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A Multiclass Hybrid Production
Center in Heavy TraflBc
Vien Nguyen
#3813-94-MSA
August 1994
A
Multiclass Hybrid Production Center in
Heavy
Traffic
Viin Nguyen
MA
Sloan School of Management, M.I.T., Cambridge,
02139
Abstract
This paper presents an analysis of a single-stage hybrid production system that nnakes
multiple types of products,
The
some of which
are
made
to-order while others are
made
analysis begins with a formal heavy traffic limit theorem of the production system,
which
is
modeled
as a
mixed queueing network. Taking
insights from the limit theorem,
the analysis continues with the development of an approximation procedure.
experiments indicate that this procedure provides good estimates
and bounds such as
fill
rates
and average inventory
KEYWORDS: multiclass queueing networks,
tion,
to-stock.
for
Numerical
performance measures
levels.
mixed queueing networks, make-to-order produc-
make-to-stock production, diffusion approximation, reflected Brownian motion, perfor-
mance
analysis.
Contents:
Introduction
1.
The Network Equations
2.
Centering and Scaling
3.
The Heavy
4.
The Approximation
5.
Numericed Results
6.
Proof of the Heavy Traffic Limit Theorem
Traffic Limit
Theorem
JUN 2
Procediu"e
8 1995
L.IBRAB>ES
References
August, 1994
This paper presents an analysis of a "hybrid" production system that makes multiple
types of products,
produced
in
some
of which are produced to inventory (make-to-stock), while others are
response to actual customer demands (make-to-order).
performance analysis of the production system depicted
in
Figure
We develop a procedure
In particular,
1.
for
we envision
the production process as a single aggregate operation with first-in-first-out (FIFO) service
discipline.
That
is,
Production of make-to-stock items follows a policy of one-for-one replenishment.
a base stock
level is specified for
each type of products;
demand
is filled
from finished-
goods-inventory; and each item pulled from inventory triggers a replenishment order to restore
the finished-goods-inventory to the desired base stock level.
Demands
met due
that cannot be
to insufficient inventory will be considered lost.
Typed+1
Make- to-order
Typed+5
requests
Make- to-stock
requests
Replenishment orders
Figure
We
1:
A
workstation with mixed jobs of multiple types
propose to study the production system depicted
network shown
in Figiire 2.
Station
in
Figure
1
via the "mixed" queueing
represents the workstation (herccifter interchangeably
signals a production request
referred to as "workcenter"):
an
service completion at station
corresponds to the production of an item. Stations
arrival at station
1
and each
to d
model
the finished-goods-inventories (FGI) for make-to-stock products: items in queue k represent the
FGI
of type k {I
demands
(i.e.,
<
k
<
d)
and service durations
inter-demand times) of product
k.
at station k correspond to intervals
Each
filled
demand
triggers a corresponding
replenishment order, so jobs that "depart" from station k are routed to station
demands that cannot be
filled
from inventory are simply
lost,
the
between
number
0.
Because
of items in
FGI
Class
d+I
Class
d+2
O
Class c
3
Make-to-order
products
Class
-0
1
Class 2
<a
Make-to-stock
products
Class
d
<Z)[
Figiire 2:
A
multiclass,
mixed queueing network
remains nonnegative; moreover, the number of items
replenishment orders at the workcenter
equals the pre-specified base stock
for
in
FGI summed with
each product type
is
constant at
the
all
number
of
times and
the language of queueing networks, make-to-order
level. In
products are "open" jobs whereas mjike-to-stock products are "closed' jobs.
In
some
settings, the nature of the
customer orders
(for ex£m:iple, items
to inventory (for example,
of operation can provide
Giillii,
more
line
may
require that
some items be made
with a high level of customization), and others be
commodity
however, a recent study by Carr,
mode
product
items).
Even
in
to
made
a nominally make-to-stock scenario,
Jackson and Muckstadt (1993) suggests that a hybrid
efficient service.
Although there
is
an abundance of
results
regarding analysis and optimization of production systems that are either make-to-order or
make-to-stock, there are few results for systems that employ both types of production (see
Basket, Chandy,
Muntz and
Palacios 1976).
approximation procedure, based on heavy
system
in
type (that
A
previous paper by
traffic theory, for
Nguyen
(1994) developed an
a single-stage hybrid production
which each make-to-order and make-to-stock category contains exactly one product
is,
d
=
1
and
c
=
2 in Figiires
1
and
2). In
the present paper, we extend the analysis
to the case in which each make-to-order/make-to-stock category
may
contain several types of
products.
We
will label
make-to-stock products as types
I
to d. This convention conveniently corre-
sponds to the numbering of stations representing finished-goods-inventories, so that make-to-
stock products of type k alternately
of type k
demand
and
visit stations
rate
+
level for
to
I
c.
(SCV) of type k demands
c^
is
service processes at stations
1
SCV
and the
demand
depletion (during which
demands
of type k processing times
may
first
by
not be a faithful representation
inter-demand time following a period of inventory
are lost) typically
is
not statistically similar to other inter-
Nonetheless, Nguyen (1994) noted that the difference
intervals.
is Cq;^-
Strictly speaking, a service process characterized
to d.
process because the
coefficient
processes of make-to-stock products by the
independent and identically distributed service times
demand
products
Let A^ be the
and mojk be the mean production time of type k products; the squared
Figure 2 suggests that we model the
demand
The base-stock
nt- Make-to-order products will be designated by types d
is
of variation
of the
k.
not significant
is
in
the sense that the two systems closely approximate each other under heavy traffic conditions
(see also Iglehart
and Whitt
convenient to define mfc
station
We
for
I
A:
for
will
<
A;
<
1
<
A:
=
d),
the "relative"
thus proceed with our setup, in which case
A^', with the interpretation that
the
rrik is
mean
it
will
be
"service" time at
d.
assume throughout
<
We
1970).
paper that
this
<
product types, A^
for all
mok < mk
(i.e.,
ttIq^'
which requires production rates to exceed those of customer demands. Define
traffic intensities at station j to
_
be
Efc=i-^fc"iofc
J
[1
J
=0
1
<
J
<
d.
Next, define
n
Given
finite
positive
base-stock levels (ni,
number
strictly less
.
= Y,nk
.
.
,
and
nj), the
fill
3k
= -.
(2)
rate of type k jobs,
<
1
than one. (For make-to-stock products, the
be the fraction of orders that are
filled
from inventory.) Denoting type
actual throughput rates are calculated from
AfcQfc
and the actual
<
fc
fill
A:
rate
fill
be some
d, will
is
defined to
rate by
traffic intensities
Qfc,
the
are given
by
^
.
^'
One
expects the
f
1^=1
\
Ofc
(AfcQk) mojk
-t-
EUd+i ^kmok
rate of type k products to approach
fill
1
j
=
1
<
>
may
(so that n*
=
/3fcn
-+
cx3
as well).
Then q*
—
1
for
(/3i
each
Qok{t),
1
<
A:
<
define Qkit) to be the
c,
be the number of type
number
of jobs at station
A;
1
jobs at station
<
fc
<
d at time
,
.
.
.
=
A;
therefore view equation (1) as an initial approximation of (3)
Let,
d.
as the type k base-stock level
increases. In other words, let us fix the proportions of closed jobs
n — oo
<
J
1,
,
.
>
3d)
.
.
,
while letting
d as n
- oo.
We
when n
is
large.
at time
t.
If in
addition we
=
nk-Qok{t)-
t,
then Qk{t)
One
interprets
Qok
of type k products.
as type
k work-in-process inventory and Q^ as the finished-goods-inventory
We
show herein that when the workstation
will
is
roughly balanced, namely,
if
c
A)
=
^><k^TTOk
^
1,
k=l
when
the following approximation holds
Oo»(-)
where Wq{-)
is
n
the base-stock level
s
„--
zQoti"'')
a reflected Brownian motion
.
=
large:
is
A.W„°((),
.
....
(RBM) on
(4)
the interval
[0,6]
having
drift 0°
and
variance a^, with
=
b
Tmn{Tni3i,...,Tndi3d}
(5)
9°
= n(A)-l)
(6)
a'
=
(7)
i:Afcm2,(c^ + c2,).
fc=i
The remainder
of the paper
is
organized as follows. Section
are of interest in the heavy traffic limit
continues to set
up
for the
3.
statement of the limit theorem.
The approximation procedure
investigated in Section
5.
is
presents the processes that
and the approximation procedure. Section
restilt
conventions as well as the assumptions of heavy
Section
1
traffic.
It
discusses centering
The formal Umit theorem
then presented in Section 4 and
its
and scaling
is
stated in
performance
main theorems
Lastly, Section 6 contains proofs of the
in the paper.
We will take as primitive the demand and service time processes for each customer type.
by Dk{t) the number of demands
m
will
for
think of Vok(m) as the
sum
one
of
is,
let
As explained
in the
SotCO
= max{m
products by services completed at stations
<
it
<
d.
think of Dfc as a
sum of the
first
:
Vokim)
<
t}.
For concreteness, one
m independent and identically distributed random variables,
introductory remarks,
to denote by Vt the ctimulative
may
Denote
Sok be the renewal process associated with the
which case So* would be a renewal process, but again, we need not
situation.
1
t:
not require this restriction. Let Voki'm) be the
cumulative sums process Vot; that
in
type k products by time
processing times of class k products, and
may
is
The Network Equations
1
renewal process although we
2
1
to
d.
we
It
will
is
restrict ourselves to this
model demands of make-to-stock
thus consistent with our notation
sums process corresponding
to the counting process
Dk
for
Let Tkit) be the amount of time the workcenter (station 0) has spent processing class k
jobs in the the interval
[0, (].
Denoting by Bj{t), the cumulative "busy" time at station
we
j,
find that
Bo{t)
= ^n{t).
(8)
k=l
Define
.
l<k<d,
d + l <k<c,
/ D,iB,{t))
_
,,,
Dkit)
[
and
Ak{t)
d+l <
For
k
<
Aok{t)
c,
is
=
l<k<d.
Sok{Tk{t)),
simply the number of demands for make-to-order products of type
k to enter the workcenter by time
For
t.
make-to-stock demands that have been
<
1
k
filled
<
d,
one interprets Aok{t) as the number type k
from inventory; equivalently,
production requests to enter the workcenter by time
to enter the type k
FGI by time
Mokit)
and
(10)
t.
=
t.
Similarly, Ak{t)
is
it is
the
the
number
number
of
of items
Next, define
Mk{t)
VokiAokit)),
=
(11)
VkiAkit)),
set
I
i<]<d.
MAO
with the interpretation of Lj{t) as the amoimt of work (or load) that has arrived to station j
by time
t.
Let us assume that initially the
FGI
of each product type
is full
and the workcenter contains
no production requests:
<3fc(0)
=
\<k<d,
nfc,
and
Qoki^)
=0,
1
<
fc
<
(13)
c.
Denote by Wj{t) the sum of aD remaining processing times associated with those jobs found
at station j at time
t
be referred to as the "wx)rk" at station
(this quantity will
j).
Define the
net flow process
Xj{t)
Letting
Ij{t)
be the cumulative
idle
=
W,{<^)
+
Lj{t)-t.
time of station
it
j,
(14)
follows from the previous definitions
that
I,{t)
= t-Bj{t)
(15)
and
Wj{t)
=
W,{0)
+
L,{t)
-
B,{t)
=
Xj{t)
+
Ij{t).
(16)
The
first-in-first-out service discipline
may make
work-conserving, hence we
is
the following
additional statements regarding the idleness processes:
Next,
is
continuous and nondecreasing with
IJ
is
/_,
increeises
only at times
t
a customer in service, and set
to
it
be
t
=
with Wj{t)
be the arrival time of the customer
let T]{t)
/_,(0)
=0,
0.
(18)
in service at station
otherwise.
(17)
at time
a consequence of the
It is
out policy that the number of type k departures from the workstation at time
^k{'n{^))
—
6k{t),
type k and
where
Sk(t)
otherwise.
is
is
1
if
=
there
if
first-in-firstt
is
given by
the job currently in service at the workstation belongs to
The job count
Qokit)
t
Ak{t)
-
processes at the workstation are thus given by
Ak{T}{t))
from which we obtain the number of items
l<k<c,
+ Skit),
(19)
in the finished-goods-inventory:
l<k<d.
Qk{t)=nk-Qok{t),
(20)
Moreover, the allocation processes Tk{t) obey a similar property:
Tkit)
where
eik{t)
f ifc(0
=
is
the
amount
=
Mokivit))
of service the current jobs has received
(21)
if
that job
is
of class k
and
otherwise. Finally, observe that
V{t)
with £2(0 being
if
VVo(t)
=
=
t-Wo{T]{t))
+
(22)
e2it),
and otherwise being equal to the remaining service time
customer currently occupying station
2
To
l<k<c,
+ eikit),
of the
0.
Centering and Scaling
state the heavy traffic theorem
we need
to express the processes defined in section
1
in
terms of processes that have been "centered" and "scaled." Denoting by [xj the integer part
of X, let us define the following centered processes for
Vokit)
Aokit)
Dkit)
=
=
=
Vofc(UJ)-mofcUJ
Aok{t)-Xkt
Dk{t)-Xkt
/c
=
1,
.
.
.
,c:
Sokit)-m^i^t
Mokit)
=
=
fkit)
=
Tkit)
Sokit)
Mokit)
- Xkmokt
- Xkmokt,
(23)
and
for
A;
=
1,
.
.
.
,d:
Vkit)
From equations
(9)-(ll), (21)
=
fk{t)
=
Vk{[t\)-mk[t\.
and (23)-(24), we
VokiDkiBkivitm +
+
XkmokWo{T]{t))
=
Vok{Dk(Bk{T,{t))))
Afcmofc [Xoiriit))
+
(24)
Ccin write the
-
rn^kDkiBkiriit)))
-
mckDk{Bkiri{t)))
Ikivit))
+
-
XkmokIk{r}{t))
+ eifc(0
Afcmofc£2(0
+
centered allocation processes as
Io{T}{t))]
+
Afcmofct2(0
+
(25)
£ifc(0-
Next, set
Efc=i [Vok{Dk{Bk{t))) +mokDk{Bk{t))]
^At)
Efc=d+i [Vok{Dk{t))
=
W,{0)
V,{Soj{Tj{t)))
4-
m,D,{B,{rj{t)))
-
+
+ mokDkit)] +{po-l)t
+
m,So:{T,{t))
^o{r]{t))
J
=
0,
1
<
J
+ ^Vo,{D,{B,{ri{tm +
+ e^it) + ^eij(i)
<
d-
(26)
With equations (25)-(26) and
and workload processes
X,{t)
=
in
the observation Io{t)
=
Io{ri(t)),
we can now
write the netflow
the following form:
eo(0
-
ELi ^kmokhit)
^j{t)
-
Io{t)
-
I,iv{i))
j=0
(27)
+ T.k=i >^kTnokIk{r]{t)),
I
<
j
<
j
=
0,
I
<
J
d,
and
W,it)
We
=
^o(0
+
hit)
^j(f)
-
Io{t)
- T.Li Xkmokhit)
+ Efc=, Afcmofc/fc(7y(t)) +
are interested in characterizing these processes
and the workcenter
is
approximately balanced. To do
[7^(0
when
so,
-
(28)
/;(7?(0)1
<
d.
the base stock levels Uk are large
we
will
prove a hmit theorem for a
sequence of networks whose parameters obey these conditions. Let us then consider a sequence
of networks that are indexed by n.
0i,--,0d, but the
all
integer values of
(so
sum
The
the relative proportions of base stock inventories
n/t,
mean
processing times and
n^^ system has a base stock level of Uk
base-stock levels
we always have
being the
fix
level of the base-stock as well as
vary along the sequence.
and the sum of
We
is
given by n.
Of course
it
— nPk
for
demand
rates
type k products,
makes sense only
to consider
and we can do so by defining Uk to be the integer part plus one of n0k
positive base-stock levels) with the corresponding total base-stock level
of all nt's.
In the interest of keeping the exposition simple, however, let us
proceed with the n^'s are originally defined while keeping
in
mind that
this
without any
is
loss
of representation power.
We
denote parameters and processes associated with the
will
and mQ^
n"" system are X]^
and
as in (1) with A^"^
n —»
oo,
respectively.
The
traffic intensities p'"^ are
mQf.' in place of A^t
and
mofe.
,
and the following condition holds
n
That
is,
at the
system by a superscript
For example, the demand rates and mean processing times of type k products
"(n)".
as
n'*"
we are
some
^
is
then defined exactly
require that A^"'
—
A^, mo^^
»
— mok
finite 0:
as
n
-^ oo.
(29)
and
interested in the regime where base stock levels are high
workcenter
The heavy
-\)
(pj,"^
for
We
in the
the traffic intensity
approximately one.
theorem described
traffic limit
next section applies to processes whose
in the
space and time dimensions have been scaled. Let us describe here the three scaling conventions
that will be used. For a "generic" process X^^\t), set
X"(0 =
We
-X^'^Hn^t),
n
will refer to the first
respectively. It
is
X"(t)
=
-X(")(nt),
n
X^{t)
for
example, X^{t)
whose state space has been aggregated by a
accelerated by a factor of n^.
One can
(30)
n'
two scaling conventions as the "diffusion
understood that,
= ^X^^^nH).
is
factor of
scale"
and the
"fluid scale,"
the process associated with
n'**
system
n and whose time dimension has been
verify that the diffusion scaled
workload processes now
take the following form:
iEit)
^;(()
+
-
/o"(<)
- E'=, Ai"^mS'/?(0
ISit)
+
/"
is
T,U Ai"^mS,"fc^/,"(^
J
(t))
+
[l^{t)
- I^iV
continuous and nondecreeising with /"(O)
/" increases only at times
t
with VV"(t)
=
0,
\<j<d,
{t))\
—
=
(31)
(32)
0,
(33)
0,
and
Et.
= n = n
V^liDk {Bk {t)))
HUd+i
V^;(0)
C(o =
{
+
V'o'i(Dfc
+
(n)
m'^,>D]^{Bk
{t))
(0)+mS'i>?(0 + n
Vp{Soj^{T,^{t)))
+
+
(p^"'
-
l)
t
J
=
0,
=n
(n)c
m';'S^,{T, {t)) +
(34)
m.
1
m 0]
<
j
<
d-
,
The Heavy
3
The
/
:
setting here
[0,
—
oo)
3?^"'"'
>
the space
is
Traffic Limit
the d
D'^"'"',
+
Theorem
1-dimensional product space of functions
that are right continuous and have
left
Hmits.
with the Skorohod J\ topology. For a sequence of processes
symbol "X^^=>X" means "X" converges to
u.o.c."
if
X"
almost surely,
X
converges to
X in distribution."
Assumption
(Ko",
D")
demand and
1
As n
— {Vo',D')
W^iO)
u.o.c, where
D*
-»
we make the
comment
D^H)
15 (0,
Moreover, we write
the
D"^"*"',
"X" —
»
X
following assumptions regarding
=
rnokt
<
I
and
j
< d
D'^{t)
=
(recall that
W^{0) =
0),
and assume that
their
and
Xkt.
meaning
will
D")=>(Vo
,
Xkcl) Brownian motion, and
hold, for example,
if
D*), where
we
will
be clear without
all
V'ofcCO i^ {^^''^ok'^k)
Recall the the definitions of
ready to state the main
Under
b, a'^
and
inter-demand times and service times of all product
from
(5), (7),
and
random
(29), respectively.
result:
the heavy traffic condition (29)
and Assumptions
1
and
2,
<
d,
where
d
W^{t)
=
W;{t)
= m,0,-W^{t),
ilit)
+
/o-(0
-
^ A,moj/;(0.
l<]<d,
and variance a^
^0 is
Brownian motion with
I* is
continuous and nondecreasing,
Ij
increases only at times
Qlkit)
=
^k^oit),
Brownian mo-
component processes are independent.
types are independent sequences of independent and identically distributed
1
e
are vector processes defined in the obvious way. Henceforth,
-^ oo, {Vq,
The assumptions above
Theorem
X
or definition.
Assumption 2 As n
tion,
mj0j for
1/ofc(0
similarly write processes in vector form
further
and
endowed
is
service processes.
-^ oo,
Here, V^, D", Vq, and
in D*^"*"'
D'^"*''
uniformly on compact sets (see Billingsley 1968).
In addition to the heavy traffic condition (29),
the primitive
X"
The space
t
drift
<
when W*{t) =
l<k<c.
j
0,
<
d,
<
j
variables.
We
are
now
Remark:
One can
where ^q and
equivalently write
are as defined in the statement of the theorem and
Iq
d
r*(i)=^A,mo,/;(0j=i
Thus
times
we
Y*
defined,
t
is
when Wq (0
=
^i0j
=
I,
.
.
.
,d.
Y
j
=
I,
.
.
.
,d.
Because Brownian motion
continuous,
is
reaches the smallest of the parameters
In this form, one recognizes
Wq
as one-dimensional reflected
will increase
[0, b]
4
now
some
at
Wq
motion on the interval
Let us
fo^
and K* increases only
0,
whenever
see that the process
rrijPj, j
^(0) =
continuous and nondecreasing with
with drift 6 and variance
a'^,
where
b
=
min{mi/?i
,
.
.
.
,
Brownian
rudPci}-
The Approximation Procedure
interpret the heavy traffic limit theorem
(Theorem
1) in
terms of the production
inventory model, which will help us to develop an approximation procedure. Recall our notation
for describing the
SCV
c^,
production process: demands for type k products occur with rate A^ and
and production times
for
type k products have
mean
rrik
and
Let Uk be the base-stock level of make-to-stock type k products {k
the
sum
of
all
suggests that
base-stock levels
when
n/t;
set
the total base-stock level
process at the production center
motion
and
(RBM) on an
is
0k to be the ratio of
n becomes
large, the
rik
=
SCV
1,
.
.
.
to the
c^^ {k
,
=
1,
d); define
sum
n.
.
n
.
.
,c).
to be
Theorem
1
behavior of the scaled workload
approximately that of a one-dimensionai reflected Brownian
interval; that
is,
n
where
Wq
is
an
RBM
on the
interval [0,6] with drift 9°
and variance
introductory remarks:
6
=
9"
=
(7^
=
min{mi/3i,...,md/3<i},
c
n(5^Afc7nofc-l),
c
5^
Afcmgfc(c^
k=\
10
-f-
cgj.
a^, defined as in the
As explained
approximation
in
for the original
procedure results
where
Wq
is
Harrison and Nguyen (1993), we simply "reverse" the scaling to obtain an
an
in
workload process Wq.
the approximation of
RBM
on the
interval
nb
whose
[0,
=
It is
straightforward to verify that this
Wq by Wq:
nb] with
min{mini,.
.
.
,Tnttnti},
drift is
c
M = Yi>'kmok-
(35)
1,
ifc=i
and whose variance
We
is
again a^ (as defined in
(7)).
an approximation that treats the workload at the production center
firrive at
bounded process, even though the actual workload may become arbitrarily
level,
=
On an
intuitive
one can justify the approximation with the following argument. For the sake of simplicity,
and without
(d
large.
as a
1).
now
consider the case with one make-to-stock product
n make-to-stock products
are queued at the production center, the finished-
loss of generality, let us for
When
all
goods-inventory must be empty. All make-to-stock demands that occur during this time are
thus
lost,
and no new production requests can be
initiated.
of time, the total rate at which work enters the workstation
and the production center no longer operates
in
the heavy
Therefore, during this period
falls
traffic
below the
regime.
critical level,
The make-to-stock
products thus have an effect of regulating the production center's workload and preventing
from becoming "too large"
reaiders
may
refer to
relative to its base-stock level. (For
Nguyen
more discussion on
it
this issue,
1994.)
Continuing with Theorem
1,
we
find that the job count process
workload where the proportionality constant
is
given by the
demand
is
proportional to the
rate; that
is
we have the
1
<
approximation
Qoki-)
^
XkWoi-).
Moreover, we find that Qk, the finished-goods-inventory of type k products,
approximated by a reflected Brownian motion on the
[
—
(mfcTifc
-
fc
<
d,
is
interval
nb),nk]
rrik
(recall that
A^
*
=
rrik)-
If
A;
corresponds to the index such that
mkUk =
nb, then this states
that the type k inventory process approximately follows that of a Brownian motion and
11
may
assume values between
and
For a type k product such that TUkTik
n^.
approximation of the inventory process
from zero. That
is,
One can explain
in
is
a Brownian motion that
the "intuition" of this heavy
the total arrival rate of work
Let us interpret
product type that
strictly
away
it
rrikTik
same way that we explained
Whenever one
of the products stocks out,
critical level;
therefore
the production center essentially
able to keep
is
stocks out
is
other inventories adequately
all
as the expected duration of time that a
full
inventory of
us refer to this duration as
let
and the eventual
In the heavy traffic scaling
first
bounded
is
the
traffic result in
items can satisfy demands without replenishment, and
A:
the "buffer time."
it is
below the
falls
has temporary "excess" capacity;
type
nb, however, the
our approximation, such a product type never stocks out!
the boundedness of the limiting workload process:
replenished.
>
limit,
it
turns out that the
the one having the smallest buffer time, and consequently,
We
the only product type to ever stock out.
will call this
the "bottleneck product type."
For performance analysis purposes, such a result appears woefully inadequate. As we
we present some numerical
see in the next section, where
approximations
results,
will
turns out that the heavy
it
a sharper result
for
throughput rates, we will devote some attention to the discussion of bounds. Before doing
so,
traffic
however,
let
is
RBM
queue lengths are quite good. In order to
us review the approximation procedure that
theorem (Theorem
an
for
I).
First, the
is
offer
suggested by the heavy
workload process at the workcenter
whose parameters we have
traffic limit
approximated by Wq,
is
specified. Second, the inventory level of
each product type
^ A^VVbC)- Third, a make-to-stock product
k will risk stocking out if and only if mfcUfc = min{mini,
man^}= 2,
It follows from
,d.
illustration, let us suppose that mini < rukUk for all
approximated via the linear relationship Qoki)
of type
For
.
.
,
fc
Theorem
1
and
otu" interpretations
^o{t)
where
.
^o
is
Brownian motion with
that the
RBM Wq
= Ut) +
drift
/i
io{t)
is
1
finished- goods- inventory.
RBM Wq
regulators, respectively, of the
Alternatively, /q
process at any other finished-goods-inventory
than one product type achieving the
interval
is
minimum
.
- Aimoi/i(0,
(defined in equation (35))
on the
.
given by
the approximating idleness process at the workcenter; and
process at type
.
[0,
I\
is
and
and variance
cr^;
/o is
the approximating idleness
I\
are the lower and upper
mini]. The approximating idleness
simply zero for
"buffer time,"
all
times
t.
If
there are
we propose that the
more
idleness
process for each of these product types be approximated in the same manner. For example,
mini = miUi = min{mini,
type
j, j
=
.
.
.
if
,m(ind}, then the approximating idleness process for product
1,2 obeys
^o{t)
=
io{t)
+
/o(0
12
-
Ajmoj/,(t).
Because characterization of
RBM,
c product types
all
namely, Wq, one can explicitly calculate
ing the
fill
and average inventory
rates
can be collapsed into a one-dimensional
many performance measures
levels of
of interest, includ-
each product type. The formulas involved
in
these calculations are those obtained from steady-state analysis of reflected Brownian motion
on an
interval,
and an excellent reference of
this material
is
Harrison (1985).
The
details for
translating those formulas into performance measures of the production system are presented
in
Nguyen
(1994).
mation method, which we
will
will
apply here
now standard
modifications are
we
Nguyen (1994)
In addition,
in
in
discusses
to the approxi-
our calculation. Because these calculations and
the literature of heavy traffic approximation of networks,
not repeat them here, and simply point readers to references such as Dai and Harrison
(1992), Harrison
and Nguyen (1990 and
section with a discussion of
Lower Bound: For
some
A
and Nguyen (1994)
1993),
illustration, let us
100%
achieved throughput to
We
=
and
2
demand
rate).
now
Let us
c
=
3),
fill
rate
is
would indicate that type
defined to be the ratio of
consider a variant of the system, called
products are modeled as make-to-order, or open, jobs,
1
and the remaining product types are
as before.
That
is,
we now have a mixed network with
one type of make-to-stock products and two types of make-to-order products. The
can be applied to obtain an approximation
1
alternate system, and
throughput
To
see
this
numbered so that mini <
traffic limit result
rate (recall that the
fill
the "siltemate system," in which type
Theorem
end
suppose that there are two make-to-stock product
straightforward application of the heavy
2 products experience
for details.
possible bounds, beginning with the lower bound.
types and one make-to-order product type (d
m2n2.
some enhancements
we propose that
this estimate
for the
fill
results of
rate of type 2 products in the
be used as a lower bound
for its actual
rate.
why such an estimate might
which work enters the workcenter
throughput rate
is
now simply
in the alternate system,
its
will
serve as a lower bound, note that the average rate at
be higher
demand
rate.
in the alternate
The
system. In particular, type
utilization of the workcenter will
and so we expect that type 2 products
will
1
be higher
achieve lower throughput
than in the original system.
We
described the above procedure in the context of a production system with two make-
to-stock products
«md one make-to-order product, but the extension to multiple make-to-order
products requires no modification. To extend the procedure to a system that serves several
(d
>
2) types of
make-to-stock jobs, one simply applies d
-
1
iterations of the procedure; at
each iteration, the bottleneck product type from the previous iteration
to-order category; this reveals a
new
is
moved
to the
make-
bottleneck product type whose throughput rate can then
be calculated.
13
Upper Bound: Again, we
will
describe the procedure hy considering the system with two
<
make-to-stock product types, one make-to-order product type, and mini
extension to the general case
essence similar to that described for the lower bound; that
is,
However, rather than using the demand rate (Ai)
The
system
1
type
1
rate
products are "open."
products, we substitute
1
in
work arrives to the workcenter
with the
it
in this alternate
approximately the same as that of the original system, but the arrival process has
is
been made
of type
rate at which
for
is
we estimate the throughput
of type 2 products by considering an alternate system in which type
estimated throughput rate.
The procedure
then done exactly as described before.)
is
(The
7712x12-
The
less variable.
jobs
of no arrivals
when the type
when type
the other hand, type
rate has been slowed
1
system experiences "bursts" of relatively quick
original
1
finished-goods-inventory
finished-goods-inventory
is
is
not empty, alternated with periods
down
to achieve the
same throughput
rate.
on
In the alternate system,
depleted.
jobs arrive "regularly" according to a renewal process, but
1
arrivals
Because the
its arrival
traffic intensity
remains the same while the total variability in the system has been decreased, we propose that
this
system be used as an upper bound
for the
throughput rate of type 2 jobs.
Numerical Results
5
This section investigates the performance of the approximation procedure described
We
previous section.
will
consider product-form networks with one make-to-order product
type and two make-to-stock product types.
form
if,
example,
for
all
demand
they
(i.e.,
known
It is
that such a mixed network
processes are Poisson,
distributed, find processing times for
mean
the
in
all
all
is
product-
processing times are exponentially
product types at the workstation have the same
have the same distribution); see Kelly (1979). For such a network, one can
£ill
derive the steady-state distribution of queue lengths, from which one can calculate performance
measures such as average queue length and throughput
The parameters
Table
in
1
we
convention,
we
will
of the systems
we study
we
will
always take mini
< m2n2. To
assume that the demand process
the workcenter.
is
It
can be
fill
all
of
in
Table
1.
For each system shown
(we thus have 24 cases
ensure that the network
open jobs
is
Poisson and that
2.
all
is
in total).
As
product-form,
service times are
job tyF>es have the same processing time distribution at
verified that the relative traffic intensity of the
0.975 and equals 0.99 for System
from one, the
shown
will consider 12 different base-stock levels
exponentially distributed. Moreover,
1
are
rate.
If
workcenter
in
System
the relative traffic intensity of the workcenter
rates of closed jobs are essentially
100% even
is
for small base-stock levels.
far
For
purposes of testing the approximation, we therefore consider the more challenging cases where
14
System
Proof of The Heavy Traffic Limit Theorem
6
Theorem
is
1
proved using the same methodology as in Sections 5 and 6 of Nguyen (1994),
provided we can establish the existence, uniqueness, and continuity of a certain mapping. To
state this result, let us
x e
set of fiinctions
^
>
0,
and
(iii)
denote by
and
C'^''"'
a
is
number
let
x(0)
(i)
number
A
and
0,
=
each j
e for
and
D'^'*"'
be the
let D^"*"'
1,
.
.
,
.
d and
We
will
Df"*"', respectively, that
be the set of functions a € D' that have the following properties:
<
<
of subintervals
—
sq
<
Si
or a{t)
=
a,
on the
t
>
xo{t) +X}{t)
0, (ii)
the subset of those functions x in
(ii)
=
>
>
e
of discontinuities over every finite time interval.
nondecreasing,
that either a{t)
is
finite
Cj"*"'
are continuous. Next,
(i)
such that
D**"*"'
x has a
introduce the following notation. Fix
first
a{t)
for all
t
< sn =
<
>
t
interval
each
for
(iii)
0,
=
and constants
t
finite
<
Oq
there
t,
<
ci
•
•
<
•
a
finite
o/v
such
is
In particular, observe that e{t)
[s,, 5,4.1).
=
t
an element of A.
Let X €
in the
D^"*"',
mapping
a e A, and
($, *)
:
—
(i, a)
wo{t) =xo{t)
>
.
.
be positive numbers with J^f
,c<i
.
Cfc
<
1.
We
are interested
{w, z), where {w, z) are defined by the following:
+yo(0 - yo{t) +
= Xj{t)
= T,Li CkVkit)
wj{t)
z{t)
ci,
z{t)
z{ait))
+
{y,{t)
-
l<j<d
yAa{t))),
(36)
nondecreasing with yj(0)
j/j
is
j/j
increases only at times
(
=
<
0,
where Wj{t)
j
=
<d
<
0,
j
<
d.
Theorem
2
and a € A,
there exists a unique pair of processes {w,z) that satisfies the set of equations (36).
If
X G
and
Cj"*"'
continuous
The mapping
—
if Xj
(j/j
Remark: Observe
processes
j/j.
and
=
Xj{t)
To
—t
=
a{t)
that
see that
for j
=
Theorem
(1994) to prove
readers to
2,
*)
well defined
is
then (<J>,*)
o a) has
We defer discussion
aid of
t,
($,
is
Dj"*"'
That
x A.
no jumps doumwards for each
j
=
1,
.
for the process z only,
are not necessarily unique, consider d
=
.
.
for each x G
is,
continuous at (i,a). Moreover,
we claim uniqueness
j/j
on
and hence
j/j,
Df"*"'
z,
are
,d.
and not
2, a{t)
for the individual
=
t,
xo{t)
=t+
1,
1,2.
of the proof of
Theorem
2 to the
end of
one can follow exactly the same methods as
Theorem
Nguyen (1994)
1.
We will only outline the
for details.
By Skorohod's
that the convergence of Assumption 2 holds u.o.c.
21
this section.
For now, with the
in Sections 5
ideas of the proof here
and
representation theorem,
The same arguments
and 6
let
as in
of
Nguyen
refer interested
us
first
assume
Nguyen (1994)
will
show that
—
that ^^
t
>
and
C
1
is
<
j
e u.o.c.
»
s
with e{t)
where ^o
u.o.c.
<
's
From
t.
(34)
and Assumptions
Brownian motion and
(0,(7^)
W",
=
2:
continuous, one then uses
Theorem
2 together with the continuous
via the
mapping
Turning to the proof of Theorem
Theorem
9.1 of
Nguyen
To
following manner.
(
/")
(1994).
We
Theorem
($, ^)
on
That
for
each
=
1,
Theorem
is
shown
Because Brownian
??").
mapping theorem
the statement
we may assume
The
via construction in the following manner.
—
I,
.
.
.
,d,
and
fixed
is
simply
2 by induction on d in the
begin, observe that without loss of generality,
D^"*"'
in
we have
is,
case d
in the
thus propose to prove
equations of (36) imply that for each j
wo{t)
mj0j
follows
1.
note that
2,
(^, '^)(^",
Next, suppose as usual that the theorem has been established on D^.
mapping
=
^qH)
Theorem
r/"
equivalent to the statements of
is
+
2, it
Observing that the workload and idleness processes can be expressed
d.
to argue that convergence indeed takes place as claimed.
which
^*{t)
and
1
of
terms ^" and
motion
—
r/"
e
=
I.
existence of a
The
first
two
t,
+ Wj{t) = Xo{t) + Xj{t) + yjit) - i/j(a(0) +
z{a{t))
-
z{t).
Set
jt
=
arg
max
{y_,(t)
-
yj{a{t))}\
(37)
l<j<a
then
d
Wo{t)+Wj,{t)
- VjMit)) - Yl
-
=
xo{t)+Xj,{t)
>
xo{t)+x„{t)+y„{t)-yu{ait)) - ^Cfc(yj.(0 -
=
xoit)
>
Xo{t)+X,,{t)
+
Vnit)
cjfe[yfc(0
ykia{t))\
d
>
where the second to
and the
1
<
jt
Wj{t)
>
such that
1/2.
We
is
Ckj
(j/j.(0
- yjMt))]
(38)
1,
last inequality
last inequality
< d
+ Xj,{t) + (i-J2
J/j.(a(f))]
is
due
because x €
to the monotonicity of
Df"*"'.
To summarize,
for
j/j
and the condition
each fixed
t,
there
is
Yi^^k
<
1.
an index
+ u;j,(t) > 1, and consequently, one can find < ; < d such that
< ; < d. We set y_,o(f) = and solve a
with u;jo(0) > 1/2 for some
vJo{t)
start
problem involving a d-dimensional mapping associated with the remaining components
22
j
/
jo-
If
we can guarantee the
existence and uniqueness of such a solution, then
the unique solution of (36) until the
our procedure
ji
first
finished. Otherwise,
is
>
such that Wjiiti)
>
t\
such that Wjo(ti)
we are guaranteed by
We now
1/2.
time
we have constructed
=0.
(38) that there
freeze the process yj,
from
ti
is
=
If <i
oo then
some component
onwards, and solve the
corresponding d-dimensional problem. Iterating in this way we can thus construct {w,
by
piece, over
The
any time interval
details necessary to
[0,T],
and concatenate the
make such a procedure
z), piece
pieces in the obvious way.
rigorous are similar to those found
in, for
example, Chen and Mandelbaum (1991) and Nguyen (1994). Therefore, rather than presenting
a complete proof, we
will
mention only the main ideas that are needed.
show that the procedure described above indeed
be done
Proposition
in
If
mapping can be guaranteed by
different
methods and ideas
Proposition
as in
then the solution
to u;o), then the existence
form and Proposition 2
Nguyen
is
the "frozen" component
if
The proof
needed.
is
(1994), so
we
will
omit
it
coincide on
T
=
sup{t
(c) holds,
r, so
[r,
>
[0, 6]
some
for
t
+ 6]
for
:
z{s)
=
hence z and
(a) let
then the
yo,
for continuity of the
{w,z) satisfying the system of equations (36),
similarly to that of Proposition 2.4 of
<5
>
<5
>
0;
z'{s) for all
z'
(b)
0; (c) if z{t)
<
5
if
=
<
z{t)
2f{t)
=
on
z'{t) at
€
t
oo. It
us consider the
[0,
t}, it follows
coincide beyond r. But this
we must conclude that t =
To prove
of such a
here.
Chen and Mandelbaum
Let {w,z) be the process constructed as described above, and
coincide on
is
the
considerably involved and uses essentially the same
pair satisfying the conditions of this proposition. First, suppose
z'
and uniqueness
when
unique.
is
The proof proceeds
Proof.
takes the form of (36) (this happens
If there exists a pair of processes
1
to
gives the unique solution to (36); this will
induction. However,
mapping under the stated conditions
(1991).
mapping
the
component does not correspond
mapping takes a
one needs
Second, the pasting procedure involves constructing solutions for
1.
certain d-dimensional mappings.
"frozen"
First,
is
let {w',z')
we can show that
interval
23
[0,
(a) z
and
> 0, then the two also
z{t) = z'{t). Then defining
some t
r) then
from
(a) that
t
in contradiction
>
si) associated
6.
If
t
<
oo,
then
with the definition of
remains only to establish conditions
first
be another
(a) - (c).
with the function
a.
If
on
this interval a{t)
a constant then
is
it
must be
= xoit) + yo{t) -
Wait)
and
0,
z{t)
l<j<d
Wj{t)=Xj{t)-yo{t)+yjit),
z{t)
The
=
T.LiCkykit)
nondecreasing with
yj
is
j/j
increases only at times
If,
on the other hand,
a{t)
=
t
<
=0,
j/j(0)
where Wj{t)
t
existence and uniqueness of (to,y), hence {w,z),
(1981).
becomes
(36)
on the interval
is
j
=
<d
<
0,
j
<
d.
guaranteed by Harrison and Reiman
[0, si),
then we have
= xo{t) + yo{t) - z{t)
Wj{t) = x^it) - yoit) + zit), l<j<d
= T.LiCkyk{t)
wo{t)
m
yj is
nondecreasing with yj{0)
yj increases only at times
Here, the existence and uniqueness of {w, z)
Mandelbaum
[0,(5]
(say 6
(1991). In either case,
=
The proof
For part
(c),
where Wj{t)
t
is
j
<
=
0,
<
=0,
d
<
a direct extension of
we have shown
j
<
d.
Theorem
that the solution
is
2.5 of
Chen and
unique over an interval
51 /2).
of (b) follows from the arguments for part (a) by shifting the initial time to r.
observe that
d
wo{t)
=
[wo{t-)+xo{t)-xoit-)]-\-[yo{t)-yo{t-)\-Y,('k[yk{t)-yk{r)]
wj{t)
=
{wj{t-)+x,{t)
-
x,it-)]
-
[yo{t)
- voir)] +
(1
-
Cj)[yj{t)
-
yj{a{t))]
-
d
-
(1
Cj)[y,{t-)
-
i/,(a(r ))]
+ YL Ck[yk{a{t)) -
i/fc(a(r
))).
fc=i
If a{t)
—
t,
the above system of equations reduces to
d
wo{t)
=
^At)
= {[rvj{n+X:{t)-Xj{t-)]-{l-c,){yj{r)-yj{a{t-))] +
[ii;o(r)+xo(t)-xo(r)]
+
[yo(t)-yo(r)l-5^Cfc[j/fc(0-!/fc(r)l
d
d
E
cfc[yfc(r )
-
yk{a{t-))]}
-
[j/o(0
-
yo(t-)]
+ Y. ^k[yk{t) -
k=i
if
otherwise a{t)
<
t,
yk{r)];
fc=i
we have
d
WQ{t)
=
[wo{t-)-\-xo{t)-xo{r)]
+ [yo{t)-yo{t-)]-Y,(^k[yk{t)-yk{r)\
k=l
24
= {K(r) + x,(t)-x,(r)]-(i-cj)[j/,(a(0)-v;(a(r))l +
w,{t)
d
-
T^'^klVkiait))
we consider the
In both cases
first
+ M\y{t) - i/(r
From Theorem
1.14 of
<
exists a
Cj
<
j
I,
=
unique pair of processes {w,
I,
Moreover, z
is
.
.
.
= xj{t) + (1 = ELiCkyk{t)
c,)[j/j(t)
nondecreasing with
j/j
increases only at times
if
t
=
(1985).
Xj{t)
If in
+ Vjit),
which
interval [5i, 52),
[0,
si.
(LCP) that
y{t)
-
y{t~)
involves finding
w'[y{t)
- yir)] =
0.
LCP
has a unique solution,
=
1,
.
.
.
.
,x<i)
.
€
D**,
a € A, there
,d,
.
^(a(0)
>
=
=
0.
[sq, s\),
=
sq
>
0, si
0,
On this interval, either a{t) takes the value
a{t) = t. In the first case where a{t) = 0, then
is
the one-dimensional regulator in Proposition 2.2.3 of Harrison
=
t,
0,
or
=
then Wj{t)
is
Xj{t)
+
and the same
z{t),
result
uniquely defined on the interval
1
to
from
[sq, s\).
show that {w,
z)
can
Shifting the initial time to Si, one can similarly construct {w,z)
and imiqueness of
=
and
0,
by construction. To begin, consider
and so on. "Pasting" these
T],
d.
must be
remains to show that y
show that
,
use the same argument as in the proof of Proposition
any time interval
It
.
£ind require that
it
the latter case a{t)
be extended uniquely to
on the
.
(xi,
- yjiam +
Harrison (1985) can be used. In either case, {w, z)
One can
.
y,(r)].
x — {yoa) has no jumps downwards.
interval associated with the function a.
is
=
with Wj{t)
the
of existence
for j
(0)
j/j
The proof
Wj{t)
1,
- yir) >
For each x
,d.
15
of a constant, in which case
=
;•
-
t.
j/j
a continuous process
Cj)\y,{t)
be shown to be positive definite, hence a P-
Proof.
first
0, y{t)
z) satisfying,
w,it)
zit)
aU
Harker (1993), we conclude that the
hence y has a unique extension from f ~ to
Proposition 2 Let
>
)]
M can
In both of the cases above, the matrix
matrix.
for
-
known quantity
bracketed term to be a
0,
(1
a linear complementarity problem
is
q
- yoif)] +
[yoit)
- vAn\wi{t) =
[y>(0
What we have in essence
y{t) - y{t~) to satisfy w =
-
Vkiait-))]}
is
this
pieces together,
procedure
is
shown
we thus construct {w,
similarly to F*roposition
continuous under the stated condition, which
for all
t
>
0.
But
y{t)
25
-
y{t~)
is
z)
is
on
1.
equivalent to
the unique solution to an
LCP
w = q + M[y{t) - y(r >
))
q
0, v/[y{t)
- yit')] =
= Wj{r) + xj{t) - xj{t-) -
(1
-
0, either
cj)[i/j(r )
- vMt-))] +
z{t-)
- z{a{r))
or
q
= Wjin + Xjit) - Xjin -
depending on whether
negative jumps, and
is
a{t)
y, z,
=
t
(1
ot a{t)
- Cj)[yj{a{t)) <
t,
yMn)] + ^(a(0) - ^(a(«"))
respectively.
and a are nondecreasing, so g >
the unique solution.
26
But Wj{t~) >
in either case.
0,
x —
(y o a) has no
Hence y{t)—yit~)
—
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27
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