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

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NAVAL POSTGRADUATE SCHOOLMONTEREY, CA 93943-5000

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This report was prepared in conjunction with research funded by the Naval

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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

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November 1993 Technical Report

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Asymptotic Properties of Stochastic Greedy Bin-Packing ORGVi

6. AUTHOR(S) 3153

Donald P. Gaver and Patricia A. Jacobs

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Naval Postgraduate School NPS-OR-93-017Monterey, CA 93943

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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

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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

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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

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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

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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

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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

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