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LIDS -P -1 30 9
December, 1983
Revised January, 1985, November, 1985, and July, 1986
An Efficient Decomposition Method
for the Approximate Evaluation
of Tandem Queues with Finite Storage Space
and Blocking
by
Stanley B. Gershwin
35-433
Laboratory for Information and Decision Systems
Massachusetts Institute of Technology
77 Massachusetts Avenue
Cambridge, Massachusetts 02139
Keywords:
343
inventory
levels
and
throughput
in
lines, 570 Markov chain model of transfer lines, 721
transfer
reliability
and storage, 683 decomposition approximation of queuing networks
To appear, Operations Research
GERSHWIN, Tandem Queue Decomposition
2
ABSTRACT
This
paper
presents
an efficient
method
performance
measures
for a class
of tandem queuing
finite
buffers
phenomena.
their
in which
These
state
large
systems
spaces
The
is based
on system
very
are
with
to
evaluation
of
systems with
are
evaluate
important
because
of
they may not be decomposed
decomposition
and
the
starvation
difficult
characteristics
exact
and
and because
approximate
exactly.
Comparisons
blocking
for
such
simulation
described
approach
as
conservation
results
of
that
indicate
here
flow.
it
is
accurate.
ACKNOWLEDGMENTS
This research
ring
has been
Laboratory
for the support
Mr. Charles
under
comments
sor
Loren
contract DAAK1 l-82-K-0018.
and encouragement
Shoemaker.
and
supported by the U. S. Army Human Enginee-
of Ms.
K. Platzman,
I
of Dr. Benjamin E.
am also
grateful
Ellen L. Hahne,
Dr. Daniel
I
for the
Professor
Heyman,
Rajan
and a
am
grateful
Cummings
and
suggestions
Suri, Profes-
reviewer.
3
GERSHWIN, Tandem Queue Decomposition
This paper presents
culation
of
tandem
throughput
queuing
difficult
to
a
The
[k-l)-buffer
parameters
of
the
ters.
Numerical
method
determines
The
of
k
flows
and
so
In
... ,
outside
forth
until
general, in
state
single
systems
are
to
average
M2,
calculate
indicate
buffer
in Figure
[M1,
and
... ,
approxiThe
by
rela-
original
the paramethe
that
levels
Mk)
finite
the
system
to
M1,
reaches
Mk,,
after
type,
this
their
accurately.
I consists of a series
of
of
the
of
are
systems.
of
B_,1
systems
systems
determined
experience
and
it
a class
spaces
buffer
developed
simulation
machines
B 2,
of
is based an a model which
is
algorithm
thoughput
from
large
cal-
(i.e., the
Such
through -the_ buffers
flows
or
(B 1,
buffers
M2,
their
tandem queuing system
servers
rial
of
method
and
levels]
finite buffers.
single-buffer
A simple
the analysis
buffer
system by k-l
tionships ...among_.the
system.
with
because
indecomposability.
mates
and average
systems
treat
for
a method
separated
by
capacity.
then
Mate-
to
B1,
which
it
the machines
then
to
leaves.
are
as-
sumed to spend a random amount of time processing
each item.
In
this
failures
and repairs
of
buffer
B1 _ tends
to
paper,
the
randomness
machines.
When
accumulate
material
this
condition
become
empty.
prevented
vented
from
from
While
is
machine
and
is
buffer
B1
Bi l
that
working
to
Mi
persists,
In
due
case,
or
Mi.,
the
down,
tends
may
to
become
machine
is
lose
full
Mil
starved
material.
or
Bi
is
blocked
and
also
If
may
and
pre-
working.
a
machine
is
working
(i. e.,
operational
and
4
GERSHWIN, Tandem Queue Decomposition
between
failures
machine
has
ri
is
time, however,
not in working
Each machine
space.
state
is
ni
Ni,
where
its
capacity.
Bi
Buffer
repair.
under
the
As
a consequence,
in
and
or
Ni
1,
....
is
representation
state space
a
buffers has
Bi
chain
Markov
the
n,=0,
states:
material
of
amount
the
of
operational
states:
Ni+ 1
in
be
can
size
great
the
of
can be in two
line with k-l
of a k-machine
time.
because
difficult
is
problem
The
clock
in
1/r,. -- This '----.is-- measured
therefore
rep-ai-r-..(MTTR)
a
to
time
mean
Its
unit.
time
a
during
repaired
being
of
After
probability
has
it
and
repair
under
is
it
failed,
1/p i.
thus
is
time
working
in
(MTBF]
is
time
mean
Its
failing.
of
Pi
probability
has
it
working,
Mi
unit when machine
a time
During
unit.
time
the
as
taken
and is
for all machines
is the same
This time
a part.
to process
required
is
time
of
amount
a fixed
blocked),
starved nor
neither
of
cardi-
nality
k2'
ll[Ni+]J.
i=O
A 20-machine
line with
ple, has over
6.41
each
19 buffers
of capacity
10,
for exam-
x 10 2 5 states.
Decomposition
We
approximate
k-l
set
of
The
buffer
machine
line.
the
two-machine
in
L[i]
Both
has
line
single k-machine
lines
the
L(i),
same
the upstream
i
=
capacity
(Mdfi))
of Figure
1, ... ,
as
k-I
1 by a
[Figure
buffer Bi of
and downstream
2).
the
k-
([Md(i)l ma-
5
GERSHWIN, Tandem Queue Decomposition
ing
with
distributed
sumption
unit
time
to
the
of
Machine
and
are
Md(i)
four
per
four
parameters.
equations
is
line
in
L.
In
Bi
close
is
the
Finally,
approxi-
L(i)
find
to
order
among
in a
LUi)
and
parameters
line.
parameters
of
of
relationships
we use
part
determination
buffer
in buffer
the
models
the
in
L.
in
re-
of
interrupted
was
to
close
is
probability
event
corresponding
models
Mu(i)
it
which
proba-
The
of) the buffer in line
during
the
transfer
a
rd(i,
ial
The
level
in L): ru(i), pu(i),
of 4(k-1)
full.
(and out
values,
parameter
measures
or
of material
material
full
empty
a period
Bi.
or
empty
after
matches
L.
being
into
closely
line
L(i)
of flow
Section
in
line
of
of
Bi
of buffer
that
in
of buffer Bi in
and out
into
flow
of
rate
The
developed
buffer
of
[downstream)
upstream
geometrically
be
[[Md[i))
Mu(i)
of machine
to
pd([i],
respectively.
algorithm
(by the
chosen
in L being
amount
average
such
rd(i),
probability
the
mates
and
the buffer
of Bi
that
ru(i)
approximates
of
bility
parameters
the
is,
L[i)
line
assumed
line
the
That
are
the performance
2] so that
of
times
are
four, parameters
that
repair
Their
respectively.
and
pu(i)
parameters
by
described
distribution,
time
a geometric work-
assumed to have
lines are
in each of the
chine
the
part
of
line
line
from
line
(i. e.,
this
in
per
two-machine
line,
end, a system of 4(k-1)
Section
2.
B1
of
upstream
Bi.
There
per
buffer
The decomposition method requires
parameters
derived
the
downstream
two-machine
pd(i).
To
of
or a
the
total
polynom-
GERSHWIN, Tandem Queue Decomposition
; Survey
Literature
The
Gershwin-Schick
line
model
this
type
of Buzacott
appear
Vanderhenst
refer
to
et
al.
Hanifin,
[1975),
disagree.
program,
lems.
for
sumptions,
the
(1983)
derive
this
system.
is
systems,
not
(1983)
Law
last
in
but
and
extendable
(1981)
authors
of exporepairs,
paper
solution
it
to
for
is
difficult
larger
prob-
for
a related
a method
does not
provides
he makes
the
until
However,
of
and
simulation
exact
describe
that no more
(1981),
assumption
a
an
transfer
for models
paper,
the
their method
Ohmi
the
results
failures
Taraman,
and
lines.
such as
In
justifies
model, but again
larger
(1979),
and
-
and Proth
for longer
al.
between
of
ill-behaved,
three-machine
thod
that
times
versions
Coillard
useful
data
is based on
Simulation
[1981).
and Schick
three-machine
model
et
Buzacott
although
to
Ho
distributed
Gershwin
(1983)
(1967).
in
historical
nentially
6
appear
to
an approximate
certain
restrictive
be
me-
as-
than one machine is down at any
time.
The
models
concept
was
Takahashi
and Diehl
merical
of
approximate
discussed
et
al.
(1983),
by Hunt
[1980),
upstream
end) to
the
served by Muth (1979)
exploited.
In
from
other.
(1956),
Altiok
and others.
method sweeps
decomposition
In
one
the
end of
tandem
queuing
and Boling
(1966),
Jafari
papers
(1982),
surveyed,
the line
The symmetry
and Ammar
addition,
Hillier
[1982),
all
of
the
(generally
of the
line
(as
Suri
nu-
the
ob-
(1980)) has not previously been
interrupted
nature
of
both
the
7
GERSHWIN, Tandem Queue Decomposition
in the
processes
and service
arrival
has not been
line
decomposed
considered.
times
with random processing
to
Gershwin
[1985).
assembly
and
It
of
tions
also
((5)
and
1 describes
two
Section
(10)).
Paper
characteristics
after
2 also
in which
a period
contains
idle
of flow
material
was
results
in Sections
appear
the performance
clusions
The
of
and new
Appendix
the
research
contains
two-machine
formulas
line.
of large
directions
for
are
the
production
lines
of
flow
2
Section
time.
into and out of a
flowing.
Section
outline of
an
3 and 4.
Sec-
method, while Section 4
tion 3 is focussed on the behavior of the
is concerned with
not
decomposition method and an
the
Numerical
algorithm.
of
conservation
and
of resumption
the probability
estimates
equa-
of two key
proofs
includes
in Gershwin
appears
technique
relationship between flow rate
and the
involving
[1986).
Gershwin
used in the decomposition method:
that are
buffer
paper
this
in
speeds
networks
to
generalized
of the decomposition
That publication
of
Outline
in
disassembly
A summary
(1984).
been
has
and
(1985),
processing
different
lines
transfer
to
and Gershwin
in Choong
that have
with machines
lines
extended
paper has been
of this
method
The
transfer
discussed
probability
lines.
in
Con-
Section
5.
distribution
8
GERSHWIN, Tandem Queue Decomposition
1. TRANSFER LINE CHARACTERISTICS
Assumptions
Model
A detailed
present
1.
method
the
ai = 1,
fixed
same,
time
is
of
time
unit.
time
2.
A machine
is
starved
It
is
if
its
blocked
buffer
is
the
of
That length
is
buffer
empty.
The
full.
first
is never blocked.
last machine
and the
called
The amount of material in a buffer at any time is n,
O < n < N.
a time
during
upstream
A buffer
machine
is working.
becoming
machine
i
operational
prob [ aoit+l)=l
One piece
is
under
inserted
[i. e.,
working
is
machine
nor blocked).
When
is
is
each
ja(t)=O ]
r,.
time
it
has
unit.
if the
the buffer
operational
removed
repair,
during
into
piece
one
at most
loses
or
gains
One piece
unit.
starved
4.
called
require
machines
upstream
its
if
If
[also
operations.
for their
downstream
machine is never starved
3.
the
M1.
machine
repair
under
operational
All
amount
the
is
it
down).
or
failed
to
relevant
operational - (sometimes
is
machine
of
state
repair
the
0a = 0,
if
up);
are
that
Gershwin
are:
indicate
ai
Let
presented in
is
model
assumptions
The
(1983).
and Schick
the
of
description
if the
and
downstream
probability
That
neither
is,
ri
of
9
GERSHWIN, Tandem Queue Decomposition
5.
When
machine
prob [ ai[t+l)=O
6.
By
the
convention,
j ni 1(t)>O,
has
it
Pi of
probability
fail-
time
of
at the
end of
ai[t)=1, ni[t)<Ni ] = Pi'
units,
the
failures
and
repairs
beginnings
place
working,
is
is,
That
ing.
i
Thus,
during periods
take
while
i,
buffer
-starvation---and-- ... blockage -do---n.ot -influence
at
occur
levels
in buffer
and changes
time units.
to
assumed
are
n,(t+l] = n,[t) + ot+l] - o,,[(t+l).
generally,
More
= n[(t) + I,[t+l] - Idt+l]),
nj(t+l]
I(, t+ 1)
where
buffer
i
the
indicator
That
from upstream.
IUI
i,[t+l) =
The
is
indicator
of
whether
flow
arrives
at
is,
I={1if ai[tl)=l and ni-l(t)>O and ni[t)<N i,
ctl)=l otherwise.
of
flow
system
is
Idl(t+l]
leaving
buffer
i
is
defined
similarly.
The
state
of the
s = n,, ... nk ls, a,
Performance
The
or
line
..., ak).
Measures
production
efficiency)
of
rate
machine
(throughput,
M,,
in
parts
flow
per
rate,
time
10
GERSHWIN, Tandem Queue Decomposition
is
unit,
average
The
([1]
> 0, ni < Ni .
Ei = prob [ a i = 1, -
i is
of buffer
level
fii = E n prob ([s)
of
Flow
there
is no
Conservation
Because
of material,
tion
the
in
presented
are
and
for these
Formulas
lines
2]
two-machine
for
Appendix.
mechanism
the creation
for
or destruc-
or
conserved,
is
flow
quantities
related
(3)
E = E l = E2 = ... = Ek.
The
Flow
Rate-Idle
Time
Relationship
Define e i to be the isolated production rate of machine Mi.
the
what
peded by
production
rate
other machines
of Mi would be
or buffers.
It
of
time
if
it
were
never
is given by
It is
im-
(Buzacott,
1967)
ei
ri + pi
represents
and
it
The
actual
king
E;
=
rate
production
starvation.
or
fraction
the
It
Ei
of
M1
is
that
less
Mi
is
because
operational.
of
bloc-
satisfies
e, prob [ n,t > 0 and ni < N, ].
5)
11
GERSHWIN, Tandem Queue Decomposition
see
For a proof,
expect
that
the
events
being
empty
or
full
Bi-I
only
Mi
when
is
of
are
independent.
become
can
until
next
the
machine
fraction
of
same as
the fraction
not
state
time
non-idle
is useful
can
change
Mi
that
observe
prob [ni- l = 0 and n i = Nit]
The probability
be
only
= Ni-1
1.
Machine
2.
ai
3.
Machine
The
Mi.
1,
=
of
this
from
reached
by means
of
l
failures
may
Bi
and
can
idle
an
event,
is
period
either
is
either
running.
is
operational
full
become
would be operational
can
be
time
The
thus
the
if it
were
and buffers.
that
0.
[6)
is small
event
states
a transition
is
is not
occur
or blockage.
starvation
Therefore,
of time it
to
However,
due to
in a system with other machines
It
buffers
in which a clock, measuring working
thought of as a hiatus
to
reason
adjacent
empty
operational.
is no
(19843.
and
failure
machine
are not idle
only while machines
Furthermore,
There
is counter-intuitive.
result
This
Gershwin
(1981),
and Berman
Gershwin
in
because
ni_
which
such
states
=
and
1
ni
in which
under
repair
or
starved,
and
and
Mll
production
rate
may
under
therefore
repair
be
Ei z e i [1 - prob ( ni_,=O ) - prob ( ni=N i )].
or
blocked.
approximated
by
(7)
12
GERSHWIN, Tandem Queue Decomposition
2. DECOMPOSITION METHOD
The
vation
decomposition method is based on the equation of conser-
of
flow
[3],
to
line
empty
in
being
full
and
L(i)
efficiency
or production
the
be
two-machine
of
the
four
are
functions
pd(i)
through
the
two-machine
Conservation
of
Flow
One set of conditions
buffer
line.
E(i),
Rate-Idle
E(i) = e
Equation
Pd[ i-l
)
i
p 5 (i),
pu(i),
and
pb[i]
rd(i),
the
Appendix.
conservation
of flow:
in
Time
[1- p[i-1)
(9),
Bi
(8)
second set of conditions
The
being
of buffer
r(i),
formulas
is related to
Bl
E(i) = E(1), i=2, ..., k-1.
Flow
the
of two-machine
probability
unknowns
line
rate
of
probability
the
pb[i]
let
that
in
be
p 5(i)
Let
L(i).
transfer
equations.
of
Let EWi] be the
line
approach
way, and to find a solution to
in a simple, approximate
set
and
(7),
The
of the
the most important features
characterize
resulting
relationship
time
and [12)) developed below.
(([[11)
a set of equations
is
flow rate-idle
the
-
p(i ),
after
some
Pu+i)
1
r 1=r,
i) =
e1
follows
from [7):
(9)
i=2, ... k-.
manipulation,
can
be
written
1
13
GERSHWIN, Tandem Queue Decomposition
This
in
described
is
of
Resumption
In
[1984).
Gershwin
Flow
following, we derive
the
of the
of equations
a set
form
(12)
rdi-1) =rd(i-) Y(i) + r1 (l-Y(i)), i = 1, ..., k-l
which
show
the
relationship
lines
two-machine
neighboring
repair
the
we
the meaning
Ml(i)
in
upstream
of
Machine
everything
and
in
of failure
line
Bi
in
probabilities
the original
in
probabilities
characterize
consider
repair
between
and
the
in
To
line.
lines,
two-machine
in
repair
L(i)
represents,
line
L.
those
to
Thus,
at
systems.
Bi,
buffer
t,
time
> 0)13
(otui] = 1) iff (a,[t) = 1) and (nj[t-1l
(au(i) = 0) iff (ai(t) = 0) or (n,(t-1] = 0)
A failure
Mi
or
B i1 ,
turn,
of
emptying
due
to
results
Mu(
tion
a
M([i]
emptying
the
in
of
of
is
due
either
Bi I.
to
failure
of
Mu[i-l].
a
failure
a
is,
a
either
buffer
That
Bi- 2 .
failure
from
represents
of
Mi
of
machine
of
the
or
1
emptying
A failure
of
emptying
The
the
of
failure
of
Mui]
therefore
Mi
or
a
termination
of
whichever
machine
is
Bi i
failure
of
i- 1).
The
repair
was
in
of
effect.
Mu(i)
is
the
Consequently,
the
probability
of
repair
condiof
GERSHWIN, Tandem Queue Decomposition
MCi)
and
is
it
ri
is
X(i)
given
that
t+l
time
t
is
cause
Mu(i)
of
t,
given
of
failure
otherwise.
the
now make
independent
time
the
ru(i-1)
which
We
if
This
conditional
is
14
is
leads
probability
the
to
failure
equation
that
of
Mi
(11),
in
is
down
Ml(i-l]
down.
this
the
more precise.
probability
that it
that
did not produce
Assuming
M
1
that
produces
a
ru(i)
is
part
at
and was not blocked at
is
ru[i) = prob [nil[t)>O, o[(t+l)=l
(nl[(t-l]=O or ot[t)=O) and n,(t-l) < Ni].
Breaking
tioning
event,
down
we
this
expression
by
decomposing
(14)
the
condi-
have
ru(i] = A[i-l) X(i) + B(i] X'(i),
where
[15]
we define
j ni_,(t-l)=O
A[i-l] = prob [nil(t) > 0, ao[t+l)=l
and n[t-1) < Ni],
(16)
X(i) = prob [nl_,(t-l)=O and n , (t-l]) < Ni
(nll(t-l]=0 or cti(t)=0) and nl(t-l] < Nil,
B(i) = prob [ni_J(t) > 0,. al[t+l)=l
(17)
ai([t)=0 and ni(t-1] < Ni],
(18)
X'(i) = prob [Ui(t)=O and n,(t-l) < Ni I
(ni_,(t-l)=O or cL,(t)=O) and n,(t-1) < Nil.
This decomposition
(al(t)=O)
are
disjoint
is possible
events.
because
(19)
(nil(t-l)=O)
We now evaluate
all
and
four condi-
15
GERSHWIN, Tandem Queue Decomposition
probabilities.
tional
The
down
in
transition
M1
when machine
occurs
[18)
from
goes
Therefore,
to up.
B[i] = r i.
first
The
Bi-I
making
Bi l
being
or
down
down.
only
after being
ately
bility
of
way
that
empty
is
event
this
become
for Mu(i-1)
by
is,
can
Bi-,
to
The
recover.
is
immedi-
non-empty
proba-
Therefore,
ru(i-1).
definition,
either
M[i-l)
that
saying
to
is
Mi1
machine
equivalent
is
This
starved.
Buffer
non-empty.
to
empty
buffer
of
probability
the
that
implies
empty
The
from
transition
the
is
A(i-l],
quantity,
A[i-l1 = rfui-l)
We now show that
= P(i-1]
p[
X(i)
=P[i)
Equation
We
r u (i)
Ei) .
(17)
can
written
prob [ni-l[t-l)=O and nitt-l] < Ni ]
prob [ (n,_[t-l]=O or aitt]=O] and n,(t-l] < Ni
have
observed
being
empty.
We
bility
of
- 1)
being
the
(21)
of
the
that
at
full
numerator
Ps (i
be
X(i])
and Bi being
the
20)
same
is
assume
empty
as
time
is
in
L(i)
B 1I
in
L,
has
so
Therefore,
probability
the
approximately
B
(17).
small
empty
being
B
of
probability
(21]
the
the
same
of
proba-
numerator
is
Bi_-
16
GERSHWIN, Tandem Queue Decomposition
denominator
The
lity
the
of
(21)
of
following
the probabi-
(13),
suppressed):
are
arguments
time
(the
event
to
according
is,
[22)
(C[i])= 0) and (n i < Ni)
can be
The probability
calculated
by making use
r(ji) prob [ (a[i) = 0) and (n < Ni)
pu(i prob [ (ali)
Schick
(See
(21)
tor of
]
(23)
= T}and (n < N) ] =-pJi]-E[i)
114.)
page
[1978),
Gershwin
and
of
the
Thus,
denomina-
is
prob [ (Co(i = 0) and (n, < N)
]
pu[i) E[(i)
rU(i)
and
(20)
is
(24
established.
and (19)
A comparison of (17)
shows
that the events
are
Therefore
complementary.
(25)
X'(i) = 1 - X[i)
All
is
result
analysis
line,
quantities
(11],
yields
in
in
(15)
which
equation
have
X(i)
(12)
is
now been
given
for the
by
evaluated,
(20).
and the
A similiar
second machine
in the
i-l'st
where
vr.(i) pb
rd([i-)
Y[i]=pdi-l
] Ei-l)
(26]
GERSHWIN, Tandem Queue Decomposition
Finally,
there
are
17
boundary
conditions:
rul) = r
rd(k-l] = rk
p (1)
p1
=
([27)
pd[k -l) = Pk
are
There
(12),
and
Pd(i],
total
(27)
i=l,
Summary
in
... ,
of
4[k-1]
4(k-1)
unknowns:
r([i],
[8].
(10].
pu(i],
(11].
rd i],
k-i.
the following
made
approximations:
We have assumed that two machines for each buffer can be
described
whose
adequately
behavior
downstream parts
of the
represented
line.
summarizes
We have
be
2.
We have made use of approximation
well
as
(10]
as
(11)
and [12).
occurs
in
the
by geometric
the
calculation
(The
of
Equations
boundary value
(8),
problem of the
in
assumed that they
This
and Y[i)
the
and
models.
affects
in
equation
equations
calculation
of
numerator
of
and
define
a
line
L[i);
X(i)
(21).)
is
Pu i)
involves
a 4-vector
rd(i),
the
of
(27)
two-point
form
xl(O), x2 (0), x3(k), x4(k] specified
(ru(i),
the
([1), (12),
(10),
f(x(i), x(i+l)) = , i=l.....
x(i)
(6).
of X(i]
approximation
simplification
further
upstream
Technique
Numerical
where
up-
the
and down-time
can
f ]H
among
equations
Approximations
of
We have
1.
a
3
the
parameters
p,1 i)].
evaluation
The
of
E[i),
of
nonlinear
ps(i),
and
x(i)
=
function
pb(i)
by
18
GERSHWIN, Tandem Queue Decomposition
of the
means
The
following
the
next
sections.
1.
Initialization.
2.
The
3.
ru(i),
of E in step
3.
Using
and
rd(i-1)
outer
loop
seeks
i=l,
.....,
k-1.
average.
their
E whose
value,
used
in
s.teps,_pro.duces--- a
nearly
equal
value
from
ECi),
of
E in place
11)
pd( l.)
(The
seeks
loop
evaluated
as
Guess
5.
Evaluate
EMl).
6.
Calculate
pu( 2 )
from
For
i=2,
close
value
recent
(10)
value
a
1, is
in Loop
most
4.
of
values
new
calculate
rU[i)
(12).
and
middle
The
that E[k-1),
placed
E,
and
pd(i),
15.
2.
Loop
rd[i],
E(k-l)
andthe _ following
_ _step_ 3
....
pu(i],
... ,
E1),
Evaluate
Loop
Guess
of
results
numerical
the
produced
algorithm
of the Appendix.
formulas
line
two-machine
with
may
of
so
to El1).
be
and
with
inner
loop
i=2
pd(l]
used.)
Eli]
re-
E.
by
Loop
1.
value
of
so
pd[i)
7.
Set i
-
1.
8.
Set
i
-
i+l.
9.
Find
the
value
... ,
k-2
the
that
Eli)
is
close
of
pd(ij
so
thila
to
Eli]
seeks
E[1).
is
suf-
a
GERSHWIN, Tandem Queue Decomposition
ficiently
close
evaluating
19
to E(l).
the
This
two-machine
is
line
done
by searching
formulas
of
where
E(i)
and
the Appen-
dix.
10.
Calculate
by
parameters
has
of
If
15.
Otherwise,
15.
been
Lfk-l]
13.
The
are
of the
value
buffer
two-machine
E(k-l)
= ...
= E[k-1)
the
previous
loop
in the
9
1
Loop
IE (i)-E [ 1)[<c
tionship
Line
L(i)
replaced
go to 8.
10.
All
other
Evaluate
to
closer
E[1),
go
to
E(1).
E(k-l].
to
step
Go
to
by
are
equation
as
follows.
held
and
new
E.
If
it
Otherwise,
loop]
pdfi)
is
are
is
modify
FI (
formulas,
when
all
those
by
the
solution
For example,
pd(i]
be
that
simply
reached.
unknown.
determines
Let
constant:
line
is characterized
in one
two-machine
E(i)
long
convergence
algorithm
innermost
between
the
E, stop.
of the
when
(the
the
is
3.
levels
lines
nonlinear
determined
close
bring
to
of a single
in
step
known.
to
of E[1]
close
average
Each
already
14.
pd(1]
Otherwise,
in
step
the old E and go to step
The
is
5.
sufficiently
16.
12.
determined
E(k-l]) is sufficiently
Modify
step
[10)
If i = k-2, go to step
pu[k-l)
14.
from
E.
11.
12.
pu(i+l)
such
the
represents
other
Step
that
function,
the
parameters
relaof
GERSHWIN, Tandem Queue Decomposition
20
(29)
EMi) - F[lpdi)])
(The
superscript
successive
ponding
_______
_-=
guesses
values
the
1 refers
Loop
pd(i
of E)i].
next
_
of
to
1.)
Let
let E (n- l )
and
p(n-l(i)d
and pn)([i)
d
and E(n) be
the
be
corres-
The Newton-Raphson method for choosing
guess
pn+U(i)
d
dII dF[pdM[i]
d
dF(Pi E)
is
i)f pin)-E(i
-
Pd
where
the
E[1]
is
treated
tive
by
derivative
the
is
evaluated
a constant here.
difference
We
n)
at
Note
approximate
the
that
deriva-
quotient
E-n)(i) - En-i)31)
j(i) - pn_)(i)
dFiPd([i)]
-dp(i
and (30)
as
(30]
)
becomes
pPn-+1)i)(E(n)(i) - E[l) - pcn)(i)(En-'')(i) - E1]))
d
E(n)[i) - E(nl)(i)
Similarly,
in Loop
IE(k-l)-E(l)c<E.
In
this
apply Newton's
method
to
2 we
seek
loop,
E(1)
that
the
iteration
is
for
pd (1) so
a variable.
Here
that
we
the equation
E[k-1)-E[1) = F 2(pd(1))
so
a value
(33)
formula
is
21
GERSHWIN, Tandem Queue Decomposition
p-[1(13n(E(m[(k-l] - E1]) - pm)[1)(E(m-l)hLk-
PM[el)]
)
d
Etn)(k-l) _ E(m)(1)
Finally,
for
and Resumption of Flow
Time
Let e (
of Flow.
as Conservation
as well
a value
determines
loop)
the Flow Rate-Idle
E that satisfies
equations
outermost
3 (the
Loop
- E(]))
E(ml)(k-l) + E(m-l)1 ]
-
be the value
of E used in Step 3 (Resumption of Flow) and Step 6 [Flow RateIdle
Time)
E(1)
to which
E = R.
in
cient.
iteration
_ g(F3(g(l)
F3(g()
that
quired.
large
In
in
in
all
the
and is
some
cases
simply
on
in which
lines in which
[36)
loops,
in the
the
iteration
process
that
when
agrees
initial
guess.
failed
had much
are
are
it
re-
E was
sections,
then reduced
algorithm
a few machines
following
values
and was
suggests
of the
initial
two
solution which
independent
to be more effi-
6
the
experience
appears
-
iterations
a unique
to
tion
that
is
examples
results
Numerical
converges
of
))
three
all
initial
Analytical
able.
- F3[gl
the
method
formula
,,( +_ =
- (e-1F3(gf())
Note
is
process
value
(35)
the Newton-Raphson
The
the
be
] C )
= F3[6
E
but again,
^(&
E
seek 6 such
We
iteration
possible
let
Then
iteration.
Loop 2 converges.
One
?*~'I=
tth
the
to
not
10- 8.
avail-
converges,
closely
There
to converge.
larger values
with
have
it
simulabeen
These were
for r and p
22
GERSHWIN, Tandem Queue Decomposition
than the
others, or where
the
lines were
very long
(more
than 20
machines].
The
speed of the
evaluations
many
cases.
algorithm is best measured by the number of
of a two-machine
line.
This
is
reported below
for
GERSHWIN, Tandem Queue Decomposition
23
3. NUMERICAL RESULTS-BEHAVIOR OF THE ALGORITHM
In
The
this
issues
effort.
section,
investigated
The
method,
is
behavior
presented
Three-Machine
It
the
is
results
Table
1A
are
of
in
those
Section
to compare
range
were also performed
of
the
algorithm
accuracy
lines,
as
and
is
described.
computational
determined
by this
4.
the
results
by using the method of
a wide
a total
of
transfer
lists the parameters
represent
of
Cases
possible
exact
behavior
of
of this algorithm
Gershwin
and Schick
of a set of cases.
three-machine
for comparison.
100,000 time units.
These
systems.
results
are
(1983).
cases
Simulations
Each simulation was
The
with
run for
shown in Table
lB.
The
close
approximate
to
the
exactly.
than
The
exact
error
0.02)} and
average
close
buffer
or
The
values
is
to
burden.
tude
less
or for the
too
large
exact
as
larger
(less
approximate
the
computer
time
simulations.
line
evaluations
results
Note
that Case
but
are
E (less
the
generally
the method is.
required
that
in
chain
as
results.
requires
that
extremely
rate
2.6Z)
is a very
than
the exact method,
are
the Markov
than
simulation
transfer
that
in the production
also shows how efficient
Eighty-eight
for
The
results
by solving
small
only a little
uation of a two-machine
nal
obtained
very
levels.
closer
table
method produces
for
small
orders
the
The evalcomputatioof magni-
exact
6 has buffers
the agreement
solution
that are
with
GERSHWIN, Tandem Queue Decomposition
simulation
is
Longer
good.
Lines
Exact methods
three
24
machines
fers.
are not available
or for
Consequently,
accuracy
of
qualitative
the
three-machine
other
The
cases
techniques
approximation.
results.
for systems
cases
with
are
They
of more than
very
required
include
considered
large
to
assess
simulation
here
The
average
results
buffer
There
cover
close
agreement
lation
results.
levels
agree
to within
large
buffer
capacities
There
is no
obvious
Cases
in
In most
which
cases
.1.
in which
to
Most
the next
ter
with
(over
100)
as
=
the
pi
evaluations
be
of
repair
the
and
indicates
O(k3 ),
buffer
rates and
where
rates
and
This
remains
long
lines
that
the
length
buffer
true even
[20
accuracy
of
the
increases.
of
two-machine
lines
increases
this for two
probabilities
are
all
the
number
of
k
the
number
of machines.
operating
for
machines.)
that
is
and simu-
.1
failure
approximation
Multics
approximation
Figure 3 shows
section were produced
the
the
and
line
=
length of the line.
Experience
pears
exact
a wide
and
of production
cases, production
trend indicating
r
of
between
a few percent.
decreases
The number
with the
range
the
levels.
is
approximation
a wide
and
represent
. .---.------ range-of---.failure-probabilities,--repair--probabilities,
sizes.
buf-
and simulation
on the MIT
system.
Honeywell
All
equal
evaluations
results
times
sets of
ap-
of this
68/DPS
to
and
compu-
indicated were
GERSHWIN, Tandem Queue Decomposition
for
runs
the
computer
performed
to be much
time
less
on
this
for
the
--
both
machine.
failure
probabilities
to
Production
5.
buffer
levels
line.
Note
production
results
the
rates
but with
Reversibility
or
authors
shown [Dattatreya,
Gershwin,
reverse
1981)
that
of one
tion (Ammar,
of
=
applies
1, ... ,
ni
to
.1
--
were
per-
computers.
average
and buffer
indicated
times
agreement
are
between
for
the
to
and
equal
average
20-machine
and simulated
3 contains
equal
all
cases;
approximate
Table
capacities
sizes
for all
indicated
levels.
buffer
similar
10.
Symmetry.
have
conjectured
1978;
Muth,
two tandem
buffers
lines
and simulation
of compatible
to
are
are
are
their
(Hillier
1979; Ammar,
queueing
the
1980; Ammar and
corresponding
property
(E)
another have
Symmetric
i
all
and
Several
equal
computer
and
appears
results of a set of cases with repair
all
close
method
that
of simulation.
approximate
rates
and
indicates
approximation
formed on an IBM PC and a variety
Table 2 shows the
Experience
analytic
than that
Other runs
25
and Boling,
1980; Ammar and
systems which
same production
Gershwin,
1981),
1977)
rates.
are
In
the
addi-
the
average
The
complementarity
levels
complementary.
own
symmetrically
reverses.
opposite
buffers.
That
is,
for
k-l,
+ n ki =
(37)
Ni = Nki.
Furthermore,
the
average
level
in the
middle
buffer
of a
GERSHWIN, Tandem Queue Decomposition
symmetric
line with an
capacity
odd number
half
the
even
and i = k/2, [37)
=
k/2
The
equal
1980).
of the
decomposition
Simulation,
agreement.
18
Table
These
2;,
the
That
method agree
however,
properties
first case
and 23; and 19
Perturb'ed
earlier
be
to
are
cases.
to
cases
nearly,
were perfect,
close
are
produces
of Table
We
that
but
these
expect
of the
line whose
two buffers
that
of
buffers
in
ap-
the
3; Cases
last
1-4; 6;
14
already
not
considered.
quite,
perfect.
cases would be
the behavior
It
If
the same
of
that
includes
those
as the
the perturbed
cases
second machine almost never fails.
Case
4,
the same.
to
Case 7 is a fourThe
of Case 7 have a total capacity which is equal
B, of
are
cases
others.
and
The
two buffers of Case
7 is
average
level
of
Bl
of
5i,
Case
7
is
not
of
these
only
exhibited
Case 7 was chosen to be close to Case 4.
machine
is
and 21.
three-machine
that
machines
k
cases
close
machines
exactly
is, if
with
Tables 4A and 4B contain a set of four-machine
are
to
implies
exactly.
of the
15;
[Ammar,
is
[38)
results
proximate
and
that buffer
of buffers
N/2
observations
case
of
26
the
other
corresponding
sum of the average
5.977.
Case
far
first
to
machines
levels of
the first
This should be compared with
4 which
from
is
n2
6.063.
of
Case
In
4.
and
addition,
the
GERSHWIN, Tandem Queue Decomposition
Similarly,
highly
Case
8 was
chosen to be close
to Case
first three
efficient
machine
is
and two buffers
of Case
8 are
are
also
buffer
in
can
close
be
Because
the
the
than
a condition
the
third machine
one
persisted,
the
do: it
is
consequence,
the
The
are
the
rate,
the
third buffer
and the
otherwise.
expected
average
is
in
simulation
this
level
fails,
If it
buffer would have
production
in the same
strate
The results
in
the
last
the
were
last
buffer
initialized with
whenever
would
exactly
If
become
one
empty.
piece
second buffer were
if
the
non-empty;
That is, it would have one
subset
was producing
parts
and
it was not.
last machine
production
the
fraction of time
the
the
never
in it.
three-machine
calculate
the
nearly
piece
were working
no pieces when
to know
Furthermore,
3.
the
machines
down or the second buffer was empty.
third
whenever
To
to Case
Here
the number of pieces would diminish
and it would be empty
piece
The
identical
last machine
third machine was
Eventually,
last.
agreement.
than one piece,
such
the
3.
explained.
never has more
more
27
level
of
number
relationship
behavior.
third buffer,
the three-machine
of parts
almost perfectly
results
the
the subset is producing
rate
parts
in
per
also
time
bear
as Cases
in the
reliable)
we need
parts.
subset.
As a
last buffer
is
equal
We
to
[when
the
unit.
this out.
Cases 4
and 9
3 and 8 and they demon-
28
GERSHWIN, Tandem Queue Decomposition
Cases
Simulations
Published
with
Comparison
10-12
SA]
(Table
and
of Ho, Eyler,
example
([c
results
(production
slightly
different
rates)
a set of
are
Chien
and
(1979)
Table
5B
are
In
each
of
these
simulation
the
(written in
theirs
of
form).
lines from
5-machine
a
cases,
r,=.005, pi=.001, i=l, ..., 5
approximate
The
12 which
for Case
greater
than
Cases
the
all
methods
for Case
results
for Case
rates
11, which are
10.
6A)
are
(Table
4-machine
in good
are
production
calculate
than those
greater
[Table
13-23
are
are
those
simulation
These
both
Furthermore,
agreement.
rates
simulation production
and
6B)
lines
from Law
taken
from
come
and again
(1981)
the
cited
paper.
in which
r,=.05, p,=.00S, i=l, ..., 4
The
lated values.
by
ported
which
tions
a
I
and
greater
are
5 to
be
is
16)
production
which is
less
However,
14 and
than
less
that
rate
than
15)
is
are
less
13-17
Cases
Law finds
indistinguishable
than Allocations
we
the
than
and
2,
3, and
find
that
of
I (Case
13).
4,
rever-
Allocation
that of Allocation 5 (Case
of Allocation
to
Alloca-
same (due to
that
simu-
not sup-
that are
results.
statistically
rate
2 and 3 (i.e., Cases
4 (Case
conclusions
with
1-5 of Table 4 of Law (1981).
indistinguishable.
Their
agreement
in close
decomposition
production
also
sibility).
which
or
reversibility
Allocations
are
Law draws
However,
are Buffer Allocations
have
results
decomposition
17)
GERSHWIN, Tandem Queue Decomposition
Cases
(1981).
chines
to
1-6
(L).
The
have
order
The parameters
L machines
capacity
of
using
in all
production
(23,22),
21,
18,
19,
be
and two have low
and r=.05.
Law
are
The
finds
p=.0025
and
three buffers
that
the patterns,
are
20.
of this paper.
distinguished,
(H)
of four ma-
the H machines
cases.
rate,
the case numbers
cannot
of
have p=.02
19
of Table 2 of Law
find the best sequence
of which two have high efficiency
r=.05.
each
7) are Patterns
Law is seeking
efficiency
in
18-23 (Table
29
but
That is,
their production
Cases 23 and 22
rate
is
less
than
that of Case 21, and so forth.
We
The
find
instead the
following
22,
(18,23),
( 19,21]),
cases
that we
find
order:
20.
indistinguishable
are
the
reverses
of one
another.
Cases
with
Many
Table
8B
Different
of
the
shows
5, doubled
The
and
simulation
treated here
the results
whose parameters
twelve-machine
lines
Machines
of two
have
had
identical
machines.
cases
with
different
machines,
are
listed in Table 8A.
cases
that consist of the
quadrupled.
The
and approximation
buffers
results
These
are
six-
three machines
are
are
all
in
of
and
of Case
capacity
close
10.
agreement.
GERSHWIN, Tandem Queue Decomposition
30
4. NUMERICAL RESULTS-BEHAVIOR OF TRANSFER LINES
Figure
the
length
4 demonstrates
of the line
sing that production
does
not,
rate
approaches
for
in
two
the
the
as
to
buffer
is
where
lines
to
how
i,
This
graph
the production
infinity.
half
the
the
average
buffer's
buffer
For
level
capacity.
This
as discussed above.
complementarity.
all
is not surpri-
grow.
to
with
is distributed in the buffers
of both cases,
for
It
of whether
goes
Note
equal
exhibit
Ni=10
as
length
cases.
rate varies
of cases.
question
how material
the symmetry
buffers
sets
the
the
twenty-machine
other
case
zero
production
decrease
settle
5 shows
tenth
is due
for two
rates
however,
Figure
how the
example,
All
in
the
fi3 +fi17 =10.
Cases 3, 4, and 6 of Section 3 show that as the buffer sizes
increase,
rage
the
buffer
general,
buffer
levels,
levels
the
p=.001
do
increases.
a fraction
rate
of
increases
not always
6 shows
effect
of
is
the
of varying
nine
identical
considered.
buffers
with
equal
buffers
with
equal
fourth
rate
In
capacities,
with
approach
addition,
buffer
the
ave-
approach
.5.
capacity,
half of buffer
In
but
capacity
as
increases.
Figure
system
as
production
capacity
and
production
machines
effect
the
of
distribution
machines
In one
and between
with
set
capacities.
capacities,
increasing
the
located
the
sixth
buffer
of buffer
parameters
of cases,
In
total
there
other,
between
there
the
and seventh
space
space.
r=.019
are
and
eight
are
third
two
and
machines.
A
31
GERSHWIN, Tandem Queue Decomposition
(Each
set
parameters
The
r=.01898
lines when
when
ing
Note
that
sets
of
is
in
machines
acts
like
may
the
cases:
be
always
the
of the
1966),
which
is
that,
for
space
should
be
allocated
with
In
latter
it
rate
with
buffer
.8636
since
the
than
the
2-buffer
levels
are
the
(due
to
same
of block-
to
the
production
middle
of
8-buffer
distribution.
in the
two
18-23, that 13
is the best of 31-33
phenomenon"
maximal
each
symmetry).
20 is the best of Cases
"bowl
the
The behavior
other models,
capacity
total
is
former case,
the
is
2-buffer
the
the production
a single machine.
better
be
average
half
instances
than
better
and 17 are the best of 13-17, and that 33
may
one machine
is small may be due to the model
that Case
The facts
are
that
the
used here;
total
to
machine.
.95;
starvation
distribution
large
is
the buffer space
and
lines
least productive
efficiency
of three
is
equivalent
p=.002997.)
8-buffer
the space
limited by the
set
and
the
reason
limiting
is
of three machines
(Hillier and Boling,
rate,
a line.
more
buffer
GERSHWIN, Tandem Queue Decomposition
32
5. CONCLUSIONS AND FURTHER RESEARCH
A new method has been found for the analysis of tandem
queuing
tant.
Exact
while
at
systems
with
and
simulation
approximate,
extending
finite buffers
this
work
in
two
directions:
reliable
and
unreliable
as
tial
other
processing
1985;
[Gershwin,
as
Gershwin,
1986).
Jackson-like
Research
this
method.
numerical
curacy
the
of
networks
is needed
That
algorithm.
method
accurate.
time
1985);
Future
experiments
the
indicate
quite
such
shwin,
results
is
behavior,
and
in which
is,
efforts
with
to
that
method,
the
research
other
service
machines
(Choong
be
devoted
exponen-
and
to
aimed
process
with
assembly/disassembly
will
is
Ger-
networks
systems
such
blocking.
determine
analytic
precisely
techniques
should be used to
and to
is impor-
Current
distributions
and
blocking
determine
the
rather
performance
than
develop bounds
convergence
of
only
on
the
properties
acof
GERSHWIN, Tandem Queue Decomposition
33
REFERENCES
T.
Altiok
Queues
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Vol.
M. H. Ammar
facturing
for
Assembly
IEEE
fer
Lines
neering
J.
Networks
of
with
A. Buzacott
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of
and Analysis
Finite
Systems
B. Gershwin
Tandem
Operations
Re-
on Decision
and
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Ph.D.
of
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LIDS-TH-1004.
"Equivalence
of
Relations
the
in
Nineteenth
Analysis
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of Automatic
Department
Trans-
of Engi-
Birmingham.
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International
MIT
Manu-
Control.
"Markov
(1967),
Report
Proceedings
(1967),
University
of Unreliable
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of Exponential
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with
and Decision
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A. Buzacott
European
"Modelling
and S.
Models
Analysis
1982.
(1980],
M. H. Ammar
J.
11,
Information
Queuing
"Approximate
Transfer
Journal
of
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Production
with
Buffer
Research,
Vol.
6.
Y. F.
for
Choong
the
Evaluation
Machines
and Random
Information
appear
S. B. Gershwin
Approximate
Unreliable
for
and
in
P. Coillard
and Decision
IIE
and
[1985),
of
"A Decomposition
Capacitated
Processing
Systems
Transfer
Times,"
Report
Lines
MIT
to
Transactions.
J.-M.
Proth
(1983),
"Sur
L'Effet
des
en Ligne,"
INRIA Rapports
cherche,
Recherche
en
National
de
with
Laboratory
LIDS-P-1476;
Tampons Dans Une Fabrication
Institut
Method
Stocks
Informatique
de Reet
en
34
GERSHWIN, Tandem Queue Decomposition
Automatique,
de
Domaine
Voluceau,
Roquencourt,
Chesnay,
Le
France.
E.
Ph.
S. B. Gershwin
Approximate
of
Evaluation
California,
Production
Lines
and
Opera-
Method
for
Berkeley.
Efficient Decomposition
"An
(1984),
Engineering
Industrial
of
University
Research,
tions
of
Department
D. thesis,
with
the
International
Sixth
with
Laboratory
to
1446;
S.
that
Machines
for
in
appear
1579,
July,
for
Proceedings
Times
13, No.
and
for
1984.
LIDS-P-
Research.
Operations
An
Systems:
Tree-Structured
and Decision
MIT
Rates,"
Report
Systems
in
Transfer
of
Analysis
Processing
"Assembly/Disassembly
Information
and 0.
Berman
(1981),
of Two
Unreliable
Machines
Finite
Storage
Buffers,"
1, March
S. B. Gershwin
Three-Stage
and
and Decision
Algorithm
Notes
Lecture
Springer-Verlag,
have Different
of
Optimi-
and
MIT
Networks,"
Report
Systems
Effi-
LIDS-P-
1986.
S. B. Gershwin
Consisting
of
Annals
(1986),
Decomposition
Laboratory
63
Sciences,
Information
B. Gershwin
cient
Volume
"Representation
(1985),
S. B. Gershwin
Lines
Analysis
Information
and
Control
on
2,
Part
Systems,
of
zation
Conference
the
Storage
Finite
------- Space-,"-- in--- Analysis--and--Optimization--of-- Systems,
of
Blocking,"
with
Systems
Queueing
"Tandem
(1978),
S. Dattatreya
"Analysis
of Transfer
Lines
with Random Processing
AIIE
Transactions
Vol.
1981.
and
Transfer
I.
C. Schick
Lines
with
(1983),
"Modeling
Unreliable
and Analysis
Machines
and Finite
of
35
GERSHWIN, Tandem Queue Decomposition
Technical
SME
20501
Engineers,
the
on
17,
F.
12,
No.
S.
and
Dearborn,
Michigan
Operation
Times,"
of
II, Marc
Reubens,
Technique
for
General
G. C. Hunt
rations
M. A. Jafari
T.
(1981),
Automatic
Transfer
Research,
Vol.
19,
the
Chien
Research,
(1979),
Design
in
Production
of
Variable
with
Proceedings
Amsterdam.
North-Holland,
Storage
Arrays
Ph.
4,
"A
Gradient
a Production
Research,
Vol.
Lines,"
No.
6,
Waiting
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proposal,
Analysis
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pp
Lines,"
Ope-
1956.
D. Thesis
"A Statistical
S. S. Law
Characterizing
1979.
Volume
(1982),
"Toward
Operations
T.
"Sequential
Research,
Variable
Vol.
in
Journal
557-580,
[1956)],
Design
Engineering,
Editor;
Buffer
International
6, pp
with
Lines
in Production Lines
Advances
M. A. Eyler, and
17, No.
of _ Some
Industrial
(1977),
of Work
in
48128.
1966.
Y. C. Ho,
Line,"
of
Transfer
of
"The _Effect
of Production
R. Boling
Allocation
EURO
Road,
Journal
December,
Hillier
Optimal
Society
Efficiency
Times,"
Operation
EM75-374,
(1966),
"A
(1975],
of Manufacturing
Paper
and R. W. Boling
S. Hillier
Factors
Ford
Models
and Simulation
of Analytical
Comparison
F.
354-380,
pp
Taraman
and K. S.
A. Buzacott,
J.
Hanifin,
Lines,"
No.2,
31,
1983.
March-April
L. E.
Vol.
Research,
Operations
Buffers,"
709-724,
Syracuse
System
Journal
1981.
University.
Parameters
of
in
Production
GERSHWIN, Tandem Queue Decomposition
E. J.
Muth
(1979),
"The
Lines,"
Management
T.
(1981)
Ohmi
Automatic
Transactions,
I. C. Schick
of
Reversibility
Science,
Lines
Volume
with
13,
Massachusetts
and S. B. Gershwin
tory
Report
Institute
of
for
No.
of Production
2.
the Production Efficiency
In-Process
No.
Unhreliable--Transfer-Lin-es--
Property
Vol.25,
"An Approximation
Transfer
36
Storages,"
1, March
(1978),
AIIE
1981.
"Modelling and Analysis
-with--Finite
Technology
of
Interstage
Electronic
Buffers,"
Systems
Labora-
ESL-FR-834-6, September, 1978.
R. Suri
and G. Diehl
(1983], "A Variable Buffer-Size Model
its Use
in Analyzing
Closed Queuing
Harvard
University,
Division
of
Networks
Applied
and
with Blocking,"
Sciences,
September,
1983.
Y. Takahashi,
H. Miyahara,
tion
for
Method
Research,
Vol.
Open
28,
and
T. Hasegawa
Restricted
No.3,
Part
Queuing
I,
(1980),
"An Approxima-
Networks,"
May-June
1980.
P. Vanderhenst, F. V. Van
Steelandt, and L. F. Gelders
"Efficiency
a Transfer
Improvement
tholieke
Universiteit
Afdeling
Industrieel
of
Line
Leuven, Faculteit
Beleid, Belgium;
(1981),
Via Simulation,"
Toegepaste
Report
Operations
Ka-
Wetenschappen
81-04, January
1981.
37
GERSHWIN, Tandem Queue Decomposition
APPENDIX
Machine
In
Performance
Measures
a l,
0 2)
is
the
buffer
and
that
and
Probabilities
Steady-State
for
Two-
Lines
the
following,
there
are
These
probabilities
n
parts
in
p(n,
the
are
taken
+ r2
-
from
Gershwin
p[O,O,O) = O
C X rI
p(oo,l) =
:0
p(0{l,0)
r 1r 2 - rip2
rIP2
p(O,l,l] =0
p[(l,OO) = C X
p(1,0,1) = C X Y2
p(1,1,O) =
p1,,l1]
=C
X rl + r 2 - rlr2 - rlp2
P2
p1 + p 2 - P I P 2 - rip 2
p(N-l,O,0] = C X-'
p(N-l,O,l) = 0
p(N-l,l,O) = C XN- ' Y 1
pC-,,l
r2
rr2
rr
_- C X N- rl +
2 P1
pi + P 2 - PIP 2 - P"r2
Mi
and
probability
is
in
Schick
that
state
(1983).
a i.
GERSHWIN, Tandem Queue Decomposition
38
p([N,O,O] = 0
p(N,O,l) = 0
p(N,l,0) = C XN -I rl + r 2 - rf r 2 - plr 2
= 0
p(N,l,1]
al
p(n, al, c 2 ) = C Xn y
Y2 2, 2 < n < N - 2
where
r1 +
-
rFr 2
- rl P 2
P1 + P2 - PP 2
- Plr 2
rl + r 2 - rlr 2
-
2P
Plr 2
+ P2 - PP2 - rip 2
X Y2
yi
and C is
a normalizing
constant.
Performance measures
by
E
=
p[n, a, a 2 )
n>O
o 2 =l
=
Sp(n, a 1, a 2 )
n<N
al=l
PS = p(O,O,1), the probability of starvation
Pb = p(N,l,0), the probability of blockage
ni= _
all n,at,
Note
that
n p(n,. o 1, a 2 )
a2
are
given
GERSHWIN, Tandem Queue Decomposition
E = e (1 - pb]
= e 2 (1 -PS).
39
40
GERSHWIN, Tandem Queue Decomposition
Case
rI pi
Nl
r2
P2
N2
r3
p3
1
.1 .05
5
.1
.05
5
.1
.05
2
.05 .002
5 .05
.002
5 .05 .002
3
.1
.1
5
.1
.1
5
.1
.1
4
.1
.1
10
.1
.1
10
.1
.1
5
.07 .01
10
.1 .013 10 .05 .007
6
.1
100
.1
.1
Table 1A.
.1
100 .1
Three-Machine Cases
.1
GERSHWIN, Tandem Queue Decomposition
41
E
2l
Evaluations
Case 1
Approximate Results
Exact Results
Simulation 1
Simulation 2
Simulation 3
.4542
.4545
.4567
.4554
.4514
3.088
3.063
3.030
3.034
3.072
1.912
1.937
1.953
1.877
1.969
70
.8993
.8992
.9031
.9030
.8883
3.001
2.987
3.087
3.135
2.998
1.999
2.013
2.024
2.056
1.982
32
.3105
.3109
.3075
.3126
.3109
3.101
3.075
3.060
3.096
3.086
1.899
1.925
1.918
1.912
1.926
68
.3563
.3556
.3544
.3572
.3552
6.063
6.035
6.089
6.109
6.011
3.937
3.965
3.908
4.024
3.903
64
.7546
.7539
.7548
.7478
.7558
6.182
6.120
6.107
6.412
6.074
3.349
3.437
3.443
3.523
3.310
86
Approximate Results .4722
Simulation
.4676
56.60
58.59
43.40
45.71
88
Case 2
Approximate Results
Exact Results
Simulation 1
Simulation 2
Simulation 3
Case 3
Approximate Results
Exact Results
Simulation 1
Simulation 2
Simulation 3
Case 4
Approximate Results
Exact Results
Simulation 1
Simulation 2
Simulation 3
Case 5
Approximate Results
Exact Results
Simulation 1
Simulation 2
Simulation 3
Case 6
Table lB. Results of Three-Machine Cases
GERSHWIN, Tandem Queue Decomposition
42
Line length, k
Quantity
Eval's
Approx.
Sim. 1
Sim. 2
4
5
6
7
8
10
13
17
E
116
238
399
583
996
2030
4110
7705
.2806
.2630
.2516
.2438
.2382
.2309
.2248
.2206
.2830
.2632
.2514
.2418
.2365
.2306
.2212
.2119
.2809
.2647
.2519
.2458
.2332
.2251
.2193
.2132
20
E
i1
9467
.2188
3.902
.2107
3.850
.2117
3.909
i5
2.935
3.106
2.856
ilo
2.500
2.684
2.543
nl9
1.098
1.086
1.119
time
6.897
248.169
247.685
Table 2. Performance of Systems with N=5.
GERSHWIN, Tandem Queue Decomposition
43
Line length, k
Quantity
Eval's
Approx.
Sim. 1
Sim. 2
4
E
fil
131
.3352
6.535
.3341
6.604
.3390
6.556
5.000
3.465
5.133
3.637
4.992
3.564
ni2
113
5
6
7
8
10
....
13
17
20
E
234
386
614
1387
1899
3521
10389
.3228
.3149
.3095
.3056
.3006
.2964
.2936
.3232
.3075
.3033
.2970
.2934
.2830
.2768
.3175
.3127
.3057
.2957
.2910
.2861
.2813
E
time
17185
.2924
12.155
.2774
262.188
.2759
261.079
Table 3. Performance of Systems with N=10.
44
GERSHWIN, Tandem Queue Decomposition
Case
Parameters
7
ri=.l i=1, ., 4
pl=p 3=p4=.1, p 2=.000001
N1=N2=5, N 3=10
8
ri=.l, i=l, ..., 4
pl=p 2=p3=.1, p4=.000001
Ni=N2 =N3 =5
9
ri=.1, i=1,.... 4
pl=P 2=P3=. , p 4=.000001
Nl=N2=N3=10
-- Table-4A.-Parameters of Perturbed Cases.
45
GERSHWIN, Tandem Queue Decomposition
Case
Quantity
Eval's
Approx.
Sint 1
Sim. 2
7
E
133
ii1
.3537
2.656
.3550
2.631
.3619
2.632
n2
3.321
3.406
3.403
53
3.876
3.986
3.975
iil
.3105
3.101
.3095
3.058
.3093
3.086
112
1.899
1.917
1.882
-n3
.3107
.3095
.3093
.3563
6.063
.3546
6.042
.3619
5.897
fi2
3.937
3.992
3.843
fl3
.3566
.3548
.3619
8
9
E
E
fll
Table 4B.
128
115
Results of Perturbed Cases.
GERSHWIN, Tandem Queue Decomposition
Table SA.
46
Case
Buffer Sizes
10
Ni=100, i=l, ..., 4
11
N 1=80, N2=122, N 3=126, N 4=72
12
N 1=60, N 2=146, N3=142, N 4=52
Parameters of Ho, Eyler, Chien (1979) Simulations.
47
GERSHWIN, Tandem Queue Decomposition
Case
Eval's
Approx.
Sim
10
293
.6007
.5927
11
289
.6034
.5948
12
295
.6045
.5958
Table SB. Production Rates from Approximation and Simulation.
48
GERSHWIN, Tandem Queue Decomposition
Case
Buffer Sizes
13
Ni=40, i=l, ..., 3
14
Nl=10, N2=40, N3 =70
15
N 1=70, N2=40, N3=10
16
N1=50, N 2=40, N3=50
17
N1=30, N 2=60, N3=30
Table 6A.
Parameters of Law (1981) Simulations.
GERSHWIN, Tandem Queue Decomposition
Table 6B.
49
Case
Eval's
Approx.
Sim.
13
14
15
16
17
128
133
94
117
137
.8337
.8202
.8202
.8273
.8336
.8380, .8344
.8212
.8219
.8200
.8364
Production Rates from Approximation and Simulation.
50
GERSHWIN, Tandem Queue Decomposition
Case
Case
Sequence
SMachince
Eval~s
Approx.
Simr
18
LLHH
898
.6363
.6534
19
LHLH
111
.6707
.6651
20
LHHL
122
.6906
.6767
21
HLHL
148
.6707
.6442
22
HLLH
137
.6351
.6192
.6363
.6173
-__23. ---- HHLL-'
146
Table 7. Sequences from Law (1981) and Results.
51
GERSHWIN, Tandem Queue Decomposition
Case Line Length, k
24
6
Parameters
r 1=r4=.07, r 2=r5=.l, r 3=r6=.05
pl=p4=.01, P 2=p5=.013, p 3=p6=.007
Ni=10, i=l, ..., 5
25
12
rx=r 4=r7=rjo=.07
r 2=r 5=r 8=rl=. 1,
r 3=r 6=r 9=rl2=.05,
P =p 4 =p'7 =p O= 0, 1
=
= l.013,
P 2=P5=P=P
8
P 3=P 6=Pg=PI2= 007,
of,Differen...t Machines.11
i=nes
AParameters
with
N=Table
Table 8A. Parameters of Lines with Different Machines.
GERSHWIN, Tandem Queue Decomposition
52
Case
Eval's
Approx
Sim
24
1582
.6692
.6684
25
4455
.6098
.5916
Table 8B.
Results from Lines with Different Machines.
Page 53
Line L
MI
8
M2
82
r,, P
N1
r2,p
N2
M3
r,
rP
M4
83
N
4
84
M5
85
M6
86
M7
N4
rS,ps
N5
r6 ,p6
N6
r7 ,P7
Figure 1
lI1-CG-m
r(:l),psl) N1
Line L(C)
rd(1),pd(1)
M,(2)
Line L(2)O
22
M (2)
:
ru(2),pu(2) N2
rd(2!,Pd- 2 )
MU3)
1-0
Line L(3)
M.(3)
83
ru(3),Pu(3) N3
r~( 3 ),pd( 3 )
M,(4)
Line L(4)
Mj(4)
E4
"--r
_
ru(4),pu(4) N4
rd(4 ),pd( 4 )
M(5)
C.
Line L(5)
MC5)
"r
',
ru(),pu,)
N
]
r),p(
Mu,()
5)
sE
Md( 6 )
Line L()
ru(6),p,(6)
Figur-
'2
N rd(6),pd(6)
Page 54
100,000
en
C
0
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10,000
-
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Page 55
Production Rate E, Parts/Cycle
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Page 56
Average Buffer Level, N i =5
Co
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Page 57
Efficiency
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