On Models for Object Lifetime Distributions

Transcription

1 On Models for Objet Lifetime Distributions Darko Stefanović Department of Eletrial Engineering Prineton University Prineton, NJ 8544 Kathryn S. MKinley Department of Computer Siene University of Massahusetts Amherst, MA 3 mkinley@s.umass.edu J. Eliot B. Moss Department of Computer Siene University of Massahusetts Amherst, MA 3 moss@s.umass.edu ABSTRACT Analytial models of memory objet lifetimes are appealing beause having them would enable mathematial analysis or fast simulation of the memory management behavior of programs. In this paper, we investigate models for objet lifetimes drawn from programs in objetoriented languages suh as Java and Smalltalk. We present ertain postulated analytial models and ompare them with observed lifetimes for 58 programs. We find that observed lifetime distributions do not math previously proposed objet lifetime models, but do agree in salient shape harateristis with the gamma distribution family used in statistial survival analysis for general populations. Categories and Subjet Desriptors D.3.4 [Programming Languages]: Proessors memory management (garbage olletion); G.3 [Mathematis of Computing]: Probability and Statistis survival analysis General Terms Measurement Keywords Objet lifetimes, lifetime distributions, garbage olletion modelling. INTRODUCTION If we an develop aurate analytial models for objet lifetimes in objet-oriented programs, they will enable faster and more thorough eploration of memory management tehniques. For instane, given a model of objet lifetimes, we ould ompute an estimate of opying osts of a generational or some other garbage olletor. If distribution models and garbage olletor models are simple enough, we may even arrive at losed-form analytial desriptions; but even if both are quite ompliated, we an use the lifetime distributions to drive simulations of a proposed garbage olletor sheme. Lifetime models are not suffiient for eploring garbage olletion, beause they do not aount for heap pointer struture effets: the diret ost of pointer maintenane (inluding write barriers), and the opying ost inrease owing to the eess retention of objets, both osts being present in generational and other heap-partitioning shemes. Nevertheless, the models an be useful as a tool for preliminary evaluation (and understanding) of olletor performane. Observed objet lifetime behaviors are inherently disrete; we measure the lifetime of eah objet and arrive at a disrete distribution. Most performane-related propositions have dealt with mortality, whih is a derivative form (in the sense of the alulus of funtions ); sine obtaining a derivative of a disrete observed funtion involves inherently arbitrary smoothing deisions, it has been diffiult to haraterize the mortality of observed distributions, let alone to math it to an analytial model. The etremely fast deay of objets eaerbates the situation: most models developed in other domains are for muh more slowly deaying populations. If we knew whih distribution family desribes typial objet behaviors, we ould fit observed lifetimes to the model of that family, and find the best mathing instane (i.e., its parameters). In fat, a running program ould reognize the lifetime distribution of objets alloated (overall, or at a partiular alloation site) and adjust olletion poliies aordingly. But it is not yet known whih family this may be. In the following, we first briefly introdue terms and notation related to lifetime distributions (with more details in the Appendi), then review what assumptions have been made impliitly (or stated epliitly) in past researh in garbage olletion. We develop models based on a plausible qualitative haraterization of lifetimes; namely, that past lifetime is a strong preditor of future lifetime. Lastly, we put the models to the test of empirial evidene against atual lifetime distributions from objet-oriented programs, using a graphial devie reommended in statistial survival analysis. We shall find that these partiular analytial models are not a good math for atual distributions. In searh of a good math, we tentatively onsider several well-known distributions families, but must onlude that none is ompletely satisfatory: objets in programs are a muh different population from those statistiians have eamined. This indiates on the one hand the need for further modelling effort to ahieve good mathes that an be validated against eperiment, and on the other, the need to derive suh models from first priniples, viz. from program semantis. 2. BACKGROUND MATERIAL The lifetime of an objet is defined as the amount of alloation that ours between the alloation of the objet and its demise. We view The atual point of demise depends on the auray of the memory management sheme. In the empirial data reported here, that sheme is an aurate-roots garbage olletor performing full-heap olletion

2 objet lifetime as a random variable. Future studies may look at objet lifetimes as stohasti proesses, and in this ontet, distinguish eah alloation site as generating a different proess. Here, we do not attempt suh fine distintions. Atual objet lifetimes are natural numbers, thus disrete probability distributions are the obvious representation. However, ontinuous models are used for mathematial ease and onveniene. Below we review some definitions and symbols from probability theory as they apply to survival analysis; further details appear in the Appendi along with a summary of properties of ommonly used analytial distribution families. Although we shall write more formulae than are typially seen in garbage olletion literature, the mathematial apparatus is elementary. The survival funtion of a random variable L is s L t L t. For objet lifetimes, it epresses what fration of original alloation volume is still live at age t. We usually drop the subsript L. The survivor funtion is a monotone non-inreasing funtion. The probability t. Oasionally we density funtion for objet lifetime is f t s also use the umulative distribution funtion F t s t. The mortality funtion is m t f t s t d dt logs t, and it epresses the age-speifi death rate. Mortality is also known as the hazard funtion (and written h t ) in the literature on lifetime analysis [4]. 3. STATEMENTS ABOUT DISTRIBUTIONS Objet lifetime distributions have been of interest to researhers of garbage olletion, espeially generational olletion: the suess of a partiular garbage olletor organization or promotion poliy depends on how well it is mathed to the behavior of typial user programs. In fat, laims have been made about lifetimes. Hayes introdued a distintion between a weak and a strong generational hypothesis [8]. Our understanding of his statement of the weak generational hypothesis is this: newly reated objets have a muh higher mortality than objets that are older. His statement of the strong generational hypothesis (whih he in fat introdues) is that even if the objets in question are not newly reated, the relatively younger objets have a higher mortality than the relatively older objets, or simply, that m t is an everywhere dereasing funtion. Baker learly pointed out that an eponential distribution of lifetimes, with m t onstant, annot be favorable to generational olletion (as opposed to whole-heap olletion), and that instead m t should be dereasing []. Nevertheless, the eponential distribution has a unique ahet among survival distributions: its mathematial simpliity and the property of lak of memory. In a garbage olletor this property assures that an objet just disovered live by the olletor has the same residual lifetime as the lifetime of a newly alloated objet, and this greatly simplifies the analysis. Thus, the eponential distribution was used by Clinger and Hansen in the analysis (and to inspire the design) of a non-preditive olletor, outside the generational realm [3]. In our eamination of a generalized form of that olletor [3], we deided to use not only the eponential distribution s t e ρt, but also a variation with dereasing mortality s t e ρt as being in agreement with the strong generational hypothesis, as well as a varia- for ontrol. In fat, these 2 tion with inreasing mortality s t e ρt three are instanes of the Weibull distribution family s t [4, ]. e ρt at eah objet alloation. Thus, demise is deteted preisely at the point when the objet beomes unreahable from the global roots. 4. PAST-IS-FUTURE MODELS A multitude of models an be developed that have dereasing mortality. But developing them e vauo, just for the simpliity of their mathematial formulation (or their use in other domains) is not satisfatory. We an base models on a broad eperimental study, and in Setion 5 we make a first attempt at that. But, our understanding would be aided more if distribution models ould be derived from ertain priniples that we epet to be naturally assoiated with program behavior. In this spirit, Appel suggested (in a personal ommuniation to us) that plausible objet lifetime distributions should satisfy the following property: () An objet s future epeted lifetime is proportional to its urrent age. Thus, past lifetime is a strong preditor of the future (residual) lifetime. This stands in stark ontrast to the eponential distribution. An objet s future epeted lifetime C is the differene between its epeted lifetime and its urrent age. The epeted lifetime (one we know the urrent age) is the onditional epeted value [2] of the lifetime random variable L, E L L, where is the urrent age. It is alulated as: E L L t f t t dt t f t dt f t dt t f t dt s Thus C EL L!, and statement () is: #" &% ψ $ C ψ with ψ a proportionality onstant between the urrent age and the, we have future epeted lifetime C. Unfortunately, with E L( E L L ) C ψ * for any ψ. We look at two ways out of the quandary: first, by letting proportionality () hold only in the limit as ; and seond, by restriting the domain of definition of the distribution to an interval. We shall find use for an alternative form of (): Let G ) t f t dt; then E L L ( G s. Let Λ + G s E, L- L. / C C 2. Then statement () is #" % ψ $ with ψ4 ψ 2. Λ 3 ψ 4. Past-is-future in the limit Let us weaken the statement () so that the proportionality holds asymptotially, for large values of. (A similar analysis was outlined previously by Pearlmutter in omp.lang.sheme, Otober 995.) We look for a distribution suh that lim 5 Λ eists and is stritly greater than. Here is one suh distribution. Let f t β t 2 τ

3 6 so that s t 3 where $ 2. Normalization: gives We find Epeted value s 3 τ 76 β t 2 τ β 8 β τ 9: G 3 β 2 2 τ 2 τβ 2 τ E L4 G β 2 β 8 2; ; If we take this value as a free parameter EL) V (as it is neessary to do in order to generate a trae for simulation, where we want to ontrol the heap size in equilibrium), we have We an then simplify: The ratio hene Λ 3 β G s V 2 < τ V τ V lim 5 Λ 3 2 $ as required. Varying hanges the value Λ uniformly for all ; lower values of produe distributions with heavier tails. 4.2 Past-is-future restrited to $ We make ondition () stritly hold for $, where $, and we define the distribution (funtions F, f, s, and m) in that interval, but we set F, f, s, and m in the interval. This formulation is intuitively appealing, sine lifetimes in pratie take on disrete values in =, and hene setting is quite natural. Condition (): that is, gives the integral equation, #" &% ψ $ Λ 3 ψ G ψ s > t f t dt ψ f t dt It is easy to obtain the orresponding differential equation: F 3 ψ F? ψ The solution, with the boundary value F 3, is ψc F 3 A@ ψc ; f 3 s 3D@ m 3 B ψ B B ψc ψc ; ψ ψ 2ψC ; ψc ; The mortality is indeed an everywhere dereasing funtion. G 3 ψe ψc F G ψ The epeted live amount in the heap is: v 3 ψ ψ E ψc ; ψc F ψc ; Let us eamine this family of distributions qualitatively. The steadystate heap volume V 5 lim v equals the epeted value of L, whih is ψ. What are reasonable parameter values? Suppose that we wish to set V 5. 2 Suppose also that we want to be. Then ψ V G 5. However, the live volume in the heap, v, approahes its limit value V at the rate of deay of the seond term, that is, as the th root of. With suh a slow approah, that is, with suh a heavy tail in the distributions, one must allow a time 3 67 * to pass before the heap is within % of equilibrium; until the heap is in equilibrium, the distribution of objets in it does not reflet the heavy tail of the soure distribution. Simulating that many objets is somewhat impratial. Moreover, an atual programs e- 5? Sup- hibit suh etremely slow heap growth as with ψ pose that a program does. We an only observe eeutions of muh shorter duration than 99997, say, up to ; but then we annot empirially distinguish the postulated ψ 5 past-is-future distribution from another distribution that agrees with it up to t, but laks the heavy tail beyond that age. Alternatively, to allow 99% of the steady-state volume to be reahed with 7 objets simulated, one must have ψih 2 5 (approimately), but then $ 2. Thus, beyond the onstrution of an elegant analytial model of objet lifetimes, we must keep in mind the need to be able to validate it against real data. This eample shows that sometimes validation may be diffiult to ahieve. 5. MODELS VS. EMPIRICAL EVIDENCE In validating lifetime models, we apply the tools of statistial analysis of survival data to the distributions of objet lifetimes, to the etent that they are appliable to our problem. (The populations traditionally studied in statistial analyses are quite different from programs 2 This number is suffiiently large so that in simulation, even when a heap is divided into J regions, eah one ontains at least J objets, whih allows us to vary the heap onfiguration widely without inurring signifiant fragmentation effets.

4 Coeffiient of skewness γ past-is-future restrited past-is-future in the limit log-normal Weibull gamma observed distributions fit to observed Coeffiient of variation γ Coeffiient of skewness γ 3 past-is-future restrited. past-is-future in the limit log-normal Weibull gamma observed distributions fit to observed... Coeffiient of variation γ Figure : Coeffiient of skewness, γ 3, versus oeffiient of variation for several analytial distribution families and for empirially observed objet lifetime distributions. Top: linear sale; bottom: logarithmi sale. (See the appendi for definitions.)

5 objets. Moreover, the relevant literature onentrates mostly on statistial K onfirmation of eplanatory variables (e.g., in linial trials), whih is different from our goal: finding distributions.) One reommended test for omparing families of distributions is based on the graph of moment ratios: the oeffiient of skewness γ 3 vs. the oeffiient of variation γ. These moment ratios are dimensionless quantities independent of time-sale [4, p.24 28] that reflet only the shape of the distribution. The oeffiient of variation is a measure of the spread of the distribution around its mean; the oeffiient of skewness is a measure of the asymmetry of the distribution, and large positive values indiate heavy tails. Eah distribution orresponds to a single point γ γ 3 ; a single-parameter family of distributions defines a parametri urve in the γ γ 3 plane. (See Appendi for definition of γ and γ 3.) Three families of distributions ommonly used in survival analysis (log-normal (Setion A.4.3), Weibull (Setion A.4.2), and gamma (Setion A.4.)), are plotted in the γ γ 3 plane in Figure. Note that the Weibull and gamma urves interset at the point 2 ; at this point eah has degenerated into the eponential distribution. The two families introdued here are also shown (past-is-future in the limit (Setion 4.) and past-is-future restrited (Setion 4.2)). Finally, the figure ontains satter points γ γ 3 of objet lifetime distributions obtained empirially, and a line of least-squares fit for these points. These empirial distributions ome from 58 Smalltalk and Java programs. (Complete objet-level traes are available at stefanov/ ISMM2objettraes-README.html.) The satter points of empirial distributions show a trend of orrelation between γ and γ 3. This trend is somewhat surprising, sine there is no a priori reason to epet it from a haphazard olletion of benhmark programs. Perhaps the presene of this trend points to fundamental properties of program behavior, and it ertainly ought to be studied further. The satter points lie for the most part well to the right and below the ommon analytial distribution families. The one eeption is the gamma family: in fat, even though most satter points are to the right and below the gamma urve, they are quite lose to it. To our hagrin, the two past-is-future families both have muh lower oeffiient of variation and muh higher oeffiient of skewness than the empirial distributions. Therefore, however intuitively plausible they are, they should not be employed to model objet lifetimes. Indeed, we must onentrate the searh for analytial distributions on those not in standard literature with muh higher oeffiient of variation; in the meantime, the gamma family is to be favored as a andidate. We see that the γ γ 3 diagram usefully summarizes the shape properties of distributions and allows us to elude ertain analytial distribution families as models for a set of observed distributions. Note, however, that the two moments displayed do not ompletely apture the shape of a distribution. For positive mathing further statistial tests are neessary. 6. SUMMARY Analytial modelling of objet lifetimes is desirable for the design, analysis, and simulation of dynami memory management systems, but it remains a diffiult problem. We eamined ertain qualitative riteria that may be imposed on lifetime distributions, and demonstrated the use of a simple graphial tehnique for (in)validating postulated distribution models against empirial evidene. 7. ACKNOWLEDGMENTS We thank Gilberto Contreras, Barak Pearlmutter, and the reviewers for their onstrutive advie. We are espeially grateful to Andrew Appel for omments and disussions that initially spurred the developments presented. 8. REFERENCES [] H. G. Baker. Infant Mortality and generational garbage olletion. SIGPLAN Noties, 28(4):55 57, 993. [2] R. S. Burrington and D. C. May, Jr. Handbook of Probability and Statistis with Tables. MGraw-Hill, New York, 2nd edition, 97. [3] W. D. Clinger and L. T. Hansen. Generational garbage olletion and the radioative deay model. SIGPLAN Noties, 32(5):97 8, May 997. Proeedings of the ACM SIGPLAN 97 Conferene on Programming Language Design and Implementation. [4] D. R. Co and D. Oakes. Analysis of Survival Data. Chapman and Hall, London, 984. [5] L. Devroye. Non-Uniform Random Variate Generation. Springer-Verlag, New York, 986. [6] R. C. Elandt-Johnson and N. L. Johnson. Survival Models and Data Analysis. Wiley, New York, 98. [7] M. Evans, N. Hastings, and B. Peaok. Statistial Distributions. John Wiley, New York, 2nd edition, 993. [8] B. Hayes. Using key objet opportunism to ollet old objets. In Proeedings of the Conferene on Objet-Oriented Programming Systems, Languages, and Appliations, pages 33 46, Phoeni, Arizona, Ot. 99. SIGPLAN Noties 26, (Nov. 99). [9] N. L. Johnson and S. Kotz. Distributions in Statistis: Continuous Univariate Distributions (vols. and 2). John Wiley, New York, 97. [] A. Lemmi. Empirial testing of the weibull distribution in the analysis of younger ages mortality. Tehnial report, Faoltà di Sienze Eonomihe e Banarie, Università di Siena, De Paper presented to a seminar on Demographi Models at the Department of Demography, University of Kinshasa, Deember 5 22, 982. [] I. Miller and J. E. Freund. Probability and Statistis for Engineers. Prentie-Hall, Englewood Cliffs, New Jersey, 965. [2] A. Papoulis. Probability, Random Variables, and Stohasti Proesses. MGraw-Hill, New York, 3rd edition, 99. [3] D. Stefanović, J. E. B. Moss, and K. S. MKinley. Oldest-first garbage olletion. Computer Siene Tehnial Report 98-8, University of Massahusetts, Amherst, Apr [4] W. Weibull. Fatigue testing and analysis of results. Pergamon Press, for the Advisory Group for Aeronautial Researh and Development, North Atlanti Treaty Organization, New York, 96.

6 APPENDIX A. ABOUT LIFETIME DISTRIBUTIONS A. Basi definitions The survival funtion of a random variable L is s L t L t. For objet lifetimes, it epresses what fration of original alloation volume is still live at age t. We usually drop the subsript L. The survivor funtion is a monotone non-inreasing funtion. The probability density funtion is f t L s t. Oasionally we also use the umulative distribution funtion F t s t. The mortality funtion is m t f t s t d dt logs t, and it epresses the age-speifi death rate. Mortality is also known as the hazard fun- tion (and written h t ) in the literature on lifetime analysis. Oasionally we also use the integrated t mortality: M t 3 m u du. Certain properties always hold: s ; t5 lim s t + ; f t dt (normalization of density); s t 3 e M t [4, p.4]. A.2 Moments The moments of a distribution of random variable L are defined as m k E L k I tk f t dt. The entral moments are defined as µ k E L m k t m k dt. Here m is the mean, or epeted value, and m 2 is the variane, or dispersion; a ommon notation is σ 2 µ 2, where σ is alled standard deviation. In alulation, we usually first find moments m, m 2, and m 3, by integration in the ase of analytial definitions, or by summation over observed disrete points in empirial distributions, and then ompute entral moments using the formulae µ 2 m 2 m 2 and µ 3 m 3 3m m 2 2 2m 3. The oeffiient of variation is γ σ m. The standardized third moment or oeffiient of skewness is γ 3 µ 3 σ 3 ; it is also written η 3 or M β. A.3 Finiteness of epeted value It is a simple eerise to show that the epeted value of the live amount in the heap at time (that is, after an amount has been alloated) is v V when they eist. s t dt, and that 5 lim v 3 s t dt t f t dt ELN We may impose on the objet lifetime distribution an additional property of finiteness (eistene) of epeted value, to ensure that a heap equilibrium is reahed in the limit. (Heap equilibrium has been the underlying assumption in some omparative analyses of garbage olletion osts [3, 3]. A relative heap size parameter is used as the basis for omparison of two olletion algorithms: heap size is a fied multiple of a steady-state live data amount.) How essential is this requirement, and ould we also onsider distributions with unbounded epeted value? On the one hand, the running time of real programs is finite, and thus f is finally-zero, hene E L is finite. On the other hand, it is theoretially plausible that we are observing initial segments of potentially infinite omputations, and so it is useful to investigate heaps that grow without bound. From a purist standpoint, many realisti programs that run indefinitely do use inreasing amounts of spae; for instane, ounting requires logarithmially inreasing spae. If the live data amount does not stabilize, but rather grows indefinitely, then the available heap size ought to grow in equal proportion if one desires measurements in terms of the relative heap size parameter. This property must be ensured with due are in analysis and simulation. A.4 Distribution families of Figure Basi definitions of distribution families, ompiled from tetbooks [2, 5, 6, 7, 9,, 2]. A.4. Gamma f t Γ b bo t b e t m k µ 2 b 2 b 2 µ 3 2 b 3 γ b γ 3 2 b b 2 2 P*Q*Q* b 2 k k Therefore γ 3 2γ is a straight line in Figure. A.4.2 Weibull f t t e t m Γ m 2 Γ 2 m 3 Γ The urve γ γ 3 is plotted parametrially with respet to in Figure. A.4.3 f t tδ 2π e SR Log-normal logt; ζt 2 2δ 2 e δ2, γ LU ω and ω 2 2 U ω, therefore γ 3 3γ 2 γ 3 in Figure. It an be shown that, with abbreviation ω γ 3