New NAVAL POSTGRADUATE SCHOOL · 2011. 5. 13. · AD-A273 378 NAVAL POSTGRADUATE SCHOOL Monterey,...
Transcript of New NAVAL POSTGRADUATE SCHOOL · 2011. 5. 13. · AD-A273 378 NAVAL POSTGRADUATE SCHOOL Monterey,...
AD-A273 378
NAVAL POSTGRADUATE SCHOOLMonterey, California
ASYMPTOTIC PROPERTIES OF STOCHASTICGREEDY BIN-PACKING
Donald P. Gaver
93-29562 Patricia A. Jacobs
INovember 1993
Approved for public release; distribution is unlimited.
Prepared for:Naval Postgraduate SchoolMonterey, CA 93943
93 12 3 006
NAVAL POSTGRADUATE SCHOOLMONTEREY, CA 93943-5000
Rear Admiral T. A. Mercer Harrison ShullSuperintendent Provost
This report was prepared in conjunction with research funded by the Naval
Postgraduate School Direct Funded Research Program.
Reproduction of all or part of this report is authorized.
This report was prepared by:
DONALD P. GAVER, JR. PATRICIA A. JA'COBS .Professor of Operations Research Professor of Operations Research
Reviewed by: Released by:
"_ __ __ _ /iPETER PURDUE L- PAUL J. IART,Professor and Chairman Dean of ResearchDepartment of Operations Research
REPORT DOCUMENTATION PAGE o1tW No. 070","Piubc •pwm bunni b l ha mhocwml ci r,*tma• m ea W~mmbd m a~rqsg i lour pp repauN, Pm ImS Brae vi vaV•bru, SW~rn9 oaSWg Oai so*co
oec ci.iftma Inn, urJirg aqpggsu wr 1~wg nxv rs bu w, Nen. ngum i* aquws Serv.i)cs ffUWW:i"00W 2&1ma WSVPcv wcrwr ni$~ 21Dea igs~~ouy.Suim 1204 MngS~nVA 2202-432 w'4D stoOF of woemne'1 -d Budge Paoe'w>*POA' eUcn P o(0704-0188)Wash' 511 DC 20603
1. AGENCY USE ONLY (Lewe bionk) 2. REPORT DATE 3. REPORT TYPE AND DATES COVERED
November 1993 Technical Report
4. TITLE AND SUBTITLE S. FUNDING NUMBERS
Asymptotic Properties of Stochastic Greedy Bin-Packing ORGVi
6. AUTHOR(S) 3153
Donald P. Gaver and Patricia A. Jacobs
7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 8. PERFORMING ORGANIZATIONREPORT NUMBER
Naval Postgraduate School NPS-OR-93-017Monterey, CA 93943
9. SPONSORING / MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSORING / MONITORINGAGENCY REPORT NUMBER
Naval Postgraduate SchoolMonterey, CA 93943
11. SUPPLEMENTARY NOTES
1 2a. DISTRIBUTION / AVAILABILITY STATEMENT 12b. DISTRIBUTION CODE
Approved for public release, distribution is unlimited.
13. ABSTRACT (Maximum ZOO words)
An adaptive or greedy policy for packing I bins, or equivalently for scheduling jobs for theattention of a number, I, of processors is studied. It is shown that the suitably normalized bincontents become nearly jointly but degenerately Gaussian/normal if the rate of approach ofjobs becomes large. Explicit and simple parameter characterizations are supplied and theasymptotics are compared with simulation. The advantage of the greedy policy over a laissez-faire policy of equal access is quantified, and seen to depend upon
`number of bins or processors.
14. SUBJECT TERMS 15. NUMBER OF PAGES
greedy algorithm, bin packing, job scheduling, asymptotic results, 27Ornstein-Uhlenbeck process 16. PRICE CODE
17. SECURITY CLASSIFICATION 18. SECURITY CLASSIFICATION 19. SECURITY CLASSIFICATION 20. LIMITATION OF ABSTRACTOF REPORT OF THIS PAGE OF ABSTRACT
Unclassified Unclassified Unclassified ULNSN 7540-01-280-5500 Standard Form 298 (Rev. 2-89)
Prescribed by ANSI Std. 239ý-18
Accesion For
NTcS 2.
0TFl .
ASYMPTOTIC PROPERTIES OF
STOCHASTIC GREEDY BIN-PACKING
D. P. Gaver -P. A. Jacobs
flTIC QUALr13i 7 T~'~ 7
Abstract _
An adaptive or greedy policy for packing I bins, or equivalentlyfor scheduling jobs for the attention of a number, I, of processors isstudied. It is shown that the suitably normalized bin contentsbecome nearly jointly but degenerately Gaussian/normal if the rateof approach of jobs becomes large. Explicit and simple parametercharacterizations are supplied and the asymptotics are comparedwith simulation. The advantage of the greedy policy over a laissez-faire policy of equal access is quantified, and seen to depend upon
* number of bins or processors.
Introduction
We study an adaptive or greedy policy for packing I bins, or equivalently for
scheduling jobs for the attention of a number, I, of processors. The connection
between bin-packing and makespan scheduling is well described in Coffman and
Lueker [1], chapter 1.
It is shown that the suitably normalized bin contents become nearly jointly
but degenerately Gaussian/normal if the rate of approach of jobs becomes large.
Explicit and simple parameter characterizations are supplied, and the asymp-
totics are compared with simulation. The advantage of the greedy policy over a
laissez-faire policy of equal access is quantified, and seen to depend upon
,number of bins or processors.
Greedy Bin-Filling
There are I bins (potential servers). Jobs arrive at the bin system according to
a homogeneous Poisson process, rate A. Each job size is independently and
identically distributed according to FB(r); a generic job size is B, a real positive
random variable. Let Nj(t) denote amount of work in bin i at time t; this is the sum
of the job sizes deposited in that bin up to time t. We refer to Ni(t) as the bin size
at time t hereafter, although other terminology is used in the scheduling
literature.
Consider Policy E (Equalization): When a new job arrives it is deposited in the
bin with smallest amount of accumulated work. N(t) = (Ni(t), t > 0, i E (1, 2, ... ,/ ))
is a Markov process with the following generator; for A > 0
Ni(t + A) = Ni(t) with probability 1 - ý`pi(Ni(t), N(t)) + 0(,6)
= Ni(t) + B(t) with probability -•pi(Ni(t), N(t)) + o(A) (0,
Define
pi(N i(t),N (t)) = -if N i(t):•N )(t) 'J ,
I0 otherwise
Consider also Probabilistic Policy E: same as above but
h(Ni (t))
j=1
where h(K) is a positive homogeneous function chosen to become large when Ni(t)
is small; e.g. h(x) = xP. Such a policy leads to workload growth very similar to
Policy E's, but can be analyzed more readily. See Gaver, Morrison and Silveira [2]
for application of such a probabilistic policy in a service-system scheduling
context. In the present context, the bins can be buffers containing jobs to be
2
processed later and the aim is to keep the total processing time short; it is
(optimistically) assumed that the contents of each bin is known at all times to the
scheduler and that each processing time is known when the job or task appears.
Joint Moment-Generating Function
Let the moment-generating function (assumed to exist, otherwise use the
characteristic function) be
,(o, t)=E[e• N/t)]- = Exp eN, (t) (3)
Condition on N1(t), i E (1, 2, ... ,I) and use the generator to obtain
E expX 6^N (t + A)', N(t), B(t)]
=[1-LXý]exp Y.ONj(t) +
+).AJ exp OI((N,(+B(t))+ XekNk(t)jPj(Nj(t);N(t)) +o(A).j=1L , k•Ij
Remove conditions, defining the m g f of the task size arriving at t, B(t), to be
b(O), to findI
v(e. t + A) = (1 - ).A)v(e,t)± + .A, b(e,)E[exp(61N1 (t)) -pj(N)(t); N(t))] + o(A).j= l
Let A -+ 0 to get
dt _ AV(9, t) + A. Xi(0,)E[exp(8,Nj(t)) pj(Nj(t);N(t))]. (4)dt
j=1
Note that nothing that follows prevents non-stationary input rates: i.g. A = Ar(t),
and the mgf of B(t) to be b(e;t) provided these do not drop quickly to zero.
3
Scaling
Put
X,(t) = N,(t)-A/3,j(t)(5
and let A - 1. It is anticipated that with suitable choice of the functions (f/3(t))
{Xj(t), j r (1, 2, M., ) should become a Gaussian process as A -~c*Let
(d)Iq,(6, t;;)= Eexp~ eX,(t)~J for Xj(t) defined as above. Note that
j==1
= V(e,t;AL)exp(Vý-T9O(t)).
Now since from (4)
dv(6//Xt )L(e / ...I,t) + ;LI a e / -IT)E ee~/~7() pj(Nj(t);N(t))]dt =
substitution of (5) yields
div(8 / Vk t 0(e,t;A)exp(N'f7e/(t))] =dp extA)p(NIF ep3(dtdt dt e )
+,(O t; A)193(~x(~Pt)
-A 08~, t; A ) exp(IXG/30(t)) +
+A b(j/ -Jr)E[exp(ex(t)). exp(1ý-eP(t)) Pj (Ljt)+ VT X~t); Ap + .,;L-x)].
4
Cancel exp(1A-eI3(t)) to obtain
d~No, t;A) + 9=,t -A(P(8, t; A)
dt
+A b~(e 1 i/ -rX)E[exp(OX(t)).- pj (.Af3 (t) + \f-Jx, (t); Afl + \fA-X)]. (7
Let
pj(j~tNt)) h(NjAO) I h(AI3A~t)+, -.KXj(t))
~hk (Nk (0) Xh(fi~k (t)+ iXXk (0)k=1 k=1
where h(.) is homogeneous: h(Ax) = APh~x). Consequently
pj;p~)+ xjt;,~ + ;,Xt)= 1h(~() ~t/~~ (8)
Xh(fOk(t) + Xk(t)/ rk)
OW %) assumed to be differentiable). Expand in inverse powers of -IL:
pj(Aflj(t)+ "ThX1i(t);kL3(t) + VX--X(t))=
k=1 k=1 kk1
Since
Pj A ij t)+ ýIAXj (t); A 0 (t) + N/lAX (t))=j=1
I= h(11 (t))
k=1 (9,a)
5
this implies that the summed coefficients of 1/ ,1/3, etc. must be indi-
vidually zero; this is easily verified for the coefficient of I / I-X.
Asymptotic Expansion For p(e, t;a.).
Put
,p(O,t;a )= (et)+ q),CO, t)+-192(et)+....
This can now be entered into (7) and evaluated by means of (9):
d9 1 1
__dt =eo
*0 )k) Ca' vi Xh(flk(t)j
k=W (10)
1 h'(Pj3(t) d~, 1 h(131(t)) 'c +~,[ , 2 h'(fl,(t)).- +oýk~~* k=11
In the above bk = E[Bk I, the kth moment of job size; of course bo = 1.
Now identify the coefficients of inverse powers of "1X, and thereby equations for
47, and 631 . From (10) for e = 0,
6
j=1 k=h(j() h(j() dgo h 2P (ht))t)-• PO(
Xh(3k(0)) j h(/3k(O) [h(Pk (t))]dk)k=1 k=1 =1
The terms of order A cancel from the r h s. The terms of order v on I h s and
r h s cancel if
efl(t)gPO b], e! "go"I h(1A (t))
= 1 Gk
In order for this to occur,
dfj _ b h(pj (t))d--t-•,~l~)"j=1 2,2...,.) (12)dt hA(0
k
the solution of which determines /At).
Next look for terms of order 1. The 1 h s provides ---.. The r h s provides thedt
t1 2 h(13j(t))terms ~o •b2Ij=1 2 (Ok(0)
k
((t)) 3go0 h(p 1(t)) IOand b, 0 h(fk(t) OJ 2 h( k (0) ) Ok
j= X(k h()k (0)) k.•1
Note that the condition (12) actually annihilates any term of order I (or higher) in
(pt for I = 1, 2, ... on the r h s, and the discussion of (9, a) shows that there is no
contribution from dgj~oi/j. Consequently
7
1
dt 2
j1 k1
Xhfk t) dO1 l
( h( k (13)
rb I I_7- Ze'pj(Pj(0t))(PO(e,t)+b1Y! d - Hjk(t)2-TOO't)
proess Simla eq=ton do ihrodrcrelton ka derie imlry
d91k
Note: Y HjtdPL _Ho
i=1 (P I (J)__ k k dk(f)k
The PIDE for pj(69,t is recognizable as that of an Ornstein -Lhlenbeck (Gaussian)
process. Similar equations for higher-order corrections can be derived similarly,
but we omit this step.
MOMENTS
The (0:h order) joint moments of X(t) = (Xj(t), i = 1, 2, ... , 1) satisfy ordinary
differential equations that are readily obtained by differentiation of (13) at 0 = 0.
If V/) = E[X?(t)], V11(t) = E[Xi (t)X (0)] for i then
•YL = b2P1 (fi(t)) +2b1 [H3 ()V2 (t)- Hik(lVik (t0dtLI
=j§ l Fi(H)+ H,(t)) Vij(t)- -'Hjk(0)VikWt)+ HikWtVjk(t0l
The above must be tailored as follows:
8
d Vi M)-b2Pi(flMt)+2bj (Hi(t)-Hii(t))Vi(t)-IHik(t)]'Iik(t)ld
dt k(,i (14)
Ht)+ H I (t))v', (t) -(H11 vl Mt + H 11V1 (t))d (t)Vik(t)+ 2.,Hik(t)Vjk(t)
(k~i)H (ksj)
Now return to specifics; consider (12): for j k
d/31(t) _dflk(t)
h(/3,(t)) h(1k(t))
if the bins are filled as suggested. This implies that
-- allj.E (1,2,...,I) (15)
and p,(/31(t)) = 1/I.
For the casc in which h(x) = xP it can be seen that
HiM -- , H,1 Mt = bP.- (16)b1t I H,,t) -
Substitute into (14) to obtain these equations for Vi(t) V(t), (Vi);
Vii(t) = W(t), (Vi# j)
d W b2 2p Idt - )(V - W) (17,a)
dI•.yV = 2p (V-W) (1 7,b)
dt -It
from which an equation for Z(t) = V(t) - W(t) emerges:
dZ + 2- Z(t) = 2 (18)dt t I
A solution to (18) over t = (L,-), L > 0 is of this form:
Z(t) = b2 t + K(L)I(l + 2p) t2-p L<t
9
From this and (17,b)dW(t) .- p Z(t)
dt It
we get
W(t)= 2p 2t+t K(L)1 - 1 (19)1+2p 12 2p •t•2 P LP L<t
and from (17,a)
dt I I
V(t)=22 +K(L) - (22+)2p+(2
1 22p+ 1 1 7}+ I /k L?(20
Now if p >>1 it is seen that
V(t) = W(t) = b2t (21)
Parenthetically, the comparable figures for independently filled bins are
V(t)= b2t and W(t) = 0. (22)I
From (15) and (20)
E[Ni(t)] = At + O(I)
Var[Ni(t)]= Cov[Nj(t),Nj(t)]= t b2 + O(1ý)L)1 1 2 (23)
The singular behavior of W(t) and V(t) for small t, as in (19) and (20), can be
attributed to the indeterminacy of the bin selection probability, (2), for Nj(0) = 0,
Vi, which was the assumed initial condition. The long-time behavior of greedy
packing, expressed by (23), is of interest: since Var[Ni(t)] = Cov[Ni(t), Nj(t)] all bin
contents are essentially perfectly correlated at any time. Consequently, to order
-F- the maximum bin contents Nm(t), are approximately normal/Gaussian with
mean Atbj/I and standard deviation A / I. If bins are filled independently
the mean is the same but now Nm(t) is distributed approximately as the
10
maximum of I independent normals, each with standard deviation A / N-' -
considerably larger in a probabilistic sense. Note that putting p = 0 in (19), (20)
yields the independent result; putting p finite yields other probabilistic options.
In the scheduling context the makespan, i.e. time to complete all tasks present at
time t, is substantially reduced by the current greedy scheme, which is equivalent
to what Coffman et al. (1991) call list scheduling (LS); our approach is on-line Est
scheduling, meaning that tasks are assigned to processors sequentially as they
appear in time. It should be pointed out that the moments obtained above can
also be derived directly from (1) and (2) by expansion of (5), rather as suggested
by Isham [3].
Simulations
Limited informal simulations were conducted in order to check the accuracy
of the proposed asymptotic approximations. The simulations were written in
APL2 and conducted on an AMDAHL 5995-700A at the Naval Postgraduate
School using the LLRANDOM random number generator; cf. Lewis et al. [4]. All
simulations were run for time t = I at the indicated A-values for two job size
distributions, both gamma: the exponential and an extended-tail highly-skewed
gamma with shape parameter one-half.
Examination of Table 1 indicates that agreement is good between the
asymptotic approximation and simulation results (based on 1000 replications) for
the marginal distributional properties of an arbitrary bin when the greedy policy
is followed. As anticipated, a considerable reduction in the variance of bin size,
and also of upper-tail percentiles, is achieved by greediness, as contrasted to a
simple random assignment.
The figures of Table 2, which describe the approximation to the maximum bin
size, or makespan in a scheduling context, are serviceable but tend to be low or
11
optimistic, especially for the smaller A-value of 50. For ;. = 300 the agreement is
better and correctly predicts the substantial reduction of mean, variance, and
upper percentiles achieved by the greedy policy. Note that numerical agreement
between simulation and our asymptotics should improve if the job sizes have
smaller variances and third moments.
It can be conjectured, and demonstrated, that a cyclic or round-robin policy of
putting every Ph arrival in the same bin will tend to reduce within-bin variance
and makespan levels. It may be advantageous that both random and round-robin
policies require no information concerning current bin size or occupancy at the
time an assignment must be made, whereas the greedy policy and others depend
on precise distributional forms do require such information. If reduction of bin
size or makespan variation is important the acquisition of the information
needed to implement a greedy policy may be well worth the cost.
Acknowledgments.
The writers are grateful to Paul Wright and Ed Coffman of AT&T Bell
Laboratories, Murray Hill, for acquainting us with the greedy algorithm
evaluation problem.
12
Table 1. ARBITRARY BIN SIZE
I=5B: G = 1: Gamma, Shape Param. = 1 (Exponential)
G = 0.5: Gamma, Shape Param. = 0.5; E[B] = 1.Number Replications = 1000
PolicyGreedy Random
Quantiles QuantilesMean Job Mean Var 80% 90% Mean Var 80% 90%
Demand(A) Size (B)
50 G = 1 App: 10.0 4.0 11.7 12.6 10.0 20.0 13.8 15.7
Sim: 10.1 5.0 11.9 13.0 10.0 19.6 13.7 16.1
G =0.5 App: 10.0 6.0 12.3 13.1 10.0 30.0 14.6 17.0
Sim: 10.0 8.5 12.1 13.8 10.2 32.6 14.8 18.0
300 G = I App: 60.0 24.0 64.1 66.3 60.0 120.0 69.2 74.0
Sim: 60.2 24.1 64.4 66.4 60.4 112.1 69.5 73.9
G = 0.5 App: 60.0 36.0 65.0 67.7 60.0 180.0 71.2 77.1
Sim: 60.2 38.9 65.4 68.0 60.0 172.7 70.7 77.1
13
Table 2. MAXIMUM BIN SIZE
1=5B: G = 1: Gamma, Shape Param. = I (Exponential)
G = 0.5: Gamma, Shape Param. = 0.5; E[B] = 1.Number Replications = 1000
PolicyGreedy Random
Quantiles Quantiles
Mean Job Mean Var 80% 90% Mean Var 80% 90%Demand(A) Size (B)
50 G = 1 App: 10.0 4.0 11.7 12.6 - - 17.6 19.1
Sim: 11.2 5.2 13.1 14.2 15.6 15.1 18.7 23.9
G = 0.5 App: 10.0 6.0 12.1 13.1 - - 19.4 21.2
Sim: 12.2 10.7 14.8 16.5 16.9 28.2 20.7 23.9
300 G = 1 App: 60.0 24.0 64.1 66.3 - - 78.7 82.3
Sim: 61.5 24.3 65.6 67.7 73.3 61.5 79.5 83.5
G = 0.5 App: 60.0 36.0 65.0 67.7 - - 82.9 87.3
Sim: 62.3 40.0 67.7 70.3 76.2 107.6 84.3 90.4
-: Not convenient to compute
14
REFERENCES
[11 Coffman Jr., E. G. and Lueker, G. S. (1991). Probabilistic Analysis of Packingand Partitioning Algorithms. Wiley-Intersc~ence. John Wiley and Sons, Inc.New York.
[21 Gayer, D. P., Morrison, J. A., and Silveira, R. (1993). Service-adaptive multityperepairman problems. SIAM J. App!. Math., Vol. 53, No. 2, pp. 459-470.
[3] Isham, V. (1991). Assessing the variability of stochastic epidemics. MathematicalBiosciences. Vol. 107, pp. 209-224.
[4] Lewis, P. A. W. and Uribe, L. (1981). "The New Naval Postgraduate SchoolRandom number generator - LLRA.NDOMIIL" Naval Postgraduate SchoolTechnical Report NI'S 55-81-005, Monterey, CA.
15
INITIAL DISTRIBUTION LIST
1. Library (C ode 052) ........................................................................................... 2Naval Postgraduate SchoolMonterey, CA 93943-5000
2. Defense Technical Information Center ....................................................... 2Cameron StationAlexandria, VA 22314
3. Research O ffice (Code 08) .............................................................................. 1Naval Postgraduate SchoolMonterey, CA 93943-5000
4. Prof. Peter P urdue ................................................................................................. 1Code OR-PdNaval Postgraduate SchoolMonterey, CA 93943-5000
5. P rof. M ich ael B ailey .............................................................................................. 1Code OR-BaNaval Postgraduate SchoolMonterey, CA 93943-5000
6. Department of Operations Research (Code OR) ........................................ 1Naval Postgraduate SchoolMonterey, CA 93943-5000
7. Prof. Donald Gaver, Code OR-Gv ............................................................... 10Naval Postgraduate SchoolMonterey, CA 93943-5000
8. Prof. Patricia Jacobs ....................................................................................... 10Code OR/JcNaval Postgraduate SchoolMonterey, CA 93943-5000
9. Center for N aval A nalyses .................................................................................. 14401 Ford AvenueAlexandria, VA 22302-0268
16
10. Dr. J. Abrahams, Code 1111, Room 607 ..................................................... 1Mathematical Sciences Division, Office of Naval Research800 North Quincy StreetArlington, VA 22217-5000
11. D r. D avid Brillinger .............................................................................................. 1Statistics DepartmentUniversity of CaliforniaBerkeley, CA 94720
12. D r. D avid Burm an .......................................................................................... IAT&T Bell Telephone Laboratories600 Mountain AvenueMurray Hill, NJ 07974
13. Prof. H . C hernoff ............................................................................................. IDepartment of StatisticsHarvard University1 Oxford StreetCambridge, MA 02138
14. D r. Edw ard G . Coffm an, Jr ................................................................................... 1AT&T Bell Telephone Laboratories600 Mountain AvenueMurray Hill, NJ 07974
15. Professor Sir D avid C ox ....................................................................................... 1Nuffield CollegeOxford, OX! INFENGLAND
16. Professor H . G . D aellenbach ................................................................................ 1Department of Operations ResearchUniversity of CanterburyChristchurch, NEW ZEALAND
17. D r. S. R . D alal ......................................................................................................... 1Bellcore445 South StreetMorristown, NJ 07962-1910
17
18. D r. D . F. D aley ................................................................................................. 1Statistic Dept. (I.A.S.)Australian National UniversityCanberra, A.C.T. 2606AUSTRALIA
19. D r. B. D oshi ...................................................................................................... IAT&T Bell LaboratoriesHO 3M-335Holmdel, NJ 07733
20. Prof. Bradley Efron ................................................................................................ IStatistics Dept.Sequoia HallStanford UniversityStanford, CA 94305
21. D r. G u y Fayolle ...................................................................................................... 1I.N.R.I.A.Dom de Voluceau-Rocquencourt78150 Le Chesnay CedexFRANCE
22. D r. M . J. Fischer ............................................................................................... 5Defense Communications Agency1860 Wiehle AvenueReston, VA 22070
23. Prof. G eorge S. Fishm an ...................................................................................... ICurr. in OR & Systems AnalysisUniversity of North CarolinaChapel Hill, NC 20742
24. D r. N eil G err .......................................................................................................... IOffice of Naval ResearchArlington, VA 22217
25. D r. R. J. G ibbens ..................................................................................................... IStatistics Laboratory16 Mill LaneCambridgeENGLAND
18
26. P rof. Peter G lynn ................................................................................................... 1Dept. of Operations ResearchStanford UniversityStanford, CA 94350
27. Prof. Linda V . G reen ............................................................................................. 1Graduate School of BusinessColumbia UniversityNew York, NY 10027
28. D r. Shlom o H alfin ................................................................................................ 1Bellcore,445 South StreetMorristown, NJ 07962-1910(MRE 2L309)
29. Prof. Bernard H arris ............................................................................................. 1Dept. of StatisticsUniversity of Wisconsin610 Walnut StreetMadison, WI 53706
30. D r. P . H eidelberger ................................................................................................ 1IBM Research LaboratoryYorktown HeightsNew York, NY 10598
31. Prof. J. M ichael H arrison ..................................................................................... 1Graduate School of BusinessStanford UniversityStanford, CA 94305-5015
32. A rthur P. H urter, Jr ....................................................................................... 1Professor and ChairmanDept. of Industrial Engineering and Management SciencesNorthwestern UniversityEvanston, IL 60201-9990
33. Prof. D . L. Iglehart .......................................................................................... 1Dept. of Operations ResearchStanford UniversityStanford, CA 94350
19
34. Institute for D efense A nalysis ............................................................................ 11800 North BeauregardAlexandria, VA 22311
35. P rof. J. B . K ad ane ................................................................................................... 1Dept. of StatisticsCarnegie-Mellon UniversityPittsburgh, PA 15213
36. D r. F . P . K elly .......................................................................................................... 1Statistics Laboratory16 Mill LaneCambridgeENGLAND
37. D r. Jon K ettenring ............................................................................................ IBellcore445 South StreetMorris Township, NJ 07962-1910
38. K oh P eng K ong ...................................................................................................... 1OA Branch, DSOMinistry of DefenseBlk 29 Middlesex RoadSINGAPORE 1024
39. D r. Prabha K um ar ................................................................................................. 1Defense Communications Agency1860 Wiehle AvenueReston, VA 22070
40. D r. A . J. Law rence ............................................................................................. 1Dept. of Mathematics,University of BirminghamP. O. Box 363Birmingham B15 2TTENGLAND
41. Prof. M . Leadbetter ................................................................................................ 1Department of StatisticsUniversity of North CarolinaChapel Hill, NC 27514
20
42. P rof. J. L eh o czky .................................................................................................... 1Department of StatisticsCarnegie-Mellon UniversityPittsburgh, PA 15213
43. D r. D . M . Lucantoni ........................................................................................ 1AT&T Bell LaboratoriesHolmdel, NJ 07733
44. D r. C olin M allow s ................................................................................................. 1AT&T Bell Telephone Laboratories600 Mountain AvenueMurray Hill, NJ 07974
45. Prof. R . D ouglas M artin ....................................................................................... 1Department of Statistics, GN-22University of WashingtonSeattle, WA 98195
46. Prof. M . M azum dar ....................................................................................... 1Dept. of Industrial EngineeringUniversity of PittsburghPittsburgh, PA 15235
47. D r. Jam es M cK enna ....................................................................................... IBell Communications Research445 South StreetMorristown, NJ 07960-1910
48. Operations Research Center, Rm E40-164 ..................................................... 1Massachusetts Institute of TechnologyAttn: R. C. Larson and J. F. ShapiroCambridge, MA 02139
49. D r. John O rav ......................................................................................................... 1Biostatistics DepartmentHarvard School of Public Health677 Huntington Ave.Boston, MA 02115
21
m mm m mm m mmm mm mo m o
50 . D r. T . J. O tt .............................................................................................................. IBellcore,445 South StreetMorristown, NJ 07962-1910(MRE 2P388)
51. Prof. Paul M oose ................................................................................................... IC31 Academic GroupNaval Postgraduate SchoolMonterey, CA 93943-5000
52. D r. John A . M orrison ........................................................................................... IAT&T Bell Telephone Laboratories600 Mountain AvenueMurray Hill, NJ 07974
53. D r. V . R am asw am i ........................................................................................... 1MRE 2Q-358Bell Communications Research, Inc.445 South StreetMorristown, NJ 07960
54. D r. M artin R eim an ........................................................................................... 1Rm #2C-117AT&T Bell labs600 Mountain Ave.Murray Hill, NJ 07974-2040
55. Prof. M aria R ied ers ...............................................................................................Dept. of Industrial Eng.Northwestern Univ.Evanston, IL 60208
56. D r. Joh n E . R olph ..................................................................................................RAND Corporation1700 Main St.Santa Monica, CA 90406
57. Prof. M . Rosenblatt ...............................................................................................Department of MathematicsUniversity of California, San DiegoLa Jolla, CA 92093
22
58. P rof. Frank Sam aniego ........................................................................................ 1Statistics DepartmentUniversity of CaliforniaDavis, CA 95616
59. Prof. W . R . Schucany ............................................................................................ 1Dept. of StatisticsSouthern Methodist UniversityDallas, TX 75222
60. D r. R honda R ighter .............................................................................................. 1Dept. of Decision & Info. SciencesSanta Clara UniversitySanta Clara, CA 95118
61. Prof. G . Shantikum ar ..................................................................................... 1The Management Science GroupSchool of Business AdministrationUniversity of CaliforniaBerkeley, CA 94720
62. P rof. D . C . Siegm und ............................................................................................ 1Dept. of StatisticsSequoia HallStanford UniversityStanford, CA 94305
63. Prof. N . D . Singpurw alla ................................................................................ 1George Washington UniversityWashington, DC 20052
64. P rof. H . Solom on .................................................................................................. 1Department of StatisticsSequoia HallStanford UniversityStanford, CA 94305
65. Prof. J. R . T hom pson ............................................................................................ 1Dept. of Mathematical ScienceRice UniversityHouston, TX 77001
23
66. P rof. J. W . T ukey ................................................................................................... 1Statistics Dept., Fine HallPrinceton UniversityPrinceton, NJ 08540
67. D r. D . V ere-Jones ................................................................................................... 1Dept. of Math, Victoria Univ. of WellingtonP. 0. Box 196WellingtonNEW ZEALAND
68. Dr. Daniel H. Wagner ..................................................................................... 1Station Square OnePaoli, PA 19301
69. D r. E d W egm an ..................................................................................................... 1George Mason UniversityFairfax, VA 22030
70. D r. A lan W eiss ................................................................................................. 1Rm. 2C-118AT&T Bell Laboratories600 Mountain AvenueMurray Hill, NJ 07974-2040
7 1. D r. P . W elch ........................................................................................................... 1IBM Research LaboratoryYorktown Heights, NY 10598
72. P rof. R oy W elsch ................................................................................................... 1Sloan SchoolM.I.T.Cambridge, MA 02139
24