August, 1982
LIDS-P- 1259
DIFFUSION APPROXIMATIONS FOR THREE-STAGE TRANSFER LINES
WITH UNRELIABLE MACHINES AND FINITE BUFFERS
by
David A. Castanon*
Bernard C. Levy
Stanley B. Gershwin
Laboratory for Information and Decision Systems
Massachusetts Institute of Technology
Cambridge, Mass.
Presented at the 21st IEEE Conference on Decision and Control,
Orlando, Florida
December, 1982
DIFFUSION APPROXIMATIONS FOR THREE-STAGE TRANSFER LINES WITH UNRELIABLE MACHINES AND FINITE BUFFERS
Dr. David A. Castanon Prof. Bernard C. Levy Dr. Stanley B. Gershwin ALPHATECH, Inc. Laboratory for Information and Decision Systems Burlington, Mass. Massachusetts Institute of Technology
Cambridge, Mass.
SUMMARY - ai_lbi_l(x, a) if xi_ 0 and (2.c)
In this paper, we derive an approximate model for the
flow of.parts in a three-stage transfer line with unre- ibi i-li-liable machines and finite storage elements. The
analy-sis of exact models of these systems was described in Then, Gershwin and Schick [1], where the basic notation for
this paper is established. In [1], it was noted that dxi -(aibitx a) - ai+lbil(x, a))dt (3) exact analysis of three-stage or higher problems is very
difficult, if not altogether impossible. In this paper
we develop an approximation methodology which is used stochastic differential equation to study the three-stage transfer line model. This
methodology can be extended to n-stage transfer lines
without introducing additional levels of complexity. da i + i(-{bi(x, -a)dR a) )dFi (4)
These extensions are discussed in [2].
Figure 1 describes the basic three-stage transfer where Ri, Fi are independent counting processes with
line. Machine i processes parts from the preceding exponential jump rates ri, fi respectively. Note that the last term on the right side of (4) implies that storage element until no parts are left in it. The out- machines cannot fail unless they are processing some
machines cannot fail unless they are processing some put of each machine goes into the subsequent buffer, flow. The presence of this coupling term implies that It is assumed that the first buffer is an infinite sinkthe a process is not a Markov process; rather, the full source, and the last buffer an infinite sink. In addi- (x
tion, an exponential failure and repair process is used ) process is Marko described by equations (3) to model the operation of the machines, with
compensa-tion for eliminating failures when the machine is not Consider now two scaled processes, yi and yl, defined in use. It is also assumed that flow is conserved, and as follows:
that parts are infinitesimal, and that machine i will process at maximum rate Ei whenever possible
(t) - x 1 (5.a)
The stochastic equations which describe the flow of i t N Xi (Nt)(5.a) parts through buffers I and 2 can be described as
follows: Let xi denote the level of flow into buffer i. 2 (t(N't) (5.b)
Let ai be a binary variable which is 0 when machine i Yi N i.b) is off, and 1 when machine i is on. Let N denote the
buffer capacities. The accumulation of flow in buffer The main results of the paper can be stated as follows:
i can be described by: L
Let
ir bi rl+l bi+l
-d (6)
dxi - O if xi N and (a +lb+l- ibi) 0 i ri +f + + f
or and let T denote the time of first exit of the process
(yI(t), y'(t)) from the unit square D - (0, 1) x (0, 1).
x1 2
xi . O and (a i+ -i+lab abi ) > 0 (1.a) Define the process z°(t) by
dxi i - i+b ) dt otherwise (I.b) zit)' (0) + dit
Equation 1 is a random evolution for the level of Theorem 1: Unbalanced line approximation
flow in each buffer, because the on-off state of each Assume di is 0(1) for some i. Then, for o<t<T, the pro-machine is a random process. Assuming that machines cess 1I(.) converges uniformly almost surely as No to can slow up their production rate 6i if the storages are the process z°(t). That is, for any e>o,
either empty or full, the production rate of machine i can be described by the function
lim sup Prof{{z0 (t) - Z(t)C)I>e} -0 O
N-' o<t<T
bi(x. a) -bi if xi < N (2.a)
Furthermore, define the process v(t) by
x 1 > O v () 1 [N (y(t) - zT(t))
ai+lbi+l(x, a) if xi -N and (2.b)
The process v(t) converges weakly to a zero-mean Wiener ai+lbi+l < Babi process w(,) with covariance.
T average lost production and other ergodic properties can
E(w(t)w (a)) - E min(t, s) be computed using this approximation, with a resulting
simplification of solving only one second order elliptic
where PDE, rather than 8 first order by perbolic PDE's. The
results presented here are part of a larger paper [ 2
1,
2(-bl2flr) 2b 2f rb 2 which contains the proofs of the main theorems.
L. L + -REFERENCES_
(f +r) 3 (f + r ) (f + r )3
1 1
2 2 2 a 1. Gershwin, S.B. and Schick, I., "Modeling and Analysis
of Two- and Three-Stage Transfer Lines with Unre-liable Machines and Finite Buffers," LIDS-R-979, Lab.
2b 2f r 2b 2f r 2b 2f r for Information and Decision Systems, MIT, Cambridge,
2 2 2 2 2 2 3 3 MA 1980.
+
2( + r2)3 (f2
r)3
2+ (f 3 + r)3 2. Castanon, D.A., Levy, B.C., and Gershwin, S.B., "Diffusion Approximations of Transfer Lines with Un-Theorem 2: Assume that fl,f2 are both of order (1/N) reliable Machines and Finite Storage Elements," ase N-- Then, for any K-~, the procests Yy_2(t) converges .LIDS R-1157, Lab. for Information and Decision Systeas,weakly on C{ [o, Kl; D} to a diffusion process on I with March 1982
instantaneous reflection on the boundary DD. The in- 3. Buzacott, J.A. and Hanifin, L.E., "Models of Auto-finitesimal generator of this process in D is given by matic Transfer Lines with Inventory Banks - Review
L, where and Comparison," AIIE Trans., Vol. 10, No. 2, 1978.
4. Harrison, J.M., "The Diffusion Approximation for
aLg()
= N\ 1 2 2 ai Tandem Queues in Heavy Traffic," Advances in Appl. \O1.) ax P + f ax 2 i CC ij a 2xj ° Probability, Vol. 10, No. 4, 1978.' 12
5. Watanabe, S., "On Stochastic Differential Equations for Multi-Dimensional Diffusion Processes with
with dense domain Boundary Conditions, I & II," J. Math. Kyoto
Uni-versity, Vol. II, 1971. Dom(L) - {g e C2(D) CCD) satisfying a-d}
where '
a)
a g(°X
)3
a gfachie Buff.er Machine MachineSufflra) -~- g(O, x) g(O, x
)
Sour;. 1x a
~~~x
2 1 2 2 3 Sinkb) (x 1) (x 1) Fr 1: Three StaSe Tranfer Ltne
x1 2
c) a (1, x ) =0
ax 2
d) ) (x, 0) O
ax
The directions of reflection of the limit process in
Theorem 2 are shown in Figure 2. Note that the limit Figure 2. Oirection- of Reflctcion for Diffuion
process has a discontinuous reflection field at (0, 0), (0, 1) and (1, 1), as well as existing in a domain with non-differentiable boundaries. Hence, the classical theory of reflected Markov processes, as developed in
[5], cannot be used to establish this theorem. Rather, the theorem is established by developing a strong characterization of the boundary process associated with the limiting diffusion process.
The properties of the limiting diffusion process in Theorem 2 can be used to approximate the properties of the correspondling exact proctas definedt by (2) - (5). In particular, the ergodic distribution of the diffusion process serves as an approximation to the ergodic dis-tribution of equations (2) - (5). Hence, long-term
