C^^^Y
HD28
.M414
no.
^1
DUE-DATE SETTING AND PRIORITY SEQUENCING
IN A MULTICLASS M/G/1 QUEUE
Lawrence M. Wein
i
WP? 2502
was 1902-89
December 1988
Please do not quote, duplicate, or distribute without the author's written permission.
Comments Are Welcome,
f^p?i?^»p?V
DUE-DATE SETTING AND PRIORITY SEQUENCING
IN
A MULTICLASS M/G/1 QUEUE
Lawrence M. Wein
Sloan School of Management, M.I.T.
Abstract
The problem
of simultEineous due-date setting
the setting of a multiclass
M/G/1
and
queueing system.
priority sequencing
The
objective
is
is
analyzed in
to minimize the
weighted average due-date lead time (due-date minus arrival date) of customers subject to
a constraint on either the fraction of tardy customers or the average customer tardiness.
Several parametric
and non-petrametric due-date
setting policies are proposed that
depend
on the class of arriving customer, the state of the queueing system at the time of customer
arrival,
and the sequencing policy (the weighted shortest expected processing time
that
used.
is
rule)
Simulation results suggest that these policies significantly outperform
ditional due-date setting policies
tra-
and that setting due-dates can have a larger impact on
performance than priority sequencing.
December 1988
DUE-DATE SETTING AND PRIORITY SEQUENCING
A MULTICLASS M/G/1 QUEUE
IN
Lawrence M. Wein
Sloan School of Management, M.I.T.
1.
Introduction and
Most
Summary
assume that the due-dates
for
The scheduling problem then becomes one of sequencing
the
of the literature on due-date scheduling problems
individual jobs axe exogenous.
jobs at the various stations in a job shop to optimize some measure of the ability to meet the
given due-dates. However, in most firms, the setting of due-dates
who have knowledge
responsibility of the marketing personnel,
negotiable and
is
the
is
of the customer's wishes,
and the majiufacturing personnel, who have knowledge of the shop
floor's capability.
If
the marketing group sets the due-dates oblivious to the shop floor's capability, then the
result
is
often an overloaded shop with a large work-in-process inventory and
past due.
On
the other hajid,
to the relative importance
if
many
jobs
the manufacturing personnel set the due-dates oblivious
and urgency of the various jobs, then the customers' wishes
not be satisfactorily addressed. Thus,
it is
verj'
will
important for due-dates to be based on the
knowledge of the status of the shop floor and the urgency and importance of the various
jobs.
In this paper
we study two problems
sequencing in a multiclass
setting,
shops.
it still
We
M/G/l
of simultemeous due-date setting and priority
queueing system. Although this system
captures the dynamic and stochastic elements that
Eire
is
an idealized
inherent in
all
job
Eissume that jobs of each class arrive to the shop according to an independent
Poisson process, and the service times for each job class are independent and identically
distributed rEindom variables.
The scheduler must
customer ajid must also dynajnically decide
Preemption of the customer
Since
we
in
assign a due date to each arriving
which order to serve the customers
eissume discretionary due dates, there axe two conflicting objectives. Let us
between a job's
system
arrival to the
£ind its
DDLT's
of jobs.
If
a firm
DDLT)
by
to be the length of time
promised delivery date. The
to set the due-dates as tight as possible; that
average
is,
However, some job
cleisses
objective
first
set the due-dates to
minimize the
reduce their DDLT's, then they can achieve a
cein
demand
competitive adveoitage and will be able to attract more business and/or
prices.
queue.
in service is not allowed.
define the due-date lead time (abbreviated hereafter
is
in
may be more
higher
importeint to the firm (due to potential
future saJes, for example) thaji others, and thus a more appropriate objective can be found
by assigning weights to the various
Once the DDLT's
their promises
classes
are set, the second
and meet the due
dates.
or the abihty to meet due-dates.
level,
job's actuaJ completion time
minus
negative,
and
lateness.
The second measure
the lateness
is
its
and minimizing the weighted average DDLT's.
and
conflicting goal of the
positive,
to back
is
One measure
due-date.
is
The
the lateness of a job, which
lateness of a job
mean and
may be
is
positive or
stajidaxd deviation of job
and equals zero otherwise. The typical objective
The
or equi\'alently, the fraction of taxdy jobs.
final
The
measure
is
the
number
typical objective here
is
focus on the latter two of these three measures, since they are used
that the two objectives of
tion are conflicting, since the shorter the
DDLT
of tardy jobs,
to maximize the
to achieve a given level of service. Since these
most
insightful to state our
that a shop quotes, the
two objectives are
two scheduling problems
2
more
minimization and service
DDLT's
in
if
in this case is
proportion of jobs that are completed on or before their due-date. In this paper, we
is
the
the tardiness of a job, which equals the job's lateness
to minimize the average tardiness of jobs.
It is clesir
up
There are three common measures of the service
typical objectives axe to minimize the
is
shop
will
in practice.
level mciximiza-
more
conflicting,
difficult
it is
it
perhaps
terms of a single objective and a
single constraint.
We
a due-date management policy as a combination of a
will refer to
due-date setting policy and a priority sequencing policy.
Problem
I) is
Our
first
problem (denoted by
to find a due-date majiagement policy that minimizes the long-run expected
weighted average
tion of tajdy jobs.
DDLT
subject to a constraint on the long-run expected average propor-
The second problem (Problem
II)
has the same objective but
is
instead
subject to a constrciint on the long run expected average tardiness of jobs.
We
will not explicitly solve
Problems
new due-date mEinagement
identify
agement
I
policies that
policies appeEiring in the literature.
sidered superior to due-date
management
satisfy the appropriate constraint
and
policy
is
instead to
outperform convent ioneJ due-date man-
B
in either
problem above
with equality, and policy
There have been several simulation
[22],
the goal of this paper
Due-date management policy
A
A
if
will
be con-
both
policies
achieves a lower objective
we wiU review the
value than policy B. Before stating our results,
Weeks
II;
studies, including Eilon
relevant literature.
and Chowdhury
[9]
and
concluding that due-dates based on job content and simple estimates of shop
congestion lead to better shop performance that due-dates based solely on job content.
The
analytical work on this
resentation of workload
Seidmann and Smith
DDLT
and machine capacity
[19],
who
[4],
who
uses a time- phased rep-
to set workload-dependent due-dates,
derive a constant due-date assignment poHcy (that
and
is,
the
equals a constant) in a dynamic job shop that minimizes a particular penalty cost.
The two
studies that
and Baker and Bertremd
single
problem include Bertrand
Eire
[2].
most
closely related to ours Eire
Bookbinder and Noor
[5]
Bookbinder and Noor
look at minimizing
machine problem subject to a constraint on the fraction
of tardy jobs,
DDLT
and
[5]
for a
set due-
dates based on shop content, job information, and the sequencing policy. However, they
assume that the FIFO
(first-in first-out) rule is
used between batches of jobs. Their due-
date setting rule appears to be the only non-parametric rule in the literature; most rules
include at least one parameter that must be adjusted (usually via simulation) in order to
satisfy
some
criterion.
Baker and Bertrand
[2]
3
compare three parametric due-date setting
machine model and a
strategies for a single
DDLT
fixed set of jobs.
The
equcJ to a constant, a constant plus the job's expected processing time, and a
constant times the job's expected processing time. (The parameter
three cases.)
The authors pose
of
is
We
dominated by the other two.
which depend
explicitly
the constant in these
is
the problem of minimizing the average
jobs being tsirdy and, under the assumption of
policy
three rules set a job's
on the status
known processing
will
DDLT
times, prove that the
be incorporating these three
shop
of the
subject to no
policies,
first
none
simulation experiment
floor, into the
in Section 7,
In
summary, aJthough there have been memy simulation
results, there
appear to be very
difficult,
DDLT. Although
the length of time
lateness,
it
it
the jinalyses of Problems
an improvement on the existing
observing the relationship between
is
and some analytic
has been no attempt to set due-dates and sequence jobs in a unified manner
that will lead to the minimization of
II
studies
DDLT
DDLT
can be made by
literature
and cycle time, where the cycle time of a job
spends in the shop. By the definitions of
follows that the
and
I
DDLT,
cycle time,
equals the cycle time minus the lateness. Thus
and
if
we
can find an accurate due-date setting policy (thereby keeping the lateness small), then the
desired sequencing policy should aim to minimize the weighted average cycle time in order
to achieve the objectives in Problems
[17],
Harrison
[11],
I
and
and Tcha and Pliska
II. It is
As mentioned
all
in
rule,
which
is
known with
(see, for
example, Klimov
M/G/l
queue
is
the weighted shortest
often referred to as the c^ rule.
Baker and Bertrcind
processing times were
known
that the sequencing pohcy that minimizes
[20])
the weighted average cycle time in a multiclass
expected processing time
well
[2],
if
a fixed set of jobs was considered zmd
certainty, then
we could sequence the jobs by the
weighted shortest processing time rule and choose due-dates so that the lateness of
jobs equalled zero. This due-date
average
DDLT
Of course,
in a
management
objective in Problems
I
and
II
policy
all
would then minimize the weighted
and no job tardiness would ever occur.
dynamic stochastic environment, one cannot choose due-dates with such
4
impressive results. But
system, since
assuming
cin
Problems
I
II,
rule be used for sequencing in the
M/G/l
DDLT
of jobs,
accurate due-date setting policy caji be found that satisfies the constraints in
and
constrciints in
Eire
Cfi
would be effective for minimizing the weighted average
it
II.
In this paper
rules
we propose that the
we propose
Problems
I
six
eind
due-date setting rules,
II;
of
which
satisfy the appropriate
two of these rules are parameterized and the other four
The two
non-parameterized.
all
parajneterized rules apply to both Problems
and, of the four non-paraxneterized rules, two apply to each problem.
the scheduler in Problems
number
service
I
and
II
We
I
and
Jissume that
can observe, at the time of each customer
arri\'al,
the
of customers of each class in queue (not including service), the class of customer in
(if
does not
any), and the length of time that the customer has been in service.
know
the times of subsequent
their realization.
cLrri\'als
The
scheduler
or the service times of customers before
Given the scheduler's knowledge of the queueing system, we define the
conditional sojourn time of an zirriving customer to be the total time the customer spends
in the
queueing system
each customer
is
if
the
c/i
rule
is
being employed. The conditional sojourn time for
a rajidom variable that depends on the class of the arriving customer and
the state of the system at the time of the customer's arrival. Using standard arguments
from the theory
Conway
et al.
of priority
[7]),
queues (see Cobham[6], Kesten and Rurmenburg
[15],
and
we derive the (state-dependent) expected value and Laplace tranform
of the conditional sojourn time.
The
first
parametric rule
is
based on the mean of the conditional sojourn time
bution, and the second parametric rule
is
the conditional sojourn time distribution.
of
Problem
I
and Problem
II)
based on the meain and
Two
distri-
stcLndctrd deviation of
of the non-paraxnetric rules (one for each
are found by analyzing the
tail
of the conditional sojourn
time distribution.
Unfortunately, the non-parametric rules axe difficult to calculate for a general multiclass
M/G/l
queue. For the simple example described in Section 5 (two customer classes
with different exponential processing times), we use Newton's method to compute the
non-parajnetric due-dates for the higher priority class and, for the lower priority class,
use an efficient algorithm recently developed for finding
transforms by Platzmaji et
such as Jaegerman
al.
[18].
Keilson et
[13] or
For more
al.
difficult exzmiples, this
may be
[14],
Notice that the rules described above do not
the due-dates that have already been
setting rules (one for
Problem
I
set.
and one
We
for
is
from Laplace
method
or others,
helpful.
taJce into
account information about
propose two other non-parametric due-date
Problem
to allow an arriving customer to
that use past due-date information
II)
by the
to exploit hot streaks (a string of fast service times)
these policies
tails of distributions
move
server.
The main
idea behind
aihead of customers of its
own
class
that are in queue (and hence to receive an earlier due-date) as long as these customers will
still
be expected to
depeurt the
system
in the desired
cimount of time.
Using a simulation model of a very simple system, we
and three proposed
(four proposed here
are used in conjvmction vnth the
rule in this exEimple)
earliest
Cfx
in [2]) for
rule (which
is
test
seven due-date setting rules
each of Problems
I
cind
II.
These
rules
the shortest expected processing time
and various non-paxametric due-date sequencing
policies (such as the
due-date rule) that appear in the literature. As mentioned above, the queueing
system has two customer classes that have different exponential processing times.
The primary
insight
from the simulation study
much larger improvement
in
setting policies reduced the
II
is
performance than priority sequencing. The proposed due-date
mean DDLT by 25-50%
in
relative to conventionzJ due-date setting policies.
the parajnetric rule based on the
reducing
DDLT
It is
I
and 50-68%
in
Problem
interesting to point out that
two moments was only
slightly
more
effective at
performance between the parametric rules and the non-parametric
rules; the peirametric rules
were
first
Problem
than the peirametric rule based on one moment. Also, there was not a
significant difference in
rules
that proper due-date setting offers a
slightly
more
were slightly more effective in Problem
effective in
Problem
6
II.
I
and the non-parametric
Thus, the parametric rule based on the
expected value of the conditional sojourn time, which
multiclass
A//G/1 system, appears to be a very
importEint implications for the
quite difficult to obtain
effective
is
very
eeisy to calculate for
any
due-date setting policy. This has
more complicated network
setting, since
good estimates of second moments or
it
appears to be
conditional sojourn
tails of
time distributions in a multiclass queueing network under various priority schemes. Caxe
must be
teiken in
drawing broad conclusions concerning the relative strength of the four
proposed due-date setting
policies, since only
M/M/1
a single instance of an
system has
been analyzed numerically.
The simulation
results also suggest that
the impact from priority sequencing
posed due-dates are
set in
is
when
the due-dates proposed here are used,
minimal. This
accordance with the
c/x
rule,
from the
date setting policies (from
priority sequencing has
[2]),
intuitively clear, because the pro-
and thus due-date based sequencing
However, when using the traditional due-
policies will not differ greatly
c/i rule.
is
some impact, but not
as the proposed due-date setting policies. Thus, although there have
tion studies (see
Baker
for due-date scheduling
[1],
for
necirly as
much
been many simula-
example) comparing various priority sequencing heuristics
problems,
it
appears that more leverage can be gained by being
concerned with the setting of due-dates, not with priority sequencing.
The
results of this simulation
study closely parallel those of Wein
lem of simultaneous input control (how to
sequencing
wafer fab.
[25])
(2)
is
It
analyzed
in
releaise
[23],
jobs onto the factory floor) and priority
the setting of a 24-station simulation model of a semiconductor
was found that (1) input control (loosely based on the analysis
provided a
much
larger
improvement
under proper input control, the
in
this
effect of priority
study and the present one
is
of
Wein
[24]-
performance than did priority sequencing,
sequencing was minimal, and (3) un-
der traditional input control, priority sequencing had a moderate impact.
between
where the prob-
The connection
that due-date setting and input control can both
be thought of as tactical design decisions that are made at a higher
sequencing decisions. Thoughtful decisions
made
7
level thain the priority
at this higher level lead to well designed
systems that
zire
much
easier to control (in that detailed control issues such as priority
sequencing do not need to be a major concern) than poorly designed systems.
This paper
is
Problems
orgaxiized as follows.
and the Laplace transform and expected value
Section
3.
Two
in Section 4.
I
and
II
axe formulated in Section 2
of the conditional sojourn time axe given in
parzmietric and two non-peirametric due-date setting policies are proposed
In Section 5, these rules axe derived for the case of two customer classes
with different exponential processing times. In Section
6,
we
describe two additional non-
parametric due-date setting policies that attempt to exploit hot
the simulation experiment
2.
We
Two Problem
is
presented in Section
streaJcs
ser\'er,
Formulations
=
consider a multiclass A//G/1 queueing system where jobs of class k
class k Eire
The
variance, general distribution Fk{t),t
must assign a due date Dk,t to a
in
>
class
0,
jfc
and Laplace transform
k job that arrives at time
t
is
Dk,t
-
t-
in queue.
Thus the
the long-run expected average
DDLT
expected average proportion of jobs that are
management
J
,
finite
t,
and must
DDLT
of class k jobs.
tEirdy.
policy.
Let
Then Problem
I is
P
For
fc
also
of a class
To repeat, a due-date management policy
combination of a due-date setting policy and a priority sequencing
Dk be
/i
The scheduler
Fl^{s).
customer who arrives at time
which order to serve the jobs
1,...,A'
service times for job
independent and identically distributed random variables with mean
dynamicedly decide
and
7.
arrive according to an independent Poisson process with rate At.
let
by the
=
1,
is
...,
a
A',
be the long-run
to find a due-date
policy to
minimize
(1)
(2)
where Ck
is
a linear cost (or weight) for job class
and p
k,
95%
T
Let
on time,
of their jobs
in
bound on
the desired upper
many companies
the proportion of jobs that are tardy. For example,
goals by desiring to deliver
is
which case p
define their service
=
.05.
be the long-run expected average job tardiness. Then Problem
II is
to find a
due-date management policy to
K
mimmize
subject to
where f
is
the desired upper
The non-parametric
Pfc
<
pt for i
=
r<
(4)
f,
bound on the average job
Problems
1, ...,i^,
k jobs that are tardy,
(3)
where
emd pt
is
I
and
Pfc is
II.
tardiness.
paper can accomodate class-dependent
rules proposed in this
service level constraints in
by
E^'^*
For example, constraint (2) can be replaced
the long-run expected average proportion of class
the desired upper
bound on
the proportion of class k jobs
that are tzurdy.
The Conditional Sojourn Time
3.
The
goal of this section
is
to derive the Laplace transform
conditional sojourn time Sk,t for a class k customer
the conditional sojourn time of an arriving customer
in the
system
and the
we
if
the
Cfi
rule
is
state of the queueing
will suppress the
system
at the
dependence on the
time of
t,
time
/.
Recall that
the total time the customer spends
Ci/ij
arrival. In order to simplify notation,
time.
EtrriN^al
>
customers in objective function (1) and
system at time
arrives at
of the
being used, conditioned on the class of arriving customer
the customer classes are ordered so that
class k
is
who
and expected value
...
(3).
>
If
Without
...c/^/i/^-,
loss of generality,
where Ck
is
the weight for
a customer arrives to the queueing
the scheduler can observe the A' — dimensional vector Q{t)
9
assume
=
{Qk{i)),
where Qk{t)
time
t,
=
n^
is
the
number
the class of customer
customers in queue (not including service)
of class k
who
currently in service at time
is
that has elapsed since this customer started service, which
is
t,
and the length
denoted by
=
a{t)
at
of time
a.
Let us
define Fq to be the distribution of the residueJ processing time of the customer currently
in service.
Then
Fo{t)
=
for
<
>
the server
if
^"W if
a class k customer
in service, for k
is
and Laplace tranform,
class k
let
and
to
in service.
l,...,K.
Let
/i^'
and Fq{s) denote the mean
=
if
the server
is idle,
is
known from
well
and
_!_
^ Jo°^fdF,ix)
no
2j^xdFk{x)
^g^
Following the notation and reasoning of Chapter 8 of
Conway
et
=
5Zj=i ^i ^^ ^^^ total arrival rate of jobs with higher priority than class
Fak{^)
=
A~^ ^i=i ^i^ii^)^i >
let F^i^{s)
for the
service plus the processing times of
denote the Laplace transform of
si,t{s)
where B*i^{s)
be their composite processing time distribution,
be the associated Laplace transform. Define Gq{s)
be the Laplace trajisform
is
=
f;{s)[gi{s
+
sum
all
Sk,t
x^k
=
FQ{s)Y[i^i
[•'^t*("5)l"*
of the processing times of the job currently in
jobs in queue of higher priority thaji class
by S^
,(5),
then
it
follows that for k
=
k. If
we
l,...,K,
- KkB:,{smF:is + Kk - KkB:,{s))r''
(?)
the solution to
B'akis)
Thus,
(^'
- n(a)
Xak
al.[7], let
fc.,
is
=
and
respectively, of the residual processing time. It
renewed theory that /i^^
if
1
is idle,
in order to obtain
=
F:,{s
+
X,k
- KkBUis)).
a closed form solution for S^
solution B*i^(s) to equation (8).
In cases
,(5),
one needs to
(8)
first find
where a solution can be found, the expected
value and standard deviation of the conditional sojourn time, denoted by
a[Sfc,j], respectively,
the
can be found by differentiating the Laplace transform S^
10
£^[5fc,t]
,{s).
and
However, a simple expression for £[St,c] can be found by using the direct expected
value procedure of Cobhain
Hcirris [10].
=
Let pk
[6];
example, Dolan
see, for
=
^k/ftk and Ok
^'
=
Hi
+
I3,=i Pt for
[8]
or pages 205-207 in Gross and
l,---,-^*
where ao
=
0.
Then
it
follows that
y''
1
=
C/ Ok.t
4.
customers in queue at time
class k
eire n/t
(9)
l-<Tk-i
/jfc
where there
_L
1
i,
and where
fiQ is
given by
(6).
Four Due-date Setting Policies
we
In this section
present four due-date setting policies,
propriate constraint (2) or (4).
to both Problems
applies to
and
I
Problem
I
cind
II.
The
Of the
two
first
last
two
edl
of
which
satisfy the ap-
policies axe paxzLmeterized rules that apply
policies,
one applies to Problem
II.
which are both pairameterized, one
The
first
parametric rule assigns the
due-date
Dk,t
and the second parameterized
pareimeters
q
eind
P
(10)
t
rule assigns the due-date
Dt,t
The
= + aE[Sk,t],
in (10)
= +
t
E[St,t]
and (11)
Eire set
+ MSk,,]'
(via simulation) in
(11)
Problems
I
and
II
so
that the constraints in these problems are satisfied.
The non-parametric
rule for
Problem
I
assigns the due-date
Dk,t=t+Pk,t,
(12)
=
(13)
where
P
P{Sk,,>Pk,t).
11
Thus,
—
pk,t is the (1
due-date setting rule
be automatically
Suppose
is
fractile of
employed
random
the distribution of the
in conjunction
with the
cfi rule,
variable Sk,(-
If
this
then constreiint (2)
will
satisfied.
now
for
p)th
that the
ramdom
non-parcimetric rule for Problem
II
Then
variable Sk,t has distribution Gk,t-
the
first
assigns the due-date
= t + rk,t,
Dk,t
(14)
where
r= /
due-date setting rule
Simil8Lrly, if this
will
be
earlier, the
employed with the
Cfi rule,
then constraint (4)
due-date setting policies described in (11)-(15) are not easy
to calculate for general multiclass
E[Sk,t]i<^[Sk,t]iPk,t,
and
Tk^t for
AI/G/1 queues.
In the next section
we
will derive
a particular simple example.
An Example
Suppose there are
rates
is
(15)
satisfied.
As mentioned
5.
{x-Tk^t)dGk,M.
fii
that the
and
c/x
/i2,
rule
K = 2 customer classes that have exponential processing times with
respectively.
Without
loss of generality,
awards higher priority to class
1
suppose that
customers.
By
fik if
class k
customer
the time of a customer arrivej
customers are in queue,
is
r'l
=
in service, for
fc
=
1, 2.
The
12
=
one
if
a
cleiss
k
customer
implies that ni
=
n2
=
is
is
exponential
state of the system at
adequately described by ("i, n2,ii,
eind ik equals
zero otherwise. Notice that
is
C2/i2, so
the memoryless property
of the exponential distribution, the residual processing time distribution Fq
with paxauneter
>
ci/ij
22)1
where
in service,
njt
and
ik
class k
equals
0.
Let us begin by analyzing the conditional sojourn time Sj^j of the higher priority
customer
class.
If 12
=
0,
then Sij has an Erlang distribution with shape parameter
12
Til
+zi
+ l and
scale parameter
/ii.
=
^^2
1,
Erlang distribution with shape parameter nj
when
distribution with parameter ^2- Thus,
and
<7[Si,j]
=
=
nipit.
Then equation
y
/ij"^
\/n\
-\-
i\
+
I'l
When
I'l
ij
_
=
^
=
1,
distributed
eis
the convolution of an
+ 1 and scale parameter /i] and an exponential
,
I'j
=
0, it
follows that
JE^[Si_,]
=
/if*(ni+ii4-l)
when
=
t2
0, let
(13) reduces to
+M +l
j-\
equation (16) has a closed form solution that leads to p^t
0,
=
5i,i is
In order to find pi^t in equation (13)
1.
"l
When
then
equation (16) can
eeisily
=
ln(p~')-
fJ-\^
be solved using Newton's method. Similarly, when
equation (15) reduces to
0,
/i;"-^"-^^e->-'-M
+
("i
'i)!
Ani+ti +
V
1)!
^"'^"''
"^'
T-t
,^
/^r""""'^'
(n, +,-^
+
1)!
J'
\
_ (m+iQ! 7'
0-1)'
(17)
Once ag£dn, a closed form solution Tit
method
When
When
be used when
caji
t2
P
=
12
1,
Z',
=
1,
we have
I'l
>
= — /xf
^
ln(/iif) exists
when
ij
=
0,
and Newton's
0.
^[5i,t]
=
/^r^C"!
+1) + /^^^ ^^^
^['^i,<]
~
y/^r^C"!
+
1)
+
-2
A*2
equation (13) reduces to
/'.e
^^^^(
1)
(^^_^,)„,-,>,^.,+(
1)
(^,_^,)"'-^'
,
(18)
and equation (15) reduces to
f^_1\"l
which
c£ui
The
difficult
p-ft\ri,t
p-fiiTij
\
both be solved by Newton's method.
analysis of the conditional sojourn time $2^ of the lower priority class
and requires the use
of the Laplace transform S^
13
,.
Since ^0*2(5)
=
is
more
Fi{s)
=
7
/ii/(5
+ /ii),
^
the solution B^2i^) 'o equation (8)
[16])
By equation
(7)
is
(see, for exaimple,
page 215 of Kleinrock
we have
^il+\l+s+ V(/ii+Ai+5)2-4/iiAiy
2/^2
2/i2
where, for
A;
=
1,2,
otherwise. Readers
(9),
and that
u =
may
-
/il
+
Ai
a class
1 if
verify that
+
(21)
+ ^{fli+Xi +5)2-4/i]
5
fc
customer
— Sj
is
in service at time
t,
and equals zero
given in equation
j(0) yields the value of E[S2,t]
differentiating (21) twice gives
E[Sl] = S^iO)
—+
=- + iny+i,)(—
2Ai
'-—)
(/^1-Ai)''/
^A'2(mi-'^i)
^2
+
(n2
+
7
+
—
2fij
+
t2)(-27
r^
- Aj)
V/i5(/ii
A'K/^l-'^l)
^-^
1
(ni
+
+
ti)(ni
—
ru)
/i2(/il-Ai)3
u2(/ii - Ai)-*/
+
ii
+
1)
+ ^(n2 +
t2)(n2
+
»2
(ni+.i)(n2+t2)y
/^2
+
1)
(22)
y
Thus,
^[S2,t]
Since
£'[5fc,j]
=y/E[Sl]-E[S2,tV
does not equal
<7[Sfc,i]
for k
=
1,2, due-date setting policies (10)
and (11)
should yield different results for our example.
In order to calculate p2,t ^ind
T2^t
in equations (13)
algorithm recently developed by Platzman
et al.
14
and
(15),
[18] for
we used an approximation
computing
tail probabilities
=
For a prespecified eiccuracy parameter A/1 eind prespecified precision
from transforms.
parameter
AP,
they showed that the
^^
""utLf
teiil
probability
TP
defined by
+ i: ^'"tc" - ^")^^-..o-))
(24)
n=l
satisfies
P{S2,t
<
as long as P(52,t
>
I/)
+
yl
-
A>1)
<< AP.
AP
TP <
<
By
=
^
JL],
equation (25),
probability P{S2,t
Ayl)
+ AP
(25)
a =
y2it'
e^^,
;9
=
L/
=
(26)
+ 2AA'
and
e>(^+^^)'^,
7
e-^^-.
can be seen that equation (24) ceJculates the appropriate
it
> A)
> ^-
Here,
AP^'
AT
P(52.,
we need
given A, whereas
probability equals the desired value of
to find the value of
such that the
(24) within a simple search algorithm allows us to ceJculate p2,t-
The
search algorithm,
given previously calculated values of >li_i,TP,_i,i4,, and TPi, calculates a
A, denoted by >l,+i
,
new
= ^-1 -
(
7^fz^)(^-i
Equation (24)
is
then used to ceJculate TP,^i given
a solution p2,t
=
-^i
^4^+1.
- ^^)-
The cJgorithm
(27)
is
stopped with
when
\TP,-p\<e,
c is
value of
by
^.+1
where
tail
However, imbedding equation
in equation (13).
p
A
tail
(28)
a small specified value.
Platzmaji et
al.
[IS]
showed more generally
that, for
any integrable function
g{S2,t)i
N
rP* = Co +
2
^
Q"'real{CnS2',,(ju;n)}
n=l
15
(29)
'
always
lies in
the range
a)|0
<
S2.,
max{g{x)
—
g{x')
conv{E[g{S2,t
where diam(^(7))
=
N
=
+
n
By
speciaJizing the function g to
we
casi
P2,i:
...,
:
obtain an approximate v'alue of
equation (29)
takes the place of
x, x'
<
/},
:
\a\
< AA,
\b\
<
diam(5(/))},
where the Fourier
(30)
series coefficients
€„
appearing in equation (29) axe given by
for
0, 1,
<U] + bAP
is
rj,*
with a similar algorithm that was used to find
simply used in place of equation (24) to find
TP
in
TP*
given
j4,,
and TP'
equation (27). For our special case of the function g in (32), the
Fourier coefficients are given by
_
^°
and, for n
=
1,
...,
A^+{AA)^+2UAA-2AAA-2AU
= U^ +
W+2AA)
N^
Cn = e>-(^-^AM)(^^-t^ +
V
where
6.
w
is
,..,
^^^^
27rjn
-L^ - e>-^(-^ + ^),
27ru>n^/
2-Kjn
(34)
27ru;n^
given in (26).
Exploiting Hot Streaks
Although the due date Dk,t
for a class k
customer
eirriving at
time
<
proposed
in
Section 4 depend) on the class k of arriving customer and on the state of the queueing
system at time
<, it
does not depend on the due dates of customers
16
who
are in
queue
at
time
and
t.
reasonable to presume that improved due-date setting policies for Problems
It is
I
could be found by allowing Dk,t to depend on past due-date information.
II
In this section,
we
and one
for
Problem
I
describe two non-parametric due-date setting policies (one for
Problem
II)
that use past due-date information to exploit hot
The
streaks (a sequence of fast service times) by the server.
to exploit
is
situation
the following: suppose a customer arrives at time
t
we
Eire
attempting
and, just prior to time
t,
the server hcis completed a sequence of services that were faster thein expected. (In the case
where machine breakdown and repair axe incorporated into the service time distributions
(see, for exEunple, Hcirrison [12]),
such a hot streak can occur when there has not been
a machine breaJvdown for an unusually long time, or
Then
quicker than expected.)
there
when
may be customers
in
a machine
repaired
is
much
queue who have particularly
slack due-dates, because their due-dates were set before the stairt of a hot streak. Indeed,
may be enough
there
slack in these due-dates so that
an arriving customer can move eihead
of these customers in queue without endangering these customers with tardiness.
With
this situation in
policy for
Problem
tomers of
its
classes
own
by the
I.
mind, we define the following non-parametric due-date setting
This policy only allows an
cleiss.
Cfi rule,
airriving
The corresponding sequencing
each customer in the queue that
is
of the
These due-dates axe computed only
This policy
same
The new due-date
time of their
arrival.
again computed according to (12)-(13), but
has just arrived and observes nt customers of
than
equals the
this
number
first
computes new due-dates
customer
retain the original due-date that
them
TTik
ranks the customer
for
(say, class k).
for the purpose of assigning a due-date to the arriving
still
and
still
class as the arriving
customer; the customers in queue
is
policy
but now customers are ranked within each class by the earliest
due-date, not by the earliest arrival date.
at the
customer to move ahead of cus-
is
class k in
customer; the extra one in addition to
customer of class k
now computed
of class k customers in
17
for a
mt
was assigned
as if the
queue, where
queue that have
njt
in
to
queue
new customer
equcds nik
ain eairlier
+
1
due-date
accounts for the arriving customer.
Thus the new due-date
is
based on a revised conditional sojourn time that assumes that
the arriving customer moves ahead of the customer in queue.
customer
in
queue
is
If
the
new due-date
than the customers original due-date, then there
earlier
slack in the customer's originzJ due-date to allow the arriving customer to
customer in queue; in
this
this case,
we say
of the
enough
is
move ahead
of
that the customer in queue can accomodate
the arriving customer.
Under our proposed
policy,
an arriving customer, rather than joining the end of the
queue of class k customers (who are ordered according to the
instead passes customers in
meets a customer
is
in
its
earliest due-date criterion),
queue (starting from the back of the queue)
he/she
until
queue who cannot accomodate him/her. The arriving job's due date
then calculated by (12)-(13), ignoring
eJl
the customers of
its class
that
it
hcis
passed in
queue.
The corresponding non-parametric
rule for
Problem
II is identical to this rule,
that equations (14) emd (15) are used in place of (12) and (13) to calculate
due-dates.
If
and
old and
these two due-date setting policies aie used in conjunction with the
where jobs are served within each
(2)
all
(4), respectively,
class
should be
by the
earliest
except
Cfi
new
rule,
due-date criterion, then constraints
satisfied.
This idea of exploiting server hot streaks cam be extended so that arriving customers
may
pass ahead of customers in queue that are of a higher class, not just of the same
class. In this case, the
corresponding sequencing policy would be according to the earliest
due-date criterion, regardless of the class of customer, although equations (12)-(15) would
still
be used to
set
new and
old due-dates.
However, the conditional sojourn time for a
customer in queue would need to take into account
all
customers in queue who have earlier
due-dates than this customer. Such a policy cannot be guaranteed to satisfy constraints
(2) or (4), since the Cfi rule
of these policies
would not be used. Due to the rather cumbersome nature
and due to the modest inprovement achieved
in the simulation
study of
the next section by the two rules described earlier in this section, this idea has not been
18
pursued
7.
einy further.
A
Simulation Study
Using the example
performance of the due-date setting
with Poisson arrival rates Aj
p
=
.8.
The weights
Problems
Cfj
/ii
I
emd
=
1
=
.4
and ^2
for the
and
=
p
=
.05 (that
=
The two
.2.
Thus, p^
=
5%
is,
1
The
jobs.
tardy jobs) and f
For each of Problems
I
ageiinst
conven-
=
cj
=
C2
and
II,
=
1
classes
have exponential service
classes
= A and
p2
two customer classes are
rule gives higher priority to class
set at
-S-
A2
minimize the long-rim expected average
II is to
and 6
policies described in Sections 4
The example queueing system has two customer
tional due-date setting policies.
times with rates
a simulation study was undertalcen to compare the
in Section 5,
the server utilization
(that
is,
is
the objective in
DDLT of jobs),
service levels for problems
and so the
Eind II
I
were
0.5.
seven due-date setting policies and five priority
se-
quencing policies were tested; thus, 35 due-date management policies were tested for each
problem. For each due-date management policy tested on each problem, 20 independent
runs were made, each consisting of 5000 customer completions. Hach. simulation run stcirted
with an empty system and had no initialization period. Five parametric due-date setting
policies, the first three of
policy (referred to as
rameter
c;
where D^j
£^[St,t];
CONSTANT
= {
t
c/i^ ^;
and policy
(11),
P
and the
±
in Tables
were tested on both problems: a constant
I
and
.0005]
II),
=
t
+
/^^
^
+
c;
=
+
c for
some pa-
proportional
(PROP),
where Dk,t
t
the policy described in equation (10), which will be referred to as
was
(.05
[2],
a slack poHcy (SLACK), where Dk,t
ciated parameter
e
which are from
which
will
be referred to as
<T[5jfc,t]-
For these
policies, the asso-
set so that the resulting average fraction of tardy jobs
resulting average job tardiness
f
satisfied
f
6
In addition, two non-parcimetric due-date setting rules were tested
19
[.5
±
P
satisfied
.005].
on each problem.
The
policy described in (12)-(13), which will be referred to as
on Problem
I,
on Problem
II.
referred to as
and policy (14)-(15), which
I
and
be referred to as
two corresponding poUcies described
Also, the
HOT
will
HOT II, were tested on Problems
I
NONPAR
I,
was tested
NONPAR
II,
weis tested
and
II,
set equal to .01
(28)
was
and
.005, respectively. Also, the
set equal to .0005 for
NONPAR II
HOT
asid
II.
NONPAR
I
and
parameter
HOT
I,
U
The upper bound parameter
will
be
respectively. For these
AP
four policies, the accuracy peirameter A^l and precision parameter
were
which
in Section 6,
e
equation (25)
in
appearing in equation
and was
appearing
set equal to .005 for
in (24)
was
set equal
to twenty times the expected value of the conditional sojourn time.
Only non-parametric
priority sequencing rules were considered in our study; readers
are referred to Vepsalainen
five priority
where
and Morton
sequencing policies are:
[21] for
work
recent
in parameterized rules.
the shortest expected processing time rule (SPT),
due date
class 1 jobs get priority over class 2 jobs; the earliest
priority
given to the job with the earliest due-date; the
is
minimum
rule
its
expected processing time minus the current time; a
where priority
given to the job with the smzJlest ratio of
is
processing time; and the modified due-date
(MDD)
its
is
the
maximum
current time plus
For the
its
SPT
within each class.
of its due-date
and
two sequencing
(S/EPT),
critical ratio rule
slack divided
its earliest
due-date
is its
by
its
expected
[3],
which
where a job's modified due-
expected completion time (that
is,
the
expected processing time).
rule,
Two
we need
to specify the
majmer
possibilities are to order
in
which customers
them by the
respectively.
Under the three
6,
the
20
ordered
date or by
SPT(FIFO)
traditional due-date setting policies, these
do not depend on the state of the
rules are identical, since the due-dates
queueing system. As mentioned in Section
Eire
earliest arrival
the earliest due-date; the two subsequent sequencing policies are denoted by
and SPT(EDD),
(SLACK),
policy of Baker and Bertrand
gives priority to the job with the earliest modified due-date,
date
(EDD), where
slack rule
which gives priority to the job with the smallest slack, where a job's slack
minus
The
HOT
I
and
HOT II due-date setting rules
SPT(EDD)
need to be run with
in order to satisfy the necessary constraint.
However, the
remaining proposed due-date setting policies were tested in conjunction with both the
SPT(FIFO) and SPT(EDD) sequencing
The
Table
policies.
results of the singulation study are
Problem
II (for
II).
policy, eind the average
summarized
in Table
As
caji
is
DDLT
of jobs
stated in Table
be seen
in
(for
Problem
In both tables, each row corresponds to a due-date
is
given for each row, along with a
interval. In addition, the average fraction of tardy jobs is stated in
job teirdiness
I
Table
II,
both with
95%
Table
I,
I)
management
95%
confidence
and the average
confidence intervals.
the four due-date setting policies easily outperformed the
I,
three traditional due-date setting policies. Also, the due-date setting poUcies have a
As
bigger impact on performance than do the priority sequencing policies.
proposed due-date setting
two parametric
rules.
outperforming the
two non-paraxnetric rules
policies, the
The
HOT
NONPAR
I
I
policy
policy.
and
it
since only a
for our four
outperformed the
policy, slightly
minor improvement
would appear that the
not often lead to a significant improvement over the
NONPAR
I
HOT
I
policy will
policy. Also, the clSt,*]
DDLT compared to E\Sk,t], suggesting that the second
of the conditional sojourn time improves performajice, but not drajnatically.
Notice that the service level target of p
=
.05
intervals of the proportion of tardy jobs observed
I,SPT) due-date management
policies; thus the
There was a negligible difference
when they were used
Problem
I.
This
and thus the SPT
in conjunction
is
rule
in
was well within the 95% confidence
under the
(NONPAR
I,SPT) and
(HOT
algorithms developed in Sections 5 and 6
for the non-parajnetric due-date setting policies axe
for
much
Since the ability to exploit hot streaks should
was obtained with exponential processing times,
policy 8wJiieved a minor reduction in
slightly
was the best due-date setting
increase with the \'ariability of the processing times,
moment
and
shown
to
be
reliable.
performance axnong the priority sequencing policies
with one of the four due-date setting policies proposed
because the due-dates were
and the due-date based
21
set in
accordance with the
SPT
rules prioritized jobs in a similar
rule,
manner.
nUF-DATF,
SFTTING
POMCY
ni;F.-r)ATF
SKTTING
POI.IO'
The only exception
the
is
proposed due-date setting
may attempt
rule
S/EPT
policies.
sequencing rule, which did not perform well with the
This
not surprising, however, since this sequencing
is
to serve class 2 customers before class
customers when there are no
1
tardy jobs in queue, and hence will counteract the proposed due-date setting policies.
(PROP,SPT) due-date management pohcy was
had a
rule
impact under a
significant
1
the only case where a priority sequencing
SPT
to Table
for this case axe quite similar to those of
due-date setting policies cut the
II,
which gives
Problem
I,
In contrast to
performed the pEirametric
rules.
[<j[Sk,t]
(NONPAR II,SPT)
of the interval in
As
in
95%
and
Problem
Also, there
rules in
SPT
II.
I,
II.
The
results
the non-parametric rules
I,
was virtually no difference
Problem
II.
HOT
The
II
in
outperformed
now
out-
performance
NONPAR
service level target f
=
.5
II,
once
(HOT
II,SPT) policies; however, the target was near the endpoint
there
was very
little
difference
policies.
among
Under the
priority sequencing policies
traditional due-date setting poH-
did not perform as well as the due-date sequencing policies. In particular, the
policy,
which had performed well
in
Also, our study agrees with the results of
rule performs better
In simimary,
all
Problem
I,
did not perform well in Prob-
Baker and Bertrand
than the other sequencing rules
in
most
[3]
in that the
MDD
cases.
the due-date setting policies described in this paper easily outper-
formed the traditional due-date setting
in
Problem
both cases.
Problem
(PROP,SPT)
lem
results for
confidence interval of the observed average job tardiness for the
under our proposed due-date setting
cies,
rule.
but are even more dramiatic: the best
but the increase in performeince was again modest.
again was within the
SPT
DDLT by a factor of two or three compared to conventional
due-date setting policies.
between the E[Sk,t] and
due-
sequencing rule in this example because class
jobs have a shorter cycle time them class 2 jobs under the
We now turn our attention
PROP
due-date setting policy. The
particuleir
date setting rule worked well with the
The
policies,
performance than did priority sequencing.
24
and provided a much larger improvement
Moreover, the proposed due-date setting
policies are very robust
with respect to the sequencing policy that was used with
it.
Acknowledgements
I
am
grateful to J. Michael Harrison for
would also
like to theink
Dimitris Bertsimas
many
helpful discussions about scheduling.
and Stephen C. Graves
25
for helpful
I
comments.
REFERENCES
[1]
Baker, K. R., "Sequencing Rules and Due-Date Assignments in a Job Shop,"
Mgmt. Sci
Baker, K. R. and
[2]
Rules,"
[3]
30, 1093-1104 (1984).
AIIE
W. M.
J.
IVans. 13, 123-131 (1981).
Baker, K. R. and
J.
W. M.
ing Against Due-Dates,"
Bertrand,
[4]
J.
Bertrand, "A Comparison of Due-Date Selection
W.
Bertrand,
J.
Bookbinder,
J.
H. and A.
Level Constraints,"
[6]
Cobham,
Mgmt.
3,
Priority Rule for Schedul-
37-42 (1982).
M., "The Effect of Workload Dependent Due-Dates on Job
Shop Performance," Mgmt.
[5]
Ops.
"A Dynamic
J.
A., "Priority
I.
Sci. 29,
799-816 (1983).
Noor, "Setting Job-Shop Due-Dates with Service
Opl. Res. Soc. 36, 1017-1026 (1985).
Assignment
in
Waiting Line Problems," Ops. Res.
2,
70-76; correction ibid. 3, 547 (1954).
[7]
Conway, R. W., Maxwell,
W.
L.
and
Miller, L.
W., Theory of Scheduling.
Addison- Wesley, Reading, Mass. (1967).
[8]
Dolan,
R
J.,
"Incentive Mechanisms for Priority Queueing Problems," Bell
Journal of Economics
[9]
Eilon, S.
J.
[10]
9,
421-436 (1978).
and L G. Chowdhury, "Due Dates
Job Shop Scheduling,"
Intl.
Prod. Res. 14, 223-237 (1976).
Gross, D. and C.
M.
Harris, Fundiimentals of Queueing Theory,
John Wiley and Sons, New York
[11]
in
Harrison, J.M.,
2nd
Edition.
(1985).
"Dynamic Scheduling
of a Multiclass
Queue: Discount
Op>-
timality," Ops. Rsch. 23, 270-282 (1975).
[12]
Harrison,
J.
M., "Brownian Models of Queueing Networks with Hetero-
geneous Customer Populations,"
in
26
W. Fleming and
P. L.
Lions
(eds.),
Stochastic Differential Systems, Stochastic Control Theory and AppUcations,
[13]
IMA Volume
Jagermaji, D. L.,
10, Springer- Verlag,
"An
New
York, 147-186 (1988).
Inversion Technique for the Laplaxre Transform with
Application to Approximation," Bell Syst. Tech.
[14] Keilson, J.,
W.
Nunn, and U. Sumita, "The
R.
form", Appl. Math. Comput.
[15]
Kesten, H. and
57, 669-710 (1978).
J.
Bilateral Laguerre Trans-
137-174 (1981).
8,
T. Runnenburg, "Priority Waiting Line Problems
J.
I, II,"
Koninki. Ned. Akad. Wetenschap., Proc. Ser. A. 60, 312-336 (1957).
[16]
Kleinrock, L., Queueing Systems Vol.
I:
Theory. John Wiley and Sons,
New
York (1975).
[17]
KHmov, G.
"Time Sharing Service Systems
P.,
I,"
Th. Prob. Appl. 19,532-
551 (1974).
[18]
Platzman,
L., J.
Ammons, and
C.
Bartholdi,
J. J. Ill,
"A Simple and
Efficient
Algorithm to Compute Tall Probabilities from Transforms," Ops. Res. 36,
137-144, (1988).
[19]
Seidmann, A. and M. L. Smith, "Due Dates Assignment for Production
Systems," Mgmt.
[20]
Tcha, D.
W. and
Sci. 27,
571-581 (1981).
R. Pliska, "Optimal Control of Single-Server Queueing
S.
Networks and Multi-Class
M/G/1 Queues
with Feedback," Ops. Rsch. 25,
248-258 (1977).
[21]
[22]
Vepsalainen, A. P.
and T.
J.
E.
Morton, "Priority Rules for Job Shops with
Weighted Tardiness Costs," Mgmt.
Sci. 33,
Weeks,
of Predictable Due-Dates,"
J. K.,
"A Simulation Study
1035-1047 (1987).
Mgmt.
Sci.,
25, 4, 363-373 (1979).
[23]
Wein, L. M., "Scheduling Semiconductor Wafer Fabrication," IEEE Trans.
Semiconductor Mfg.
I,
115-130 (1988).
27
[24]
Wein, L. M., "Optimal Control of a Two-Station Brownian Network," to
appeeir in Math, of Oper. Res. (1988).
[25]
Wein, L. M., "Scheduling Networks of Queues: Heavy
Traffic Anadysis of
a
Two-Station Network With Controllable Inputs," to appear in Ops. Res.
(1988).
28
3 '//'?>
;(//
/
'Iks
Date Due
SEP.
.
1
1992
itlt^
Lib-26-67
LIBRARIES DUPI.
3
1060 DD7267-=I5