J. Appl. Prob. 29, 667-681 (1992)
Printedin Israel
Trust 1992
Applied
Probability
?
ON THE OPTIMALITYOF LEPTAND cp
RULES FOR MACHINESIN PARALLEL
CHENG-SHANGCHANG,* T. J. WatsonResearchCenter
XIULI CHAO,**New JerseyInstituteof Technology
MICHAELPINEDO,***ColumbiaUniversity
RICHARDWEBER,****Universityof Cambridge
Abstract
We consider schedulingproblemswith m machines in paralleland n jobs. The
machinesare subjectto breakdownand repair.Jobs have exponentiallydistributed
processingtimes and possibly random release dates. For cost functions that only
depend on the set of uncompletedjobs at time t we provide necessaryand sufficient
conditions for the LEPTruleto minimizethe expectedcost at all t withinthe class of
preemptivepolicies. This encompassesresults that are known for makespan,and
providesnew resultsfor the workremainingat time t. An applicationis that if the cyi
rule has the same priority assignment as the LEPT rule then it minimizes the
expected weighted number of jobs in the system for all t. Given appropriate
conditions, we also show that the cy rule minimizes the expected value of other
objectivefunctions,suchas weightedsum ofjob completiontimes,weightednumber
of late jobs, or weightedsum of job tardinesses,whenjobs have a common random
due date.
EXPONENTIAL
ING; WEIGHTED
DISTRIBUTION;
FLOWTIME
MAKESPAN;
AMS 1991 SUBJECT CLASSIFICATION:
PRIORITY POLICIES; STOCHASTIC SCHEDUL-
PRIMARY 90B35
1. Introduction
Considerm machinesthatoperatein parallelandaresubjectto breakdownandrepair.
Their up times and down times are arbitrarilydistributed.The machinesare used to
process n jobs, whose release dates, R1,- - *, R, are also arbitrarilydistributed. Jobj has
a processingtime Pj, that is an exponentiallydistributedrandomvariablewith rate yj.
Received 6 December 1989; revision received 28 June 1991.
* Postal address:IBM Research Division, T.J. Watson Research Center, PO. Box 704, Yorktown
Heights, NY 10598, USA.
** Postal address: Division of Industrial and ManagementEngineering,New Jersey Institute of
Technology,Newark,NJ 07102, USA.
*** Postal address: Department of Industrial Engineering and Operations Research, Columbia
University, New York, NY 10027, USA.
The researchof this authorwas partiallysupportedby the NSF under grantECS 86-14689.
**** Postal address:CambridgeUniversity EngineeringDepartment, Mill Lane,
Cambridge,CB2
1RX, UK.
667
668
C.-S. CHANG, X. CHAO, M. PINEDO AND R. WEBER
Eachprocessingtime is distributedindependentlyof all other randomvariablesin the
model. Withoutloss of generality,we makethe followingassumption.
(Al)
0
1
1
2
'
n.
Let Q1,tbe the indicatorfunctionforthe eventthatjobj hasbeen releasedby time t butis
not yet complete,i.e. Qj,t is 1 or 0 as job j is or is not in the system at time t. Define
Q,= (Q1,t,Q2,t,' ' ', Qn,) as the state of the jobs at time t. A cost g(Qt) is associated
with the system at time t, whereg: (0, 1)"
'-> R. This cost dependsonly on the set of
in
the
at
t.
jobs
system
Our main resultis the followingtheorem.
Theorem 1.1. Suppose(A1)holds.Let 7r be thepolicythatschedulesjobs accordingto
prioritiesthataredecreasingin theirindices.Thenwithintheclassofpreemptive
policies,
7r minimizesEg(Q,), the expectedcost at time t, for all t, and all processesof arrivals,
machinebreakdownsand repairs,if and only if the costfunctiong(x) satisfies
(A2)
(A3)
g(x) - g(x - ep),
gg(x - e,) > #pg(x - ep)+ (Cu- p )g(x)
for all a > fl,
whereejis a rowvectorof n components,whosejthcomponentis 1 andothercomponents
are 0.
Note that since (Al) is assumedto hold, r is equivalentto the LongestExpectedjob
ProcessingTime (LEPT)firstrule.This ruleis wellknownto stochasticallyminimizethe
makespanof jobs with exponentiallydistributedprocessingtimes that areprocessedon
parallelmachines;moreover,it is knownto hold foran arbitraryarrivalprocess(see Van
Der Heyden(1981),Fredericksonet al. (1981), Weiss(1982)and Weber(1982), (1983)).
This result reappearsas an applicationof Theorem 1.1 at the start of Section 2. Our
proof of Theorem 1.1 is based on certain coupling techniquesand is similar to the
approachof Van Der Heyden.In Section2 we shedsome lighton conditions(A2)-(A3),
by providingfurtherexamplesof cost functionsfor whichthey hold.
In Section3, we considerthe specialcaseg(x) -=
This is the caseof weighted
I'.. cjxj.
in
for
time
which
a
cost
is
incurred
each
unit
job j remainingin the
holding cost,
cj
system. It is easy to checkthat this cost functionsatisfies(A2)-(A3) if and only if
(A4)
pici i 92C2i ' ' ' i
i
-nCn
0.
Condition(A4)saysthatthe orderof increasingexpectedprocessingtimes is the sameas
the order of increasing values of cy. So if both (Al) and (A4) hold, nrand LEPT are the
same and equivalent to the cp rule, which is defined as the preemptive rule that always
processes a set of jobs whose values of cp are greatest. In the literature, a combination of
conditions such as (Al) and (A4) is often called an 'agreeability condition'. In Section 3
we show that the cp rule minimizes various objective functions, including
(i) the expected weightednumberof jobs in the systemat an arbitrarytime t;
(ii) the expected weighted sum of job completion times;
On theoptimality
of LEPTandcyirulesfor machinesinparallel
669
(iii) the expectedweightednumberof latejobs whenthejobs havea commonrandom
due date;
(iv) the expectedweightedsum of job tardinesseswhenthejobs have a commondue
date.
The LEPT and cp rules have received a great deal of attention in the stochastic
schedulingliterature.If there is only a single machine, then there is no need for an
agreeabilitycondition. Variousauthorshave consideredthe single-machinecase and
shownthat the cp ruleminimizesobjectivefunctionssuch as (i)-(iv) above;see Pinedo
(1983), Baraset al. (1985), Buyukkocet al. (1985) and Shanthikumarand Yao (1991).
Optimalityof the cjprule for parallelmachinesrequiresan agreeabilitycondition.
Ross (1983) consideredtwo machinesin parallel,n jobs availableat time 0 and no
arrivalsafterwards.For this setup he showed that under the agreeabilityconditions
(Al) and (A4), the cuirule minimizesthe expectedweightedsum of completiontimes.
Kampke(1987) extendedRoss's resultto m machines,and weakenedthe agreeability
conditions to c,
c, and (A4). Under the agreeability conditions cl ... ~
...
and p,(t,)c, > pn(tn)c,for all t,, - - -, tn,Weber(1988) generalizedKampke'sresultto
??
models in which the job processing times have hazard rates, p,(t),.
. .
, p,(t). Corollary
3.1 in this paper generalizesRoss's result to m machines and an arbitraryarrival
process.
For moregeneralcost functions,conditions(A2) and (A3) arisequite naturally.They
have been consideredby Weissand Pinedo(1980), who investigatedsimilarscheduling
problemsto those in this paper,but underthe assumptionthatalljobs arepresentat the
start.Theyshowedthata policyminimizesthe totalholdingcost incurredby time t if the
expectedtotalcost underthatpolicy,say G(x), satisfies(A2)-(A3),withg replacedby G.
They statedconditionson g thatwouldbe sufficientto guaranteeoptimalityof the LEPT
rule;however,these wereincompleteand correctedby Kampke(1987), (1989) through
the additionof(A3). (Wenote thatKampkewas concernedwithgenerallist policies,not
only LEPT.He thereforealso requireda submodularityconditionthat is not neededin
Theorem 1.1.) The contributionof the presentpaperis to show that conditions(A2)(A3) aresufficientto guaranteeoptimalityof LEPTwhentherearearrivalsandmachine
breakdownsand repairs.Also, conditions(A2)-(A3)arenecessary,in the sensethatthey
must hold if LEPTis to be optimalfor all t and for everypossibleprocessof machine
breakdownand repair. Our results also apply to models in which machines have
differentspeeds,since this scenariocan be approximatedby rapidlyalternatingperiods
of breakdownand repair.This generalizationhas previouslybeen made by Weiss and
Pinedo, and also Kampke.
2. The optimality of the LEPT rule
To shed some light on the conditions (A2)-(A3) in Theorem 1.1, we first provide some
examples that satisfy these conditions. We then prove Theorem 1.1.
Example 2.1.
Consider the indicator function
gj(x) =
{xji, x,>o)*-
670
C.-S. CHANG, X. CHAO, M. PINEDO AND R. WEBER
It is clearthatthis functionsatisfies(A2)-(A3).It thenfollowsfromTheorem1.1thatthe
LEPT rule minimizes P(j:.1 Qi,t > 0). Now let Zj be the completiontime of job j.
Conditioning on the event {Rj = rj,j = 1, 2, ..., n), we note that
Qi,t >
P(i
0)=
1 -P(Q,
=0,i=
1,2,...,j)
(Zil{r<t) t)
=1-P (max
where 1(r,;t) equals 1 or 0 as the event [ri ~ t] does or does not occur.Consideringthe
casej = n, we have that the LEPTrule stochasticallyminimizesthe makespanof jobs
arrivingbeforet. Note that since the makespanis largerthan t if thereis a job arriving
after t,
P(
max (Zi)~ t)=P(
max (Zi) < t, t
Unconditioningthe event {Rj = rj,j = 1, 2,
cally minimizesthe makespan.
limax(ri)).
., n })yields that the LEPTrulestochasti-
Example 2.2. Let V1,... , V,, be independentexponentialrandomvariableswith
mean 1/z1,. * , 1/n,. Consider the function g(x)= Ef(V1x,
x,, V2x 2,
, VnXn), where
f: R
u {0}
~()
R.
Theorem2.3. Iff is
(i) increasing,i.e. f(z')
f(z2)for all z' z2 componentwise;
(ii) arrangement increasing, i.e. f( . ., z1, . z,*
f(. . ., zj, .
z,. . . ) for all
..
?,
,..)
i, j and zi zj;
•_ in each variable,i.e. f(z +
+ f(z + c2ei)
l
(iii) convex
(6, + 62)ei)+ f(z) f(z + c5ei)
for all i and 65,62
i
0;
(iv) submodular,i.e. f(z + ei +62ej) + f(z)
and 6,, 2
f(z + 61ej)+ f(z + 2ej)for all i #j
0;
>--
theng(x) satisfies(A2)-(A3).
Proof. It is clear that the increasingpropertyoff implies condition (A2). Chang
(1992) has shown that (i)-(iv) imply (A3) when f is symmetric.His proof is easily
adaptedto the case thatf is arrangementincreasing.
Let Vi,,be the remainingworkof job i at time t. Since the job processingtimes are
Q.
memoryless, V,t has the same distribution as the quantity V Q,
Using Theorem 2.3,
we have the following corollary for the work remaining at time t.
Corollary 2.4. Suppose (A 1) holds and fsatisfies (i)-(iv) of Theorem 2.3. Then the
LEPT rule minimizes Ef(V, t, V2,,,
,
t
"? V,,t).
Examples of functions that satisfy these four conditions are (i) f(z)=
a2
' ' ' an, and zi;> 0 for all i, and (ii) f(z)=
max(alz,,* , az,), with a
>-
Ontheoptimalityof LEPTandcprulesfor machinesinparallel
S
671
h,,(z) for all z (see Definition3.4) andh,(z) increasingconvex
hi(z,),with h,(z) >%,
-pIall i. For more examples,see Chang(1992) and Changand Yao (1990).
for
In fact,the LEPTrulenot onlyminimizesthe totalamountof workat time t, 2:i=1
V,t,,
in expectation,but also in the senseof stochasticordering.The argumentgoesas follows.
For a given problem,consideran auxiliaryproblemin which the numberof available
machines is truncatedto 1 past t. Let M be the makespanof the auxiliaryproblem.
Clearly,the total amountsof workare the same at time t, if in both problemsthe same
schedulingpolicy has been used up to time t. Moreover,the total amount of work
remainingat time t is max(M- t, 0). Sincethe LEPTrulestochasticallyminimizesthe
makespanfor the auxiliaryproblem,it also stochasticallyminimizesthe totalamountof
workremainingat time t forthe originalproblem,takinginto accountthatthe maximum
functionis increasing.
The rest of this section is devotedto the prooffor Theorem1.1. The prooftakesthe
same approachas Van Der Heyden. It uses the uniformizationtechnique and an
inductivemethodthathasbeen called'forwardinduction'(see Walrand(1988), Section
8.3). Using the well-knownuniformizationtechnique,our continuous-timeoptimization problemis transformedinto a discrete-timeone. We firstshowthat i is optimalin
one step and then take as an inductivehypothesisthat 7r is optimalin k - 1 steps. A
problemwith k steps has k decisionepochs.Denote the policiesappliedbetweenthese
If we are able to show that (r, r,. . 7r) is better than
epochs as (a0,
. , ak-_).
a1,..
.,
for
admissible
(a, nt,..., rt),
any
policy a and any initial state, it follows from the
induction hypothesisthat 7r is optimal over k steps. As a resultof applyingdifferent
policies 7rand a at time 0, the states at step 1 are different. To show that (7r, r, * *,Ir) is
better than (a, tr,- *, rt),we must show that starting from time 0 the states reached after
applying7r are better than those reachedafter applyinga. Thus, the proof requiresa
partialorderingamongstthe states.The appropriatepartialorderingis containedin the
followingdefinition.Using it, we can establishinequalitiesthat are similarto (3.3) and
(3.4) in Van Der Heyden'spaper.
Definition 2.5 (partial sum ordering). Let x' = (x , x.,
.
x ), i = 1, 2, be two
?,denotethisx'
vectors.Wesay thatx' is smallerthanx2 underpartialsum, and
is
x l
x2, for all
1, 2, - -, n.
iji
l1=
,
x2,
if
..
The partialsum orderingis very similarto the weakmajorizationordering(Marshall
and Olkin(1979)).However,the weakmajorizationorderingof x' andx2 requirestheir
components to be in decreasingorder. The following propertyof the partial sum
orderingis easyto prove(see Ross(1983),Lemma3.4). Note thatit does not requirexjto
be either0 or 1.
Lemma 2.6. If
Cf;_l
,tj
x'
s x2,
=
1
Z = 1 x/
and
i
p2 < ?...
<,,
then
z '_1u,
!,jxjK.
-
The next lemma gives a constructive characterization of the partial sum ordering.
Consider two integer-valued vectors that are partial sum ordered. We show that the
greater of the two vectors can be transformed into the lesser by a number of canonical
steps, each of which either reduces some component by 1, or makes a transfer of 1 from a
C.-S. CHANG,X. CHAO,M. PINEDOAND R. WEBER
672
lesserto a greatercomponent.In the firstcase,we thinkof I as being'transferredout' of
the vector;in the secondcase 1 is transferredbetweentwo components.Informally,we
call this the 'transferproperty'of the partialsum ordering.A similarpropertyholdsfor
the weakmajorizationordering(Muirhead(1903)).
Lemma 2.7. Ifx' andx2 are two integer-valuedvectorsandx' -, x2, thenx2 can be
transformedinto x' by a finite numberof applicationsof the following two types of
transfers:
(i) x
H
x - ep, where xf > 0;
(ii) x x + ea- ef, wherea > fl and xf > 0.
Moreover,if T is the combinationof afinite numberof transfersof theforms in (i) and
(ii), then T(x) <P,,x.
x is clear.Conversely,the transferpropertyis clearfor n = 1,
Proof. That T(x)
P•,
in one dimensionmeansx1 x22,and we need only make
since the partialsum ordering
inductionhypothesisthatx2
reductionsof 1 to x1 to obtainx,. Thus,we can takeas our<_
can be transformedinto x' when these are vectorsof n - 1 components.If :1%1xJ <
x x<xf
. We can use
then thereis a constantl such that
. x2,
-~2i' x?
=•need to consider
the transferx -* x - ef, fl l until i" xi = f=, x. Thus,we only
•
x2 impliesx' > 2. If xI = x2, then
the case that IP., xf =
x2. Nothathatx'
_p, n - 1 components.If x) > x2, then
can be appliedto the first
the inductionhypothesisI.,we can find a constantI such that X2 > 0 and xj = 0, i = 1 + 1,..., n and repeatedly
apply the transfer x2
h
x2 + en - el until the nth component of the resulting vector
equalsxI. Once again,this leaves a case to which the inductionhypothesisfor n - 1
applies.This completesthe proof.
In the followinglemma,we show that the partialsum orderingis preservedunderIt
(the LEPTrule).
Lemma 2.8 (n-monotone). Considertwo systems with differentinitial conditions.
Let Qj,Lbe the indicatorfunctionfor the eventthatjobj is in the systemi at time t, and
Q i = 1,2. Assume(Al) andQ <,<Q02.Thenthereexist two
Q =
Q2t,',,.*,and,,t),
(Q,/,
random
vectors
e2 such that under nr we have (i) i =st Qi, i= 1, 2, and
Q^t
Proof. Let m(t) denote the number of machines available at time t and let
(Mk, k 1} be the set of epochsthat dmi(t)# 0. At each time Mk,one machinebreaks
down or-one machinebecomesavailableagainfollowingits repair.Considerthe event
{(R = r, j = I, .., n andMk= mk, k 1. Let to= 0 and (tk,k - I} be the sequenceof
{r1,j = 1,. ., n and mk, k > })after sorting in time. Clearly, between tk and tk+l there
are no arrivals and the number of machines is constant. First, we show that we can
construct two processes between toand t1 such that they satisfy conditions (i) and (ii) of
this lemma. Using the standard coupling and uniformization technique (Keilson
(1979)), we generate a Poisson process with rate A= m(to)j,,. Let {Zk, k ? 1} be its
arrival epochs and define -o = 0. Generate a sequence of independent and identically
Ontheoptimality
of LEPTandcl rulesfor machinesinparallel
673
distributed (i.i.d.) random variables (Uk, k > 1) uniformly distributedover [0, 1].
Constructthe uniformizedMarkovchainsas follows:
Qi= Q,
i = 1,2,
and
Qj,Tk+I= Qj,
where
=i {kA1
-
for all Tk+1
jk
IXk
k
A
<
tl
X
and Xj,k = 1 ifjobj is served in system i at time Tkand XJ k = 0 otherwise. As desired, it
2.Under
is clearthatQ =tQ,Q, i
,k
, k m(to)andX k = 0
j, k if =
otherwise.Thus,
= min(m(to),2
)}.From the induction hypothesis
X2k
2=Qk
.~l,
it
that
I
follows
I2 k
n. Sincethere
, k < =1 X2k, j =
1,Q
21=1X,
Tk
^
=
^2
most one
at Tk+1,
is atI-1l,
.,
departure
,
Tk
Tk
Ql, Tk+1, =QTTk+1
if 21
l
"2
In
Therefore,we need only considerthe
?=1 case,
=-1 case that
1 ik
I1
= fsuch
Tk.
P= Xk , P
II X/,k
1 X, kand
~f= X1, k
p = I1 ,' j - 1.FromLemma2.6, it
= ,k
follows that
and thus CIJ
. Therefore, we
=1
=1
IXk
_1,k
liXJ
<
concludethat k1
2 Observethat the partialsum orderingis also preserved
whenthereis an arrivalof the sametypeat bothsystems.By induction,we canconstruct
two processessuch that Q1
PsQt.
Let
be the expectedcost at time t from the initial state Q0
Jt(Qo)d E(g(Q) IQo)
under the policy
7r.
Corollary2.9. Assume(A1)-(A3) hold, and Q1<s Qj. Thenschedulingundern we
have
for all t ? 0.
t
Jt(Q1) Jt(Q2)
Proof. From (A2) and (A3), it follows that g(x - e,) E g(x - ef) for all a > p. Using
the transferpropertyin Lemma2.7 yields thatg(x1) ? g(x2) ifx1
It then follows
,ps x2.
from the ir-monotone propertyin Lemma2.8 that J(Q ) ? Jt(Q2)if Q1< Q2.
Lemma 2.10. Under(A1)-(A3) and n,
(1)
it,Jt(Qo- e,,)
pJt(Qo - ep) + (i, - pfl)J (Qo),
for any initial state Q0 with Q,,o = Ql,o = 1 and a > p.
= e, + e . We use the
e,.
same construction as in the proof of Lemma 2.8. First, we show that (1) holds for all
= 0. Now take as an induction
t E [to, t1). From(A3), it followsthat (1) is satisfiedat mo
Proof. To simplifythe notationsin the proof, we denote
hypothesis that (1) holds for tE
Tk].Let S(or Sa, Sp) be the set ofjobs served at time 0
under at given the initial state[m0,
Q0 (or Q0 - ea, Q0 - ep). Note that {rk+1 - Tk) is a
of
i.i.d.
random
variables. Analogous to Kolmogorov's backward
sequence
exponential
equation, we have
C.-S. CHANG,X. CHAO,M. PINEDOAND R. WEBER
674
X
J'k+I(Qo)
Similarly,
- e)
and
=
J'*k+ 1(eO
Jtk+(Qo - efi)
=
j
- ej)A
Jrk(Qo
(Q)
jES
A
Es
SJ
- ea,j)
Jk (Qo-e,)
+J,,(Qo-e.)
kj+
Jtk(Qo - ef,j)
+
-a
jES,
(eo
Jk,,(Qo
-
ea)
AjES'
Considerthe followingfour cases.
Case 1:
Q, o > m(to). In this case, S = S = S . Thus,
if
) - uJQ+,l(Qo
ZJ,,+l(Qo e(
=
+
-
- aJ,,
((u Ji(Qo e.)
(S aJ,
+
ef) - (eU- i
a
)J•-+l(Q0)
-
(Qo e.iJ)
- (Cu
--
)J (Qo - ej))
U-f)J(Q))
-)(Qo- e) - J, j(Qo
>0.
Case 2: Ifl. Qj,0,5 m(to)and 21. 1Qj,0> m(to).In this case, there exists a y such
that i y<a and =I Q,:o< m(to), Y' ,0> m(to).Clearly,S = S = S U (}) \
{y + 1). Thus,
eQ aJ,,+,(eo
=
eai)
)
)Jk+l(Qo)
-u1J,,?(Qo
XI
(kuGJk(Qo- ea,j)- lJk(Qo -e e,j )-(au -
jeS \ (p)f
A-
- (~U-iti
X
)Jk(Qo -
))
i.,
+ A (USaJk(Qo
- ea) -
Jk(Qo - ef) -
(Uui
JT,1(Qo- ea,8) - ,A+1],,(Qo
- eay+
ef+)) )-
-
(U -
)J,(Q))
A+,
)Jr,(Qo -
>0.
Case 3: CIf.
m(to) and f. ,Qj,, > m(to). Analogous to the proof of Case 2,
Qj,o,
that a 7 < n and CI_
there exists a 7 such
o m(to), f:2~'Q, o > m(to). Clearly,
<
S = S, u {a} \ {y + 1} = Sp U {fl} \ {y +1}). Qj,
Thus,-
675
On the optimalityof LEPT and cprulesfor machinesin parallel
rk+,(Qoa•J
- ep) - (Ya- l )Jk+(Qo)
ea)- tJfjrk+,(Qo
S-
u
+in
S
jo-s A
+
tak
+
- efq)- (ora
2U)Jk(Qo))
- e)( aJrk(QO
JLkk(QO
a(JtCu(Qon- eo,)ti+
hlJak(Qo - el,
,)- La-4
-
I(QkeP).(Q
A foeYP+
(+lJ
e,+)-
J
)Jk(Qo - eY+o))
-f)Jrk(Qo))
(
(/r+
>0.
Cwhease
4:(t)
followsClearly, all
same
, then itm(t).
the
samethe
argument, withfrom
+
ereplaced by(Qo
e.) -
AjE
(!JYaJrk(Q?_
+ I'a l
A
e.)
- Jk(QO
with and S =replaced by
argument,
served
e, tha- () holds)J(Q
for all
- efi)J,,Jk(Qo
[t,
(Pa -Ufl)Jrk(Qo))
ep)).
i(J,(Q0e-
J,(
- - e) from Corollary
It
e e)(
Since Q0- e2.9. then follows
)e>_from these fourcases that (1)
holds for t
t).
0.[t0,
Now take as the inductionhypothesisthat (1) holds for all t E [t0, tk). If tk is an epoch
=# then it followsfromthe sameargument,with
where
me(to)beingreplacedby
dmn(t) 0,
E
If
t
is
the
arrival
that
holds
for
all
epochof job]j, then it follows
[t0, tk + ,). tk
m (tk),
(1)
from the sameargument,with replacedby Qo+ ej, that (1) holds for all t E [to,tk +1).
Q0
Thus, (1) holds for any t 0.
_->
Proof of Theorem1.1 (sufficientcondition). Again, we transformthe continuoustime problem into a discrete-timeone by using uniformization.Let {tk, k > 0} be
defined as in the proof of Lemmas 2.8 and 2.10. Recall that tkis either an arrival epoch or
an epoch when a machine breaks down or becomes available. Generate a sequence of
independent Poisson processes, N1(t), 1 > 0 with rates Ai(y) = ym(t)lu, for some y 1.
Let (rt,, k 1} be the arrival epochs of the lth Poisson processes and define o,0=_ 0.
Observe that- the ZI,k'Sare epochs when a job completion may occur in the coupling
scheme we presented in the proof of Lemmas 2.8 and 2.10. Analogous to the proof of
676
C.-S. CHANG, X. CHAO, M. PINEDO AND R. WEBER
Pinedo (1983), Theorem 1, we firstshow that 7ris optimalwithin the class of policies
W(y),where W(y) consists of all policies that allow preemptiononly at time epochs
{tt + Tl,k, k _>0, 1 0) with tt + zT,k<t+,1. Then it followsfroma continuityargument
as y -- oo that 7ris Optimalamongall the policies.
To simplifythe notation, let {(;', k 0} be the sequenceof {t1 + TIk,k,k O0,
1 0)
aftersorting.Weuse inductionon ({ }).First,
we showInis optimalin a singlestepfor all
initial statesQ0.If rT is the arrivalepochofjobj, thenthe stateofjobs at Tris Q0+ ejfor
any policy.This impliesthe costsat z arethe sameforany policy.If z is an epochthata
machinebreaksdownor becomesavailable,i.e. dm(t) z 0, then the stateofjobs at Ti is
Qofor anypolicy.Again,this impliesthe costs at T" arethe samefor any policy.Thus,it
sufficesto consider the case that Tz is not a point of {tk, k i 1). Considera policy
a EEW(y).Let Xj0 be the indicatorfunctionof the event thatjob j is servedat time 0
under a policy a and let Xg = (X10,... Xn,0). Let XKbe the correspondingquantity
underIn. Since the policy n schedulesjobs accordingto priorityof the lowestindex, we
have thatXg <,, Xg for anypolicya in W(y).FromLemma2.7, it followsthatXg canbe
reachedfrom Xg througha finite numberof applicationsof the followingtwo types of
transfers:(i) x + x - epand (ii) x
-
x + e, - ep, a > f. Thus, we only need to compare
the expectedcost at z for two policies a' and a2 with X' = T(Xg), where T is of the
form (i) or (ii). Now let Q1''be the states of jobs at ZTunderthe policy a', i = 1, 2. It
sufficesto show that
(2)
Qo)> E(g(Q 02) Q0).
E(g(Q
•')
Case 1: The first type of transfer. In this case, X'l = X'2 - ef for some fi. This is
equivalentto insertingidle time into a machine.Clearly,we have Q`>
Q)'.
follows from (A2) that (2) holds.
It then
Case 2: The secondtypeof transfer.In this case,Xa =
somea > f.
X•2 + ea epfor
=
=
we
1
and
The
assume
difference
a'
between
and a2 is
only
(Clearly,
Xa,o 0.)
X,0'o
that a2puts the low-indexjob fl into serviceinsteadof the high-indexjob a. Now define
Si = (j : Xj' = 1} as the set of jobs served at time 0 under policy ai. Then for i = 1, 2,
(3)
P(Q•f
= Qo- e
=
IQ0)
,
E
jEs' ,
Ao(Y)-A0o(y)
/I
jS'
j 4 Si.
Thus, we have from (A3) that
E(g(QI')Qo)- E(g(Q2)IQo)
=
> 0.
1 (uag(Qo - e,) - gpg(Qo ep) - (J. - /p )g(Q0))
677
On the optimalityof LEPT and cp rulesfor machinesin parallel
Therefore,7 is optimalin a singlestep.
Take as the inductionhypothesisthat 7ris optimalover k - 1 steps. Now we would
like to showthat the policythat appliesa2 at time 0 and then appliesInafterTris better
thanthe policythatappliesal at time 0 andthenappliesInafterTz.Now let Qa, i = 1, 2,
be the statesof jobs at zr underthese two policies.It sufficesto showthat
(5)
E(g(Q
E(g(Qa'Qo)
I
fz
).
Again,we only haveto considerthe casethatTris not a point of {tk,k > 1). Forthe first
It then followsfrom
type of transfer,we have that Q,' > Q1. This impliesQ, >
the transitionprobabilities
Corollary2.9 that (5) holds.Analogousto (4), we have fromQ-,.
of the uniformizedMarkovchainsthat for the secondtype of transfer,
E(g(Qa')IQo)- E(g(Q'2) IQo)
= zIk(y)(i.J,;;(Qo
Ao
- e.) - JiJ ,',(Q0 - ep) - (.a - Ip,)Jr;,,(Q0))
=
where
IQ,,). From Lemma2.10 and the inductionhypothesis,it
J,,,
E(g(Q,,)
,(Qo)
followsthat 7ris optimalwithin W(y).A continuityargumentthen completesthe proof
that the conditionsin Theorem1.1 are sufficientfor optimalityof r.
(Necessarycondition). We would like to show that if the policy r minimizes the
expectedcost for all t and any realizationof machinebreakdownsand repairsthen (A2)
and (A3)mustholdforall initialstatesQ0.Thismeanswe mayconsidera problemwith1
the policy
machines available at time 0, where I=/C,=,
Qi,. Clearly,
=
7r
begins by
assigningto machinesall jobs whose index is not largerthanft. Call this set of jobs S.
Then the expectedcost of the policy n aftera very smallamountof time 3t is
(6)
E g(Q0- e)uidt + (1-
iEpidt) g(Qo) + o(3t).
Now considera policy a' whichschedulesjobs havingindicessmallerthanfl on to I - 1
machinesand keepsthe Ith machineidle. Let a' be the policythat is the same as 7r but
schedulesjob a insteadofjob fl, a > ft. The expectedcostsof the policiesa' anda2,after
a very small amountof time St, can be computedsimilarly.It is easy to see that the
differencebetweenthe expectedcosts of the policies n and a' at time 3t is
(7)
g(Qo - ep),p&t- g(Qo),pdt + o(St).
The optimalityof n for all t, whenthereare I machinesavailableat time 0, impliesthat
the quantity in (7) must not be greater than 0. Letting 3t -* 0 yields (A2). Similarly, the
difference between the expected costs of the policy ir and a2 at time st is
g(Qo - ep)lpuSt- g(Qo)up3St- g(Qo - ea)uaOt + g(Qo)jpS3t+ o(3t).
The optimality of r implies that the quantity in (8) must not be greater than 0. Letting
6t -* 0 yields (A3).
(8)
678
C.-S.CHANG,X. CHAO,M. PINEDOAND R. WEBER
3. The optimalityof the cy rule
In this section,we considerthe cost functiong(x) =
cjxj. In otherwords,for each
•_,1
unit time thatjobj remainsin the system,a cost cjis incurred.Assumingthe agreeability
condition(A4)of Section1,we use Theorem1.1to establishoptimalitycriteriaforthe cy
rule. Recall that the cy rule is the preemptivepolicy that ordersjobs in increasing
priority according to increasing values of cy. Under (A4), the cy rule results in the same
order as the LEPT rule. The objective functions that are consideredin this section
include the expected values of (i) the weightednumberof jobs in the system at an
arbitrarytime t, (ii) the weighted sum of job completion times, (iii) the weighted
number of late jobs when the jobs have a common random due date, and (iv) the
weightedsum of job tardinesseswhen the jobs have a commondue date.
Corollary3.1 (minimizationof theexpectedweightednumberofjobsat arbitrarytime
t). Assume(A1)holds.Let Irbe thepolicythatschedulesjobs accordingtoprioritiesthat
are decreasingin theirindices. Then r minimizesE[ I ~, cjQ, t], the expectedweighted
numberofjobs, for all t (t > 0), and all processesof arrivals,breakdownsand repairs,if
and only if (A4) is satisfied.Preemptionneedonly occurat the releaseof a newjob.
Proof. For g(x) =
I..",
cixi, it is easy to see that (A2)*c >-c 0 and that (A3)c c,: >
In fact, Corollary3.1 also follows immediatelyfrom the known result that LEPT
minimizesthe makespanstochastically.Recall,Vj,, = Qj,,/j. Observethatthe objective
functionis an arrangement
increasingfunctionof the expectedworkremainingat time t,
i.e.
= E
EEcjQj,
j-1
t
j-1 c
jt
V;,
c
i-i (cipi -
E
j-1
,
,
where we take cn+ ,n+, = 0. Since LEPT minimizes the expected work remaining at
everytime t, it is clearthatthe summationon the right-handside aboveis minimizedby
LEPT.
It is clear that without the agreeabilitycondition (Al), the ci rule itself cannot be
optimal. Counterexamplescan be found, even under Kampke'sweaker agreeability
condition of cl, >.. c, and (A4). Consider the following set of jobs: cjyj= 1,
9 ,=? and cj=1I(n-1),
j=n-1,
j=
j=l,*..,n-1,
j=1,***,n-l,1,
n
two
machines
and
all
1, 2, ... , n. Thereare
jobs have a common(fixed)due date at 2.
Take n very large.The firstn - 1 jobs requirea total amountof processingequalto 1.
The variance in this total amount goes to 0 as n tends to 0o. It is clear that job n will start
under the ci rule at time ?. It is also clear that starting job n (the large job) at time 0
significantly increases the probability that it is completed by time 2, and therefore
maximizes the expected number of jobs completed by the due date. For an example in
which the ci rule does not minimize the expected sum of the weighted completion times,
consider again two machines and the same set of jobs as the one described above and all
n jobs available at time 0. Now, instead of a due date at time 2, a second batch of jobs
Ontheoptimality
of LEPTandcyirulesfor machinesin parallel
679
arrivesat time 2. It is clearthat the cy ruledoes not minimizethe expectedsum of the
weightedcompletiontimes. It is necessaryto start the long job at time 0 in orderto
maximizethe expectedmachineutilizationbeforetime 2.
Corollary3.2 (minimizationof the expectedweightednumberof latejobs). Assume
(A 1) and (A4) hold, and thejobs havea commondue date, D. Let Zj be the completion
time ofjob j. Let Ujbe the indicatorfunctionfor the event[Zj Dj]. Then7 minimizes
>=
, cjUj ].
E[
•-.
Proof. Consider the event (Rj = rj,j = 1,... , n, D = d). On this event, Uj = Q, dif
rj< d and 1 if rj d. ApplyingCorollary3.1 completesthe proof.
_
Remark 3.3. Corollary 3.2 is an extension of Pinedo and Rammouz (1988),
Theorem 2, to parallel machines. Pinedo (1983), Theorem 8, also showed that if
cn, and the common due date D has a concave
Jl
cl c2
"-,,
_:2 _-...
_->
SEPT
ruleminimizesE[
distribution
then
the
statesa
function,
i=1, cjUj].Corollary3.2,
differentset of conditionsunderwhichthe LEPTruleis optimal.
The followingdefinitionis useful in obtaininga result for more generalweighted
functionsof completiontimes.The subsequentcorollaryextendsPinedoand Rammouz
(1988), Theorem1, to parallelmachines.
Definition 3.4 (Pinedoand Rammouz(1988)). Let F,(t) and F2(t)be two increasing
functions. F, is said to be steeperthanF2 if F,(t2) - F1(t1)- F2(t2) - F2(tl)for all t, < t2.
Wedenotethis by F1 > F2.
Corollary3.5 (minimizationof the expectedweightedsum ofjob completiontimes).
n are increasingfunctions,such that
Suppose(A1) and (A4) hold, and Fj(t),j =
- .
Then 7rminimizes 1,.-,
Fl
, F2
s
> , F,.
E[ 1j,
cjFj(Zj)].
Proof. It is notedin Pinedoand Rammouz(1988) thatCorollary3.2 is equivalentto
minimizingE[
cjF(Zj)],whereF(t) is the distributionfunctionof the commondue
if.•,7t minimizes E[2
date D. Thus,
cjF(Zj)] for all F increasing. Note that
CI
=
- Fn+2_i(Zj)], and taking
c=E[Fn+,_i(Zj)
Bi with B, =
E[;-1
cFj(Z1)]-•I1
=
0.
Now
that
is
Zn•1i-i
Combinedwith the fact
implies
(t)
increasing.
+
F,n
Fj , Fj+,
Fj Fj1,
that F, is increasing, this implies that 7t minimizes B,, i = 1,.
minimizesthe sum of the B 's.
.,
n, and thus that it
Corollary3.6 (minimizationof the expectedweightedsum ofjob tardinesses). The
tardinessof job j, Tj, is definedas max(Zj- Dj, 0). Suppose(A1) and (A4) hold, and
5
D, : D2
D, a.s. Thenn minimizesE[ I=1 cjTj].
_
???
Proof. Consider the event (D1 = d), j = 1,. ., n). The objective function is equivalent to E[2J.. c?FI(Z,)],with F1(t)= max(t - di, 0) (Pinedo and Rammouz (1988)).
Clearly, the Fj(t)'s are increasing and F1 >, FJ, . An application of Corollary 3.5
completes the proof.
Remark 3.7. Pinedo (1983), Theorem 3, showed that ifD, D2
D, a.s. and
? ?? _
(A4) is satisfied, then rminimizes E[ I,
c?T ] for a single-machine problem with all the
C.-S. CHANG, X. CHAO, M. PINEDO AND R. WEBER
680
jobs presentat time 0. Corollary3.6 is an extensionof thattheoremto parallelmachines
with releasedates.
In the followingtwo corollaries,we considerthe specialcasethatalljobs arepresentat
time 0. A static list policy is one that assignsjobs to machinesin the orderof a fixed
permutationof their indices. Clearly,the set of static list policies is a subset of the
dynamicpolicies we have consideredthus far, and n, whichunder(Al) assignsjobs in
the order 1,. *.., n, is optimal amongstall static list policies when (Al) and (A4) are
satisfied.In the followingtwo corollariesthe conditionsneededin Corollaries3.2 and
3.6 are relaxed from the strong sense (a.s.) to the weak sense (distribution).These
corollariesgeneralizePinedo(1983),Theorems4 and 2, to parallelmachines.Recallthat
a randomvariableX is stochasticallysmaller than Yif P(X > t) < P(Y > t) forall t.
- s
Corollary3.8 (minimizationof the expectedweightednumberof latejobs). Assume
function
(A 1) and (A4) hold, and duedatesDj,j = 1,- - , n, havea commondistribution
F(t). Let Ujbe the indicatorfunctionfor the eventthatjob j is late. Then7rminimizes
E[ J', cjUj] amongst all the static list policies.
Proof. Let a =
a2,. ., a,)} be a permutationof (1, 2, - , n } and supposethe
?
accordingto the static list policy that assignsjobs to machinesin
jobs are scheduled(al,
priority order a,, q2,.. , a,. Under these circumstances,let Q1, be the indicator
function for the event that job j is in the system at time t and let Uf'be the indicator
functionthatjob j is late. FromCorollary3.1, it followsthat under(Al) and (A4)
n
(9)
Z
j=1
n
<
EcjfQ
j=1
EcjQj,.
t_
All the jobs are presentat time 0, so U;J= Q J. Since the orderin whichjobs will be
assignedto machinesis determinedat time 0, it followsthat QTt,is independentof the
due date Dj. (This is not true if the policy is dynamic.)We have
(10)
=
EQO
f
EQtdF(t).
Combining(9) and (10) completesthe proof.
Corollary3.9 (minimizationof the expectedweightedsum of job tardinesses). Assume (A1) and (A4) hold and due dates satisfy <st D2st '
stDn. Let Tj be the
D1
all
list
static
tardinessofjob j. Then7r minimizesE[ 1~ , cjTj]
policies.
among
Proof. Again,let a be a permutationof {1, 2, . , n } anddefineQ7,as above.Let Tf
be the tardinessof jobj whenjobs areprocessedunderthe staticlist policydefinedby a.
All the jobs are present at time 0, so T7 = SJ Qr,dt. Again, Qf, is independent of DI and
we have
ETJ =
f
f
EQ dtdF(s)
,t
=
•
Fl(t)EQftdt,
On the optimalityof LEPT and cy rulesfor machinesin parallel
681
where Fj(t) is the distributionfunctionof the due date of job j, Dj. Analogousto the
proof of Corollary 3.5, E[ Cj., cjTf] = i= B,, where
n++-i
0
B=
0
0(Fn+-i(t)-
Fn+2-i(t))j=1 cEQtdt
St
and we define F+,,(t) = 0. The assumption D, st D2 st<
Dn implies F,(t) >
>
0
It
7t
from
3.1
that
minimizes
follows
Corollary
Bi, i = 1,* *, n,
F2(t) >
Fn(t)
O.
? ?
and thus minimizesthe sum of the Bi's.
References
A. M. (1985) Two competing queues with linearcosts
BARAS,J. S., DORSEY,A. J. AND MAKOWSKI,
and geometricservice requirements:the uc-rule is often optimal.Adv.Appl. Prob. 17, 186-209.
BUYUKKOC,C., VARAIYA,P. AND WALRAND,J. (1985) The cui rule revisited. Adv. Appl. Prob. 17,
237-238.
CHANG,C. S. (1992) Orderingfor stochasticmajorization:theoryand applications.Adv.Appl.Prob.
24(3).
CHANG, C. S. AND YAO, D. D. (1990) Rearrangement,majorizationand stochasticscheduling.
G. N., BRUNO, J. AND DOWNEY,P. (1981) Sequencingtasks with exponential
FREDERICKSON,
service times to minimize the expectedflow time or makespan.J. Assoc. Comput.Mach. 28, 100-113.
T. (1987) On the optimalityof static prioritypolicies in stochasticschedulingon parallel
KAMPKE,
machines.J. Appl. Prob. 24, 430-448.
KAMPKE,T. (1989) Optimalschedulingof jobs with exponentialservice times on identicalparallel
processors.Operat.Res. 37, 126-133.
T. (1979) MarkovChain Models - Rarity and Exponentiality. Springer-Verlag,New
KEILSON,
York.
MARSHALL, A. W. AND OLKIN, I. (1979) Inequalities:Theoryof Majorizationand its Applications.
Academic Press, New York.
MUIRHEAD,R. F. (1903) Some methods applicable to identities and inequalities of symmetric
algebraicfunctionswith n letters.Proc. EdinburghMath. Soc. 21, 144-157.
PINEDO,M. (1983) Stochasticschedulingwith releasedatesanddue dates. Operat.Res. 31, 559-572.
M. ANDRAMMOUZ,
E. (1988) A note on stochasticschedulingon a singlemachinesubjectto
PINEDO,
breakdown and repair. Prob. Eng. Inf. Sci. 2, 41-49.
M. ANDWEIss,G. (1979) Schedulingstochastictaskson two parallelprocessors.NavalRes.
PINEDO,
Log. Quart. 26, 527-535.
Ross, S. M. (1983) Introductionto StochasticDynamicProgramming.AcademicPress, New York.
J. G. ANDYAO,D. D. W. (1991) Multiclass queueing systems: polymatroidal
SHANTHIKUMAR,
structureand optimal schedulingcontrol. Operat.Res.
VAN DER HEYDEN, L. (1981) Schedulingjobs with exponential processingand arrival times on
identical processorsso as to minimize expectedmakespan.Math. Operat.Res. 6, 305-312.
WALRAND, J. (1988) An Introductionto QueueingNetworks.PrenticeHall, EnglewoodCliffs,NJ.
WEBER,R. R. (1982) Schedulingjobs with stochasticprocessingrequirementson parallelmachines
to minimize makespanor flowtime.J. Appl. Prob. 19, 167-182.
WEBER,R. R. (1983) Schedulingstochasticjobs on parallel machines to minimize makespanor
flowtime.In AppliedProbability- ComputerScience: TheInterface,ed. R. Disney and T. Ott, pp. 327338. Birkhauser,Boston, MA.
WEBER,R. R. (1988) Stochastic schedulingon parallel processorsand minimization of concave
functions of completion times. In Stochastic Differential Systems, Stochastic Control Theory and
New York.
Applications,10, ed. W. Fleming and P. L. Lions, pp. 601-609. Springer-Verlag,
WEISS,G. (1982) Multiserver stochastic scheduling. In Deterministicand StochasticScheduling,ed.
M. A. H. Dempster, J. K. Lenstra, and A. H. G. Rinnooy Kan, pp. 157-179. Reidel, Dordrecht.
WEISS,G. AND PINEDO,M. (1980) Scheduling tasks with exponential service times on non-identical
processorsto minimize various cost functions.J. Appl. Prob. 17, 187-202.