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Single machine scheduling when processing times are correlated normal random variables
EUROPEAN JOURNAL OF OPERATIONAL RESEARCH ELSEVIER
European Journal of Operational Research 102 (1997) 111-123
Theory and Methodology
Single machine scheduling when processing times are correlated normal random variables R.J. B o y s , K . D . G l a z e b r o o k , C . M . M c C r o n e Department of Mathematics & Statistics, University of Newcastle upon Tyne, Newcastle upon Tyne, NE1 7RU, UK
Received 1 July 1995; accepted 1 April 1996
Abstract
A single machine is available to process a collection of jobs whose processing times are jointly multivariate normal. Processing is nonpreemptive. We show that in the equicorrelation case, the permutation policy which schedules the jobs in ascending order of their (marginal) mean processing times is optimal for general order-specific costs and also for job-specific costs under an agreeability condition. Suboptimality bounds on the performance of Smith's rule are obtained for the weighted flow-time criterion. A computational study shows that a dynamic version of Smith's rule comes very close to optimality. (~) 1997 Elsevier Science B.V. Keywords: Agreeability;Dependentprocessing times; Multivariatenormal distribution; Optimal policy; Smith's rule; Stochastic scheduling
1. I n t r o d u c t i o n
A single machine is available to process N jobs, the jth of which has random processing requirement X j , j = 1,2 . . . . . N. Processing is nonpreemptive, i.e. interruptions of the processing of a job are not allowed. At time 0 and at all subsequent times at which jobs complete, a decision has to be made concerning which job to process next. The objective is to choose a policy to minimise some overall measure of total cost. This measure is always a non-decreasing function of the job completion times. Such models have been used to study problems inter alia in research planning (see, for example, Bergman and Gittins, 1985), computer scheduling (Bruno and Hofi, 1975) and the sequential design of experiments (Glazebrook, 1978). See Gittins (1989) for a general theoretical account. A key simplifying feature shared by almost all such stochastic scheduling models considered to date concerns the independence of the processing requirements X j , j = 1 , 2 . . . . . N. Glazebrook and Whitaker (1992) argue that there are situations in which the independence assumption is unwarranted. They take over concepts from the reliability literature to develop models which allow for dependencies between the jobs and seek conditions which yield optimal scheduling policies of simple structure. However, the development of optimal scheduling policies for models with dependence is extremely difficult. Although the models considered by Glazebrook and Whitaker (1992) are quite general, the above sufficient conditions are very severe. This theoretical paper aims to promote further understanding of scheduling models incorporating dependence by consideration of the important special case in which X = (X1, X2 . . . . . XN) T is 0377-2217/97/$17.00 (~) 1997 Elsevier Science B.V. All rights reserved. PII S0377-2217(96) 00211-1
R.J. Boys et aL /European Journal of Operational Research 102 (1997) 111-123
112
multivariate normal. This crucially allows us to model the appropriate dependence structure simply through a covariance matrix __£. Note that Section 5 of Hodgson, King and Stanfield (1996) makes a contribution to production scheduling in which job service times are assumed to be normally distributed. See also Robb (1992). In Section 2, we consider the order-specific costs model in which the overall cost of processing is a general function of the ordered completion times. We show that in the equicorrelation case, the permutation policy which processes jobs in the order of increasing mean processing times (i.e. shortest first) is optimal. This result is generalised to a job-specific costs model in Section 3 when an agreeability condition holds. In Sections 4 and 5 we focus on the Classical weighted flow-time model for which Smith's rule is optimal in the independence case. See Smith (1956). In Section 4, we obtain two suboptimality bounds on the performance of Smith's rule. These bounds are, respectively, natural measures of the spread of X (reflecting the fact that Smith's rule is optimal for the deterministic case in which processing times are known in advance) and the degree of dependence in X (reflecting the fact that Smith's rule is optimal for the independence case). Section 5 contains a computational study. Here we see that almost all of the suboptimality in Smith's rule is eliminated by implementing a dynamic version in which the mean processing times of uncompleted jobs are updated by suitably conditioning on past outcomes at each decision epoch.
2. The order-specific costs model Consider a single machine scheduling model in which job j has processing requirement Xj, j = 1,2 . . . . . N. We assume X = ( X l , X2 . . . . . XN) T to be multivariate normal with mean vector/~ and covariance matrix ~. An important special case is when
___~=o-2{(1 --p)/+pl!T},
(1)
where / is the N x N identity matrix, 1_ is the N × 1 vector of ones, o-2 > 0 is a common variance and -1/(N1 ) < p < 1 is a common correlation. This is the equicorrelation case - see Mardia, Kent and Bibby (1979). Positive dependence seems natural for the scheduling context and so the restriction on the choice of negative p need not overly concern us. See Glazebrook and Whitaker (1992). A nonpreemptive scheduling policy ~ has a free choice of which job to schedule first, designated ~-( 1 ). Once the first (n - 1) jobs are scheduled by ~-, the choice of ~r(n) is allowed to depend upon the realised values of X~r(j), j = 1,2 . . . . . n - 1. A job, once chosen, is processed through to completion without interruption. Any nonpreemptive policy which always (i.e. with probability 1) schedules the jobs in the same order is called a permutation policy. The identity permutation ~" schedules the jobs in numerical order, i.e. ~-'(j) = j, j = 1,2 . . . . . N. Under nonpreemptive policy ~-, F(j) (Tr) is the completion time of ~r(j), defined as
J F(j) (~-) = ~
X~r(k),
j = 1,2 . . . . . X.
k=-I
Costs are incurred according to measurable function h : N N ~ N, assumed increasing in each of its arguments. An optimal scheduling policy ~r* satisfies E { C ( ~ - * ) } = i n f E {C(~r)},
(2)
qT
where C(q'l') ~ h {F(I)(qT),F(2)(Tg) . . . . .
F(N) ("B-)},
and the infimum in (2) is over all nonpreemptive policies and is assumed to be finite. The above is the order-specific costs model, see Glazebrook and Whitaker (1992).
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Suppose now that the jobs are numbered such that /-'1
(3)
T h e o r e m 1. When ~ is an equicorrelation covariance matrix, the identity permutation is optimal in the class o f nonpreemptive policies f o r the order-specific costs model. Proof. We use an induction on N, the number of jobs. The result holds trivially for N = 1. We suppose that it holds for N < N and consider a problem,/6 say, with N = ,~. Let # be a nonpreemptive policy for P such that # ( 1 ) = k, where /2~ > /21. We will demonstrate the existence of a nonpreemptive policy ~ for which Or( 1 ) = 1 and
£{C(~)} < £{C(~)}.
(4)
First, consider ~ at the point when X~(1) = Xk has been realised, say, at Xk = x~. By standard theory, the remaining processing times Xk ~
(XI,X 2 .....
Xk-l,Xk+l ..... XN) T
are conditionally multivariate normal with mean vector +
p(x
(5)
and equicorrelation covariance matrix 02 {(1 - p ) / +
p(1 - p ) l l T } .
(6)
In (5) and (6), / and 1 are ( N - 1) x ( f i / - 1) and ( N -
1) x 1 respectively. We note from (3) that
/21 "q- p ( X k -- /2k) ~ /22 "Jr-p ( X k -- /2k) ~ . . . ~ /2~ + p ( x k - - / 2 k )
and hence from (5) we deduce that conditioning upon Xk = xk does not change the ordering of the means of the unrealised processing times. We also note that the function hxk : ~ - l ~ 1~ defined by hxk ( Yl, Y2 . . . . . Y?7-1 ) = h ( Xk, Xk + Yl, Xk + Y2 . . . . . X~ + Y?7-1 )'
(7)
is increasing in each of its arguments. In the light of the above, we infer from the inductive hypothesis that if ~" denotes the permutation policy (k, 1,2 . . . . . k - 1 , k + 1 . . . . . ,~), then =xk}
<
E{C( r)IX
=
x,, e R,
and hence that E {C(~-)} _< E { C ( ~ - ) } .
(8)
Now consider ¢r, the permutation policy ( l, k, 2 . . . . . k - 1, k + 1 . . . . . ,Q) obtained from ~" by interchanging the first two jobs. We wish to show that
E{C(?r)} < E{C(~)}.
(9)
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In what follows, if x is an 19-vector with 19 _> 2, then we use x lk to denote the (19 - 2)-vector obtained by omitting the first and kth components and x lk the N-vector obtained from x by interchanging the first and kth components. Let ~b(x) denote the multivariate normal density of X, i.e.
4,(~)
=
(2~-)-~nl.Sl-'/2exp {-½ (x - ~)Tx-I (X --/x) },
where
X-'
{o-2(I - p ) } - '
{L-p{1+
(19- l)p}-'_]_]T}.
By direct computation we have that ~b(x) = exp { (xl - xk) (/x, - / x k ) }
.
(10)
From (10) we deduce that ~b(x_) _> ~b(X_lk)
~
xl _< xk.
(11)
It is trivial from the increasing nature of h that ~x~(Xk,X jk) - = h ( x l , x l + x ~ , x l + x k + x 2
< h~,(xl,x lk)
-', '-
. . . . . x l + x k + x2 +
.. + x o )
xl _< xk.
(].2)
Compare with Eq, (7). We now have that E{C(qr)}
-- E{C(9~/") I ( X 1 < Xk) } + E { C ( " ~ ) I ( X 1 > Xk) }
= f
.....
XI~Xk
"~ h(Xk, Xl "~ Xk, Xl "4- Xk "~- X2 . . . . . Xl + Xk + X2 + . - .
/
+
x#) q~(xlk) } dx_
{L, (x~,x ~k) ~b(x) + hxk(X~,X ~) ~(Xlk) } dx
351 (Xk
{hx, (xk, x lk) ~k(xl,) + hx,(Xl ,x_lk) 0~(x)} dx_
(13)
Xl (Xk
= E {C(~') I ( X , > Xk)} + E {C(~') I ( X 1 < X~)}
which establishes (9). Note that inequal!ty (13) uses both (11) and (12), and that I stands for the indicator function in the above. From (8) and (9) we infer that
E { c ( ~ ) } _< E { c ( # ) } , which establishes (4). Hence ~- is no better for P than a nonpreemptive policy, optimal among those which schedule job 1 first. Any such policy must therefore be optimal among all nonpreemptive policies. If we utilise the inductive hypothesis in the manner of the argument leading to (8) (with X1 replacing X~), then it is easy to establish that the identity permutation is indeed optimal among nonpreemptive policies which schedule job 1 first. Hence the identity permutation is optimal for/5 and the induction goes through. This concludes the proof.
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3. The job-specific costs model In the model considered here, the stochastic structure for the processing requirements X and the class of nonpreemptive scheduling policies are both as in Section 2, However, now the costs incurred are job-specific as follows: let F1(7r) denote the completion time of job j under nonpreemptive policy zr, i.e.
Fi(Ir) =
~
xk,
j = 1,2 . . . . . N.
~r(k)<~r(j) Costs are incurred according to the measurable function h : R N --~ JR, assumed increasing in each of its arguments. An optimal scheduling policy ~-* satisfies E{C(~*)} = infE{C(~-)},
(14)
qr
where
C ( ¢r) =~ h { Fl ( cr), F2( cr) . . . . . FN(Tr)} and the infimum in (14) is over all nonpreemptive policies. Suppose now that the jobs are numbered according to increasing values of their mean processing requirements, as in (3). Recall that in the important special case N
h( F1,F2 . . . . . FN) = ~
wiFj,
(15)
j=l
where wj > 0, j = 1,2 . . . . . N, discussed at some length later, the policy which orders the jobs in decreasing order of the index wj/tzj is optimal when p = 0 and all means are positive. We shall refer to this policy as Smith's rule (SR), although Smith's (1956) original paper concerned only deterministic models. Smith's rule will coincide with the identity permutation when the weights w i are agreeable with the m e a n s / z j, such that Wl ~ W2 ~ . . . ~ WN.
(16)
The following definition expresses the general notation of agreeability between cost rates and processing times which we require. Definition. Cost function h is agreeable with the mean vector/z, satisfying (3), if for any x E ~ v satisfying x i > x j for s o m e i < j h(x_) _< h(x~j), where xij is obtained from x by interchanging the ith and jth components, Theorem 2. When 2_ is an equicorrelation covariance matrix and h is agreeable with iz, the identity permutation
is optimal in the class of nonpreemptive policies for the job specific costs model. The proof involves minor adjustments to that of Theorem 1 and is omitted. The agreeability condition is necessary for an appropriate form of inequality (12). Examples. To obtain cost functions which are agreeable with /x, satisfying (3), suppose that ~i : ~ ~ ]~, j = 1,2 . . . . . N, are increasing and differentiable. The cost function
R.J. Boys et al./European Journal of Operational Research 102 (1997) 111-123
116 N
g ( x ) ~. ~-'~l[Ij(Xj),
X ~ R N,
(17)
j=l
is agreeable when
¢~(x) ~ ¢ ~ ( x ) ~ ... ~ ¢ ~ ( x ) ,
x c R.
(18)
The Smith's rule example discussed in (15) and (16) above is obtained when the ~/,j are all linear. Hence we obtain from Theorem 2 the optimality of Smith's rule for the cost function h satisfying (15) and (16) when __Xis an equicorrelation covariance matrix. Suppose now that r/j : R ---+R +, j = 1,2 . . . . . N, are increasing and differentiable. The cost function N
h(x) = I-Irlj(xj),
(19)
x E I~N,
j=l
is agreeable when
n ~ ( x ) / m ( x ) ~ n ~ ( x ) / ~ 2 ( x ) ~ ... ~ n ~ ( x ) / n N f x ) ,
x ~ ~.
(20)
Note also that any monotone increasing transformation of an agreeable cost function is also agreeable. In particular h in (19) is agreeable if logh is agreeable. Condition (20) then follows from the analysis of function g in (17) and (18).
4. Suboptimality bounds for Smith's rule To take the analysis further, we focus on the weighted flow time criterion in (15). We saw in Section 3 that this criterion (or, indeed, any monotone increasing transform of it) is optimised in the class of nonpreemptive policies by Smith's rule under the agreeability conditions (16) and when Z is an equicorrelation covariance matrix. However, we suspect (and our simulation study in Section 5 bears this out) that Smith's rule performs well for criterion (15) under quite general conditions. Our goal here is to bound the degree of suboptimality of Smith's rule in terms of important system parameters. Note firstly that when X is deterministic, with X = (xl,x2 . . . . . xu) T, i.e. when processing times are known in advance, then Smith's rule defined in terms of wj/xj is certainly optimal. In the light of that, we would expect Smith's rule to perform well when the spread of the distribution of X (suitably characterised) is small. In the equicorrelation case in (1) the obvious requirement is that o- be small. Hence it is natural to seek bounds on the suboptimality of Smith's rule which are measures of the spread of X. We also note that when the processing requirements are independent random variables (achieved by taking p = 0 in the equicorrelation case), then Smith's rule is optimal for criterion (15) irrespective of whether the agreeability conditions hold. Hence it is also natural to seek bounds on the suboptimality of Smith's rule which are natural measures of correlation in X. Although our approach to both of these problems holds good for general X we shall concentrate in the second case on developing results for the equicorrelation case where correlation in X is expressed through the single parameter p. For both of these analyses we require the following result: suppose that the jobs are numbered according to decreasing values of wj/Izj and hence that Smith's rule (SR) is the identity permutation. Recall that in the restriction from nonpreemptive to permutation policies in which an immutable ordering of the jobs is chosen at t = 0, Smith's rule is certainly optimal for the weighted flow-time criterion (15) for any /L and ~. This is easily seen from the fact that for any permutation policy ~-
R.J. Boys et al./EuropeanJournalof OperationalResearch102 (1997)111-123 N
117
j
j=l
We shall suppose that /zj > 0, j = 1,2 . . . . . N throughout. In Lemma 3, which is a version of Lemma 4 in Glazebrook and Owen (1995), ¢r is a general permutation policy. L e m m a 3. (Suboptimality bounds for permutation policies)
N E {C ( qT") - C ( NR ) } ~ J~-i { [W~(j) /IJ'rt'(j) - ll~n ]{ Wrr(k) /l'l'rr(k, ) ] l'Zrr(j) ~'~ k=l lL~rr(k) }
(21)
It is natural to consider the r.h.s, of (21) as a measure of the extent to which permutation ~ differs from Smith's rule. We now proceed to the first of our analyses in which we bound the suboptimality of Smith's rule for the weighted flow-time criterion (15) by a natural measure of the spread of X for general ~ . Suppose that Xi ~ N(tzi, o'2), i = 1,2 . . . . . N, marginally. In Theorem 4, 0 stands for the N-vector of zeros and 7r* is, as usual, an optimal nonpreemptive policy. Theorem 4. (Suboptimality bound for Smith's Rule) [{c(sm
-
> o)
<
--.
In order to prove Theorem 4 we consider an artificial construct - namely, the so-called super-optimal rule (SO). The rule SO (which is inadmissible) is allowed to observe the realised value of X = (xl, x2 . . . . . xN) T in advance of any processing and schedules the jobs in decreasing order of wj/xj. Since SO is optimal for each realisation x > 0, then plainly for any such x, ~-(-~-~
X =
_< E k
C(SR)
X =
.
(22)
Our first goal in the proof of Theorem 4 is to hound the right hand side of (22). This we do in Lemma 5. L e m m a 5.
--(-~
c(so,
l(x_ > O) ~ ~
=
I~.l -- xjl/I,Zj.
j=l
Proof. We condition on X = x > 0 and suppose that the jobs are numbered according to decreasing values o f wj/tzj. Applying Lemma 3 to the problem with processing times x and where permutation ~r is the identity (i.e. Smith's rule), we obtain E{C(SR)-C(SO)IX=x
wj/xj-
} <_ .-
Now, let ) be such that
w)/x)= min wk Xk .
min -
-
-
-
Wk Xk} Xj~'~Xk k=l
•
(23)
R.J. Boys et al. /European Journal of Operational Research 102 (1997) 111-123
118
Plainly
x.i < wjx:Jw>
wg/x9 < wj/xj
(24)
Further
wjlx; - w~lx~ = (Wl/X.i - wjl m ) + (w//~.j - w~lm) + (w~/m - w~lx~)
(25)
<_ (wj/xj - w d m ) + (w~/m - w~/xD, since
w~/m >_ Wl/m by Smith's rule. Combining (24) and (25), we obtain
wjlxj
-
min {wkl~k}'] ~; = (wjlxj - w~/x~) xj
1 ~k~_.i
)
<_ (wjlm - xA) l m + (wjlm
x)l) l m
-
J <_ wi ~
(26)
I ~ - xkll~k.
k=l
Inequalities (23) and (26) yield N
E{C(SR)-C(SO)[X=x}
j
_< ~-~wj(~_l
~
j
llZk xkl/tzk] ~-~xk
j=l
k=l
N
N
j
x
k=l
/
~ (j~I ]]'Zj--XJl/~LI'J(ZWJ ) ~_jxkI •-
" j=l
x, ll ,
=
The result now follows easily. P r o o f of T h e o r e m 4. Combining (22) and Lemma 5, we obtain that
N
<_ ~-] e(Im - xjl)/m j=l N
j=l
as required. We now seek suboptimality bounds for Smith's rule in terms of some measure of correlation in X. For simplicity, in this problem we restrict to the equicorrelation case in (1) where 0 < p < 1 and seek bounds
R.J. Boys et aL/European Journal of Operational Research 102 (1997) 111-123
119
expressed in terms of the common correlation p. Again, we focus on the weighted flow-time criterion (15). Theorem 6 is an example of what can be achieved. T h e o r e m 6. (Suboptimality bound for Smith's rule)
where ~ = ~-~=1 wj/N, fz = ~-~jN=, txj/N, f4 = EjU=l Xj/N, and u - = m a x ( 0 , - u ) . In order to obtain (27), we first transform X as follows:
X ~ MVN(I~,2__)
~
o'~_-1/2X ~ MVN(~r2_-I/2~_, o'2/).
Hence the components of ~r2-V2X_ _ are independent. It is not difficult to show that o'__.~-1/2 = ( 1 - p) -1/2L + b(p) 11T ,
(28)
where b(p) < 0 is given by
b(p)
p)-l/2 [1 - (1 - p)1/2{1 + ( U - 1)p} -1/2]/N.
(29)
Some straightforward algebra yields
- b ( p ) < p/2 + o(p).
(30)
Write
o'~__-l/2x = P = (P1,/°2 . . . . . PN) T, where, by (28), N
Pj = ( 1 -- p) -1/2Xj + b(p) Z
Xk,
j = 1,2 . . . . . N.
Denote by /3 the scheduling problem with weighted flow-time criterion (15), but with processing times X replaced by P_. Since the Pj are independent, P will be optimised by the version of Smith's Rule which orders the jobs according to decreasing values of N
wj / E ( Pj ) = wj/ { ( l -- p)-l/2ixj + b(p) ~ ixk }, k=l provided that E(Pj) > 0, j = 1,2 . . . . . N. We shall refer to this rule as a modified Smith's Rule, abbreviated to M S R ( p ) to emphasise its p-dependence. However, since b(p) < 0, it may be that some of the Pj have negative expectation. This problem is resolved as follows: define a new scheduling problem P(A), identical to t3 save only that the processing times are
Pj(A) = P: + Awj,
j
--
1,2 . . . . . N,
where A _> 0. Provided only that wj > O, j = 1,2 . . . . . N, we can choose A such that E{Pj(A)} >_ O, j = 1,2 . . . . . N. Let 7rl, ~r2 be two permutation policies and denote by C. C(-, A) costs computed for/~ and /3(a) respectively. Now
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N
j
j=l
k=l
± }
= E{C(Tri,0)} + AE{w~.,(.i) wm(k) , ,/=1 k=l
i= 1,2,
(31)
where the second term in (31 ) is permutation invariant. Hence E {C (7/'1, ,~) -C(q't-2, ~ ) } = E { C ( ~ ' , , O ) - C(¢r2,0)}
(32)
and so the permutation optimising P ( ~ ) optimises/3(0) ~ / 3 also. Hence the modified Smith's rule MSR(p, ~) which orders the jobs according to decreasing values of
wj/E{Pj(,~) }
= wj/ { ( l -
p)-l/2ld,-j + b(p) ZN tzk + Awj ) k=l
optimises /3. We are now ready to give an outline proof of Theorem 6. Outline proof of Theorem 6. Consider the scheduling problem P in which all jobs are subject to an initial delay from time zero of N 2 (b(p)X} +, where u + = max(u, 0), and the processing times are (1 - p)-U2Xi, j = 1,2 . . . . . N. It is plain that the total expected cost for P under some general nonpreemptive policy is at least as great as the expected cost for P under the optimal policy MSR(p, ~). From this it in turn follows that for all ¢r
N3#E [{b(p) X}+] + (1 - p)-U2E{C(zr)} _> E [C {MSR(p, ])}] and hence that
N3#E [{b(p) 2}+] + (1 - p)-'/2E{C(cr*)} > E [C {MSR(p,,~)}].
(33)
We infer from inequality (33) that E{C(SR) - C(~-*)} = E {C(SR) - (1 - p ) U 2 d ( S R ) }
+ ( l - p)t/2E [C(SR) - (~ {MSR(p, A) }] + _< g
[(1 -
{C(SR) -(1-
{MSR(,,,
} -
p)U2O(SR)}
+ (l - p)U2E [C(SR) - d {MSR(p,.~) }] + (l - p)'/~N~®~ [{b(p)~?}] + .
(34)
Direct computation of the costs involved yields E {C(SR) - (1 - p)I/gC(SR)} _< (1 -- p)1/2Nayv{-b(p)}fx.
(35)
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121
Further, we note from (32) that E [C (SR, A) - C {MSR(p, A), A}] = E [C(SR) - C {MSR(p, ~) }].
(36)
We now use a suitable version of Lemma 3, applied to scheduling problem/3 (~), to bound the left hand side of (36). Calculations similar to those in the proof of Lemma 5 yield E [C(SR) - d {MSR(p, A)}] < 2pN3#fz + o(p).
(37)
To obtain inequality (27), combine (34), (35) and (37) with inequality (30). This completes the proof.
5. Computational study For the weighted flow-time cost model there is an attractive dynamic analogue of Smith's rule which modifies the mean processing times of uncompleted jobs by conditioning on the realised values of the processing requirements of completed jobs. More specifically, suppose that this dynamic version of Smith's rule (DSR) z/, say, has already chosen jobs zr' (1), z / ( 2 ) . . . . . ¢rI (j) for processing and that these jobs have been completed with realised processing requirements x~,(1), x~,(2~ . . . . . x~,(.i~. In order to choose the next job for processing, 7rI ( j + 1), we first compute the conditional means Ixk(j,x)=--E(XklX~,(i)=x,~,(O, i = 1 , 2 . . . . . j ) ,
k~z/(i),
i = 1 , 2 . . . . . j.
The DSR rule chooses z / ( j + 1) to satisfy
W~'(j+I )/]-t~Tr'(j+l)(j, x) = mkax{wk/Ixk(j, x) },
(38)
where in (38), the maximisation is over k 4= ~ ( i ) , i = 1,2 . . . . . j. Our computational study of the weighted flow-time problems shows that Smith's rule performs very well. In all our simulation studies, the mean cost incurred by Smith's rule was within 0.5% of that incurred by an optimal policy. Where suboptimalities have occurred, these have almost all been eliminated by the adoption of DSR, the dynamic version of Smith's rule described above. So well did DSR perform that it was a challenge to design a simulation study which demonstrated any suboptimality in it. We did so by generating job characteristics in such a way that the weights wj and marginal means/zj were tied together in a manner likely to induce changes in job ordering (i.e. from the static Smith's rule) when the conditioning leading to (38) was applied. In Table 1 we report one such study whose results were typical. For each problem generated we took N = 5. The size of problem was delimited by the need to compute the optimal policy in each case via dynamic programming. Job weights were generated by sampling independently from the continuous uniform U[20, 30] distribution. Given job weight wj, we obtained/xj by adding to wj the value of a U[0,2] random variable. Hence all values of wj/tzj were in the range [0.9, 1.0]. Covariance matrices for X = ( X 1 , X 2 , X 3 , X 4 , X s ) T were obtained by sampling independently from Wishart distributions. In the usual notation, the distributional assumption was 4 ~ ~-, W5 {4, B ( P ) } ,
(39)
where R(p) denotes the equicorrelation matrix in (1) with o.2 = 1. Note that, while the covariance matrices generated by (39) do not have all correlations equal, we do have E(E) = R(p) and so we are considering random departures from the equicorrelation case. Having generated/z and __X,processing times were obtained by sampling independently from the corresponding multivariate normal distribution.
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Table 1 A computational study of Smith's rule and a dynamic version of it p
E{C(SR)}
E{C(DSR)}
E{C(~-*)}
/~{ZlsR}
~7"[ADSR}
0 0.3 0.5 0.7
9695.6 ( I0.1 ) 9683.4 (10.3) 9694.6 (10.4) 9691.9 (10.5)
9670.7 ( 10.1 ) 9664.6 (10.3) 9680.1 (10.4) 9681.6 (10.5)
9669.8 ( 10.1) 9664.1 (10.3) 9680.0 (10.4) 9681.4 (10.5)
0.270 (0.007) 0.201 (0.005) 0.153 (0.004) 0.109 (0.003)
0.010 (0.004) 0.005 (0.003) 0.001 (0.002) 0.002 (0.001)
In Table 1 below, we report the results for the cases p = 0,0.3,0.5,0.7. For each row of the table, 10000 problems were generated as above and the costs C ( S R ) , C ( D S R ) and C(~-*) incurred by the operation of Smith's rule, the dynamic version of Smith's rule, and an optimal policy were obtained, as were the percentage differences ,asR = l o o { C ( S R )
- C(¢r*)}
~COt*)
and ADSR = 100 { C ( D S R ) - C ( ~ * ) } / C ( ~ r * ) . In the table we have estimates of the means of these quantities, with standard errors in parentheses. We see from the table that Smith's rule performs very well, coming within a fraction of 1% of optimality on average. The dynamic version of Smith's rule eliminates almost all of this suboptimality and is virtually indistinguishable from an optimal policy.
6. Summary We have considered the task of determining the optimal nonpreemptive policy for scheduling a collection of jobs whose processing times follow a normal distribution with equicorrelated components. We show that the permutation policy which orders the jobs in ascending value of their mean processing times is optimal within the class of nonpreemptive policies for general order-specific costs. This policy is also optimal for job-specific costs when the cost function satisfies an agreeability condition. Suboptimality bounds are given for the relative performance of Smith's rule for the weighted flow-time criterion. These bounds explore regions of the parameter space around points where Smith's rule is known to be optimal. Simulation studies have shown that Smith's rule performs well, even in situations which are at some distance from those for which it can be shown to be optimal. A dynamic version eliminates almost all suboptimality in Smith's rule.
Acknowledgements The authors would like to express their gratitude to an anonymous referee for pointing out an error in an earlier version o f the proof of Theorem 1.
References Bergman, S.W., and Gittins, J.C. (1985), Statistical Methods for Planning Pharmaceutical Research, Marcel Dekker, New York.
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Bruno, J., and Hofri, M. (1975), "On scheduling chains of jobs on one processor with limited preemption", SIAM Journal on Computing 4, 478-490. Gittins, J.C. (1989), Multi-Armed Bandit Allocation Indices, Wiley, New York. Glazebrook, K.D. (1978), "On the optimal allocation of two or more treatments in a controlled clinical trial", Biometrika 65, 335-340. Glazebrook, K.D., and Owen, R.W. (1995), "On the value of adaptive solutions to stochastic scheduling problems", Mathematics of Operations Research 20, 65-89. Glazebrook, K.D., and Whitaker, L.R. (1992), "Single-machine stochastic scheduling with dependent processing times", Advances in Applied Probability 24, 635-652. Hodgson, T.J., King, R.E., and Stanfield, P.M. (1996), "Common ready time scheduling with stochastic service times", to appear in Operations Research. Mardia, K.V., Kent, J.T., and Bibby, J.M. (1979), Multivariate Analysis, Academic Press, London. Robb, D.J. (1992), "Scheduling in a management context: stochastic tasks and non-regular performance measures", Doctoral thesis, University of Calgary. Smith, W.E. (1956), "'Various optimizers for single-stage production", Naval Research Logistics Quarterly 3, 59-66.
European Journal of Operational Research 102 (1997) 111-123
Theory and Methodology
Single machine scheduling when processing times are correlated normal random variables R.J. B o y s , K . D . G l a z e b r o o k , C . M . M c C r o n e Department of Mathematics & Statistics, University of Newcastle upon Tyne, Newcastle upon Tyne, NE1 7RU, UK
Received 1 July 1995; accepted 1 April 1996
Abstract
A single machine is available to process a collection of jobs whose processing times are jointly multivariate normal. Processing is nonpreemptive. We show that in the equicorrelation case, the permutation policy which schedules the jobs in ascending order of their (marginal) mean processing times is optimal for general order-specific costs and also for job-specific costs under an agreeability condition. Suboptimality bounds on the performance of Smith's rule are obtained for the weighted flow-time criterion. A computational study shows that a dynamic version of Smith's rule comes very close to optimality. (~) 1997 Elsevier Science B.V. Keywords: Agreeability;Dependentprocessing times; Multivariatenormal distribution; Optimal policy; Smith's rule; Stochastic scheduling
1. I n t r o d u c t i o n
A single machine is available to process N jobs, the jth of which has random processing requirement X j , j = 1,2 . . . . . N. Processing is nonpreemptive, i.e. interruptions of the processing of a job are not allowed. At time 0 and at all subsequent times at which jobs complete, a decision has to be made concerning which job to process next. The objective is to choose a policy to minimise some overall measure of total cost. This measure is always a non-decreasing function of the job completion times. Such models have been used to study problems inter alia in research planning (see, for example, Bergman and Gittins, 1985), computer scheduling (Bruno and Hofi, 1975) and the sequential design of experiments (Glazebrook, 1978). See Gittins (1989) for a general theoretical account. A key simplifying feature shared by almost all such stochastic scheduling models considered to date concerns the independence of the processing requirements X j , j = 1 , 2 . . . . . N. Glazebrook and Whitaker (1992) argue that there are situations in which the independence assumption is unwarranted. They take over concepts from the reliability literature to develop models which allow for dependencies between the jobs and seek conditions which yield optimal scheduling policies of simple structure. However, the development of optimal scheduling policies for models with dependence is extremely difficult. Although the models considered by Glazebrook and Whitaker (1992) are quite general, the above sufficient conditions are very severe. This theoretical paper aims to promote further understanding of scheduling models incorporating dependence by consideration of the important special case in which X = (X1, X2 . . . . . XN) T is 0377-2217/97/$17.00 (~) 1997 Elsevier Science B.V. All rights reserved. PII S0377-2217(96) 00211-1
R.J. Boys et aL /European Journal of Operational Research 102 (1997) 111-123
112
multivariate normal. This crucially allows us to model the appropriate dependence structure simply through a covariance matrix __£. Note that Section 5 of Hodgson, King and Stanfield (1996) makes a contribution to production scheduling in which job service times are assumed to be normally distributed. See also Robb (1992). In Section 2, we consider the order-specific costs model in which the overall cost of processing is a general function of the ordered completion times. We show that in the equicorrelation case, the permutation policy which processes jobs in the order of increasing mean processing times (i.e. shortest first) is optimal. This result is generalised to a job-specific costs model in Section 3 when an agreeability condition holds. In Sections 4 and 5 we focus on the Classical weighted flow-time model for which Smith's rule is optimal in the independence case. See Smith (1956). In Section 4, we obtain two suboptimality bounds on the performance of Smith's rule. These bounds are, respectively, natural measures of the spread of X (reflecting the fact that Smith's rule is optimal for the deterministic case in which processing times are known in advance) and the degree of dependence in X (reflecting the fact that Smith's rule is optimal for the independence case). Section 5 contains a computational study. Here we see that almost all of the suboptimality in Smith's rule is eliminated by implementing a dynamic version in which the mean processing times of uncompleted jobs are updated by suitably conditioning on past outcomes at each decision epoch.
2. The order-specific costs model Consider a single machine scheduling model in which job j has processing requirement Xj, j = 1,2 . . . . . N. We assume X = ( X l , X2 . . . . . XN) T to be multivariate normal with mean vector/~ and covariance matrix ~. An important special case is when
___~=o-2{(1 --p)/+pl!T},
(1)
where / is the N x N identity matrix, 1_ is the N × 1 vector of ones, o-2 > 0 is a common variance and -1/(N1 ) < p < 1 is a common correlation. This is the equicorrelation case - see Mardia, Kent and Bibby (1979). Positive dependence seems natural for the scheduling context and so the restriction on the choice of negative p need not overly concern us. See Glazebrook and Whitaker (1992). A nonpreemptive scheduling policy ~ has a free choice of which job to schedule first, designated ~-( 1 ). Once the first (n - 1) jobs are scheduled by ~-, the choice of ~r(n) is allowed to depend upon the realised values of X~r(j), j = 1,2 . . . . . n - 1. A job, once chosen, is processed through to completion without interruption. Any nonpreemptive policy which always (i.e. with probability 1) schedules the jobs in the same order is called a permutation policy. The identity permutation ~" schedules the jobs in numerical order, i.e. ~-'(j) = j, j = 1,2 . . . . . N. Under nonpreemptive policy ~-, F(j) (Tr) is the completion time of ~r(j), defined as
J F(j) (~-) = ~
X~r(k),
j = 1,2 . . . . . X.
k=-I
Costs are incurred according to measurable function h : N N ~ N, assumed increasing in each of its arguments. An optimal scheduling policy ~r* satisfies E { C ( ~ - * ) } = i n f E {C(~r)},
(2)
qT
where C(q'l') ~ h {F(I)(qT),F(2)(Tg) . . . . .
F(N) ("B-)},
and the infimum in (2) is over all nonpreemptive policies and is assumed to be finite. The above is the order-specific costs model, see Glazebrook and Whitaker (1992).
R.J. Boys et al./European Journal of Operational Research 102 (1997) 111-123
113
Suppose now that the jobs are numbered such that /-'1
(3)
T h e o r e m 1. When ~ is an equicorrelation covariance matrix, the identity permutation is optimal in the class o f nonpreemptive policies f o r the order-specific costs model. Proof. We use an induction on N, the number of jobs. The result holds trivially for N = 1. We suppose that it holds for N < N and consider a problem,/6 say, with N = ,~. Let # be a nonpreemptive policy for P such that # ( 1 ) = k, where /2~ > /21. We will demonstrate the existence of a nonpreemptive policy ~ for which Or( 1 ) = 1 and
£{C(~)} < £{C(~)}.
(4)
First, consider ~ at the point when X~(1) = Xk has been realised, say, at Xk = x~. By standard theory, the remaining processing times Xk ~
(XI,X 2 .....
Xk-l,Xk+l ..... XN) T
are conditionally multivariate normal with mean vector +
p(x
(5)
and equicorrelation covariance matrix 02 {(1 - p ) / +
p(1 - p ) l l T } .
(6)
In (5) and (6), / and 1 are ( N - 1) x ( f i / - 1) and ( N -
1) x 1 respectively. We note from (3) that
/21 "q- p ( X k -- /2k) ~ /22 "Jr-p ( X k -- /2k) ~ . . . ~ /2~ + p ( x k - - / 2 k )
and hence from (5) we deduce that conditioning upon Xk = xk does not change the ordering of the means of the unrealised processing times. We also note that the function hxk : ~ - l ~ 1~ defined by hxk ( Yl, Y2 . . . . . Y?7-1 ) = h ( Xk, Xk + Yl, Xk + Y2 . . . . . X~ + Y?7-1 )'
(7)
is increasing in each of its arguments. In the light of the above, we infer from the inductive hypothesis that if ~" denotes the permutation policy (k, 1,2 . . . . . k - 1 , k + 1 . . . . . ,~), then =xk}
<
E{C( r)IX
=
x,, e R,
and hence that E {C(~-)} _< E { C ( ~ - ) } .
(8)
Now consider ¢r, the permutation policy ( l, k, 2 . . . . . k - 1, k + 1 . . . . . ,Q) obtained from ~" by interchanging the first two jobs. We wish to show that
E{C(?r)} < E{C(~)}.
(9)
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In what follows, if x is an 19-vector with 19 _> 2, then we use x lk to denote the (19 - 2)-vector obtained by omitting the first and kth components and x lk the N-vector obtained from x by interchanging the first and kth components. Let ~b(x) denote the multivariate normal density of X, i.e.
4,(~)
=
(2~-)-~nl.Sl-'/2exp {-½ (x - ~)Tx-I (X --/x) },
where
X-'
{o-2(I - p ) } - '
{L-p{1+
(19- l)p}-'_]_]T}.
By direct computation we have that ~b(x) = exp { (xl - xk) (/x, - / x k ) }
.
(10)
From (10) we deduce that ~b(x_) _> ~b(X_lk)
~
xl _< xk.
(11)
It is trivial from the increasing nature of h that ~x~(Xk,X jk) - = h ( x l , x l + x ~ , x l + x k + x 2
< h~,(xl,x lk)
-', '-
. . . . . x l + x k + x2 +
.. + x o )
xl _< xk.
(].2)
Compare with Eq, (7). We now have that E{C(qr)}
-- E{C(9~/") I ( X 1 < Xk) } + E { C ( " ~ ) I ( X 1 > Xk) }
= f
.....
XI~Xk
"~ h(Xk, Xl "~ Xk, Xl "4- Xk "~- X2 . . . . . Xl + Xk + X2 + . - .
/
+
x#) q~(xlk) } dx_
{L, (x~,x ~k) ~b(x) + hxk(X~,X ~) ~(Xlk) } dx
351 (Xk
{hx, (xk, x lk) ~k(xl,) + hx,(Xl ,x_lk) 0~(x)} dx_
(13)
Xl (Xk
= E {C(~') I ( X , > Xk)} + E {C(~') I ( X 1 < X~)}
which establishes (9). Note that inequal!ty (13) uses both (11) and (12), and that I stands for the indicator function in the above. From (8) and (9) we infer that
E { c ( ~ ) } _< E { c ( # ) } , which establishes (4). Hence ~- is no better for P than a nonpreemptive policy, optimal among those which schedule job 1 first. Any such policy must therefore be optimal among all nonpreemptive policies. If we utilise the inductive hypothesis in the manner of the argument leading to (8) (with X1 replacing X~), then it is easy to establish that the identity permutation is indeed optimal among nonpreemptive policies which schedule job 1 first. Hence the identity permutation is optimal for/5 and the induction goes through. This concludes the proof.
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115
3. The job-specific costs model In the model considered here, the stochastic structure for the processing requirements X and the class of nonpreemptive scheduling policies are both as in Section 2, However, now the costs incurred are job-specific as follows: let F1(7r) denote the completion time of job j under nonpreemptive policy zr, i.e.
Fi(Ir) =
~
xk,
j = 1,2 . . . . . N.
~r(k)<~r(j) Costs are incurred according to the measurable function h : R N --~ JR, assumed increasing in each of its arguments. An optimal scheduling policy ~-* satisfies E{C(~*)} = infE{C(~-)},
(14)
qr
where
C ( ¢r) =~ h { Fl ( cr), F2( cr) . . . . . FN(Tr)} and the infimum in (14) is over all nonpreemptive policies. Suppose now that the jobs are numbered according to increasing values of their mean processing requirements, as in (3). Recall that in the important special case N
h( F1,F2 . . . . . FN) = ~
wiFj,
(15)
j=l
where wj > 0, j = 1,2 . . . . . N, discussed at some length later, the policy which orders the jobs in decreasing order of the index wj/tzj is optimal when p = 0 and all means are positive. We shall refer to this policy as Smith's rule (SR), although Smith's (1956) original paper concerned only deterministic models. Smith's rule will coincide with the identity permutation when the weights w i are agreeable with the m e a n s / z j, such that Wl ~ W2 ~ . . . ~ WN.
(16)
The following definition expresses the general notation of agreeability between cost rates and processing times which we require. Definition. Cost function h is agreeable with the mean vector/z, satisfying (3), if for any x E ~ v satisfying x i > x j for s o m e i < j h(x_) _< h(x~j), where xij is obtained from x by interchanging the ith and jth components, Theorem 2. When 2_ is an equicorrelation covariance matrix and h is agreeable with iz, the identity permutation
is optimal in the class of nonpreemptive policies for the job specific costs model. The proof involves minor adjustments to that of Theorem 1 and is omitted. The agreeability condition is necessary for an appropriate form of inequality (12). Examples. To obtain cost functions which are agreeable with /x, satisfying (3), suppose that ~i : ~ ~ ]~, j = 1,2 . . . . . N, are increasing and differentiable. The cost function
R.J. Boys et al./European Journal of Operational Research 102 (1997) 111-123
116 N
g ( x ) ~. ~-'~l[Ij(Xj),
X ~ R N,
(17)
j=l
is agreeable when
¢~(x) ~ ¢ ~ ( x ) ~ ... ~ ¢ ~ ( x ) ,
x c R.
(18)
The Smith's rule example discussed in (15) and (16) above is obtained when the ~/,j are all linear. Hence we obtain from Theorem 2 the optimality of Smith's rule for the cost function h satisfying (15) and (16) when __Xis an equicorrelation covariance matrix. Suppose now that r/j : R ---+R +, j = 1,2 . . . . . N, are increasing and differentiable. The cost function N
h(x) = I-Irlj(xj),
(19)
x E I~N,
j=l
is agreeable when
n ~ ( x ) / m ( x ) ~ n ~ ( x ) / ~ 2 ( x ) ~ ... ~ n ~ ( x ) / n N f x ) ,
x ~ ~.
(20)
Note also that any monotone increasing transformation of an agreeable cost function is also agreeable. In particular h in (19) is agreeable if logh is agreeable. Condition (20) then follows from the analysis of function g in (17) and (18).
4. Suboptimality bounds for Smith's rule To take the analysis further, we focus on the weighted flow time criterion in (15). We saw in Section 3 that this criterion (or, indeed, any monotone increasing transform of it) is optimised in the class of nonpreemptive policies by Smith's rule under the agreeability conditions (16) and when Z is an equicorrelation covariance matrix. However, we suspect (and our simulation study in Section 5 bears this out) that Smith's rule performs well for criterion (15) under quite general conditions. Our goal here is to bound the degree of suboptimality of Smith's rule in terms of important system parameters. Note firstly that when X is deterministic, with X = (xl,x2 . . . . . xu) T, i.e. when processing times are known in advance, then Smith's rule defined in terms of wj/xj is certainly optimal. In the light of that, we would expect Smith's rule to perform well when the spread of the distribution of X (suitably characterised) is small. In the equicorrelation case in (1) the obvious requirement is that o- be small. Hence it is natural to seek bounds on the suboptimality of Smith's rule which are measures of the spread of X. We also note that when the processing requirements are independent random variables (achieved by taking p = 0 in the equicorrelation case), then Smith's rule is optimal for criterion (15) irrespective of whether the agreeability conditions hold. Hence it is also natural to seek bounds on the suboptimality of Smith's rule which are natural measures of correlation in X. Although our approach to both of these problems holds good for general X we shall concentrate in the second case on developing results for the equicorrelation case where correlation in X is expressed through the single parameter p. For both of these analyses we require the following result: suppose that the jobs are numbered according to decreasing values of wj/Izj and hence that Smith's rule (SR) is the identity permutation. Recall that in the restriction from nonpreemptive to permutation policies in which an immutable ordering of the jobs is chosen at t = 0, Smith's rule is certainly optimal for the weighted flow-time criterion (15) for any /L and ~. This is easily seen from the fact that for any permutation policy ~-
R.J. Boys et al./EuropeanJournalof OperationalResearch102 (1997)111-123 N
117
j
j=l
We shall suppose that /zj > 0, j = 1,2 . . . . . N throughout. In Lemma 3, which is a version of Lemma 4 in Glazebrook and Owen (1995), ¢r is a general permutation policy. L e m m a 3. (Suboptimality bounds for permutation policies)
N E {C ( qT") - C ( NR ) } ~ J~-i { [W~(j) /IJ'rt'(j) - ll~n ]{ Wrr(k) /l'l'rr(k, ) ] l'Zrr(j) ~'~ k=l lL~rr(k) }
(21)
It is natural to consider the r.h.s, of (21) as a measure of the extent to which permutation ~ differs from Smith's rule. We now proceed to the first of our analyses in which we bound the suboptimality of Smith's rule for the weighted flow-time criterion (15) by a natural measure of the spread of X for general ~ . Suppose that Xi ~ N(tzi, o'2), i = 1,2 . . . . . N, marginally. In Theorem 4, 0 stands for the N-vector of zeros and 7r* is, as usual, an optimal nonpreemptive policy. Theorem 4. (Suboptimality bound for Smith's Rule) [{c(sm
-
> o)
<
--.
In order to prove Theorem 4 we consider an artificial construct - namely, the so-called super-optimal rule (SO). The rule SO (which is inadmissible) is allowed to observe the realised value of X = (xl, x2 . . . . . xN) T in advance of any processing and schedules the jobs in decreasing order of wj/xj. Since SO is optimal for each realisation x > 0, then plainly for any such x, ~-(-~-~
X =
_< E k
C(SR)
X =
.
(22)
Our first goal in the proof of Theorem 4 is to hound the right hand side of (22). This we do in Lemma 5. L e m m a 5.
--(-~
c(so,
l(x_ > O) ~ ~
=
I~.l -- xjl/I,Zj.
j=l
Proof. We condition on X = x > 0 and suppose that the jobs are numbered according to decreasing values o f wj/tzj. Applying Lemma 3 to the problem with processing times x and where permutation ~r is the identity (i.e. Smith's rule), we obtain E{C(SR)-C(SO)IX=x
wj/xj-
} <_ .-
Now, let ) be such that
w)/x)= min wk Xk .
min -
-
-
-
Wk Xk} Xj~'~Xk k=l
•
(23)
R.J. Boys et al. /European Journal of Operational Research 102 (1997) 111-123
118
Plainly
x.i < wjx:Jw>
wg/x9 < wj/xj
(24)
Further
wjlx; - w~lx~ = (Wl/X.i - wjl m ) + (w//~.j - w~lm) + (w~/m - w~lx~)
(25)
<_ (wj/xj - w d m ) + (w~/m - w~/xD, since
w~/m >_ Wl/m by Smith's rule. Combining (24) and (25), we obtain
wjlxj
-
min {wkl~k}'] ~; = (wjlxj - w~/x~) xj
1 ~k~_.i
)
<_ (wjlm - xA) l m + (wjlm
x)l) l m
-
J <_ wi ~
(26)
I ~ - xkll~k.
k=l
Inequalities (23) and (26) yield N
E{C(SR)-C(SO)[X=x}
j
_< ~-~wj(~_l
~
j
llZk xkl/tzk] ~-~xk
j=l
k=l
N
N
j
x
k=l
/
~ (j~I ]]'Zj--XJl/~LI'J(ZWJ ) ~_jxkI •-
" j=l
x, ll ,
=
The result now follows easily. P r o o f of T h e o r e m 4. Combining (22) and Lemma 5, we obtain that
N
<_ ~-] e(Im - xjl)/m j=l N
j=l
as required. We now seek suboptimality bounds for Smith's rule in terms of some measure of correlation in X. For simplicity, in this problem we restrict to the equicorrelation case in (1) where 0 < p < 1 and seek bounds
R.J. Boys et aL/European Journal of Operational Research 102 (1997) 111-123
119
expressed in terms of the common correlation p. Again, we focus on the weighted flow-time criterion (15). Theorem 6 is an example of what can be achieved. T h e o r e m 6. (Suboptimality bound for Smith's rule)
where ~ = ~-~=1 wj/N, fz = ~-~jN=, txj/N, f4 = EjU=l Xj/N, and u - = m a x ( 0 , - u ) . In order to obtain (27), we first transform X as follows:
X ~ MVN(I~,2__)
~
o'~_-1/2X ~ MVN(~r2_-I/2~_, o'2/).
Hence the components of ~r2-V2X_ _ are independent. It is not difficult to show that o'__.~-1/2 = ( 1 - p) -1/2L + b(p) 11T ,
(28)
where b(p) < 0 is given by
b(p)
p)-l/2 [1 - (1 - p)1/2{1 + ( U - 1)p} -1/2]/N.
(29)
Some straightforward algebra yields
- b ( p ) < p/2 + o(p).
(30)
Write
o'~__-l/2x = P = (P1,/°2 . . . . . PN) T, where, by (28), N
Pj = ( 1 -- p) -1/2Xj + b(p) Z
Xk,
j = 1,2 . . . . . N.
Denote by /3 the scheduling problem with weighted flow-time criterion (15), but with processing times X replaced by P_. Since the Pj are independent, P will be optimised by the version of Smith's Rule which orders the jobs according to decreasing values of N
wj / E ( Pj ) = wj/ { ( l -- p)-l/2ixj + b(p) ~ ixk }, k=l provided that E(Pj) > 0, j = 1,2 . . . . . N. We shall refer to this rule as a modified Smith's Rule, abbreviated to M S R ( p ) to emphasise its p-dependence. However, since b(p) < 0, it may be that some of the Pj have negative expectation. This problem is resolved as follows: define a new scheduling problem P(A), identical to t3 save only that the processing times are
Pj(A) = P: + Awj,
j
--
1,2 . . . . . N,
where A _> 0. Provided only that wj > O, j = 1,2 . . . . . N, we can choose A such that E{Pj(A)} >_ O, j = 1,2 . . . . . N. Let 7rl, ~r2 be two permutation policies and denote by C. C(-, A) costs computed for/~ and /3(a) respectively. Now
120
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N
j
j=l
k=l
± }
= E{C(Tri,0)} + AE{w~.,(.i) wm(k) , ,/=1 k=l
i= 1,2,
(31)
where the second term in (31 ) is permutation invariant. Hence E {C (7/'1, ,~) -C(q't-2, ~ ) } = E { C ( ~ ' , , O ) - C(¢r2,0)}
(32)
and so the permutation optimising P ( ~ ) optimises/3(0) ~ / 3 also. Hence the modified Smith's rule MSR(p, ~) which orders the jobs according to decreasing values of
wj/E{Pj(,~) }
= wj/ { ( l -
p)-l/2ld,-j + b(p) ZN tzk + Awj ) k=l
optimises /3. We are now ready to give an outline proof of Theorem 6. Outline proof of Theorem 6. Consider the scheduling problem P in which all jobs are subject to an initial delay from time zero of N 2 (b(p)X} +, where u + = max(u, 0), and the processing times are (1 - p)-U2Xi, j = 1,2 . . . . . N. It is plain that the total expected cost for P under some general nonpreemptive policy is at least as great as the expected cost for P under the optimal policy MSR(p, ~). From this it in turn follows that for all ¢r
N3#E [{b(p) X}+] + (1 - p)-U2E{C(zr)} _> E [C {MSR(p, ])}] and hence that
N3#E [{b(p) 2}+] + (1 - p)-'/2E{C(cr*)} > E [C {MSR(p,,~)}].
(33)
We infer from inequality (33) that E{C(SR) - C(~-*)} = E {C(SR) - (1 - p ) U 2 d ( S R ) }
+ ( l - p)t/2E [C(SR) - (~ {MSR(p, A) }] + _< g
[(1 -
{C(SR) -(1-
{MSR(,,,
} -
p)U2O(SR)}
+ (l - p)U2E [C(SR) - d {MSR(p,.~) }] + (l - p)'/~N~®~ [{b(p)~?}] + .
(34)
Direct computation of the costs involved yields E {C(SR) - (1 - p)I/gC(SR)} _< (1 -- p)1/2Nayv{-b(p)}fx.
(35)
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121
Further, we note from (32) that E [C (SR, A) - C {MSR(p, A), A}] = E [C(SR) - C {MSR(p, ~) }].
(36)
We now use a suitable version of Lemma 3, applied to scheduling problem/3 (~), to bound the left hand side of (36). Calculations similar to those in the proof of Lemma 5 yield E [C(SR) - d {MSR(p, A)}] < 2pN3#fz + o(p).
(37)
To obtain inequality (27), combine (34), (35) and (37) with inequality (30). This completes the proof.
5. Computational study For the weighted flow-time cost model there is an attractive dynamic analogue of Smith's rule which modifies the mean processing times of uncompleted jobs by conditioning on the realised values of the processing requirements of completed jobs. More specifically, suppose that this dynamic version of Smith's rule (DSR) z/, say, has already chosen jobs zr' (1), z / ( 2 ) . . . . . ¢rI (j) for processing and that these jobs have been completed with realised processing requirements x~,(1), x~,(2~ . . . . . x~,(.i~. In order to choose the next job for processing, 7rI ( j + 1), we first compute the conditional means Ixk(j,x)=--E(XklX~,(i)=x,~,(O, i = 1 , 2 . . . . . j ) ,
k~z/(i),
i = 1 , 2 . . . . . j.
The DSR rule chooses z / ( j + 1) to satisfy
W~'(j+I )/]-t~Tr'(j+l)(j, x) = mkax{wk/Ixk(j, x) },
(38)
where in (38), the maximisation is over k 4= ~ ( i ) , i = 1,2 . . . . . j. Our computational study of the weighted flow-time problems shows that Smith's rule performs very well. In all our simulation studies, the mean cost incurred by Smith's rule was within 0.5% of that incurred by an optimal policy. Where suboptimalities have occurred, these have almost all been eliminated by the adoption of DSR, the dynamic version of Smith's rule described above. So well did DSR perform that it was a challenge to design a simulation study which demonstrated any suboptimality in it. We did so by generating job characteristics in such a way that the weights wj and marginal means/zj were tied together in a manner likely to induce changes in job ordering (i.e. from the static Smith's rule) when the conditioning leading to (38) was applied. In Table 1 we report one such study whose results were typical. For each problem generated we took N = 5. The size of problem was delimited by the need to compute the optimal policy in each case via dynamic programming. Job weights were generated by sampling independently from the continuous uniform U[20, 30] distribution. Given job weight wj, we obtained/xj by adding to wj the value of a U[0,2] random variable. Hence all values of wj/tzj were in the range [0.9, 1.0]. Covariance matrices for X = ( X 1 , X 2 , X 3 , X 4 , X s ) T were obtained by sampling independently from Wishart distributions. In the usual notation, the distributional assumption was 4 ~ ~-, W5 {4, B ( P ) } ,
(39)
where R(p) denotes the equicorrelation matrix in (1) with o.2 = 1. Note that, while the covariance matrices generated by (39) do not have all correlations equal, we do have E(E) = R(p) and so we are considering random departures from the equicorrelation case. Having generated/z and __X,processing times were obtained by sampling independently from the corresponding multivariate normal distribution.
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Table 1 A computational study of Smith's rule and a dynamic version of it p
E{C(SR)}
E{C(DSR)}
E{C(~-*)}
/~{ZlsR}
~7"[ADSR}
0 0.3 0.5 0.7
9695.6 ( I0.1 ) 9683.4 (10.3) 9694.6 (10.4) 9691.9 (10.5)
9670.7 ( 10.1 ) 9664.6 (10.3) 9680.1 (10.4) 9681.6 (10.5)
9669.8 ( 10.1) 9664.1 (10.3) 9680.0 (10.4) 9681.4 (10.5)
0.270 (0.007) 0.201 (0.005) 0.153 (0.004) 0.109 (0.003)
0.010 (0.004) 0.005 (0.003) 0.001 (0.002) 0.002 (0.001)
In Table 1 below, we report the results for the cases p = 0,0.3,0.5,0.7. For each row of the table, 10000 problems were generated as above and the costs C ( S R ) , C ( D S R ) and C(~-*) incurred by the operation of Smith's rule, the dynamic version of Smith's rule, and an optimal policy were obtained, as were the percentage differences ,asR = l o o { C ( S R )
- C(¢r*)}
~COt*)
and ADSR = 100 { C ( D S R ) - C ( ~ * ) } / C ( ~ r * ) . In the table we have estimates of the means of these quantities, with standard errors in parentheses. We see from the table that Smith's rule performs very well, coming within a fraction of 1% of optimality on average. The dynamic version of Smith's rule eliminates almost all of this suboptimality and is virtually indistinguishable from an optimal policy.
6. Summary We have considered the task of determining the optimal nonpreemptive policy for scheduling a collection of jobs whose processing times follow a normal distribution with equicorrelated components. We show that the permutation policy which orders the jobs in ascending value of their mean processing times is optimal within the class of nonpreemptive policies for general order-specific costs. This policy is also optimal for job-specific costs when the cost function satisfies an agreeability condition. Suboptimality bounds are given for the relative performance of Smith's rule for the weighted flow-time criterion. These bounds explore regions of the parameter space around points where Smith's rule is known to be optimal. Simulation studies have shown that Smith's rule performs well, even in situations which are at some distance from those for which it can be shown to be optimal. A dynamic version eliminates almost all suboptimality in Smith's rule.
Acknowledgements The authors would like to express their gratitude to an anonymous referee for pointing out an error in an earlier version o f the proof of Theorem 1.
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