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ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
Product-Form Make-to-Stock Queueing
Networks
Rodrigo Rubio
Lawrence M. Wein
#3672-94-MSA
April 1994
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
Product-Form Make-to-Stock Queueing
Networks
Rodrigo Rubio
Lawrence M. Wein
#3672-94-MSA
April 1994
Product-Form Make-to-Stock Queueing Networks
Rodrigo Rubio
Operations Research Center, M.I.T.
and
Lawrence M. Wein
Sloan School of Management, M.I.T
Abstract
A
manufacturing
unsatisfied
amount
falls
demand
is
facility
produces multiple products in a make-to-stock manner, and
backordered.
A
simple production control policy
is
analyzed:
when
the
of work-in-process inventory plus finished goods inventory for a particular product
below a base stock
level,
then release another unit of product onto the shop
floor.
Under
a set of stationarity assumptions, we show that the cost minimizing base stock level for each
product
satisfies
a critical fractile expression for the steady state distribution of the products
total work-in-process inventory.
By
exploiting the relationship between the make-to-stock
system and an open queueing network, we identify
specific formulas for the base stock levels
under standard product-form assumptions. For the
lost sales case, a similar relation to a
closed queueing network can be used to characterize the optimal control parameters.
April 1994
1.
We
Introduction
consider a manufacturing system that produces a variety of products in a make-
to-stock mode; that
is,
completed units enter a finished goods inventory that services ex-
ogenous customer demand, and unsatisfied demand
this
backordered.
is
Production control
in
production- inventory environment has been moving away from the push philosophy em-
bodied
in materials
typified
ment
by kanban systems.
for
(MRP)
requirement planning
systems, and towards the pull philosophy
Unfortunately, kanban control policies are difficult to imple-
multi-product systems, particularly
if
a different buffer
level is set for
each prod-
uct at each production stage. Moreover, even single-product kanban-controlled production-
inventory systems axe difficult to analyze
(see, e.g.,
Mitra and Mitrani 1991), and to our
Mo-
knowledge, multi-product kanban systems have not been analyzed in the literature.
tivated by the difficulty of implementing
and analyzing kanban
policies in multi-product
production-inventory systems, we propose and analyze a very simple pull-type production
Our
control policy for a multi-product make-to-stock environment.
simpler or more natural: release a
total work-in-process
(WIP) plus
new
unit of a product onto the
finished
falls
1967, Solberg 1977
et al. 1990,
floor
whenever the
below a specified base stock
This policy can be viewed as a make-to-stock analog of the
Spearman
shop
goods (FG) inventory (where backordered demand
represents negative inventory) for this product
policy (see
policy could not be
CONWIP
(constant
.Xewell
and
Scarf's
for earlier studies),
and has
its
roots in Clark
(1960) echelon inventory policy for multistage uncapacitated inventory systems.
enjoys some of the same advantages as the traditional
al.
WIP)
and Koenigsberg 1958, Jackson 1963, Gordon and
and Whitt 1984
Whitt, Spearman et
level.
and Muckstadt and Tayur
robust (since controlling inventory
is
CONWIP
(1993):
it
is
1
policy
policy, as articulated
by
very easy to implement,
easier than controlling throughput),
tleneck starvation.
The
and prevents bot-
Readers are referred to Buzacott and Shanthikumax (1992)
survey of various production control strategies
our analysis
is
Buzacott et
performance measures
each station
is
(1992)
al.
for a
one
is
effect that,
when
In the initial
the base stock level of
this observation in a
more general
In Section 2, a multi-product production-inventory
proposed base stock
for
policy.
We
all
setting.
system
is
each product
level for
of the steady state distribution of the total
the optimal base stock levels are independent of the
WIP
WIP
considered under the
WIP
is
WIP
and FG
characterized as a
for that product;
moreover,
holding costs. Since the shop floor
viewed here as a black box, rather than as a network of queues,
as the total
their work,
stations except the
each product to minimize the long run expected average cost of holding
critical fractile
which
address the problem of choosing the optimal base stock level
and backordering FG. The optimal base stock
is
development of
in
system can be analyzed by means of a traditional open queueing
Our paper employs
network.
starting point for
and Lee and Zipkin (1992), who develop approximate
controlled by a base stock policy.
set to zero, the
The
multistage systems.
tgmdem single-product production-inventory system,
both make observations to the
last
for
an excellent literature
for
this result holds as long
process for each product possesses a steady state distribution with finite
mean, regardless of distributional assumptions, setup times, scheduling policy or routing
structure. In Section 3, the result
Jackson queueing network; that
demand
is
is
is,
specialized to the case where the shop
is
employed
at each station.
formula characterizing the optimal base stock
for the
optimal base stock
levels for
level
is
modeled
as a
each machine has exponential service times, exogenous
Poisson, job routing between workstations
served (FCFS) policy
is
level
is
Markovian, and the first-come
first-
Under these product-form assumptions, a
is
derived. Since a closed form expression
not available, we numerically
compute optimal base stock
some balanced production-inventory systems. Of particular
the optimal base stock level appears to grow almost linearly in the
interest
number
is
the fact that
of stations; this
linearity is partially explained
by an application of Chernoff 's exponential bound.
By modeling
Section 4 briefly discusses three generalizations.
work of quasireversible queues, we show how the powerful
queueing networks (see Baskett
results for
the shop floor as a net-
product-form multiclass
1975 and Kelly 1979) can be employed to determine
et al.
the optimal base stock level for each product in a multi-product system.
The remainder
of
the section deals with the single-product and multi-product cases for production-inventory
systems in which unserved demand
fact that the related
is
queueing network
lost.
is
Although additional
difficulties arise
closed, our analysis allows for a practical
from the
means
to
find the optimal base stock policy.
There axe several straightforward generalizations that we do not investigate
example
is
a mixed system (see Carr et
where some products are produced
mode. Another example
distributions to
is
in
al.
One
here.
1993 and Nguyen 1993 for the single queue case),
a make-to-stock fashion and others in a make-to-stock
to incorporate infinite server queues with general service time
model the transportation delays incurred
in material
handling systems on
the shop floor or in production-distribution systems.
2.
Base Stock Control of Make-to-Stock Systems
Consider a manufacturing
Completed units enter a
i
items demanded
starts with
a
new
Zi
units in
produces
finished goods inventory that
demand, and unsatisfied demand
of type
facility that
FG
in [0,i].
for
backordered.
is
Without
product
i
I
is
products indexed by
represents backordered
i,
let Zi{t)
demand, and
=
Let A^{t) denote the cumulative
loss of generahty,
let A^,(i)
FG
.
.
,
.
/.
number
we assume that the system
and no WIP. The proposed control policy
denote the
1,
depleted by exogenous customer
unit of a product onto the shop floor whenever a unit of that product
by a customer. For product
i
is
releases
demanded
inventory, where a negative quantity
be the amount of
WIP
inventory on the shop
floor at
FG
time
t.
demand
Since the
WIP
inventory and increases the
Ni{t)
+
Zi{t) for product
constant at
high
which
value of
its initial
the base stock level
i,
2,,
one unit of product
for
inventory by one unit, the total inventory position
the total physical inventory net backorders,
is
FG
Although product vs
z,.
simultaneously depletes the
i
backordered demand allows the
inventory
WIP
is
never larger than
is
inventory to attain arbitrarily
levels.
manufacturing
In Section 3, the
individual workcenters
demand
facility will
and corresponding
structure and queueing discipline.
be modeled as a queueing network with
service time distributions,
and a
specific routing
made
Here, no specific assumptions are
regarding the
distributions, the routing structure of the facihty, the scheduling policy, the service
time distributions, machine breaicdowns and repair, or setups. Instead, we treat the
as a black
Di{t)
held
is
the
box that
number
is
facility
characterized by a vector of stationary departure processes, where
of completed units of product
departure processes axe a function of the
demand
i
delivered to
FG
by time
t;
of course, the
processes, which act as the input processes
to the black box.
Under the proposed
Di{t)
—
{Nt{t),t
Ai{t) for
>
z
=
1,
.
.
.
policy,
,
/
and
it
t
>
follows that N^i^t)
0.
=
^4,(0
Our primary assumption
0} possesses a steady state distribution with finite
class of stochastic
models
a steady state distribution with
station, then this property
is
for the black
finite
mean.
possessed by
and ergodic inputs, services and routing
all
If
box that give
is
Z,(i)
=
is
that the
WIP
process
for
class of
rise to
the traffic intensity
is
a
=
i
1,
.
demand
WIP
less
.
.
.
/.
^,
+
We
processcn^
process having
than one
ai
each
Jackson-type networks with jointly slalintiarv
(Baccelli
multicla^s queueing networks (Kelly, Chapter 3).
multiclass queueing networks
and
A(0
mean
do not delve into the fundamental problem of characterizing the
and the
-
and Foss 1994) and
all
prodiici-lnrrn
However, the characterization of
an open problem, and
is
--lalili'
currently the subject of intiri>nf
Rybko and
research activity (see, for example,
Kumar and Meyn
Let
FG
time, hi be the corresponding
unit of product
holding cost rate, and
(zi,.
.
,zi) to
.
WIP
inventory for product
to find the base stock levels
is
then the problem
A^,,
is
If
Zi
we
to choose
/
^a(z.)=i]
1=1 \
t=l
"^(^' =
qE
n=0
'^)
+ ^'E ^^(^' = z,-k)-b,Y. kP{N, = z,*:=-oo
fc=0
=E
(^>
+
<
z,P{N,
^0
z,
-
+ Y.^P{N, =
I)
suffices to
The main
Proposition
study the steady state
result of this section
1
/
n)
b,z,
+ {c-K)E[NM.{\)
n=2.
t=i
it
k)]
oo
/
z*.
by
inventory per unit of
minimize
/
Thus,
z
WIP
and backordering inventory.
to minimize the long run expected average cost of holding
denote the steady state
in
i
be the cost of backordering one
6j
per unit time. The optimization problem
i
1993, Dai 1993 and
1993).
denote the cost of holding one unit of product
Ct
Bramson
Stolyar 1992,
is
WIP
Ni to determine the optimal base stock level
the following.
The optimal base stock
level z* is the smallest integer that satisfies
// this condition is satisfied with equality, then the expected cost obtained by using the optimal
base stock level
is
E
C^'iz*)
=
Proof. Equation
>
Ci{zi),
and so
(c
(1) implies
Ci{zi)
sumption, so that C,(0)
convex fimction of
f
+
k)E[N,] - {K
+
k)
2,
=
is
that | {Q{z^
-
1)
a convex function of
(1 4-
E nP(7V. = n))
n=0
1=1 \
i=l
Zi)
E
bi)E[Ni]
<
that must increase at
+
Q{z,
2,.
+
1))
= C,{z^)+^{k+K)P{N^ =
Furthermore, E[Ni\
oo and C,(oo)
some
.
/
=
oo.
<
oo by as-
Since the cost
is
a
point, the optimal value for the base
stock level
C,(z,
+
1)
A
level as
is
-
found by increasing
Cx{zi)
=
(/i,
+
bi)P{Ni
until C,(2,
2,
<
-
Zx)
6,,
+
a
critical fractile of the
level,
it
condition
is
^;
[(/i,
is
WIP
holding cost
is
<
but
z;)
will
-
b,]
>
which
0,
is
in genergd only a lower
is
1
obtained by substituting z' into equation
satisfied with equality,
1
(1) gives
result.
Also, because the optimal value for z'
.
+6,)P(iV,
Although Proposition
Equation
0.
characterizes the optimal base stock
1
not surprising that the
is
to be integer, the cost J2i=i C^'^iz') in Proposition
- Ct\z*) =
>
Ci{z,)
W'lP distribution. Since the W'lP process {Ni{t),t
not appear in the expression for z'
true optimal cost J2i=i Ci{z')
—
which proves the desired
few remarks are in order. First, Proposition
independent of the base stock
C,{z')
1)
zero
is
when
otherwise yield a value that
easy to derive, the result
is
(1).
is
>
0}
c,
does
is
restricted
bound; the
Notice that
the optimality
linear in z*.
deceptively powerful, because
very few conditions are imposed on the production-inventory system; for example, no as-
sumptions are made regarding the distributions of the service and demand processes, the
routing structure, or the queueing discipline.
Consequently, factories can gather historiced
data to generate an empirical distribution of the total
then employ Proposition
1
WIP
inventory for each product, and
to find the optimal base stock levels.
Monte Carlo simulation model could be used
to generate the
Alternatively, a detailed
WIP
distributions.
In
next section, we model the shop floor as a queueing network to obtain an analytical
the
WIP
distribution.
3.
Make-to-Stock Jackson Networks
Because each demand simultaneously depletes the
to the shop, the
shop
floor
inventory and triggers an arri\al
can be modeled as an open queueing network
corresponding to exogenous demands.
given by the steady state
FG
number
Therefore, the steady state
of type
i
customers
in
WIP
willi
arrivals
inventory
an open queueing network,
,V,
is
hi ihis
section,
we look
=
at the special case /
and model the shop
1,
floor as
an open Jackson
network; to simplify notation, the subscript denoting the product type will be omitted for
the remainder of this section.
K
Consider
where the server
single-server stations,
service time distribution with service rate
product-form solution, we
demand
Let the
Suppose the corresponding
A.
an exponential
although multiserver stations also give
restrict ourselves to single-server stations to
analysis relatively simple.
with rate
/i^;
at station j has
for the single
arrivals to the
product follow a Poisson process
shop
floor are
routed randomly to the
so that the arrivals are independent Poisson processes with rate \j,j
where SjLi
Aj
A.
After completing
station j with probability
— Y^f=\
with probability
1
P —
than
[pij]
is less
effective arrival rates
The
1
Ptj
and
,
.
.
.
,
service discipline
is
all
floor
(and hence enters the
items eventually exit the system.
j
is
we assume that
Pj
p^
=
<
u-,/ p,^
1,
j
v^
—
X^-^-
K
We
will consider
of
are distinct,
inventory)
Yl!k.=
Define the vector of
\
^kPkj,
=
1,...,/C.
Finally,
J
—
I,.
.
,
K.
assume that the FCFS
floor is
modeled
the two extreme cases where the
and where they are
all
and the
as a Jackson network,
WIP
distribution
independent geometric random variables with parameters
sum
FG
visits
where, to guarantee the existence of a steady
product-form solution implies that the steady state total
the
an item
used throughout the network.
Under these assumptions, the shop
classic
shop
v^) to be the solution to
traffic intensity at station
state distribution,
exits the
z,
= \...,K
where we assume that the spectral radius of the routing matrix
Pij,
so that
(i^i,
service requirement at station
its
a
keep the subsequent
K stations,
=
rise to
traffic intensities at
1
—
pi,
.
is
.
.
given by
,
1
—
pK-
the various stations
equal; our approach can easily be extended to any
intermediate case.
Without further
P\
<
P2
<
< pK <
loss of generality,
^-
If
we
assume that the stations are numbered so that
define the z-transform of the steady state total
WIP
A^ as
Pj;{z)
= E[z%
then
pT(. _ nf=.(i-P.)
n;=i(l - zp])
A
partial fractions expansion gives
a,
P(N =
=
=
n)
-^'-r:
ctjp^,
HJLi
;
-.
where
.
Ue^j{pj-Pe)
Hence, the cumulative distribution function of
N
is
P{N<n) = EY^c^jP]
:=U j=l
i-p;
^1
and substitution into the condition of Proposition
z'
is
this condition
is
Now
sum
I
-
=
l
Pj
^
+
b
Pj
satisfied with equality, then the
j
the
implies that the optimal base stock level
the smallest integer satisfying
P
If
1
'
j
=
\
h
optimal cost
vi
P])
consider the case of a balanced network, where
of iid shifted geometric
random
variables
given by
is
is
0<p\ —
=
pk~P<
1-
Since
a negative binomial random variable,
we obtain
P{N = n)=
An
^^l~^ ](1-P)V
I
application of Proposition
1
for
n
=
0,l,2...
shows that the optimal base stock
(3)
level z* is the smallest
integer such that
tr^"-'
n
n=0
and,
if
(4) is satisfied
\
with equality, then
C«V) =
(c
l(>-p)V>r^.
h + h'
/
th(>
optimal cost
+ fe)T^-(/. + fc)f:nf
1
-P
n=0
V
is
^^"-' ](l_p)V
"
(4)
Notice that z* in
(2)
and
increasing in the backorder-to-holding cost ratio |.
(4) is
Also, elementary queueing theory suggests that z* should be increasing in the pi's in (2)
and
in
p
in (4).
While the
we have only numerical
and
by a simple differentiation with respect to
latter result follows
results to support the former.
When
K—
both expressions
\,
p,
(2)
(4) yield
r
1
Inp
xi
/i
.
K^)
1
(5)
and
C«7(^*)
^
_^
P_
+
[(5
-
h)[{\
p)z*p^'
+ p''] -
/I
+
c]
,
P
where the integer
restriction
on
z*
is
ignored; these results are consistent with the single-
server results presented on page 105 of Buzacott
and Shanthikumax (1993).
Since (4) does not yield a closed form solution in general, we apply Chernoff 's bound
on the
tail
behavior of sums of random variables
149) to obtain an upper
bound on
variables satisfying E[Xi\
common
logarithmic
<
moment
z* for the
and P{Xi >
...
n
application of this
bound
any z > Kp/{1
—
p).
z*<
We
is
0,
and
Then
let
M{s)
Chernoff's
inf
.
=
.
,
X^ be
iid
random
\ogE{e^^') be their
bound
states thai
+ X„>0) <infM(5).
^
(6)
^~^
inf
ae(0,-lnp)
f
yi
-
1
pt°-
e-"^
(7)
)
This bound, used together with Proposition
1,
gives
(a-Mnf-i^^W
+ a-Mnf^H,
-
-ae{o,-inp)[
so that z*
>
0)
.
to (3) yields
PiN >z)<
for
example, see Billingsley 1986, pp. 147-
balanced case. Let Xi,
generating function.
-logF(Xi +
An
(for
upper-bounded by a
can tighten the bound
minimization of the right side of
\
h
by optimizing over a 6
(0,
\\
pe"
linear function of
in (8)
(8) is difficult
9
(8)
)]
K.
because
it is
-
Inp). Straight torw.ird
not convex in
a.
Howcwr.
ihc
value of a thai minimizes the
bound
in (7) is
Since a*
is
2
=
a
in
(9)
unknown base
a function of the
stock level
procedure to find the optimal value of a to employ
do
e (0,-lnp) and substitute
Then use
level zq.
Oo
= — lnp/2,
determine a
(9) to get
it
an updated value
a\,
and
so on.
after a\.
If
In all
120 cases, we used
we use
this
procedure to
then the large deviation approximation implies that this bound
Although a closed form solution
are easy to solve numerically,
investigation.
K
Start with an initial value
(8).
tight
is
— oo.
as /C
as
in
iterative
into the right side of (8) to obtain an initial base stock
and the value of a did not change
in (8),
we use the following
z,
,
the
The primary
number
objective
is
of stations, increases.
and two backorder-to-holding
1
and plotted
in
Figures
not available in general, both (2) and (4)
is
and the remainder
to explore
To
balanced Jackson networks, using
for 120
Table
for z'
and
2.
how
this end,
=
3
the optimal base stock level z' varies
z*
=
10.
These
number
NumericaJ calculations of the upper bound
very tight, even for
K
as large as 40.
in (8)
for z' in (8)
with (10) replacing
(8).
The
show that
with the exact result
p
1
of a in (10),
that the optimal base
of stations, as suggested by (8).
l+a-Mn(-^
To compute the value
is
this
bound
Hence, we also consider the base stock
combines the slope of the Chernoff bound
In
results are reported in
grows considerably as the utilization
approaches unity. However, the most striking feature of the results
stock level grows almost Unearly with the
we employ the
level
values of p, 12 different network sizes
and ^
As expected,
devoted to a numericaJ
is
we compute the optimal base stock
five different
cost ratios, ^
1
of this section
-
for A"
^\{K-\).
=
1
is
not
level that
in (5):
(10)
pe"
iterative procedure described earlier, but
resulting base stock levels are displayed in Table 2 £Lnd, as can
10
z*
^
=
3
Base 250
Stock
'--*'200
.
150
.
100
-
50
.
.
!
=
3
I
350
-
300
-.
p =0.9
= 0.85
=0.8
-•—
=0.7
=0.5
5
10
Number
Figure
s.
-
-i
6:
15
of Stations
Optimal cost C{z*)
for
j-
=
10.
Extensions
4.
we
In this section,
briefly describe the generalizations of this
approach lo
product queueing networks, and to the single-product and mulu-producl
lost sales cases.
Multi-Product Systems
4.1.
The powerful theory
sider
for
product-form rnulticlass queueing networks allows us to con-
any network of quasireversible queues. For example, consider a multi-product system
where each product, or customer type
mand
process with rate
A,,
z
=
1, 2,
.
terminology, has an independent Poisson de-
in Kelly's
.
.
.
and
/
its
own
arbitrary deterministic route through
the network of workstations. Suppose each workstation, indexed hy j
of a single machine, service times of
random
variables with
2* for
z
=
As
1,
.
.
.
,
it
1,2, ....
the
FCFS
queueing discipline
is
iid
K
,
consists
exponential
employed. Theorem
be used to numerically find the optimal base stock
will
level
/.
in Kelly, let
be the stage
1
=
operations at the workstation are
all
mean ^j\ and
and Proposition
3.1 in Kelly
this
rnulli-
i_,(/)
be the type of the customer
has reached along
its
route.
customer, and the state of queue j
is
Then
in position
=
Cj{l)
/
in
(ij(/), «_,(/))
queue
number
of type
i
and
let Sj{l)
denotes the class of
given by the vector {cj{l),Cj{2),
chajacterize the steady state distribution of the
j,
.
.
.
customers
,c-,{N.,{t))).
in
To
the sysu^m.
define
.... —
dj{l, s)
A,
J
Vj
let pj
=
Xl'^i ZljJi Oj{i, s),
= ^,
where S{i)
then by Corollary 3.4
n customers
stage s of customer type
i
is
at station j,
otherwise,
I
and
if
<
in
is
the total
number
of stages
on type
z's
rouuv
If
we
Kdly, the steady state probability that queue j coniains
is
P(N,=n) =
(l-p,)p';,
n
=
0,l,...
Furthermore, in steady state queue lengths at each station are independent, and
is
in position
/
The z-transform
E[z'^'], is
queue j then
in
it is
a type
customer with probability
i
number
of the steady state total
partial fractions expansion gives P(A^i
a,
P^^-
=
1
-
Pj
+
=
+
p]
=
n)
and
q,j
^
au
=
"^ H^.d - a.)
=
for all j y^
I;
-
ail)
the approach
is
easily
extended
For this case, the cumulative distribution
is
j=i
and Proposition
1
- a"+i
1
"'J
^
implies that the optimal base stock level z*
'
fr",
finite
defined as Pjj {z)
,
where
n;#j(atj
'^
a
yv,
-zp]ii])
Y.f=\ ^jOi^j,
^^
where some of these values are equal.
function for N^
pjixj
Pjli]
This corresponds to the case where
A
customers
Yls=\ 8j{i,s)/i/i.
nf=i(i-P.)
nj=i(i -
to cases
i
=
a customer
thus given by
pT ,. _
A
of type
7,_,
if
i-a,,
- k +
is
the smallest integer satisfying
^
h,
'
similar analysis carries through for the case where each workstation consists of
number
of identical machines; since the derivation
insights are developed,
we omit the
details.
is
rather tedious and no
new
Readers are referred to Chapter 3 of Kelly and
to Serfozo (1993) for other types of queueing networks that yield product-form solutions.
Also, the heavy traffic approximations in
estimate the
4.2.
WIP
distribution
The Single-product
For certain systems,
arrives to find
Nguyen and Harrison (1993) can be employed
when the exponential assumptions need
to
to
be relaxed.
Lost-sales Case
it
may be more
no inventory available
realistic to
in finished
17
assume that when a customer demand
goods, the sale
is lost.
If
the
manager
of a
single-product system
still
follows a control policy of establishing a hase stock level z
then releeising a production order every time a customer
model the system
the shop has
station,
A.
K
as a closed queueing network.
is
K
Customers queued
+
K
network
will
+
K
+
1
correspond to units
depletes the
1
unit onto the shop floor. This observation
FG
was
in
FG
possible to
is
Figure
inventory,
demand
made by Nguyen
(1993),
rate
and a service
inventory and triggers a release of a
first
7. if
have an additional
with a single server whose service rate equals the
I,
at station
completion at station
it
In particular, as illustrated in
stations, then the resulting closed queueing
denoted by
served, then
and
who
new
considers a
queueing system that employs a combination of make-to-order and make-to-stock operations.
She also notes that the
by an excess
is
life
service time in a busy period at station A"
first
and consequently we need
distribution,
to
-I-
1
is
characterized
assume that the demand process
exponentiaJ to ensure that the two systems in Figure 7 are statistically indistinguishable.
Unlike the backorder case, the steady state distribution of the total
depends upon the base stock
level z; hence, there is
we model the shop
black box, and instead
entire production-inventory system can be
stations
and
no advantage
WIP now explicitly
in viewing the facility as
as a Jackson network, as in Section 3.
modeled
a
Thus, the
as a closed Jackson network with A"
-I-
1
z customers.
We now
backorder cost
turn to the problem of finding the optimal base stock
6,
a cost
/
is
Instead of a
incurred for every lost sale. Let T{z) denote the throughput rate
of the closed queueing network; that
any given node per unit time
in
rate at which sales are lost.
we
If
is,
the average
steady state.
let
It
number
of customers passing through
follows that A
- T{z)
is
the steady state
Nj{z) denote the steady state average queue length at
station j, then the steady state cost function
C{z)
level.
is
= hNK+i{z) + cf;
A^;(z)
+
/
(A
- r{z))
(12)
.'Mthough closed form expressions for thr throughput and expected queue lengths of closed
18
Lost-Sales Make-to-Stock System:
Production
Finished
orders
Job Shop
K
Manager
<^
Stations
yv
Jackson networks are unavailable, several
algorithms exist to compute them (see
efficient
VValrand 1988 and references therein).
The second-order
properties of the cost function (12) need to be investigated to nu-
merically determine the optimal base stock level
shown that T{z)
nondecreasing and concave for closed Jackson queueing networks when
is
the service rate at each station
If
the restriction h
convex
in z'.
Shanthikumar and Yao (1988) have
z'.
=
c
However,
\s
is
nondecreasing and concave in the station's queue length.
imposed, then the concavity of T{z) implies that equation (12)
it is
easy to find instances where h
>
c
>
Q
and
(12)
is
is
not convex.
.Although in these examples the cost function appears to be quasi-convex, and hence well
behaved, we have been unable to prove the quasi-convexity of (12) for closed Jackson networks. Nevertheless, for
all
we have
the cases that
considered,
the optimal base stock level z' by calculating C{z) for ^
We
0, 1, 2,
A"
all
-I-
station equals the rate of customer
j
=
1,2,
.
.
,
.
/C
-t-
1
and T{z)
_,
demand.
—
,
7^
Differentiating equation (13) twice proves
=
if
.9
.
until
C{z'
-I-
1)
>
C{z').
>
It is
in
same
traffic intensity.
which the capacity of each
known (Walrand,
pg.
190) that A^j(2)
=
-^^, and hence
hz
^W =
z*
.
single-server stations have the
1
This case represents a balanced production-inventory system
-j^,
.
conclude this subsection with a special case that admits a closed form expression
Let us assume that
for z*.
=
has been possible to find
it
cKz
\IK
+ TcTT + TTl?-
its
convexity,
and the
,,„,
<"'
first-order condition yields
and
'_
z
=
/•
-f
/ n - An'
+ VP-^^y
-U
,
\^g<0,
(14)
2
where f
= 2K +
\
and
"=(>-^)'^('^^'>From equation
roughly linear
(14)
in
the
it
follows that z'
number
= 0{K)
of stations.
20
and hence the optimal base stock
level
is
The Multi-Product Lost
4.3.
One
system producing / products.
a product
Ai,
i
is
sold.
is
=
z
1,
the
If
.
.
,
.
/,
Case
where a base stock policy
Finally, consider the case
each product type
Sales
possible policy
and
demand
for
product
i
is
used to manage a
i
z^
for
item into the shop every time such
an independent Poisson process with rate
then this policy leads to a multichain, multiclass closed queueing network, where chain
populated by
customers
z,
for
i
=
1,...,/.
If
the shop consists of
station
K+
i
has a single exponential server with rate
inventory for product
Let z
of type
=
{z\,
.
.
=
z
1,
.
.
,
.
/,
stations, then
stations,
where
and represents the
i.
,zi)
be the vector of base stock
this multiclass
throughput rate T{z)
=
and %{z) be the throughput
levels,
items corresponding to the base stock control.
i
form assumptions,
total
.
A, for
K
K+I
the queueing network for the production-inventory system will have
FG
lost sales
to establish a base stock level
is
to release a type
is
Under the
traditional product-
queueing network has the undesirable property that the
Y.[=\
%{z) need not increase
in Zi, as
noted
in Section 8.6 of
Buzacott and Shanthikumar (1993). This property highlights the need to exert a different
sort of control
An
upon the product type
of items that are released into the shop.
alternative policy consists of setting an aggregate base stock level z
releasing a production order into the shop every time an item
being for a type
relative
7
=
(71,
i
item with probability
throughput rates
.
.
.
,
7/). If the
process with rate
At,
for the various
demand
for
7t,
where
Z]t^=i 7i
is
=
sold, the
1.
production order
Under
products are determined by the
and then
this policy,
the
relesise probabilities
each product arrives according to an independent Poisson
then this policy leads to a single-chain, multiclass queueing network.
This network model has a product-form distribution under the usuaJ assumptions; see Section
3.4 of Kelly for details.
The
decision variables for the base stock policy are the aggregate^
base stock level z and the release probabilities
21
7.
The
single-chain
of each product
model has two advantages over the multichain model: the throughput
monotone nondecreasing
is
in
the aggregate base stock level
any arbitrary
release probabilities 7 cam achieve
set of relative
2,
throughput rates
emd the
for the
various products; the multichain model, in contrast, can only achieve a discrete set of relative
throughput rates via the integer choices of
To
random
customer types. Let
Upon completing
service at station j, a type
FG
K+
i
=
1,
.
.
,
.
/.
For the closed network,
at station j a type
and
all
type
I
K+
where station
I stations,
i
item
will
next
P]k
=
P)k>
p"k+x
=
P),K+i:
P'K+x,k
=
7.'Pofc,
let
i
p^jj.'
i=
i,i
aJ =
ll[=\ a}. J
=
Jackson network with
•
•
•
K+
,
^+
/,
will
next
first
/ different
operation
the
is
products, or
at station j.
k with probability
visit station
Under the aggregate hase
a closed queueing network
A, for
be the probability that upon completing service
J
/c
as a type
=
i'
item; then
l,---,K; k=l.,...,K,
=
l,...,K,
If
sold,
we
lei
then Q|c+,
tor
I
=
number
a^ be the expected
=
1
for
1,...,/; J
i
=
and T{z) be the throughput
I stations,
is
=\,...J] k=\,...,K,
'-
1,
item
l,.-..,I: J
it is
j
where Nj{t)
has a single exponential server with rate
i=l,---,I;
j before
visits station
i
visit station
a =
Let
populated by
may be modeled by
other transfer probabilities are zero.
product
is
and service rates
example, a multiserver queue.
for
with probability p]^k+i-
stock policy considered here, the system
with
can model,
be the probability thai product Vs
p^.
or will exit the shop and enter
pjfc,
rj(-)
and the shop
policy,
stations,
variables with rale fj,jrj{Nj{t)),
queue length at station j and the function
Each server uses the FCFS
K
suppose the shop consists of
illustrate the procedure,
at station j are iid exponential
z.
z customers, the
same
=
1
,
.
.
.
,
1,...,K.
/ ajid
of limes a
a^ satisfies
(lo)
of a corresponding closed
service rates as the single-
chain multiclass network, ajid relative
visit rates aj.
rate of the single-chain multiclass network,
Then T{z)
and since orders are released into the system
according to the probabilities 7, the throughput rate for type
for
z
=
1,
.
.
,
.
Let N^{z) denote the steady state
/.
also the total throughput
is
number
products
i
of type
i
is
%{z)
cost parameters for
z
=
1,
.
,
.
.
/.
Then
and the vector 7
positive integer z
and
/,
computed
be the product dependent
the optimal base stock policy
found by choosing a
is
to
^ KN'^^,{z) + ^ ^
min
let h^, c,
'^/iT{z)
customers at station
j in the single-chain multiclass network with z customers in total, which can be
with standard algorithms (see Walrand). Finally,
=
1=1
c.N;{z)
+ Y^ h iK -
l^T{z))
!=1
Z=l J=l
/
subject to
y^7t
=
1
!=1
7,
As
>0, z=l,...,/.
in the single-product lost sales case, the objective function of this
though generally well behaved (quasi-convex). In the case where
/ij
=
problem
Cj
=
/i
not convex,
is
for
i
=
1,
.
,
.
.
/,
the objective becomes
min hz-T{z)Yk'y, +
!=1
which
convex
is
this case
if
Tj{x)
is
Yk\,
1=1
increasing and concave in x for
all
stations j
=
1,
.
.
,
.
K; hence,
can easily be solved.
A ck now ledgment
This research
at
MIT
is
supported by a grant from the Leaders
and National Science Foundation Grant DDM-9057297.
23
for
Manufacturing Program
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MIT LIBRARIES
3
'loao
ooa43tbq
i
3248 02
Date Due
JUL.
etc:
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2a
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