARKOV RENEWAL PROCESSES AT APPROPRIATE LEVELS OF OBSERVATIONS The merit of proposed method is in the use of multiple appropriate models with different levels of what could be observed. A key work in the modelling process to a given system is the choice of the level of observation to describe the structure and behaviours of the given system and to construct a stochastic model. At level 1, we observe events about which we have postulated to the transaction of packages in the production units and discuss the structure of a Markov renewal process associated with the obtained sample description space. Although the process keeps moving all the states on the state space, we may wish to study a certain subset of states and observe the process only when it is in those states on the subset for the purpose of obtaining key properties. This is used to construct an appropriate model at level 2. We reduce the MRP at level 1. It will be used as the underlying structure of a stochastic process at level 2. It would not be enough to specify the number of packages in the buffer for obtaining ordinarily important indices of performance such as throughput, busy period of the production units, the frequency that the testing unit starts a test to a new package, and so on. The underlying structure of the reduced MRP at level 1 and the results of observations at level 2 will be used to construct an appropriate stochastic model at level 2. We deal with the union of these events and the number of packages as events at level 2. Furthermore, a reduced MRP at level 2 will be used to analyze indices due to the slow transitions on the process. At level 3, of particular interest is the property of overflows of packages arrived at the testing unit. For the analysis of overflows we consider occurrences of overflows during each transaction of packages by the production units as an event in addition to the previous one at level 2. 3.1. Markov
Renewal
Process
(MRP)
at Level
1
We must first decide upon appropriate sample description space at level 1. Assume that there is always at least one package in the buffer. When the process is processing a package, the elapsed time depends only on the time interval to complete transaction of the package and does not depend on the number of packages in the buffer. We observe events described the following descriptions. El(&):
Testing unit starts a test to a package at the head of buffer immediately after that good package was transferred to taping unit (immediately after that no-good package was removed). Es: Testing unit starts to retest the package immediately after primary test is failure. E4: Removing unit starts to remove the no-good package.
A set of all possible events, EL = {Ei 1 i = 1,2,3,4}, is considered as a sample description space. Let i be the epoch at which an event Ei occurs. The process is said to be in state i between the successive epochs i and j. State space Sr, of the process at level 1 is given by SL = {1,2,3,4}. Let Qij(t) be the probability that the process makes a transition into state j once it enters state i. For i, j E SL, Qij(t) is given by (1 - p)(l
Qij(t) =
-p)Bo(t),
for i = 1, j = 1; i = 2, j = 1;
PBo(t),
for i = 1, j = 3; i = 2, j = 3;
(1 - P)PBo(th
for i = 1, j = 4; i = 2, j = 4;
(1 -
for i = 3, j = 1;
PPz(%
p&(t),
for i = 3, j = 4;
Bl @I,
for i = 4, j = 2;
. 0,
for otherwise.
(1)
1350
K. NAKADA AND T. YONEYAMA
Let qij(s) denote the corresponding Laplace-Stieltjes transform (LST) of Qij(t). [Qij (t>l (4 J’ E S L and t E R+) be the semi-Markov kernel over SL. 3.2. Reduced
MRP
at Level
Also let Q =
1
Figure 2a shows the state transition diagram of this fundamental model. Although the process keeps moving through all states on the state space SL, we now meet such a situation that we may wish to study a subset of states Sr c SL, and observe the process only when it is in those states.
(a) Original
MRP at level 1.
Figure 2. Transition
(b) Reduced MRP at level 1
diagrams of original
MRP and reduced MRP at level 1.
The notion of conditional first-passage time is useful to these kinds of situations. Suppose $41= {1,2}, since we pay our attention to occurrences of events El and E2, thus epochs 1 and 2. Let (l)&j(t), where i, j E Sr, be the distribution of conditional first-passage time from state i to state j for the first time, given that the process does not visit any state on Sr during (O,t]. Let (‘)hij(s) be the corresponding Laplace-Stieltjes transform (LST) of (l)Hij(t). For fixed j, we have the following simultaneous equations [3]:
(1)b(4
= %j(S) -c
c qim(S)%mj(s). mESL-S~
(2)
Similarly, we have (2hrLj(s)
= q??&) +
c
Qme(s)(2hj(s).
(3)
LESL-Sl
Noting q12(s) = q22(s) = qdl(s) = 0, (‘)hij(s)
can be given by
(l)Hds)‘= [:::g;:I+[q23(s) ,,,(sJ [0 q13(s)
Q14(S)
Q31(S)
434(s)442(s) 442(s)
1y
(4)
where i,j E 11s = {1,2}. Since (l)Htij,(0) are considered as a Markov chain, ll)Hlij,(s) can be considered as the LST of a semi-Markov kernel over Si. State transition diagram of the reduced MRP over Sr is shown in Figure 2b. 3.3. MRP
at Level
2
At level 2, the state of the given system must include more information than merely the number of packages in the buffer. We pay our attention to observe event Ei (i E I = { -2, -1, 0, 1,2}) and the number of packages T (E R = {O,l, . . . , N}) when an event Ei occurs, where all possible Ei are described by the following statements.
A Stochastic Model
&(Ez): E- 1 (E-2): &:
1351
Testing unit starts test to a package immediately after that good package was transferred to taping unit (immediately after that no-good package was removed). Testing unit enters an idle period immediately after that good package was transferred to taping unit (immediately after that no-good package was removed). Testing unit starts test followed by an idle period to the package at the head of the buffer.
Imagine an event described by the statement that “there are r packages in the system when an event Ei occurs”. We deal with the union of event Ei and the number of packages when E, occurs as an event at level 2. The event described by the statement can be specified by 2-tuple (Ei, T). Let E be the set of all possible events (E,, r). Let (i,~) be the epoch at which an event (Ei,r) E E occurs. The process is said to be in state (i,r) between the successive possible epochs (i,r) and (j,~‘). State space Su of the process corresponding to model at level 2 is given by SU = &I SU,, where Sv-, = { (-1, O)}? SU-, = {(-2,0)}, &J, = {(O,a)} and for fixed m E 112, SU_ = {(m, r) \ T E R}, respectively. Let the mass function Qti,r),(j,rJ)(t) be the probability that after making transition into state (i, r) at time t = 0, the process makes a transition into state (j, r’) in amount of time less than or equal to t. The elapsed time to complete a transaction to a package is independent of arrivals of packages during (O,t]. Also the event occurring at the next epoch depends only on the outcome, that is, good package or no-good package, from the production units at the end of the transaction. (l)&(t), i,j E 112, derived in equation (4), can be used as the conditional probability that the sojourn time in state (i,r) is less than or equal to t, irrespective of T and r’, given that the process makes a transition into state (j, r’). Since only one package in the buffer may be entered to the testing unit at a time, the number of packages changes from T to T’ = T - 1 + k when the system has k arrivals of packages during (0, t] (i.e., t ime interval between successive epochs (i, r) and (j, T’)). The probability of k arrivals X,(t) during (0, t] is given by
(5) Let Q~~,~J(~,,J)(s)be the corresponding LST of Q(,,,,cj,,o(t). For the notational define 1,~= {0}, and R, = {a}, respectively. We have q(z,r)(j,r/) (s) as follows.
convenience
(1) For i, j E 112, and T E R - IO; and for i E 10, j E 112, and T E R,;
I w=o , Q(i,r)(j,r+) = (lhj(S + 4,
O
(2) For i, j E 112, and T E R; and for i E IO, j E 112, and T E R,; N-r+1 q(W,(j,N)b)
=
%j(4
-
c
Q(i,r)(j,r-m+l)(S),
m=O
k>N-r+l.
(8)
(3) For i E 112, j E I - IOU 112; Q(i,O)(j,O)(s) = %,j,
(s + A)
(9)
(4) For i E I - 112 - IO; Q(i,O)(O,&) = g(s).
(10)
4(i,r),(j,r~) (s) = 0.
(11)
(5) For otherwise
1352
K. NAKADA AND T. YONEYAMA
We shall first derive the first-passage time distribution Hc,,r,cj,,lI(t). Let /I~~,,)(~,,,)(s) and q(i,,-)(j,+)(s) be the Laplace-Stieltjes transforms of Hc~,~)(~,~,J(~)and Q (2,r)(j,r~)(t), respectively. We have h(i,r)(j,?+)
= 4J(v)(W)(4
+ &
(1 - &(j,T’,) Q(i,r)y(S)h,(j,r’)(S).
(12)
Of particular interest is the mean first-passage time Lca,,)(j,,J). A set of simultaneous for L~~,~)(j,,r) is given by’ J%?-)(A~~)= c dwr 7ESU
equations
+ c (1 - ~-d.wJ Qr(w)w,(j,rJ)~ YS%l
(13)
where Lc~,,), = [-ah(ii;~Y(s)]s=c and @i,r)r = [- aq(i$~(s)]a,O. Let M(i,,.)(j,+)(t) be the expected number of visits to state (j,r’) during the interval (0, t], starting from the epoch at which the process makes a transition into state (i,r) at time t = 0, where the initial state (i, r) is not counted for the number of visits to state (j, P’) if (i, r) = (j, r’). Let rr~(+)(j,~j)(s) be the LST of M(,,t.)(j,T,)(t). We have
(14) Furthermore, M(j,rl), state, is given by
the expected number of visits to state (j, T’) per unit of time in steady
(15) 3.4. Reduction
of the MRP
at Level 2
Consider a collection {S UE, &NE} SUE = {(h>,
W,O>,
of nonempty
(-2, O)),
subsets of SU, where
%.I = SUE u &WE,
and
SUE n SrrNE =
0.
Let us pay our attention to states on SUE. An argument similar to that in the previous section can be made for obtaining a reduced MRP from the original one at level 2. Let (‘)h(i,,~(j,,l)(s), where (i, r), (j, r’) E SUE, be the LSTs of conditional first-passage distributions, thus mass functions, in the reduced MRP. The state transition diagram of the original process shown in Figure 3a can be reduced as shown in Figure 3b. We have the following LST of semi-Markov kernel over SUE represented in matrix form:
(a) Original
MRP at level 2. Figure 3. State transition
(b) Fleduced MRP at level 2. diagram at level 2.
A Stochastic
1353
Model
%u&) = [(l)fqi,rj(j,r~~(s)]
(16)
(‘)hSUE(0) is considered as the transition matrix of a Markov chain with state space SUE. Let x(~,~J) be the state probability that the process is in state (j,r’) ((j,r’) E SUE) in steady state. The LST of the first-passage time distribution from subset SA (C SUE) to subset SB (c SUE), kAsB(s), is given by
c c n(k,r,,)h(k,rll)(e,rl)(s) (e,r’)ESB (k,r”)ESA
hS,Se(s)
=
(17) c
n(v9
(fW’)ESA
3.5. Markov
Renewal
Process
at Level 3
For the analysis of overflows we pay attention to the following events in addition to the previous one at level 2. That is, E,ci,r) (Eb(i,r)): the first overflow of package occurs before the process ma,kes a transition into state (1, N) ((2, N)), g iven that the process made a transition into state (i,r) at t = 0, where i E IO, r E R, and i E 112, T E R. All of the events added to those in the set E at level 2 can be represented by Er = {E~c~,~) ( l?(i,r) = a(i,r), b(i,r); i E I is, r E R; i E IO, r E {a, a + 1,. . . , N}}. All of the events that we pay our attention to at level 3 are represented by E U Er, where E n Er = 0. Suppose that a(i, r) (b(i, T-)) is the epoch at which the event E,(i, T) (&(i, r)) occurs. The process is said to be in state a(i, r) (b(i, r)) b et ween the successive epochs a(i, r) (b(i, T)) and (1, N) ((2, N)). Note that epochs a(i, r) and b(i, r) are not regeneration points. There are two possible ways of transitions from state (i, T) to state (j, N). One is the direct transition from state (i, T) to state (j, N) and the other is the transition from state (i, r) to state (j, N) via state a(i,r) (b(i,r)) if j = 1 (j = 2). Define nq(i,r)(j,N)(s) and qF$ki,Nj(s) as the LSTs of mass functions of these transitions, respectively. For i, j E 112, r E R and r’ = N; and for i = 0, j E 112, and r E R,; we have
1w=ok=N-r+l,(18)
n4(i,r),(j,N)(4 = -$S $ (‘+-qi,r)(j,r~){s + (1- wP1 7 N-r+1 q;r:;‘(j,N)
(‘)
=
(1)hd4
-
c k=O
Q(i,r)(j,r-k+l)bL
(19)
and q(i,r)(j,N)(S)
=
%(i,r)(j,N)(s)
+ q;f$&J,)(s)~
(20)
Also Qti,r)r(i,r)(t), the mass function that after making a transition into state (i, r) at t = 0, the process next makes a transition into state I’(i,r) in an amount of time less than or equal to t, is given by
Q(i,r)r(i,r)(t) =
’ (%!&j(~) - (%&j(T)] xx,(T) d,-. S[0
(21)
Let q(i,r)r(i,r)(s) be the LST of Q (z,r)r(i,r)(t). Define qr(+)(s) as the LST of the probability that after making a transition into state I’(i,r), the process does not leave the state up to time t. It follows that qr(i,T) cs) = q(i,i-)r(i,r) cs) - $$)j,N)
cs)+
(22)
Note that it does not need for the analysis of H(i,r)(j,N)(t) to distinguish these two ways of transitions from state (i, r) to state (j, N). Consider Mci,r)r(i,,)(t). Since epoch I’(i, r) is not a regeneration point, we need to consider the way of transition that after the direct visit to state I’(i,r), the process makes a transition into
1354
K.
NAKADA
AND T. YONEYAMA
state (j, N) up to a certain time ‘u. and then the process does not leave from state (j, N). fixed j, we have m(i,r)r(i,r) (s) = Furthermore,
Q(z,r)r(i,r) Cs) { 1 - k?-)(i,T) (5) > .
For
(23)
Mr(i,t-), i.e., the expected number of visits to state I’(& r) per unit of time, is given
by MIT(~,~) = !io ~m(i,~)r(i,~)(s) = L(i~~)(i,r)q(i,r)r(i,r)(0).
4. INDICES
FOR EVALUATING
(24)
PERFORMANCE
Of particular interest are the busy period and idle period of the production system. We deal with the busy period as the time spent by the system to execute successively repeated transactions to packages followed by an idle period. The busy period terminates when the buffer becomes empty. Thus, it begins when the process makes a transition into state ((0, u)} and ends when the process goes to any of the states on subset ((-1, 0), (-2,O)) for the first time, given that the process passes through none of the states on subset SUE during the transitions. Let Hbusy be the mean busy period. In this case, subsets SA and Sg in equation (17) can be considered as SA = ((0, u)} and Sg = {(-l,O), (-2,0)}, respectively. We have
fbu.y = L(o,a)(o,a) - 9.
(25)
A similar argument can be made for the idle period. It is equivalent to the first-passage time from state(~,O)onsubset{(-1,0),(-2,0)}tostate(O,a)onsubset{(O,a)}. Subsets{(-l,O),(-2,O)) and ((0, u)} can be considered as SA and SB in equation (17), respectively. The LST of the distribution function of idle period, hi&(d), follows that
c hi&(s) = hS~,(O,a)(s)
=
~(/c,0)~(~,0)(0,&)
(k’o)ESA
(26)
c ~‘(i?O) (i,O)ESA
(see equation
(17)). S’mce h~-r,o)(c,,)(s)
= ht-z,o)(o,a)(s)
HidIe= [-““i;$‘]szo
= g(s), the mean idle time is given by
=9*
(27)
Throughput is defined by the number of good packages transferred from the testing unit to taping unit per unit of time. When the process visits state (1, r) and (-1, 0), a good package was transferred immediately before making the transition into state (1, r) and (-l,O). Thus, equivalently we can consider the process has the transfer of a good package as an output when the process makes a transition into state (1, r) and (- 1,O). The average expected number of visits to state (1,~) is given by Mci,,) = lims+,-, sm(i,,)(r,,)(s) = L$.)(,,,), where (i,~) E SU-, U Su,. Throughput is irrespective of the number of packages in the buffer. We have M gout =
as the throughput of the production units. A similar argument no-good packages from the removing unit. We have M ngout =
W!
c “s”, L&(iA (i,r)ES”-,
c (i,r)ESu-,USu2
can then be made for removing
L-1 (%r)(ivr)’
(29)
A Stochastic
1355
Model
Define Mpin as the frequency that a testing unit starts its test to a new package. When the process makes a transition into state (0, a) and (i, T), where (i, T-) E S,~J~U Suz, a test will be started. Thus, we have
zis easily verified that L&Jt,,aJ ngout
= L;_‘,,OJC-,,,j + L;_‘,,,~~-,,,~.
It is clear that Mpin = Mgout +
.
Define MO,,, as the expected number of visits to overflowing periods per unit of time in steady Let SuB = {(i,~) 1 i,j E 112, T E R; i = 0, T E R,}. state. M,,,,, is irrespective of (i,~), r(i,r). M over is given by M over =
1 {Ma(i,r) (0)ESu,
+ Mb(i,r,}
=
c L;,:,(i,r) (i,r)ESu,
{Cf(i,r)a(i,r)(“) + q(WGdO)~
c31)
5. CONCLUSION An automated testing and taping process on an LSI production system has been modeled, and indices for evaluating performance such as mean busy period, mean idle period, throughput, and the expected number of visits to overflow periods have successfully been obtained. We employed three different levels of observations to describe the structure and behaviors of the given system. We showed that the merit of proposed method is in the use of multiple appropriate models with different levels of what could be observed and in the use of the reduction of constructed MRP at each level if necessary. The underlying structure of the reduced MRP at level 1 and the results of observations at level 2 were used to construct an appropriate stochastic model at level 2. Also, the reduced MRP at level 2 was used to analyse indices due to the slow transitions on the process. The method proposed in this paper can be used as a cost-effective tool for system evaluation.
REFERENCES 1. S. Liu and J.L. Hammond, Jr., A method for modeling and analysis of the reframing performance synchronous time division multiplex hierarchies, IEEE Duns. Commun., COM-28 8, 1219-1227, 2. P. Courtois and P. Semal, Bounds on conditional steady-state distributions in large Markovian models, In Int. Seminar on Teletmfic Analysis and Computer Performance Evaluation, The (1986). 3. K. Nakada, T. Yoneyama and T. Iseki, An algorithm to compute the moment of the first passage Markov renewal process finite states, ‘i?mns. ZEICE 71, 664-678, (1988). 4. K. Nakada and T. Yoneyama, An algorithm to absorb inaccessible markings in reachability tree Petri net, Proc. of the Fnd Australia-Japan Workshop, 423-432, (1996).
of multilevel (1980). and queuing Netherlands, time
in the
of stochastic
MATHEMATICAL
www.ElsevierMathematics.com d l ONSIIED
Mathematical
and
.v
Computer
COMPUTER MODELLING
DIRECT*
ecleNce
Modelling
38 (2003)
1347-1355 www.elsevier.com/locate/mcm
A Stochastic Model to Evaluate Performance of Testing and Taping Processes on an LSI Production System KATSUHIRO Department Tamagawa
of Information and University, Machida-shi,
Department
of Electronics Machida-shi,
TADAAKI
NAKADA Communication Engineering Tokyo
194-8610,
Japan
YONEYAMA Engineering, Tarnagawa Tokyo 194-8610, Japan
University
Abstract-h
automated testing and taping process on a LSI production system is modelled as a variant of M/G/l finite capacity queue with multiple transactions, where each package receives either a normal service or normal service plus postservice. The system has N spaces for waiting. We successfully define states of system to regard as those of a Markov renewal process even though some time points are not regeneration points. Using the model, we derive properties of stochastic behaviour of the system, in particular, throughput, process of overflows, busy period, and idle period as cost-effective tools for system evaluation. @ 2003 Elsevier Ltd. All rights reserved.
Keywords-Evaluation, system.
Markov
renewal
process,
First-passage
time,
Reduction,
LSI
production
1. INTRODUCTION An automated testing and taping process on an LSI production system is modeled. If we wish to represent faithfully the actual operation of the system, the resulting Markovian model would involve a large number of states. Although the process keeps moving all the states on the state space of the model, we may wish to study a certain subset of states and observe the states in the subset for the purpose of obtaining key properties. Useful results can be achieved by using a reduction method of Markov renewal process. Liu and Hammond [l] use state reduction method for mean first-passage time analysis in a Markov chain. Courtois and Semal [2] have proposed a bounded aggregation to compute lower and upper bounds on the conditional equilibrium probabilities in a subset of the states of a large Markov process. A reduction method of Markov renewal process (MRP) [3] and its application [4] have also been proposed. In this paper, a key work in the modelling process is the choice of the level of observations to describe the structure and behaviour of the given system. The merit of the proposed method is in the use of an aggregated model with different levels of what could be observed. At level 1, we construct an MRP and reduce it. The underlying structure of the reduced MRP is used to obtain mass functions of an MRP constructed at level 2, where we deal with the union of the number of packages and observed events, and decide on an appropriate state space of the model. 08957177/03/t - see front doi: 10.1016/S0895-7177(03)00348-O
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1348
NAKADA
AND T. YONEYAMA
The reduced MRP at level 2 is more powerful for deriving indices of performance. deal with the overflow problem.
2. MODELLING
A TESTING
AND TAPING
At level 3 we
PROCESS
It is important in the design and process management of production systems such as packaging of large scale integrated circuit (LSI), to provide cost-effective tools for system evaluation. Packaging, or assembling die, significantly affects the overall cost, performance, and reliability of the packaged die. Overall integrated automation of assembly process; not only die-bonding, wire-bonding, and molding steps but also testing and taping process, is required. In this section, we set up an abstract model to represent behaviors of a testing and taping unit of LSI packages. As shown in Figure 1, the testing unit considered here normally tests whether every package satisfies its specification or not. When a package satisfies given specifications by the test, then the result of the test is “good”-the package is transferred from testing unit to taping unit, where the package is packed on tape. While the result of the test is “no-good”, the package is removed from the testing unit by the action of removing the unit. A retest, after the primary test, is made when the primary test ends in failure. The retest is made only once.
Overflow
Figure
1. Schematic
diagram
of testing
and
taping
unit.
Whenever a package supplied from the package handler finds the testing unit busy, it is stored in the buffer. Let N denote the buffer capacity. A control device in the testing unit that finds all N spaces in the buffer full refuses to store the package. The package must go away. The package is then said to have overflowed. Packages arrive according to a Poisson process with rate X. Only one package may be tested at a time. Packages in the buffer are tested in the order of arrival, so that the testing discipline is FIFO. Packages are judged as “no-good” with probability p and the primary test ends in failure with probability CL. Normal service times are independently and identically distributed random variables according to a general distribution function Bs(t) with mean E(s(O)), and so are post-time Is (I = 1,2) to a general distribution function Bl(t) with mean E(s(‘)), respectively. The operation discipline is as follows: whenever the buffer is empty, the production unit stops and warns the operator. He supplies packages to the package handler and stores “a” pieces of packages on the buffer. The operator starts the production units again. Suppose that these elapsed times to restart are independently and identically distributed random variables according to a general distribution function G(t) with mean g.
A Stochastic Model
1349
3. CONSTRUCTION OF MARKOV RENEWAL PROCESSES AT APPROPRIATE LEVELS OF OBSERVATIONS The merit of proposed method is in the use of multiple appropriate models with different levels of what could be observed. A key work in the modelling process to a given system is the choice of the level of observation to describe the structure and behaviours of the given system and to construct a stochastic model. At level 1, we observe events about which we have postulated to the transaction of packages in the production units and discuss the structure of a Markov renewal process associated with the obtained sample description space. Although the process keeps moving all the states on the state space, we may wish to study a certain subset of states and observe the process only when it is in those states on the subset for the purpose of obtaining key properties. This is used to construct an appropriate model at level 2. We reduce the MRP at level 1. It will be used as the underlying structure of a stochastic process at level 2. It would not be enough to specify the number of packages in the buffer for obtaining ordinarily important indices of performance such as throughput, busy period of the production units, the frequency that the testing unit starts a test to a new package, and so on. The underlying structure of the reduced MRP at level 1 and the results of observations at level 2 will be used to construct an appropriate stochastic model at level 2. We deal with the union of these events and the number of packages as events at level 2. Furthermore, a reduced MRP at level 2 will be used to analyze indices due to the slow transitions on the process. At level 3, of particular interest is the property of overflows of packages arrived at the testing unit. For the analysis of overflows we consider occurrences of overflows during each transaction of packages by the production units as an event in addition to the previous one at level 2. 3.1. Markov
Renewal
Process
(MRP)
at Level
1
We must first decide upon appropriate sample description space at level 1. Assume that there is always at least one package in the buffer. When the process is processing a package, the elapsed time depends only on the time interval to complete transaction of the package and does not depend on the number of packages in the buffer. We observe events described the following descriptions. El(&):
Testing unit starts a test to a package at the head of buffer immediately after that good package was transferred to taping unit (immediately after that no-good package was removed). Es: Testing unit starts to retest the package immediately after primary test is failure. E4: Removing unit starts to remove the no-good package.
A set of all possible events, EL = {Ei 1 i = 1,2,3,4}, is considered as a sample description space. Let i be the epoch at which an event Ei occurs. The process is said to be in state i between the successive epochs i and j. State space Sr, of the process at level 1 is given by SL = {1,2,3,4}. Let Qij(t) be the probability that the process makes a transition into state j once it enters state i. For i, j E SL, Qij(t) is given by (1 - p)(l
Qij(t) =
-p)Bo(t),
for i = 1, j = 1; i = 2, j = 1;
PBo(t),
for i = 1, j = 3; i = 2, j = 3;
(1 - P)PBo(th
for i = 1, j = 4; i = 2, j = 4;
(1 -
for i = 3, j = 1;
PPz(%
p&(t),
for i = 3, j = 4;
Bl @I,
for i = 4, j = 2;
. 0,
for otherwise.
(1)
1350
K. NAKADA AND T. YONEYAMA
Let qij(s) denote the corresponding Laplace-Stieltjes transform (LST) of Qij(t). [Qij (t>l (4 J’ E S L and t E R+) be the semi-Markov kernel over SL. 3.2. Reduced
MRP
at Level
Also let Q =
1
Figure 2a shows the state transition diagram of this fundamental model. Although the process keeps moving through all states on the state space SL, we now meet such a situation that we may wish to study a subset of states Sr c SL, and observe the process only when it is in those states.
(a) Original
MRP at level 1.
Figure 2. Transition
(b) Reduced MRP at level 1
diagrams of original
MRP and reduced MRP at level 1.
The notion of conditional first-passage time is useful to these kinds of situations. Suppose $41= {1,2}, since we pay our attention to occurrences of events El and E2, thus epochs 1 and 2. Let (l)&j(t), where i, j E Sr, be the distribution of conditional first-passage time from state i to state j for the first time, given that the process does not visit any state on Sr during (O,t]. Let (‘)hij(s) be the corresponding Laplace-Stieltjes transform (LST) of (l)Hij(t). For fixed j, we have the following simultaneous equations [3]:
(1)b(4
= %j(S) -c
c qim(S)%mj(s). mESL-S~
(2)
Similarly, we have (2hrLj(s)
= q??&) +
c
Qme(s)(2hj(s).
(3)
LESL-Sl
Noting q12(s) = q22(s) = qdl(s) = 0, (‘)hij(s)
can be given by
(l)Hds)‘= [:::g;:I+[q23(s) ,,,(sJ [0 q13(s)
Q14(S)
Q31(S)
434(s)442(s) 442(s)
1y
(4)
where i,j E 11s = {1,2}. Since (l)Htij,(0) are considered as a Markov chain, ll)Hlij,(s) can be considered as the LST of a semi-Markov kernel over Si. State transition diagram of the reduced MRP over Sr is shown in Figure 2b. 3.3. MRP
at Level
2
At level 2, the state of the given system must include more information than merely the number of packages in the buffer. We pay our attention to observe event Ei (i E I = { -2, -1, 0, 1,2}) and the number of packages T (E R = {O,l, . . . , N}) when an event Ei occurs, where all possible Ei are described by the following statements.
A Stochastic Model
&(Ez): E- 1 (E-2): &:
1351
Testing unit starts test to a package immediately after that good package was transferred to taping unit (immediately after that no-good package was removed). Testing unit enters an idle period immediately after that good package was transferred to taping unit (immediately after that no-good package was removed). Testing unit starts test followed by an idle period to the package at the head of the buffer.
Imagine an event described by the statement that “there are r packages in the system when an event Ei occurs”. We deal with the union of event Ei and the number of packages when E, occurs as an event at level 2. The event described by the statement can be specified by 2-tuple (Ei, T). Let E be the set of all possible events (E,, r). Let (i,~) be the epoch at which an event (Ei,r) E E occurs. The process is said to be in state (i,r) between the successive possible epochs (i,r) and (j,~‘). State space Su of the process corresponding to model at level 2 is given by SU = &I SU,, where Sv-, = { (-1, O)}? SU-, = {(-2,0)}, &J, = {(O,a)} and for fixed m E 112, SU_ = {(m, r) \ T E R}, respectively. Let the mass function Qti,r),(j,rJ)(t) be the probability that after making transition into state (i, r) at time t = 0, the process makes a transition into state (j, r’) in amount of time less than or equal to t. The elapsed time to complete a transaction to a package is independent of arrivals of packages during (O,t]. Also the event occurring at the next epoch depends only on the outcome, that is, good package or no-good package, from the production units at the end of the transaction. (l)&(t), i,j E 112, derived in equation (4), can be used as the conditional probability that the sojourn time in state (i,r) is less than or equal to t, irrespective of T and r’, given that the process makes a transition into state (j, r’). Since only one package in the buffer may be entered to the testing unit at a time, the number of packages changes from T to T’ = T - 1 + k when the system has k arrivals of packages during (0, t] (i.e., t ime interval between successive epochs (i, r) and (j, T’)). The probability of k arrivals X,(t) during (0, t] is given by
(5) Let Q~~,~J(~,,J)(s)be the corresponding LST of Q(,,,,cj,,o(t). For the notational define 1,~= {0}, and R, = {a}, respectively. We have q(z,r)(j,r/) (s) as follows.
convenience
(1) For i, j E 112, and T E R - IO; and for i E 10, j E 112, and T E R,;
I w=o , Q(i,r)(j,r+) = (lhj(S + 4,
O
(2) For i, j E 112, and T E R; and for i E IO, j E 112, and T E R,; N-r+1 q(W,(j,N)b)
=
%j(4
-
c
Q(i,r)(j,r-m+l)(S),
m=O
k>N-r+l.
(8)
(3) For i E 112, j E I - IOU 112; Q(i,O)(j,O)(s) = %,j,
(s + A)
(9)
(4) For i E I - 112 - IO; Q(i,O)(O,&) = g(s).
(10)
4(i,r),(j,r~) (s) = 0.
(11)
(5) For otherwise
1352
K. NAKADA AND T. YONEYAMA
We shall first derive the first-passage time distribution Hc,,r,cj,,lI(t). Let /I~~,,)(~,,,)(s) and q(i,,-)(j,+)(s) be the Laplace-Stieltjes transforms of Hc~,~)(~,~,J(~)and Q (2,r)(j,r~)(t), respectively. We have h(i,r)(j,?+)
= 4J(v)(W)(4
+ &
(1 - &(j,T’,) Q(i,r)y(S)h,(j,r’)(S).
(12)
Of particular interest is the mean first-passage time Lca,,)(j,,J). A set of simultaneous for L~~,~)(j,,r) is given by’ J%?-)(A~~)= c dwr 7ESU
equations
+ c (1 - ~-d.wJ Qr(w)w,(j,rJ)~ YS%l
(13)
where Lc~,,), = [-ah(ii;~Y(s)]s=c and @i,r)r = [- aq(i$~(s)]a,O. Let M(i,,.)(j,+)(t) be the expected number of visits to state (j,r’) during the interval (0, t], starting from the epoch at which the process makes a transition into state (i,r) at time t = 0, where the initial state (i, r) is not counted for the number of visits to state (j, P’) if (i, r) = (j, r’). Let rr~(+)(j,~j)(s) be the LST of M(,,t.)(j,T,)(t). We have
(14) Furthermore, M(j,rl), state, is given by
the expected number of visits to state (j, T’) per unit of time in steady
(15) 3.4. Reduction
of the MRP
at Level 2
Consider a collection {S UE, &NE} SUE = {(h>,
W,O>,
of nonempty
(-2, O)),
subsets of SU, where
%.I = SUE u &WE,
and
SUE n SrrNE =
0.
Let us pay our attention to states on SUE. An argument similar to that in the previous section can be made for obtaining a reduced MRP from the original one at level 2. Let (‘)h(i,,~(j,,l)(s), where (i, r), (j, r’) E SUE, be the LSTs of conditional first-passage distributions, thus mass functions, in the reduced MRP. The state transition diagram of the original process shown in Figure 3a can be reduced as shown in Figure 3b. We have the following LST of semi-Markov kernel over SUE represented in matrix form:
(a) Original
MRP at level 2. Figure 3. State transition
(b) Fleduced MRP at level 2. diagram at level 2.
A Stochastic
1353
Model
%u&) = [(l)fqi,rj(j,r~~(s)]
(16)
(‘)hSUE(0) is considered as the transition matrix of a Markov chain with state space SUE. Let x(~,~J) be the state probability that the process is in state (j,r’) ((j,r’) E SUE) in steady state. The LST of the first-passage time distribution from subset SA (C SUE) to subset SB (c SUE), kAsB(s), is given by
c c n(k,r,,)h(k,rll)(e,rl)(s) (e,r’)ESB (k,r”)ESA
hS,Se(s)
=
(17) c
n(v9
(fW’)ESA
3.5. Markov
Renewal
Process
at Level 3
For the analysis of overflows we pay attention to the following events in addition to the previous one at level 2. That is, E,ci,r) (Eb(i,r)): the first overflow of package occurs before the process ma,kes a transition into state (1, N) ((2, N)), g iven that the process made a transition into state (i,r) at t = 0, where i E IO, r E R, and i E 112, T E R. All of the events added to those in the set E at level 2 can be represented by Er = {E~c~,~) ( l?(i,r) = a(i,r), b(i,r); i E I is, r E R; i E IO, r E {a, a + 1,. . . , N}}. All of the events that we pay our attention to at level 3 are represented by E U Er, where E n Er = 0. Suppose that a(i, r) (b(i, T-)) is the epoch at which the event E,(i, T) (&(i, r)) occurs. The process is said to be in state a(i, r) (b(i, r)) b et ween the successive epochs a(i, r) (b(i, T)) and (1, N) ((2, N)). Note that epochs a(i, r) and b(i, r) are not regeneration points. There are two possible ways of transitions from state (i, T) to state (j, N). One is the direct transition from state (i, T) to state (j, N) and the other is the transition from state (i, r) to state (j, N) via state a(i,r) (b(i,r)) if j = 1 (j = 2). Define nq(i,r)(j,N)(s) and qF$ki,Nj(s) as the LSTs of mass functions of these transitions, respectively. For i, j E 112, r E R and r’ = N; and for i = 0, j E 112, and r E R,; we have
1w=ok=N-r+l,(18)
n4(i,r),(j,N)(4 = -$S $ (‘+-qi,r)(j,r~){s + (1- wP1 7 N-r+1 q;r:;‘(j,N)
(‘)
=
(1)hd4
-
c k=O
Q(i,r)(j,r-k+l)bL
(19)
and q(i,r)(j,N)(S)
=
%(i,r)(j,N)(s)
+ q;f$&J,)(s)~
(20)
Also Qti,r)r(i,r)(t), the mass function that after making a transition into state (i, r) at t = 0, the process next makes a transition into state I’(i,r) in an amount of time less than or equal to t, is given by
Q(i,r)r(i,r)(t) =
’ (%!&j(~) - (%&j(T)] xx,(T) d,-. S[0
(21)
Let q(i,r)r(i,r)(s) be the LST of Q (z,r)r(i,r)(t). Define qr(+)(s) as the LST of the probability that after making a transition into state I’(i,r), the process does not leave the state up to time t. It follows that qr(i,T) cs) = q(i,i-)r(i,r) cs) - $$)j,N)
cs)+
(22)
Note that it does not need for the analysis of H(i,r)(j,N)(t) to distinguish these two ways of transitions from state (i, r) to state (j, N). Consider Mci,r)r(i,,)(t). Since epoch I’(i, r) is not a regeneration point, we need to consider the way of transition that after the direct visit to state I’(i,r), the process makes a transition into
1354
K.
NAKADA
AND T. YONEYAMA
state (j, N) up to a certain time ‘u. and then the process does not leave from state (j, N). fixed j, we have m(i,r)r(i,r) (s) = Furthermore,
Q(z,r)r(i,r) Cs) { 1 - k?-)(i,T) (5) > .
For
(23)
Mr(i,t-), i.e., the expected number of visits to state I’(& r) per unit of time, is given
by MIT(~,~) = !io ~m(i,~)r(i,~)(s) = L(i~~)(i,r)q(i,r)r(i,r)(0).
4. INDICES
FOR EVALUATING
(24)
PERFORMANCE
Of particular interest are the busy period and idle period of the production system. We deal with the busy period as the time spent by the system to execute successively repeated transactions to packages followed by an idle period. The busy period terminates when the buffer becomes empty. Thus, it begins when the process makes a transition into state ((0, u)} and ends when the process goes to any of the states on subset ((-1, 0), (-2,O)) for the first time, given that the process passes through none of the states on subset SUE during the transitions. Let Hbusy be the mean busy period. In this case, subsets SA and Sg in equation (17) can be considered as SA = ((0, u)} and Sg = {(-l,O), (-2,0)}, respectively. We have
fbu.y = L(o,a)(o,a) - 9.
(25)
A similar argument can be made for the idle period. It is equivalent to the first-passage time from state(~,O)onsubset{(-1,0),(-2,0)}tostate(O,a)onsubset{(O,a)}. Subsets{(-l,O),(-2,O)) and ((0, u)} can be considered as SA and SB in equation (17), respectively. The LST of the distribution function of idle period, hi&(d), follows that
c hi&(s) = hS~,(O,a)(s)
=
~(/c,0)~(~,0)(0,&)
(k’o)ESA
(26)
c ~‘(i?O) (i,O)ESA
(see equation
(17)). S’mce h~-r,o)(c,,)(s)
= ht-z,o)(o,a)(s)
HidIe= [-““i;$‘]szo
= g(s), the mean idle time is given by
=9*
(27)
Throughput is defined by the number of good packages transferred from the testing unit to taping unit per unit of time. When the process visits state (1, r) and (-1, 0), a good package was transferred immediately before making the transition into state (1, r) and (-l,O). Thus, equivalently we can consider the process has the transfer of a good package as an output when the process makes a transition into state (1, r) and (- 1,O). The average expected number of visits to state (1,~) is given by Mci,,) = lims+,-, sm(i,,)(r,,)(s) = L$.)(,,,), where (i,~) E SU-, U Su,. Throughput is irrespective of the number of packages in the buffer. We have M gout =
as the throughput of the production units. A similar argument no-good packages from the removing unit. We have M ngout =
W!
c “s”, L&(iA (i,r)ES”-,
c (i,r)ESu-,USu2
can then be made for removing
L-1 (%r)(ivr)’
(29)
A Stochastic
1355
Model
Define Mpin as the frequency that a testing unit starts its test to a new package. When the process makes a transition into state (0, a) and (i, T), where (i, T-) E S,~J~U Suz, a test will be started. Thus, we have
zis easily verified that L&Jt,,aJ ngout
= L;_‘,,OJC-,,,j + L;_‘,,,~~-,,,~.
It is clear that Mpin = Mgout +
.
Define MO,,, as the expected number of visits to overflowing periods per unit of time in steady Let SuB = {(i,~) 1 i,j E 112, T E R; i = 0, T E R,}. state. M,,,,, is irrespective of (i,~), r(i,r). M over is given by M over =
1 {Ma(i,r) (0)ESu,
+ Mb(i,r,}
=
c L;,:,(i,r) (i,r)ESu,
{Cf(i,r)a(i,r)(“) + q(WGdO)~
c31)
5. CONCLUSION An automated testing and taping process on an LSI production system has been modeled, and indices for evaluating performance such as mean busy period, mean idle period, throughput, and the expected number of visits to overflow periods have successfully been obtained. We employed three different levels of observations to describe the structure and behaviors of the given system. We showed that the merit of proposed method is in the use of multiple appropriate models with different levels of what could be observed and in the use of the reduction of constructed MRP at each level if necessary. The underlying structure of the reduced MRP at level 1 and the results of observations at level 2 were used to construct an appropriate stochastic model at level 2. Also, the reduced MRP at level 2 was used to analyse indices due to the slow transitions on the process. The method proposed in this paper can be used as a cost-effective tool for system evaluation.
REFERENCES 1. S. Liu and J.L. Hammond, Jr., A method for modeling and analysis of the reframing performance synchronous time division multiplex hierarchies, IEEE Duns. Commun., COM-28 8, 1219-1227, 2. P. Courtois and P. Semal, Bounds on conditional steady-state distributions in large Markovian models, In Int. Seminar on Teletmfic Analysis and Computer Performance Evaluation, The (1986). 3. K. Nakada, T. Yoneyama and T. Iseki, An algorithm to compute the moment of the first passage Markov renewal process finite states, ‘i?mns. ZEICE 71, 664-678, (1988). 4. K. Nakada and T. Yoneyama, An algorithm to absorb inaccessible markings in reachability tree Petri net, Proc. of the Fnd Australia-Japan Workshop, 423-432, (1996).
of multilevel (1980). and queuing Netherlands, time
in the
of stochastic