Optimal Tenuring Collection Times for a Generational Garbage Collector based on Continuous Damage Model

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1 International Journal of Performability Engineering Vol. 9, o. 5, eptember 3, p RA Consultants Printed in India Optimal enuring Collection imes for a Generational Garbage Collector based on Continuous Damage odel XUFEG ZHAO, and OHIO AAGAWA chool of Economics and anagement, anjing University of echnology 3 Puzhu Road, anjing 86, China Graduate chool of anagement and Information ciences, Aichi Institute of echnology 47 Yachigusa, Yakusa-cho, oyota 47-39, Japan (Received on February 6, 3, revised on June 6, 3) Abstract: he processing time intervals for a generational garbage collector may be ephemeral enough to consider the objects that would survive increase with time continuously according to some probabilistic law. From such a viewpoint, this paper firstly answers for the weak points of cumulative damage model whose damage is additive at discrete times. econdly, we take up a continuous damage model and apply such model to garbage collection policies. Costs for garbage collections are estimated and two models with tenuring collection times, where tenuring collection is made at time or at level for random collections and at the th collection or at level for periodic collections, are proposed. Four cases of optimal tenuring collection times for every model are discussed analytically and numerically, and comparisons of these policies and some useful results are given. eywords: Degradation process, continuous damage, garbage collection, tenuring collection, minor collection.. Introduction Cumulative damage models [] play an important role in reliability theory: hese models are considered as a sequence of shocks that occur randomly and give some amount of damage to a unit. he damage is accumulated to the current damage level and weakens the unit gradually, and the unit fails when the total damage exceeds a failure level. he reliability properties and various maintenance policies for damage models were summarized sufficiently in []. A variety of maintenance models subjected to shocks were studied extensively in [3-8]. ome applications of these damage models into computer science, such as backup policies for a database [9, ], garbage collection policies for the memory management [-4], were explored. otivated by [9-4], we found that the hypothesis of cumulative damage models may be so strong that they are not so practicable for some applications: (a) an amount of damage due to every shock may not have an identical distribution, and the effect of later shocks may be much more serious than former ones. In other words, expected number of shocks until failure in practice is much less than that in theory; (b) it is more practical to suppose that the total damage stored within the unit would increase with time itself, but not be in a constant level until next shock occurs; (c) referring to these applications studied in [9-4], due to high frequency of computer processes in the modern society, it may be not so valid to assume that update of data or garbage occurrence would follow an nonhomogeneous Poisson process, because time intervals of procedures might be very short and unclear enough. Corresponding author s toshi-nakagawa@aitech.ac.jp 539

2 54 Xufeng Zhao and oshio akagawa Firstly, to answer the above two weak points (a) and (b) of cumulative damage models, we consider the model where the total damage increases with time itself continuously according to some probabilistic law, which could be named as continuous damage model. By modifying Xt () = x + µ t+ σwt () [5, p.6], which is a Gaussian process, where Wt () denotes a standard Brownian motion, x is some initial degradation level, and µ and σ are the drift and the variance, respectively, we suppose that the path of the total damage with time t is Xt ( ) = αt+ σb t ( ασ, > ), () where x = means that the initial state of the unit is as good as new and B t is distributed exponentially. o that we can know that total damage Xt () increases with time t strictly and undergoes some random positive change. We make such an assumption also because we cannot neglect the dependence between damage stored within the unit and damage caused by shocks, although some research papers combined the degradation process and shock process [6, 7], but they are independent. econdly, to answer (c), this paper focuses on optimization problems of a generational garbage collector using the continuous damage model obtained in equation (). In memory management in computer science, generational garbage collection [8, 9] has been popular with programmers for the reason that it can be made more efficient. However, for every garbage collection, the manner of stop and copy pause all application threads to collect the garbage. he duration of time in which a garbage collector has worked is called pause time [, p.48], which is an important parameter for interactive systems and depends largely upon the volume of surviving objects and the type of collections. he latest technical report [9] proposed some models by estimating the lifetimes of all objects, however, this kind of technique may be not so easy in practice, because for high frequency of random computer processes, the number of objects in the heap may be too great to count, also because of random factors of object lifetimes. References [-4] improved above method by estimating the survivor rate of one group of objects rather than focusing on every object. o make the estimation cost for collections more easily, we get inspiration from [9] which asserted that garbage increases with time continuously, and apply equation () to estimate surviving objects at some time by statistical methods. It is true that increase in objects might be very unclear at discrete times for the high frequency of computer processes in the modern society. his paper considers a pause time goal that is called time cost or cost for simplicity for a generational garbage collector. Our problem is to obtain optimal tenuring collection times that minimize the expected cost rates. Following sections are organized as: ection gives working schemes and costs for a generational garbage collector. ection 3 discusses the case when garbage collections occur at a nonhomogeneous Poisson process. ection 4 discusses models when collections occur at periodic times. umerical examples are computed in ection 5 and concluding remarks are given in ection 6.. Working cheme and Cost For a generational garbage collector, detailed working schemes that have been introduced in [-4] are given as following steps:. ew objects are allocated in Eden space.. When the first minor collection occurs, surviving objects are copied from Eden into a survivor space. 3. When the second minor collection occurs, surviving objects from Eden and

3 Optimum Replacement Policy for Continuous Damage odel and its Application 54 to Garbage Collection survivor spaces are copied into the other survivor space. 4. In the fashions of -3, minor collection copies surviving objects between the two survivor spaces until they become tenured, i.e., tenuring collection occurs when some parameter meets the tenuring threshold, and then, the older or oldest objects are copied into Old space. 5. When Old space fills up, major collection of the whole heap occurs, and surviving objects from Old space are kept in Old, while objects from Eden and survivor spaces are kept in a survivor space. Although major collection spends much time, Old space will be filled with tenured objects slowly, especially when the tenuring threshold is high and the survivor rate is low, that is, major collection occurs rarely in this case. o that, this paper concentrates only on minor and tenuring collections and considers tenuring collections as renewal points of the collection processes, that is, after tenuring collection, the same collection cycle begins with step to to to 3 to 4. From [-4], we have known that surviving objects that should be copied increase with the number of minor collections. hat is, we can define:. he total surviving objects in Eden and survivor space at time t is Xt () = αt+ σbt with distribution Pr{ Xt ( ) x} = Wtx (, ).. c + c( x) is the cost suffered for every minor collection, where c is the constant cost suffered for scanning surviving objects and x is the amount of copied objects, c( c > c + c( )) is the cost suffered for tenuring collection at level. 3. he expected cost of minor collection at time t is given by C( t, ) = [ c ( )] (, ), (, ) + c x dw t x Wt () where C(, ). It is clear that Ct (, ) increases with t for any. 3. Random Collections It is assumed that garbage collections occur at a nonhomogeneous Poisson process with an intensity function λ () t and a mean-value function that collections occur exactly j times in (, t ] is t R() t λ( u) du. hen, the probability Letting Fj () tdenote the probability that collections occur at least j times in (, t ]. hen, where F ( t). uppose that minor collections are made when the garbage collector begins to work, tenuring collection is made at a planned time ( < ), or when surviving objects have exceeded a threshold level ( < ), whichever occurs first. hen, the mean time to tenuring collection is W (, ) + tdwt (, ) = Wtdt (, ), (3) and the expected cost suffered for minor collections until tenuring collection is

4 54 Xufeng Zhao and oshio akagawa herefore, the expected collection cost rate is c ( c c) W (, ) + λ( tctwtdt ) (, ) (, ) = W (, t ) dt (, ). C It can be seen that C (, ) includes the following collection polices:. enuring collection is made at time for a given, the reason why making such a policy is c < c.. enuring collection is made at level for a given. In this case, c > c. 3. enuring collection is made only at time or only at level. In these two cases, c = c. () Optimal When c < c, we find an optimal which minimizes C (, ) in (5) for a given. Differentiating C (, ) with respect to and setting it equal to zero, ( c c ) h(, ) W( t, ) dt W(, ) + [ λ( C ) (, ) λ( tct ) (, )] Wtdt (, ) = c. Letting L (, ) be the left-hand side of (6), and (4) (5) (6) hus, if λ () t increases with t, then the left-hand side of (6) increases with t from. herefore, there exists a unique optimal ( < ) which satisfies (6). () Optimal When c > c, we find an optimal Letting wt (, x ) be the density function of (, ), which minimizes C (, ) in (5) for a given. Wtx i.e., w(, t x) dw (, t x)/ dx. hen, differentiating C (, ) with respect to and setting it equal to zero, where ( c c ) Q (, ) W( t, ) dt W(, ) + [ Q(, ) λ( tct ) (, )] Wtdt (, ) = c. (7)

5 Optimum Replacement Policy for Continuous Damage odel and its Application 543 to Garbage Collection w (, ) [ c + c ( )] λ( t) w( t, ) dt w(, t ) dt w(, t ) dt (, ), Q(, ). Q Letting L (, ) be the left-hand side of (7), and hus, if Q (, ) and Q (, ) increases with, then the left-hand side of (7) increases with from. herefore, there exists a unique optimal (7). (3) Optimal ( < ) which satisfies When c = c, putting that = in (5), the expected cost rate is (8) where C(, t ) = [ c + c ( x)] dw (, t x) = c + W (, t x) dc ( x). From (6), if λ () t increases with t, then an optimal tenuring collection time minimizes (8) is given by a unique solution of the equation which (9) (4) Optimal When c = c, putting that = in (5), the expected cost rate is () From (7), if Q (, ) increases with, then an optimal tenuring collection time which minimizes () is given by a unique solution of the equation 4. Periodic Collections It is assumed that garbage collections occur at periodic times j ( j =,, ) for a given time interval ( < < ) minor collections are made when the garbage collector begins to work, tenuring collection is made at the th ( =,, ) collection, or at first () collection time when surviving objects have exceeded a threshold level ( < ), whichever occurs first. hen, the mean time to tenuring collection is ( j+ ) W (, ) + ( j + ) dw ( t, ) = W ( j, ). () j he expected cost suffered for minor collections until tenuring collection is

6 544 Xufeng Zhao and oshio akagawa (3) herefore, the expected cost rate is c ( c c ) W(, ) + C( j, ) W( j, ) = C (, ). W ( j, ) (4) () Optimal When c < c, we find an optimal which minimizes C (, ) in (4) for a given. From the inequality C( +, ) C(, ), where ( c c) Q3(, ) W( j, ) W(, ) + [ C(, ) C( j, )] W( j, ) c, Q W(, ) W(( + ), ) 3(, ). W (, ) Letting L (, ) 3 be the left-hand side of (5), (5) hus, if Q3( j, ) increases with and L (, ), 3 > c then there exists a finite and unique minimum () Optimal ( < ) which satisfies (5). When c > c, we find an optimal which minimizes C (, ) in (4) for a given. Differentiating C (, ) with respect to and setting it equal to zero, where ( c c) Q4(, ) W( j, ) W(, ) + W ( j, )[ c + c ( ) C( j, )] = c, Q w(, ) 4(, ). w( j, ) (6)

7 Optimum Replacement Policy for Continuous Damage odel and its Application 545 to Garbage Collection Letting L (, ) 4 be the left-hand side of (6), and hus, if Q4(, ) increases with, then the left-hand side of (6) increases with from. herefore, there exists a unique optimal ( < ) which satisfies (6). (3) Optimal When c = c, putting that = in (4), the expected cost rate is From (5), an optimal tenuring collection time unique solution of the equation (4) Optimal (7) which minimizes (7) is given by a [ C(, ) C( j, )] c. (8) When c = c, putting that = in (4), the expected cost rate is (9) From (6), an optimal tenuring collection time unique solution of the equation which minimizes (9) is given by a 5. umerical Examples We compute numerical examples of the models discussed above when Xt () is given by () for µ = σ = and B t has an exponential distribution e x t. ext, select suitable parameters which can affect the optimal policies and resulting cost rates to compute each model numerically: (i) Initial parameters and are given to compute and and are given to compute and. (ii) /, c c c, c and c mean that the effects of tenuring collection costs at time and level for random collection models and at th collection and level for periodic collection models. (ii) c ( i =,, ) which mean that the effects of minor collection costs are selected to compute i,,,, and From ables -6, we can obtain the following results:. Optimal tenuring collection times increase with the initial parameters and decrease,

8 546 Xufeng Zhao and oshio akagawa with minor or tenuring collection costs, however, the resulting cost rates have the opposite tendencies, that is, they decrease with the initial parameters and increase with minor or tenuring collection costs. ake C, / c in able for and ( ) example: increases with and decreases with c and c. Increasing in, c and c mean tenuring collection time made at a given level is postponed, tenuring and minor collection costs are increased, respectively, so that tenuring collection times should be postponed for and be advanced for c and c to decrease the frequency of tenuring collections and to decrease the total minor collection costs. ( ) C, / c decreases with and increases with c and c for the reason that the frequencies of tenuring collections are decreased and tenuring and minor collection costs are increased.. Compared with ables and, we can derive that and C(, ) C(, ), however, ( ) than and sometimes less than C(, ). and ( ) than C(, ). 3. Compared with ables 3 and 4, C c are sometimes greater C, / c are always greater 4. Compared with ables, and 3, 4, it is clearly that random collections are better than periodic collections. 5. Compared with ables 5 and 6, optimal collection times and the resulting cost rates are almost at the same level for four models, that is, and It is clearly that when / λ =, and become the same model. According to [-4] and step 4 in ection, we also can name and as tenuring threshold of the collection processes. able : Optimal c 5 C, / c when c =. and λ = α = σ =. and ( ) c..5. (, ) / C c (, ) / C c ( ) C, / c

9 Optimum Replacement Policy for Continuous Damage odel and its Application 547 to Garbage Collection able : Optimal c 5 C, when c =. and λ = α = σ =. and ( ) c..5. C(, ) C(, ) ( ) C, able 3: Optimal C, / c when c =. and = α = σ =. and ( ) c c..5. (, ) / C c (, ) / C c ( ) C, / c able 4: Optimal c 5 C, when c =. and λ = α = σ =. and ( ) c..5. C(, ) C(, ) ( ) C,

10 548 Xufeng Zhao and oshio akagawa able 5: Optimal and when λ = α = σ =. c i =, i c =. c = able 6: Optimal and when = α = σ =. c i =, i c =. c =. 6. Conclusions We have firstly pointed out three weak points of cumulative damage models whose damage is only caused by random shocks. o answer these weak points, continuous damage model in equation () has been proposed, which shows that damage stored in the units would increase with time itself and undergoes some random positive change due to shocks. he idea has been motivated by application of damage models into garbage collections in computer science. From ection to ection 5, this paper has focused on optimization problems of tenuring collection times for a generational garbage collector, using the models given in equation (). Obviously, we could monitor the increasing path of surviving objects in practice, then an interesting point is to estimate collection cost in equation (). We have discussed random and periodic collection models for a generational garbage collector according to its working schemes. Optimal tenuring collection times have been obtained and their numerical examples have been given to explain model formulations. Acknowledgement: his work is partially supported by the Grant-in-Aid for cientific Research (C) of Japan ociety for the Promotion of cience under Grant o. 5897;

11 Optimum Replacement Policy for Continuous Damage odel and its Application 549 to Garbage Collection ational atural cience Foundation of China (7477, 7836); Humanities and ocial cience Research Foundation of China (5JA637). References [] Cox, D. R. Renewal heory. etheun, London, 96. [] akagawa,. hock and Damage odels in Reliability heory. pringer, London, 7. [3] Wortman,. A., G.A. lutke, and H. Ayhan. A aintenance trategy for ystems subjected to Deterioration governed by Random hocks. IEEE ransactions on Reliability, 994; 43: [4] heu,.h., and W.. Griffith. Optimal umber of inimal Repairs before Replacement of a ystem subject to hocks. aval Research Logistics, 996; 43(3): [5] heu,.h. A Generalized Age and Block Replacement of a ystem subject to hocks. European Journal of Operational Research, 998; 8(): [6] Zhao, X.F., H.. Zhang, C.H. Qian,. akagawa, and. akamura. Replacement odels for Combining Additive with Independent Damages. International Journal of Performability Engineering, ; 8(): 9-. [7] Zhao, X.F.,. akagawa, and C.H. Qian. Optimal Imperfect Preventive aintenance Policies for a Used ystem. International Journal of ystems cience, ; 43(9): [8] Zhao, X.F., C.H. Qian, and. akagawa. Optimal Policies for Cumulative Damage odels with aintenance Last and First. Reliability Engineering & ystem afety, Feb. 3; :5-59. [9] Qian, C. H.,. akamura, and. akagawa. Cumulative Damage odel with wo inds of hocks and its Application to the Backup Policy. Journal of the Operations Research ociety of Japan, 999; 4(4): 5 5. [] Qian, C.H., Y. Pan, and. akagawa. Optimal Policies for a Database ystem with wo Backup chemes. RAIRO Operations Research, ; 36(3): [] Zhao, X.F.,. akamura,. akagawa. Optimal Policies for Random and Periodic Garbage Collections with enuring hreshold. Communications in Computers and Information cience (Editors: G.. omar, et al.), pringer, Berlin,, pp [] Zhao, X.F.,. akamura, and. akagawa. wo Generational Garbage Collection odels with ajor Collection ime. IEICE ransactions on Fundamentals of Electronics Communications and Computer ciences, ; E94-A (7): [3] Zhao, X.F.,. akamura, and. akagawa. Optimal enuring and ajor Collection imes for a Generational Garbage Collector. Asia-Pacific Journal of Operational Research, ; 9: 48 (7 pages). [4] Zhao, X.F.,. akamura, C.H. Qian. Generational Garbage Collection Policies. tochastic Reliability and aintenance odeling (Editors:. Dohi and. akagawa), pringer, 3, pp [5] ikulin,.., and. Balakrishnan. Advances in Degradation odeling: Applications to Reliability, urvival Analysis, and Finanace. Birkhouser, Boston,. [6] Li, W., and H. Pham. An Inspection-aintenance odel for ystems with ultiple Competing Processed. IEEE ransactions on Reliability, 5; 54(): [7] Lehmann, A. Joint odeling of Degradation and Failure ime Data. Journal of tatistical Planning and Inference, 9; 39(5): [8] Appel, A.W. imple Generational Garbage Collection and Fast Allocation. oftware Practice and Experience, 989; 9(): [9] Vengerov, D. odeling, Analysis and hroughput Optimization of a Generational Garbage Collector. echnical Report, un Labs, 9; R [] Jones, R., and R. Lins. Garbage Collection: Algorithms for Automatic Dynamic emory anagement. John Wiley & ons, Chichester, 996.

12 55 Xufeng Zhao and oshio akagawa Xufeng Zhao received his B.. degree in Information Engineering in 6 and.. Degree in ystem Engineering in 9, both from anjing University of echnology, China, and Ph.D. degree in 3 from Aichi Institute of echnology, Japan. He is now a postdoctoral fellow at Graduate chool of anagement and Information ciences, Aichi Institute of echnology, Japan, and also serves at chool of Economics and anagement, anjing University of echnology, China. He is currently interested in reliability theory and maintenance policies of stochastic systems, shock models and their applications in computer science. Dr. Zhao is a member of ORJ. oshio akagawa received B..E. and.. degrees from agoya Institute of echnology in 965 and 967, respectively, and Ph.D. degree from yoto University in 977. He worked as a Research Associate at yracuse University for two years from 97 to 973. He is now a Guest Professor at the Department of Business Administration, Aichi Institute of echnology, Japan. He has published 5 books from pringer and about journal papers. His research interests are optimization problems in operations research and management science, and analysis for stochastic and computer systems in reliability and maintenance theory.