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-
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oewey.
HD28
.M414
no.
ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
Heavy Traffic Analysis of a ProductionDistribution
System
Elena V. Krichagina
of Control Sciences
Moscow, Russia
Institute
Lawrence M. Wein
#3697-94-MSA
June 1994
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
Heavy Traffic Analysis of a ProductionDistribution System
Elena V. Krichagina
of Control Sciences
Moscow, Russia
Institute
Lawrence M. Wein
#3697-94-MSA
June 1994
SEP 2 1 1994
UBBABIES
HEAVY TRAFFIC ANALYSIS OF A
PRODUCTION-DISTRIBUTION SYSTEM
Elena V. Krichagina
Institute of Control Sciences
Moscow, Russia
and
Lawrence M. Wein
Sloan School of Management, M.I.T.
Cambridge, Massachusetts
June
13,
1994
Abstract
We
consider an optimization problem for a multiclass queueing model of a de-
A
produce a variety of
products, and completed units pass through an infinite server delay node on their
way to a warehouse inventory. Each product has its own warehouse, demand for each
centralized production-distribution system:
product
is
satisfied
by
its
single server can
associated inventory, and unsatisfied
demand
is
backordered.
Each warehouse operates independently under its own base stock policy: The initial
is set at a base stock level, and each unit of demand triggers an order with
inventory
the server.
The
server processes the orders in a first-come first-served manner,
the optimization problem
is
to find the base stock level for each product to
the expected average inventory holding and backordering cost.
A
heavy
and
minimize
traffic
approx-
imation of this problem yields an inventory process that possesses both diffusion and
Gaussian components. Computational results are performed to
our analysis.
cissess
the accuracy of
Introduction
1
between nodes, or queues,
in
a queueing network model
be instantaneous. However, customers
in
most
The
to
travel time
node to another.
traveling from one
congestion in the network, as
infinite server
is
When
real
is,
often the case, the delays are appropriately
nodes within the queueing network. Examples include the delay
move between workstations
for
line.
An
in
for a vehicle
in a factory,
infinite server
a large factory, such as a wafer fabrication
approach developed by Harrison (1988) has been used
traffic
and
node
facility.
The
in recent years (see,
example, Wein 1992a, Kelly and Laws 1993 and references therein) to control queueing
networks that possess no
The aim
infinite server nodes.
of this paper
Towards
this end,
idealized queueing
we
focus on a specific problem that
facility
and
K
warehouses.
and a completed unit of a given product
is
modeled
distribution
facility to
unsatisfied
demand
is
The
its
infinite server nodes.
K
is
a
random
consists of
respective warehouse; hence,
In our queueing model, the production
and each product has
variable.
An
different products,
its
own
general service time
and renewal demand process. The delay incurred by a unit to
a warehouse
step
These inventories service the actual customer
backordered.
as a single server,
first
simple but well motivated:
facihty produces
transported to
inventories are held at the warehouses.
facility is
make a
model of a multi-product production-distribution system that
a single production
demand, and
is
to
is
towards generalizing this approach to queueing networks that contain
all
modeled as
be used as an aggregate representation of a large number of non-bottleneck (that
low utilization) workcenters
heavy
assumed
queueing systems incur delays when
data to be transmitted over a dedicated low speed transmission
also
typically
these delays are essentially independent of the
to travel between traffic intersections, a job to
may
is
travel
As alluded to above, one could
from the
also interpret
the model as a production-inventory system with a single bottleneck workstation, where the
delay represents the aggregate sojourn time through the workstations
downstream
of the
bottleneck.
The performance
is
employed.
We
of such a system relies heavily
upon the production
control policy that
derive a control policy for a decentralized system: Each warehouse has an
initial
base stock level of inventory, and whenever a
inventory at warehouse
The
facihty.
no orders are
i
is
for a unit of
product
i
occurs, the
depleted and the warehouse places an order with the production
facility serves orders in
in its queue.
demand
a first-come first-served (FCFS) fashion and
The optimization problem
for the
is
idle
decentrahzed problem
is
if
to
derive a base stock level for each warehouse to minimize the average cost of holding and
backordering inventory.
Since the optimization problem appears to be difficult to analyze without making additional distributional assumptions,
we
resort to a
heavy
traffic analysis.
This approprimation
procedure requires the server to be busy the great majority of time to meet average demand,
and assumes that the
travel delays are lengthy.
Under these assumptions, the inventory pro-
cess for each product, appropriately normalized, has a diffusion
and a Gaussian component.
Although we have been unable to derive the stationary distribution of
cess,
this inventory pro-
our analysis reveals the structure of the hmiting behavior of the production-distribution
system. Under additional assumptions, we employ our analysis to numerically derive base
stock levels, and find that they are very close in value to the optimal levels obtained via
simulation.
We
initially
planned to also analyze a centralized system, where a controller has global
information about the inventories of each product, and decides in a dynamic fashion which
product,
if
any, to produce next.
Comparison of these two systems would allow us to
assess
the value of centraHzed information in production-distribution systems; see Zheng and Zipkin
(1990) and
Wein (1992b)
for similar
comparisons for production-inventory systems, which
are essentially production-distribution systems with no travel delays.
During the course of
our analysis, we developed a variant of a myopic policy proposed by Pena and Zipkin (1992).
Although numerical
al.
results in
Pena and Zipkin, Veatch and Wein (1993) and Krichagina
(1992) show that this policy performs well in a production-inventory setting,
et
we found
that our variant of the policy did not perform well for the production-distribution system;
in fact,
the system did not stabilize under this policy in our simulation study.
that the long travel delays lead to a system that
is
not easily controllable.
It
appears
Consequently,
paper does not contain an analysis of the centralized problem.
this
In
summary, control
policies for
very difficult to analyze;
if
queueing networks with
infinite server delay
nodes are
the delays are not exponential, then the cardinality of the state
space can become unwieldy. Unfortunately, our analysis suggests that these types of problems
not yield easily to heavy
will
The remainder
of this paper
the decentralized model
heavy
traffic is
traffic analysis.
is
organized as follows.
defined in Section
is
introduced in Section
The queueing
and a sequence of systems approaching
2,
An approximating
3.
fluid control
approximating diffusion-Gaussian control problem are formulated
analysis
is
a prerequisite to the more elaborate heavy
the controls in the heavy
traffic
controls. Further analysis of the
model
model
traffic
study that assesses the accuracy of our
undertaken
problem and an
in Section 4;
the fluid
approximation, and allows
to be expressed as deviations
is
control problem for
from the optimal
in Section 5, along
fluid
with a computational
results.
The Problem Formulation
2
The
decentralized production-distribution system pictured in Figure
machine producing
K
products. Units of product
distribution with service rate
max{j
:
{i^j}i>i-
^1
+
•
•
+
^j
<
//,
and squared
=
I,
.
.
.
,K have
{i]j}j>i
be the sequence of
define the continuous distribution function Ff{t)
+
di
denote the demand rate and
Cj
^ 0'
P{ti[
<
t).
iid
finished
good inventory
delays for product
We
which
is
the
number
i/J,
of product
be the square
i
units
=
let Ai{t)
service times
i,
and
denote the mean of this
Each product has an independent renewal process Di{t)
C\
+
=
iid
We
i/^,.
denote the renewal process associated with the
warehouse. Let
distribution by A~^.
consists of a single
a general service time
coefficient of variation
Completed units incur a random delay before entering the
at the appropriate
•
i
1
demanded up
= max{j
to time
coefficient of variation of the
t.
We
:
let
interdemand
times.
We
assume that warehouses operate independently.
units of inventory, and each unit of
demand
for
product
i
Each warehouse
initially
has C,
simultaneously triggers two events:
Product Vs inventory
is
machine serves orders
in
Tj{t) represent the total
is
depleted and a production order
a
FCFS
lime
in [0,/]
the total numbe'- of product
of product
i
fashion and
i
is
idle
if
is
placed with the machine.
no orders are
i's
units produced
delay node.
the queue.
devoted to the production of product
up
to time
t.
t.
z,
then
If
.4
we
let
i/^/))
Q,{t) the
number
Let Zt{t) be the
r.r,'.Qber
Denote
orders in queue or in service at the machine at time
of unite- in product
in
The
l>y
Then
Q.(0
= A(0 - MT,{t))
[•2.r
and
(2.2)
where
r'
when
the
is
the time of the j'^
j*-^
DEMAND
unit of product
1
i
jump
of the process A,{T,{-));
completes production.
it
corresponds to the epoch
product
i,
and define the
cost function
= ^H,ix),
H{x)
1=1
=
where Hi{x)
problem
is
hixf
+
biX~
to choose Ci
,
.
.
.
,
,
J(Ci,
=
x'^
Ck
.
.
.
max(x,0) and x~
—
min(a:,0).
The production
control
to minimize the expected average cost
,
Ck) = lim sup
T— oo
^E
J
f H{X{s))ds
(2.4)
.
^0
Since the optimization problem (2.1)-(2.4) does not appear to be tractable without additional
distributional assumptions,
system operates
in
heavy
we turn
traffic;
to the aid of asymptotic analysis
that
the
is,
P
=
traffic intensity
E^<1
«=i
is
close to
1.
Assumption
of time to satisfy
demand
and assume that the
(2.5)
f^'
(2.5) requires that the
machine be working the great majority
over the long run. Hereafter, the production-distribution system
described in this section will be referred to as the original system S.
For subsequent use, we introduce the large parameter
no
=
(1
-
/>)-'
(2.6)
and the rescaled distribution functions of delay times
F.{t)
3
A
= Ff{t{l-p)-')^F^{not).
Sequence of Systems
To implement the asymptotic
analysis,
in
Heavy
(2.7)
Traffic
we introduce a sequence
of systems
indexed by
S'„
the parameter n {n —^ oo). Each system has the same structure as the original system S.
We
fix {Cj}j>i?
{?/"'} j>i to
^
=
^^---if^i
depend on
n.
which represent the interdemand
intervals,
and allow {^"'}j>i
Thus, the parameters associated with the demand
not depend on n, while the parameters
/i", u^,
and the function Fp{-),
i
=
1,
.
c?,
,
.
.
and
i/^,-
do
A' vary with
n.
To emphasize
T"
yl",
etc.
procedure
The
will
Let p"
=
this
dependence we attach an upper index n to the appropriate processes
initial inventories
C",
be implemented can
dil ji^
=
and p"
=
i
also
1,
.
,
.
.
A' with respect to
depend on
which the optimization
n.
Our main assumption on the sequence Sn
^j-xP".
are the
following:
^r^/i.,
K^^~^a^,
riT
/j"
<
1
I
= «"oS;
1
Ff{not/n), where F,(-)
is
defined in (2.7).
sequence Sn, n
>
1,
•
,
•
/T,
,
(3.1)
(3.2)
=
r/"'
is
1
(3.3)
.
given by P(7;"'
Assumption
which requires the
•
,
and v/^(l - p")
Observe that the distribution function F"(<) of
for the
=
(3.3)
<
the heavy
is
=
traffic
Fi{t/n)
condition
p^ to converge to
traffic intensity
=
1.
As a
consequence of (3.1)-(3.3) we have
p"^
—^ Pi
=
^
d,
^
as
—
n
>
oo,
^
and
/^'
Assumption
=
p,
1
(3.4)
.
.=1
n
(3.2) implies that the delays are increasing as
—
>
oo.
Let the processes Q^{-), Z^{-),X^{-) associated with system Sn be defined similarly to
(2.1)-(2.3):
and
Xnt) = Cr - Q^t) - Znt)
(3.5)
.
For the processes X", Q", Z" and the cost functional associated with the n^^ system, we
will consider
two types of
X"(0 =
rescaling: Fluid rescaling
-J^"(ni)
n
,
J(Ci",...,Q)
Q^{t)
=
-Q'^int)
n
=
,
Z"(0 = -Z'^int)
n
n~'limsup;J;£; /
T— oo T
Jo
c{X''{t))dt
,
(3.6)
(3.7)
we
of tMs sect.on,
remainder
I„ ,,e
wHh product ,.
processes
;
..e
associated
Q,
™acH.e on a FCFS
.e.
n,.U-,dasssysten,s.Let
n,.,Uiclass systems,
mach,ne
the production
U..
descnbe
^^
.
^
=
";-
Then
^^^
^^^^^^^^^ ^^^^^^
^^^
^^„,,
,,,„,. for
^^^^ ^^^^^ ,^„,
^_„,. «n,e du.n, ICl .-
^/^^^^^ ^^^
^U
w
,s .die.
^^^^
,,fined by
defined
P
e's
h-
queue
at
time
t.
Denote the
O^ = "-'H-^'"'' ' *""
atnount of «o.. P^;;;;;;;^^;;;:^
,e,ese„ts the total
^"(0 = "
of W^ by
_^^
^^ ,,^
.escaied version
space DlO.oo)
the
x(.) Iron,
any funcfon
_,„ensionaf
teflect.on
Fo.
capping
(1985));
(see Harrison
that
is,
process
Brownian motion
Define the
WW
W^
W. and
,
„here
W/
^^^^^
= -' + «'.^w + ^'
»"'*''''•
IS
"
*
a retlectea
=
*(IV)()
,
d
^^^^^
t=l
The
+
(3.11)
B(.)
B
'
'
'
"'"
„« with variances Sf^i-'kC'
processes
Wiener
independent
are two
respectively. Set
Ef-,..^,<i^*i;'''3'
SO that
-
'''
following result
is
.f
of
a consequence
Theorem
Theore
8
1
(1991).
of Peterson
Proposition
—
where
>
we have
(3.1)-(3.3)
stands for weak convergence in D^[0,oc).
The Limiting Optimization Problem
4
As mentioned
is
Under assumptions
3.1.
in
the Introduction, a rather crude fluid approximation to the control problem
required as a precursor to the heavy traflBc diffusion-Gaussian approximation.
4.1.
Fluid approximation. For brevity of notation, we introduce G,(<)
Proposition
Assume
4.1.
and n-^C^ ->
(3.1)-(3.3)
c\^\
=
i
=
1
—
Fi{t).
Then for any
1,...,A'.
C >0,
lim sup IX"
-
P-
=
Xi(t)\
a.s.
,
where
Xi{t)
=
c^"^
-
d,
f G,{u)ds
(4.1)
.
JO
Proof. Proposition
3.1 implies that
Or(0^0, n-^oo
uniformly on finite time intervals.
By
1975, page 188), n-'^D,{nt) -^ d,t as n
=
i
,
l,...,K
(4.2)
the elementary renewal theorem (Karlin and Taylor
^
oo.
0^(0 = n-^[D,{nt) -
Since
A^{Tp{nt))],
we
have
n-'^A'^{Tp{nt))
Equation
(3.5)
^ d,t
n
,
^ oo.
impHes
A^(Tr(nt))
xnt) = n-'cr - Qnt) -
n-^
y:
i=l
Observe that n~^Tf\ j
>
I
are the
the distribution function F-'{nt)
number
(4.3)
=
jump moments
F,(<).
w
+ ^r > "O
of y4"(r"(n-))
and
(4.4)
•
n"^?;"*
Therefore, the last term in (4.4)
is
=
tiq
Sj bas
the rescaled
of customers in an infinite server queue with service time distribution i^(-)
and
arrival
stream
A'^ {T-^ (nt))
Theorem
(1984).
in
1
Limit theorems for such models were considered by Borovkov
.
Chapter 2 of Borovkov implies that
A?(Tr(nt))
y
^
''
uniformly on
JivT
>
finite
t
t
+
>
T?'
J
- u)du =
nt) -^ d, / G,{t
Jo
J{C^,
.
fluid rescaling
.
,
.
G,{u)du
P
(4.2)
completes the proof.
can be written as
^E„
/-^
r H{X^{t))dt
1
=
Q)
/
JQ
time intervals, which together with
Observe that the cost with
d,
lim sup
T— oo'
Proposition 4.1 suggests that the following deterministic optimization problem should be
considered as the limiting fluid problem: Choose
Cj
1
fT
„ yj
1
Jo
minimize
lize iimsup— £^
T-.00
where
a;,(-)
are given by (4.1).
If
we
the limit of
3:,(<)
=
n'^C^
as
=
—
cx),
>
we
to
c{xi{i))dt
/
r G,{u)du
d,
and the associated cost
n
,Cj^-
(4.5)
,
set
cp)
then lim(_oo
,...
set
C" =
equal to zero and
is
'
nc,-
(4.6)
,
is
minimal. Since
+ AC", where AC" =
o[n)^
is
c-
and turn our
attention to the next level of approximation.
4.2.
Diffusion-Gaussian approximation.
y;"(0
=
^."(0
- K"(0
=
For each fixed n, define the processes y"(<)
,
<
>
^
,
=
l,...,/^,
(4.7)
where
V;"(<)
=
nc^P
-ndi f
"
G,{u)du
Jo
and
c,-
'
are given by (4.6). Since V-"(<)
i"(Cr,...,C^)
= n-^/Mimsup^£;
7'_oo
1
/
—*
as
<
^
cx)
//(X"(0)d<
Jo
for each n,
we have
= n-^/Mimsup^£; /
T-.00
1
H{Y''{t))dt
.
Jo
(4.8)
10
.
Thus, our goal
minimize the right-hand side of
to
is
Consider the rescaled process ¥"{1)
=
t
>
=
1,
n-^/^Y"-{nt),
that
n~^/^AC" —*
f Fi{t
- u)dB{u) -
Proposition 4.2 Assume
(4.8).
c,
',
/i.
/' G,{t
^0
i
0.
.
.
,
.
Then
A'.
where
Yi{t)
=
cp^
-
d,
Jo
and
Oi{-) is
=
r Fi{s - u)G,{t - u)du
d,
Oiit)
(4.9)
,
mean and covariance function
a Gaussian process with zero
E0,{t)e,{s)
- u)dWf{u) +
<t
s
,
(4.10)
Jo
independent of B{-), Wf{-) and
Proof. Definitions
y;."(<)
= n-v2Acr -
Since n~^T'^\ j
RCLL
(3.5)
=
1,2,
..
.
Oj{-), j
and
(4.7)
n-^/2
/
{ij
i
=
1,.
.
.
,
A')-
imply that
Yl
are sequential
A'/r'
+
^r'
>
"<)
jump moments
- ^r(o +d,
G,it
- u)du
of the process A^{TJ^{n-)), for any
function /(•) one can write
K{Tr(nt))
E
fin-'r-)
=
which
fact
is
and
f
f{u)dAnTnnu))
,
•'^
j=i
a Lebesgue-Stieltjes integral with respect to a nondecreasing function. Using this
we derive
(3.2)
y;"(0
=
n-'/'AC."
-
f F,{t-u)dQ'i[u)- JofG,{t-u)db^iu) + ent),
(4.11)
Jo
where
A?iTp{nt))
0?{t)
=
n-'/'
j]
[/(noS]
+ n-V;- <
t)
-
-
F,{t
n-'r^')]
(4.12)
.
«=i
The
process 6" can be represented as an integral with respect to a
case
is
we
random
field,
which
in this
a stochastic process with a two-dimensional variable. To derive such a representation,
define the
random
fields
[nt]
UJ'it.x)
=
n"'/^ II[/(^.(«oSi)
<
3=1
11
a:)
- max(x,
1)]
,
^
>
,
x
>
and
=
Vrit,x)
n-'/'
<
[/(no^*
Y.
-
x)
F,(x)]
>
i
,
x>0.
,
Hence,
Vnt^x) = C/r(n-Mr(7;"(nO),F.(x))
(4.13)
.
Observe that the random variable F,(no^7/p has a uniform distribution on
and Stute (1979),
statistical literature (see Gaenssler
The
ferred to as the stochastic empirical process.
[0,1].
example), the process
for
trajectories of [/"(•,
t/"
In the
is
re-
belong to the space
•)
D^[0,oo)^ (see Pardjanadze and Hmaladze (1986) and Gaenssler and Stute), which
is
the
space of functions u{t^x) defined on [0,oo) x [0,oo) such that the limits
u{t+,x+) = lim u{s,y)
=
u{t+,x-)
,
lim u{s,y)
u{t—,x+)= hm
u{s,y)
,
u{t — ,x—)=
and u{t+,x+)
=
u{t,x).
The space
rohod's Ji topology for Z)[0,oo)
for details).
An
lim u{s,y)
s]t,y]x
s]t,ylx
exist
,
ait,y]x
slt,ylx
i)^[0,oo)^
endowed with topology
similar to Sko-
a complete metric space (see Pardjanadze and Hmaladze
is
increment of a function u € D^[0,oo)^
=
u((r,y),(/,x))
u{T,y)
-
u{t,x)
is
-
defined by
u{t,y)
+
u{t,x)
.
Using this concept one can define the space of functions with bounded variation
usual
way
(see
(V^"(s,x),5
>
Kolmogorov and Fomin
0,x
>
0)
(1975)). Observe that the trajectories of the process
belong to Z)^[0,oo)^ and have bounded variation.
Lebesgue-Stieltjes integral below
is
e^{t)
well defined
=
in the
f' I{s
Therefore the
and we can write
+ x<
t)dVi^{s,
t)
(4.14)
.
Jo
Using the definition of increments given above, one can easily check that (4.14)
to (4.12). Bickel
weakly
is
equivalent
and Wichura (1971) proved that the empirical process Up{s,x) converges
in D^[0, oo)^ as
n -^ oo to the Gaussian random
field
U{s,x) with zero mean and
covariance function
EU{s,x)U{t,y)
= sAt{x Ay 12
xy)
,
x
<
1,
j/
<
1
.
(4.15)
For fixed
fixed
s.
x,
a standard Brownian Motion, and U{s,-)
is
Since t/"(-,)
(4.3)
3.1,
U{-^x)
is
independent of (5"() and
and (4.13) give the
in (4.11).
=
i
=
1,..,A', Proposition
joint convergence
^
(Or(-), A"(-),K"(-,-))
where Vi{t,x)
each
£>"(•) for
a Brownian Bridge for
is
{d,B{-),fl,Wf{-),V,i;.))
To complete the proof we need
U{dit,Fi{x)).
,
n -. oo
to take the
(4.16)
,
Hmit as n —+ oo
Applying the integration by parts formula and the continuous mapping theorem,
one can easily verify that the second and third terms
in (4.11)
converges to those in
(4.9).
Define
=
Oi{t)
f I{s
+
x
<
t)dV,{s,x)
,
.70
where the integral
Then
(4.14)
and
is
interpreted as the mean-square integral with respect to a Gaussian
field.
(4.16) imply that
e^
^e,
n^oo
,
(4.17)
.
Unfortunately, the integration by parts formula cannot be applied directly in this case, since
I{s
+
X
<
t)
as
a.
function of {s,x) does not belong to D^[0,oo)^. Hence, a careful analysis
is
required to justify this convergence.
it
was carried out
in
Theorem
We
do not provide here the proof of
this result, since
2 of Borovkov by different techniques. Readers are referred to
Krichagina and Puhalskii (1993) for an alternative proof based on the representation (4.14).
The joint convergence
obtain the desired result.
the process K(-,
•)
=
It
=
(4.17) allows us to take the
hmit as n
—
>
oo in (4.11) and
follows from (4.15) that the covariance function Ki{u,v,x,y) of
Ui{dit, Fi{x))
Ki{u,v,x,y)
The
and
of (4.16)
is
given by
EVi{u,x)Vi{v,y)
=
di{u
A v){F{x) A F(y) - F{x)F{y))
.
properties of the mean-square integral imply that
I{u
/
'
^oo
TOO
Jroo
+ x<s)dV,{u,x)
^0
roo
=
/
Jo
roo
/
Jo
Iiv
+ y<t)dV,{v,y)
Jo
^oo
/
Jo
roo
/
I{u
+ x<s)I{v + y<t)dKi{u,v,x,y).
Jo
13
duAv =
Since
=
I{u
the right-hand side ior s
/OO
<
t
/
/
Jo
Jo
Jo
/•TO
/-co
+ x<s)dudF,{x)-
I{u
Calculating the integrals
For each fixed
we obtain
let F,{t),
i,
I{x
=
y)dF,{x) and dFi{x)F,iy)
=
dF,{x)dF,{y),
equal to
is
/OO
(fj/
=
v)du, dF,{x)AF,{y)
/
/(ii
+ a-<5)/(w +
j/<0<iuc?F,(x)(fF.(r/)}
.
Jo
(4.10).
<
i
Jo
/-OO
be the distribution on
(
— oo,0]
generated by F,(-) as
follows:
F,{t)
Denote by
and
'yf
=
1
- F{-t) =
G,{-t)
(the superscripts represent
'yf^'
t<0
,
demand and
.
delay, respectively)
dependent Gaussian random variables with zero means and variances
di
and parameters
that
RBM
Fi{u)Gi{u)du, respectively. Let 5*'(-) be the stationary
f^
it is
as B{-) (see (3.10)-(3.12)). Since B^*
defined for
is
Gi{—t).
f^ G^{u)du and
with the same structure
a stationary process
is
we can assume
f
d,
B'\u)dF{u)
(4.18)
,
J — OO
the stationary Brownian motion averaged by the distribution function Fi{t)
The
in-
^ (— oo,+oo). Define
/
7f =
which
diV^-
two
=
following proposition describes the steady-state distribution of yi(-).
Proposition
4.3.
Under
the assumptions
and
definitions above, Yi{t)
—
>
5^(00) as
—^ 00,
t
where
y;(oo)
Proof. To prove the
result
= cp^-7f-7f +
we take the
7-^'.
limit in (4.9) as
<
—
(4.19)
>
00.
Our
first
step
is
to
show
that
R,{t)
= r F,{t -
u)dB{u)
Jo
An
4 J-/
B''{u)dF,{u)
(4.20)
.
00
application of the integration by parts formula yields
R-{t)
=
/'
Jo
For yV such that
FAt - u)dB{u)
=-
f B{u)dFi{t -u)= Jof B{u)dG,{t - u)
.
(4.21)
Jo
<
TV
<
<,
define
R'^{t)
=
f
B{u)dG,it
Ji-N
14
-
u)
.
(4.22)
By Theorem
from the statements below:
4.2 of Billingsley, (4.20) will follow
R^{t) -^ I B''(u)dF,{u)
J-N
°
L-N B'\u)dFi{u)
f
-i
t^+oo
,
B''\u)dF,{u)
N
lim \im sup P{\R,{t)-R^{t)\
t—oo
we
(4.23),
(4.24)
;
>
=
e)
,
e>0.
(4.25)
= J^ B{t - N + u)dG,{N - u)
that B{t + )—> B^\-) as < —> cx).
^f (<) —
the continuous mapping theorem gives
we have
a stationary process,
+00
by deriving R^{t)
start
Standard arguments can be used to show
is
AT -^
,
(4.23)
;
J — oo
N^co
To prove
N
for fixed
>
/q
B'\u)dGi{N —
/q
B"\u)dGi{N —
u)
=
u),
t
from
(4.22).
For fixed
—^ oo.
A^,
Since B'*
J_^ B'\u)dGi{—u), and
(4.23)
is
proved.
Because EB'\u)
{const)GiiN) ->
=
as TV
const,
it
^ oo.
follows that
Thus, (4.24)
E jZ^ B'\u)dF,{u) =
EB{t) <
for the distribution of B{t),
inequahty and allowing
zero
i
>
mean and
0,
the
i
B
—
>
variances
dii^j-
The
variables
f^'^ B{u)dGi{t-u). The closed
(see p. 15 of Harrison 1985) implies that
Jq Gi{t
jQG]{u)du and
of $i
Gi{t)).
The
(f,
— u)dWf{u) and
0i{t) are
Gaussian with
/q F,(u)G,(u)c?u, respectively.
joint convergence follows
and B, Wf. Thus, one can take the
As
—
<
>
oo,
from the structure
limit in (4.9) as
i
^ 00
result.
cost functional given by (4.8) can be expressed in terms of
limsup;^;^ /
this fact
and Propositions
lim sup
T—.00
»oo
is
^E
-I
as
.
4.2-4.3, the limiting optimization
to the diffusion-Gaussian approximation
mmimize
H{Y''{s))ds
¥"
Jo
J
r-.oo
Based on
Applying Chebyshev's
—^ oo yields (4.25), and hence (4.20).
//;
7^^', respectively.
and the independence
and obtain the
N
oo and then
random
they converge to ff and
of
>
=
Thus E\R,{t) - R^{t)\ < {const){G,iN) -
const.
For each
t
=
verified.
is
Equations (4.21) and (4.22) imply that R,{t)-R^{t)
form expression
{const)Fi{-N)
to choose c[
,
.
[ H{Y{s))ds =
Jo
15
,
.
.
c^-
problem corresponding
to
EH{Y {00))
(4.26)
=
where V'(oo)
(y'i(oo),
,
.
.
^^-(oo))
individually. Thus, (4.26)
i
given by (4.19).
is
The
additive structure of the cost
equalities (4.19) imply that the minimization can
and steady-state
component
.
is
be implemented
for
equivalent to finding
min EH,{Y,{oo))
for each
i
=
1,
.
,
.
.
K
.
Let
c|
,
z
=
1,
.
.
.
each
,
A'
(4.27)
be the solution to
(4.27).
Then our two-stage
asymptotic analysis implies that nearly optimal values for the parameters C",...,C^- are
given by
C^ =
ncl'^
+
V^c*''
i
= l,...,K
Two-Stage Approximation
5
In this section,
we employ the
limiting fluid
tions for the original optimization problem,
additional assumptions.
n >
if
,
j/^-,
that the dynamics of
1 is
we
i
set
—
The
—
//,
1,.
.
.
^i^°
,K
and
i/^,
—
relation
S
for the Original
System
and diffusion-Gaussian problems as approximaand provide numerical solutions under certain
between the original system S and the sequence Sn,
can be identified with that of
{i^aiY-
(4.28)
.
S'no,
where no
is
given by (2.6),
Notice, however, that the hmiting parameters
/i,
and
are not directly connected with those of the original system S; they were
introduced to justify the Hmiting procedure in Propositions 3.1 and 4.1-4.2. Therefore, to
obtain the diffusion-Gaussian approximation for the original system 5, we substitute limiting
"tilde"
The
parameters by the corresponding parameters of the original system.
original optimization
problem with the cost functional
Uq^ can be approximated by the fluid model
(4.1), (4.5).
As
(2.4)
normalized by the factor
the optimal parameters
in (4.6),
are
cS'^
=
d,
r G,{u)du = T^
z
,
=
1,
.
.
.
,
A'
.
A, no
Jo
In accordance with (4.28), the nearly optimal parameters at the fluid level of approximation
are
C.«nocS')
16
= ^.
(5.1)
Since
d
The
are the initial inventories, this expression admits the following explanation.
system operates
in
a balanced regime and the average production rate
demand
rate d, for each product
number
in
inventory
is
number
Therefore, the
i.
approximately equal to the
Over the long run, the average number of product
and therefore product's Vs inventory
of units in the delay node plus the
units at the delay
roughly C,
is
—
equal to the average
inventory C, for any
initial
i
is
If
di/\,.
node
i
=
1,.
.
.
^K.
equal to di/\i,
is
C, differs from (5.1), then
there would be a constant backlog or a constant excess of inventory over the long run, which
would increase the cost functional.
The
diffusion-Gaussian approximation corresponds to the original problem with the cost
functional (2.4) normalized by the factor no
(4.28),
we must
solve the minimization
To
.
find the corresponding
term y/noc-
problem
min EH,{Yi{oo))ds
for each
i
=
!,...,/<',
= c<^)-7f-7f + 7f^
7f is
the stationary
and ff and
ances diU^-
ff^'
RBM
f^ G^{u)du and
is
C,
fluid
on
[0,
d,
f
B'\u)dF,{u)
di
J^
,
oo) with drift
—p and
random
variance X^aLi
+
'^dk)i
mean and
vari-
^ArA'fc ^('^ait
variables with zero
F,{u)Gi{u)du, respectively. Once the parameter
found,
=
(5.3)
J — CO
are two independent Gaussian
minimizes the cost (5.2)
The
(5.2)
where
K(cx))
B^^
ci'^no
we use
+
and the
c-
'
that
(4.28) to set
cj"v/n^
=
d,/X,
+
c!'V(l
-
(5-4)
p)
approximation yields the crude deterministic description of the system
of the average values,
in
diffusion- Gaussian
in
terms
approximation provides the distribution of
fluctuations around these average values. Thus, the second term on the right-hand side of
(5.3) characterizes the steady-state inventory's fluctuations
size of the
caused by the variation in the
machine's queue. The third term represents the impact of
17
demand
variabihty,
and
By
the fourth term quantifies the effect of fluctuations in delay.
and
distribution function of the delay times,
Due
these quantities.
it
to the rescaHng procedure
as the length of delays, in the
compressed time
(2.7), F,(-)
—
{t
>
nt),
we observe
random
easily derived.
variables
7f and ^f
all
Thus, our analysis
scale.
is
processes, as well
oriented towards
"wide" distribution functions
is,
Unfortunately, problem (5.2) does not admit an exphcit solution.
that the
the resettled
plays an integral role in the definition of
systems with long and significantly uncertain delays (that
is
is
The main
difficulty
are dependent and their joint distribution
However, the expression for the steady-state distribution
not
is
(5.3) reveals the
structure of the limiting behavior of production-distribution systems with long delays, and
can be used
that
F^
is
for their analysis.
To
illustrate the use of
a uniform distribution on the interval
our analysis, we assume for simplicity
- af''')/\,,
[(1
(1 -f af^')/A,],
where
af^'
<
characterizes the uncertainty of delays. In this case, the rescaled distribution function Fi
uniform on an interval of length
/,
=
2af^'{l
^ Jo/' B'\u)du
—
RBM
on
[0,
Define the unitless constant I,
=
d,h
equal
is
is
in distribution to
(5.5)
.
Jo
oo) with drift
=
and ff
C d,B'\hu)du
=
/;
Let B-*{-) be the unitless
p)^/\,,
1
—p and
variance
- pYidJX,).
2af''{l
Since d,B'\l,-)
=
Bf{Li-),
equation (5.5) gives
/'
7f = Jo Bf{L,u)du = ]-
r Bf{u)du
(5.6)
.
Li Jo
The
variances of ff and jf^' are equal to
4L,/3 and
respectively.
In the remainder of this paper,
magnitude of
Z/,.
Case
1.
Suppose that the parameter L,
enough or have low
variability, or the
is
we
L./6,
investigate three cases according to the
small. This
demand
18
(5.7)
rate
is
happens when the delays are not long
small.
Then the
distribution of
7,-
is
close to that of Bf'(O), which
variables 7/
and
7^^'
have
is
sniall variances
and can be neglected.
V;2
jf).
Thus,
/
1
ip
—
+
h,
^^
(2)
minimizes EH,{c\''
By
exponential with parameter 'Ip/E^.
-
(5.7), the
random
easy to calculate that
It is
b,
h,
for small L,, the variability in the size of the
machine
s
queue has the largest impact on the fluctuations of the inventory.
w— *•
.i.- .- ;:.. -t. ^
'
J
L=0.1
L=1.0
L=5.0
EXP. DISTR.
EXP. MEAN
0.6
0.4
0.2
0.5
0.4
Figure
Case
By
2.
2.
The simulated
Suppose that L,
is
large.
distributions of
By
f^
0.6
for various value.s of L,.
(5.7), the variances of
7/ and
-ff^'
are large a? well.
the Birkhoff- ivhinchinc theorem, we have
-- /"" Bfiu)du
^ EBf {00)
=
L, Jo
Hence, by
(5.6),
7,^ Ri TJlj2p.
the distribution of ^f for large
Since
'^f
and
7,*^^'
//,
are independent, their
19
^
Ip
,
I,
^
00
.
has small variability, and one
sum
.,an let
has a Gaussian distribati:'n with
zero
mean and
variance Li{i/jj3
cS''
+
1/6).
The
value of the parameter
= argmin,//.(c-S +
2p
is
+
^f
^_<^e/)
easy to find numerically.
Case
3.
Problem
(5.2)
is
more
further progress in this case,
neglected.
That
assumption
is
is, i/J, is
difficult
we make the
small and the
when
L,
neither small nor large.
is
additional assumption that
demand
is
'yf
«
To make
and can be
nearly constant. As alluded to earlier, this
required because jf and 7,^ are dependent
random
variables.
The
distribution
of 'yf also cannot be calculated explicitly. However, a one-dimensional reflected diffusion
is
easily simulated.
The simulated
predicted by Case
distributions of
1,
we observe
jf
that for small values of L, the distribution
exponential with parameter 2p/S^ shown by the dashed
line.
mean
of the exponential distribution
shown by the
vertical line.
function Hi{x) averaged by the convolution of the distribution of
and a normal distribution with mean
for
any given
c,
and
find the
('2\
c\
is
is
As
2.
close to the
In accordance with
Figure 2 also indicates that for larger values of L, the distribution
the
Figure
for various values of Li are presented in
Case
2,
concentrated around
By
calculating the cost
7P found by simulation
and variance Li/G, we were able to obtain the cost
minimum. Tables
1
and
2
compare our
with the results obtained via simulation of the original system
results
(COST SIM)
(COST TH)
for one-
and
two-dimensional problems. The service times were assumed to be uniformly distributed on
[(1
—
Q,)///i, (1 -f Qi)///,],
i
=
1,
.
,
.
.
A'.
In all four problems, our cost minimizing values of
Ci are very close to those obtained via simulation.
20
Problem
Number
parameters: Hi =
of simulated jobs
1
:
Parameters:
/Xi
=
1.0, //2
Qi =0.6,a2
Number
COST
= 0.8, d^ = 0.5, (ij = 0.32,
= 0.8, af = Q^ = 0.8.
of simulated jobs: 2.0 x 10^.
SIM:
'
{p
=
0.9), l/A^
=
I/A2
=
50.0,
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24
G178 i6:
MIT LIBRARIES
3
TOflQ
DDfi2MD7T
S
Date Due
Lib-26-67