Idriss Fontaine
email: idriss.fontaine974@gmail.com

The interaction between labour force participation of older men and their wife: lessons from France

Keywords: Labor force participation, elderly, cohort effects, France JEL classifications: J14, J21, F42 Word count: 8939

Since the mid-1990s, the labor force participation of older men and women increased substantially in France. In this paper, we investigate the causal impact of having a participating wife on the labor market behavior of the elderly husband. Working with data from the French Labour Force Survey and using the cohort-specific participation rate of women at age 40 as an instrument for their current participation, we find that the magnitude of the causal relationship is strong. The likelihood of husbands' participation increases of about 28 points when their wives are currently active on the labor market. Such findings support the view that some complementarities in leisure exist so that French married men attribute a higher value to leisure when it is shared with their wife. It also suggests that policy makers should take into account both direct and indirect effects when they implement a change in the economic environment of elderly.

Introduction

In many OECD countries, we observe that the labor market attachment, measured by the participation rate or the employment rate, of men aged 50-64 substantially increased since at least the mid-1990s. However, such an increase in labor market attachment was preceded by an important downward trend throughout the post WWII. In the meantime, the labor supply of women of the same age group followed a clear upward trend and increased sharply. The latter increase is one of the reflections of important structural changes that occurred during this period: access to the birth control pill [START_REF] Bailey | More power to the pill: the impact of contraceptive freedom on women's life cycle labr supply[END_REF], a general increase of wives' bargaining power within the household, and so on. Figure 1 illustrates this evolution for the French economy, the country under scrutiny in this paper. In France, the labor force participation rate (LFPR, thereafter) of older men fell from 68% to less than 60% over the 1983-1995 periods while the one for older women increased of about 6 point over the same period. After the mid-1990s, the LFPR of French older men (re-)increased to reach 67% in 2016.1 At the end of the sample period studied here, the gap between LFPR of French older men and women was approximately equal to 5 points, whereas it was of about 25 points in 1983 2 . Overall, from these figures it seems that there is a positive association between the participation of women and men, and it is reasonable to imagine that this link is stronger within couples. The main objective of the present paper is to estimate to what extent the labor supply of wives causally impacts the labor supply of their husbands in France.

The causes and consequences of these shifts in labor market attachment of elderly have attracted the attention of policy makers and researchers alike, particularly in light of recent reforms that affect pension schemes or the minimum age at retirement. It is crucial to understand the reason explaining labor supply decisions of older individuals. For instance, if the recent increase in older men's activity is not related to the labor supply of their wives, but rather a consequence of other factors, such as educational attainment, then policies aiming at changing the path of labor market activity at the end of the active life have any externalities on the partner. By contrast, if husband's participation is, at least in part, causally driven by his wife own participation, then policy makers should design their toolkit on the modeling of the joint labor market participation decision within couples. In such a context, the decision to work or not is interdependent and any reforms changing the economic environment at the end of the active life could potentially affect both partners.

The current economic literature presents some piece of evidence indicating that married couples are likely to coordinate their retirements [START_REF] Hurd | The Joint Retirement Decision of Husbands and Wivespp[END_REF], [START_REF] Gustman | Retirement in Dual-Career Families: A Structural Model[END_REF]). More specifically, in the US context it is shown that a husband responds more to the retirement of his wife than she does. In this paper, we do not focus on retirement per se but rather on the joint labor supply decisions that occurs just before the end of the active life. In this respect, [START_REF] Schirle | Why Have the Labor Force Participation Rates of Older Men Increased since the Mid-1990s?[END_REF] shows that having an active wife increases significantly the likelihood of participation of her older husband, in the US, Canada and the UK. Focusing on younger people, [START_REF] Hamermesh | Togetherness: Spouses' Synchronous Leisure, and the Impact of Children[END_REF] show that couples change their working day schedule in order to enjoy leisure together. Overall, two economic interpretations are invoked to explain how wives' participation influences husband labor market attachment. The first channel is an income effect implying that, all else being equal, having a participating wife increases non-labor income of her husband. This extra-revenue is able to act as a preference shifter, decreasing the marginal utility of consumption leading ultimately to an increase in leisure and so non-participation. The second channel is a shared leisure effect operating if they are some complementarities so that couple members have a preference to spend their leisure time together. In this context, a husband does not evaluate as much its leisure time if it is spent alone. Therefore, having a participating spouse decreases the marginal utility of leisure (and so non-participation) leading ultimately to an increase in older men labor supply. The empirical model developed in this paper will allow us to conclude about which of the two effects prevails over the other one.

The concrete measure of the relationship between spouses' labor supply is however not straightforward. Although prima facie a positive correlation between husbands and wives participation is observed (e.g. figure 1), it remains quite hard to disentangle a causal relationship for at least two reasons. First, the two decisions are influenced by common factors so that it is necessary to control for observable characteristics in the empirical model. Second and perhaps most importantly, the labor market behaviors of husbands and wives are likely to be simultaneous. As a consequence of these shortcomings, standard models as Ordinary Least Square (OLS, thereafter) or Probit are biased when the wife's labor market status appears as an explanatory variable in husband's equation. In order to identify a causal effect, an exogenous source of variation of wives' participation is needed. In this respect, we follow [START_REF] Schirle | Why Have the Labor Force Participation Rates of Older Men Increased since the Mid-1990s?[END_REF] by using a measure of cohort-specific participation rate of women at age 40 as instrument, a cohort being defined as women's year of birth. This instrument should capture the general increase in female participation implying that women belonging to recent cohort are much more likely to participate than those belonging to older cohort. Thus, we expect that the cohort LFPR of women at age 40 to be strongly related to wives' current labor market status.3 Given the choice of the instrument, different econometric models could be used to estimate the marginal effect of interest. Here, we employ a Two-Stage Least Squares strategy but also a Bivariate Probit model. In doing so, we test for the robustness of the empirical finding.

We make use of the two available versions of the French Labour Force Survey (FLFS, thereafter) in our investigation. The first is the quarterly version of the FLFS that exists since 2003. This database has several practical advantages for the problem in hand: it is representative of the French population, it contains a large number of observations allowing us to be immune from sampling errors when focusing on elderly and it provides information about individuals' characteristics but also about members within the same household. The quarterly FLFS covers the 2003-2016 period and is used to construct the sample on which the econometric model will be estimated. The second database employed is the previous annual version of the FLFS that exists since 1983. This source is particularly suitable for the construction of the cohort-specific participation rate of women at age 40. 4 Finally, the availability of the data implies that the sample of interest contains married men aged between 50-64 years old with a wife born between 1943 (the first cohort available from the 1983 version of the FLFS) and 1976 (the last cohort available from the 2016 version of the FLFS).

Our key findings can be summarized as follows. The first stage estimation confirms that our instrument has a significant impact on wives' current participation. Specifically, a 1 point increase in the cohort-specific participation rate at 40 induces an increase in women likelihood of participation of about 2.5 points. Moreover, some clues suggest the appropriateness of the instrumentation strategy leading us to be confident about the interpretation of the second stage equation. OLS regressions of the husbands' equation confirm a positive correlation between wives' labor market status and husbands' participation. However, the size of the impact changes when the former is instrumented. In particular, it is shown that OLS (or Probit) estimated coefficients are biased downward. Overall, our favorite specification indicates that having a participating wife leads to an increase in husband's participation of 28 points when Two-Stage Least Squares are used. These findings confirm that the shared leisure effect is stronger than any income effects suggesting that French older married men are quite sensitive to the current labor market behavior of their wives. Our conclusions constitute another piece of evidence for policy makers indicating that any modifications of the economic environment of elderly could have important effects not only on the single individ- -France.

Sources: French Labour Force Survey (1983-2016), author's own calculations.

ual but also on the joint participation decision within households. The paper proceeds in 6 sections including the introduction. In section 2, we provide a brief overview of the literature and we present in more depth the data. Section 3 presents both the theoretical and the empirical framework. The 4th section presents the main result of the paper. In section 5, an array of robustness check is conducted. Section 6 concludes.

Preliminaries

Literature background

An important US literature suggests that couple members coordinate their retirement decisions so that both the husband and its wife retire close in time, often during the same year [START_REF] Hurd | The Joint Retirement Decision of Husbands and Wivespp[END_REF]. Some evidence of joint retirement has been put in evidence by [START_REF] Blau | Labor Force Dynamics of Older Married Couples[END_REF]. Using the Retirement History Survey, he finds strong associations between the participation probabilities of one spouse and the labor force status of the other spouse. In particular, living with a non-participant partner increases the likelihood of retirement of the other couple member. From a different data source, the Health and Retirement Study, [START_REF] Gustman | Retirement in Dual-Career Families: A Structural Model[END_REF] find that the retirement probability of husbands is substantially affected by the retirement status of their wives. However, they find no significant effect in the opposite direction suggesting that wives do not confer the same value to the retirement status of their partners than husbands do. In order to explain why couple members adjust their retirement strategies, the shared leisure effect is often put in evidence. In this respect, [START_REF] Blau | Labor Force Dynamics of Older Married Couples[END_REF] affirms that to understand the incidence of joint retirements, it is necessary to go beyond the analysis of financial incentives. The existing literature provides strong evidence suggesting that the root of the shared leisure hypothesis is the notion of complementarity between wives and husbands leisure time. In this respect, [START_REF] Maestas | Labor, Love and Leisure: Complementarity and the Timing of Retirement by Working Couples[END_REF] reports that 62% of older men look forward to retiring only if their wives are able to retire as well. Studying the labor market behavior of older men, [START_REF] Schirle | Why Have the Labor Force Participation Rates of Older Men Increased since the Mid-1990s?[END_REF] indicates that having a participating wife pushes the likelihood of older husband's participation up of about 20 points. An interesting point about her results is that the magnitude of the effect is very similar for countries such as the US, the UK and Canada.

By contrast to the US, the French case is relatively less documented. To the best of our knowledge, the first study working with French data is the one of [START_REF] Sédillot | La cessation d'activité au sein des couples : y-at-il interdépendance des choix ? Suivi d'un commentaire d'Antoine Bommier[END_REF]. They show that French couples are also likely to take joint retirement decisions. However, and by contrast to the US case, they find that French women's retirement behaviors are more affected by the labor market status of their husbands. In a recent contribution, [START_REF] Stancanelli | Couples' retirement under individual pension design: A regression discontinuity study for France[END_REF] estimates the direct and indirect effects of a reform of the French pension on the retirement probability of both spouses.5 From a regression discontinuity design, she finds that the husband's probability falls of about 1 point if his wife is affected by the reform, while she does not respond if her husband is affected, at least in the short run. In France, the implementation of the survey "Retirement motivation" in 2008 allows us to have further clues about the strength of the interactions between the retirement behaviors within couples. More specifically, 18% of retired elderly interviewed claim that having a retired partner was important for their own retirement decision. As for the US, the shared leisure hypothesis is viewed as a potential determinant of the willingness to retire together. [START_REF] Stancanelli | Partners' leisure time truly together upon retirement[END_REF] exploit diary data to investigate to what extent partners spend more leisure time together upon retirement. They find that wife's retirement increases significantly couple's joint leisure time. Their findings reinforce the idea that leisure complementarities are significant, at least in France. Our study complements the existing French evidence by examining to what extent the labor supply of wives significantly affects the labor supply of their husband, just before the effective age of retirement.

Data

We exploit the two available versions of the French Labour Force Survey (FLFS): the quarterly version over the 2003-2016 period and the annual version over the 1983-2002 period.

The quarterly FLFS Since 2003, the FLFS is a quarterly survey of about 100 000 observations, which is conducted by the French National Institute of Statistics and Economic Studies. Before 2003, the survey was annual. The survey's sample is a rotating panel divided into 6 subgroups (or wave). Each quarter, 1/6 of a sample is renewed: a new wave enters while the oldest wave leaves the sample. In this research, we only work with the entering wave of each quarterly sample. This choice can be justified by the fact that considering each quarterly sample entirely could induce repeated observations of the same individuals. The FLFS has several advantages compared to others databases. The first advantage relies on its representativeness of the French population and the very large number of individuals in the sample. This means that the FLFS is particularly suitable to eliminate most of sampling errors when one computes statistics on a particular subgroup, as it is done here with older married men. The second advantage is the wealth of information about people's demographics, labor market state or education. In particular, as the sample unit is the household, we are able to recover information about the household head (generally the man or the husband) but also all other members of the household (his wife, his children and so on). Thus, for each man living with a partner (or loosely speaking the husband),we know if his partner (or loosely speaking his wife) is in or out of the labor force. Another potential advantage of the FLFS is that it addresses some issues related to the measurement of the labor market states. Indeed, as definitions of the International Labour Office (ILO) are used, individuals participating to the labor market are those employed or unemployed, namely those that are actively searching for a job and available to take a job in a short delay (2 weeks). This distinction between unemployed and inactive people is not trivial since other databases, e.g. the French Census, use declarative measures that are more likely to be concerned by mismeasurement issues.

The annual FLFS As the instrumental variable used in the empirical application is a cohort measure of female's LFPR at age 40, in addition to the quarterly FLFS, we make use of the annual version of the FLFS available for the 1983-2002 period. We define a cohort in a restrictive way: a woman's cohort is defined according to her exact year of birth. In a robustness check, we will test the sensitivity of our results to a broader definition of cohorts. Given the availability of the data, we drop observations for which the computation of cohortspecific participation rate at 40 is not possible. Indeed, the first available cohort-specific LFPR is from the 1983 version of the annual FLFS and corresponds to women born in 1943. The last cohorts is from the 2016 version of the FLFS and corresponds to women born in 1976. Finally, our main sample of interest contains married men aged 50-64 years old, over the 2003-2016 period, with a partner born between 1943 and 1976.

Some stylized facts

Figure 1 shows that two distinct periods are observable over the post-2000s, especially for older men participation rate. In the first part of this subsample, until 2007, the LFPR of older men decreased of about 5 points. Then, a trend break is perceptible in 2008 and the LFPR (re-)increased to reach 67% in 2016. By contrast, for older women the trend is upward but accelerated from 2008. One could argue that such movements in older men participation rate could be a concern for our econometric models because they may reflect some cyclical feature of the French economy. To address this issue, we add annual fixed effects as control variables to capture some cyclical features of the data. Moreover, in a step of robustness the model is estimated by considering only the post-2008 period.

Table 1 reports a set of summary statistics about married older men of our sample. We could observe that the participation rate of French married men increases slightly between 2003 and 2016. However, this general picture hides some heterogeneity. Indeed, during this period, we witness an increase of about 3 percentage points in the LFPR of married men with a participating wife. By contrast, the LFPR of married men with a wife out of the labor force fall of about 4 points. Overall, in 2016 there is a difference in participation of about 30 points between older men with an active wife and older men with an inactive wife. This suggests that participation behaviors of husbands and wives are positively correlated. Focusing now only on the husband's wives, it is shown that their participation rate increase of 3 points between 2003 and 2016.

Table 1 also describes married men according to some demographics that enters as control variables in our empirical model. It is of interest to observe that the average age of married men belonging to our sample increases of 1.5 years old between 2003 and 2016. The age variable appears as an important control because we could expect that it plays an important role in participation/retirement behaviors. An increase in age could shift individuals' preference for leisure by, for example, decreasing the marginal utility of consumption and increasing the marginal utility of leisure. Furthermore, [START_REF] Hairault | Distance to retirement and older worker's employment: the case for delaying the retirement age[END_REF] show that the likelihood of participation is significantly affected by the distance to retirement. Specifically, on a frictional labor market, the returns to job of older workers depends on its expected duration. The shorter the time to retirement is, the lower the probability of getting a new job when unemployed is. In this application, we consider the age variable as a "catchall" of this effect. The average age of women also increases. However, the speed of the rise is higher for wives than what it is for their husband. This suggests that the average age gap within couple decreases markedly during the sample period. In the context of this paper, it should be observed that the age gap within couples could capture some income effects due to the fact that, all else being equal, a younger wife is more likely to participate but also more able to support the retirement decision of her husband. Finally, whatever the participation of the wife, we observe a substantial increase in educational attainment of older married men. For example, the share of men without any degree falls of 13 points while the share of those having a high school degree increases of 2 points between 2003 and 2016. As a result, controlling for such structural trends is of particular importance because the likelihood of participation increases with the education level.

3 The framework

Theoretical elements

The empirical model of this paper can find its rational with a simple static framework in which both the husband and his wife maximize their own utility independently and consider some proportion of the partner's income as non-labor income. It should be observed that some more sophisticated framework could be employed. However, relying on such a simple model allows us to link its theoretical prescription with its data-driven reduced-form counterpart. Let husband's labor supply behavior be driven by the following latent variable Y H * , where

Y H * it = U H (C it , Y H it , X H it , Y W it |Y H it = 1) -U H (C it , Y H it , X H it , Y W it |Y H it = 0) (1)
The first term of the right hand side of this equation is the utility obtained when the individual participates to the labor market (Y H it = 1) whereas the second term represents the utility obtained when the individual does not supply his labor (Y H it = 0). These two utility gains depend on individual's consumption of goods C it , individual's participation Y H it , his own characteristics X H it such as demographics or educational attainment, and the labor supply of his partner Y W it . In this context, a husband will supply his labor when the utility obtained when participating on the labor market is higher than the utility obtained when being outside of the labor market. In other words, an husband participates when Y H * it > 0. From this simple decision rule, wife's participation could change husband's own participation by two main channels. First, there is an income effect suggesting that a husband is less likely to participate if his wife supplies her labor. Indeed, as wife's participation acts as a preference shifters and increases to some extent the household's income available for consumption, all else being equal, the husband will be likely to enjoy more leisure. The income effect is always negative. Second, there is a substitution effect that requires two conditions for being effective. On the one hand, couples should make interdependent leisure decisions. On the other hand, it should exist some complementarity in leisure time. Put differently, this condition states that both the husband and the wife should prefer spending part of their leisure time together. As demonstrated by [START_REF] Stancanelli | Partners' leisure time truly together upon retirement[END_REF], this statement is empirically supported, at least on French data. More specifically, they show that wife's retirement, a close cousin of the notion of non-participation used in this paper, significantly increases couple's joint leisure time. Under these two conditions, the substitution effect implies that wife's participation pushes the marginal utility of husband's leisure time down leading ultimately to an increase in his labor supply. In contrast to the income effect, the substitution effect is expected to be positive.

Empirical model

Moving now to the data, the empirical model to be estimated can be written in linear terms as:

Y H it = γ H Y W it + X it β H + ε H it (2) Y W it = X it β W + δ W Z W it + ε W it (3)
with, X it a vector of covariates such as husband's age, the age gap between the husband and his wife, dummies for husband's education level, 6 the number of children at home, annual fixed effects and dummies indicating the region of residence. β is a vector of coefficients associated to covariates while ε it are standard error terms. The coefficient of main interest is the one associated to wife's participation γ H . It is a reflection of which of the two effects mentioned previously, namely the substitution effect and the income effect, prevails over the other one. If the sum of these two effects is negative (γ H < 0), then the income effect is stronger than the substitution effect and wife's activity decreases husband's participation. By contrast, if the sum of these two effects is positive (γ H > 0), then the substitution effect prevails over the income effect, and wife's activity is likely to increase husband own activity.

The estimated coefficient associated to wife's participation in equation ( 2) (γ H ) is probably endogenous because labor supply decisions within the couple is likely to be simultaneous. As a result, we estimate this system of two equations by the Instrumental Variable (IV) method and Bivariate Probit, the used instrument being Z W it . In order to be transparent, we use both strategies to estimate the marginal effect of interest. However, our favorite specification is the IV method. The IV estimator, which uses a Two-Stage Least Squares strategy, has several practical advantages for the problem in hand. In contrast to maximum likelihood models, it does not specify the distribution of error terms. It is relatively simple to implement and, as argued by [START_REF] Angrist | Mostly Harmless Econometrics: An empiricist's Companion[END_REF], when it comes to approximate marginal effects it works well. Then, this model does not specify any types of distribution for the endogenous variables. They may be censored, discrete or continuous [START_REF] Lewbel | Comparing features of convenient estimators for binary choice models with endogenous regressors[END_REF].

In the context of the IV model, two general assumptions are required for having valid instrument. First, it should have a significant predictive power on the endogenous variable. This assumption implies that the correlation between Z it and wife's participation should be different from 0 so that a difference in labor supply likelihood can be measured. Estimates of the first stage provide information about the validity of this assumption. The second assumption states that Z it should not be correlated with the error terms of equation (2) (exclusion restriction). Put differently, Z it must be itself exogenous and its effect on husband's labor supply acts only indirectly through its (direct) effect on wife's participation.

Along the lines of [START_REF] Schirle | Why Have the Labor Force Participation Rates of Older Men Increased since the Mid-1990s?[END_REF], Z it corresponds to a measure of cohort-specific participation rates of the wife at age 40, a cohort being defined by the wife's year of birth. Given the general increase in female's participation over the second half of the XXth century, it is expected that women belonging to cohort with high level of participation at age 40 are more likely to be "currently" active. In regards to our sample of married women, we could observe that those belonging to the very first cohorts did not have a full control of their motherhood. Until 1967 the access to the birth pill control were not free and abortion has been legalized (only) in 1975 with the "Veil law". By contrast, women of younger cohorts had free access to the "pill" and abortion so that they were more able to control when and how much children they want. Such changes across cohorts are probably useful for predicting 1945 1950 1955 1960 1965 1970 1975 the likelihood of current participation. In a seminal contribution [START_REF] Bailey | More power to the pill: the impact of contraceptive freedom on women's life cycle labr supply[END_REF] finds that the liberalization of contraceptive has a causal impact on female's labor market outcomes and especially their participation rate. Figure 2 shows the evolution of participation rates at 40 for women born between 1943 and 1976. Clearly, the likelihood of participating follows an upward trend suggesting that women born in the latter cohorts were more often active in the labor market at 40 than those born in the first cohorts. However, around the clear upward trend, we observe some "noisy" movements that could be a consequence of sample fluctuations. In section 5, we construct another measure of cohort-specific LFPR and we test the robustness of our results to this new instrument. Unfortunately, we do not have in hand any valid instruments to study the reversed effect, namely how husband's participation impacts wife's participation. Indeed, the same measure of cohort-specific participation rate for men does not display any trend and do not have any impact on his current likelihood of participating.

Results

When presenting the results, we follow the traditional approach which consists first in a description of the first stage equation, namely the wife's equation, and then we show results relative to the estimated equation of main interest.

The wife's equation

Results of the estimated equation linking wife's labor supply to covariates and the instrument Z it are reported in table 2. Several features of this table merits attention. It is shown that the cohort-specific participation rate induces a significant increase in female's current participation. On average, a 1 point increase in the cohort participation rate at age 40 pushes women likelihood of participation up of about 2.6 points. The estimated marginal effect is precise, significant and of the same order in both specifications. Moreover, including female's education level in the vector of control variables does not change the magnitude of the estimated effect. Even if there are no direct test for the validity of the instrumentation strategy, some clues suggest its appropriateness. First, F-statistics of first stage equations are largely above the threshold of 10 suggested by Stock and Yogo (2005). 7 This indicates that the instrument is non-weak. Second, as a further check for the validity of the exclusion restriction, we adopt a similar strategy than [START_REF] Fernández | Mothers and Sons: Preference Formation and Female Labor Force Dynamics[END_REF]. More specifically, to test whether the wife's cohort participation rate at age 40 has an impact on current husband's labor supply, we estimate both a Probit model and an OLS model of equation ( 2) but we add Z it as an additional covariate. A violation of the exclusion restriction will be suspicious in the event that, even after controlling for our set of covariates, Z it has a significant impact on older participation. In this case, a significant relationship reveals a direct channel which is beyond the impact of the instrument on wife's activity. Estimation results unveil that the estimated coefficient associated to the cohort-specific participation rate is 0.8

As regard to covariates, our estimates indicate that all of them have the expected sign. On average, the husband's age decreases wife's likelihood of participating. Combined with the fact that the estimated coefficient associated to the age differential within couples is significantly positive, this could be a piece of evidence in favor of a shared leisure effect. Living in an area other than Paris and having children at home decrease female's participation. Finally, it should be observed that the effect of husband's education level is significant only for the dummy indicating that he has no degree. As expected, wife's educational attainment is an important determinant of her participation to the labor force.

OLS

The husband's equation

The second stage estimation results linking participation of older married men with wives' participation and control variables are displayed in table 3. Due to space limitations, we do not report results using the Probit specification.9 The two OLS columns of the table consistently show a positive association between wife and husband own participation. In the other column, we use Z it as an instrument for female's participation. Such a strategy leads to a larger effect suggesting that OLS models are biased downward. This finding is quite surprising. Indeed, because of the assortative matching in the marriage market hypothesis, one could argue that individuals with similar preferences are more likely to match, leading ultimately to an upward bias in OLS estimates. To note, [START_REF] Schirle | Why Have the Labor Force Participation Rates of Older Men Increased since the Mid-1990s?[END_REF] finds that estimation results obtained without using the instrument are biased upward in the UK while any significant differences between specifications are observed in Canada and the US. Overall, in France, the estimated marginal effect of having a participating wife on husband's participation ranges between 22 and 28 points depending on the specification used. Again, the estimated effect is precise, significant and adding wife's education level as covariates does not change the magnitude of the marginal effect. Unambiguously, our estimates suggest that the shared leisure effect dominates over any income effects. This finding is in line with Stancanelli and Van Soest (2016) who finds, using French data, that spouses coordinate their retirement. Our findings could be seen as a refinement of their conclusion. Indeed, we show that interactions exist not only at the retirement stage but also at the end of the active life, just before the retirement. Results of table 3 also show that control variables also have a significant impact on the labor supply decisions of older married men. In particular, increasing the age of 1-year leads to a fall in man's activity of about 3 percentage points. This finding confirms that the age shifts individuals' preference in favor of leisure leading ultimately to an increase in the retirement probability. In contrast to their wife, the age gap between spouses has any incidence on husband's participation since the estimated coefficients are 0 in nearly all specifications. If the number of children at home decreases wife's participation, the opposite is observed for the husband. Educational attainment is another important factor in understanding participation behavior. In particular, older men with the highest education level, probably those with the highest opportunity cost of staying at home, are more likely to supply their labor. Finally, our model estimates suggest that wife's education level does not have strong effects on husband's participation decision. Indeed, estimated coefficients associated to dummies for wife's education are in general non-significant and when they are significant, the marginal effect is estimated to be low (see the last 3 columns of table 3).

OLS

Sensitivity analysis

Overall, estimates reported in table 3 indicate that the shared leisure effect dominate so that when his wife participate to the labor marker the husband is also more likely to participate. However, these findings could be sensitive to different modeling choices. In this subsection, we apply a battery of robustness checks to confirm the phenomena unveiled previously. Table 4 reports the results of our alternative estimations. It should be observed that to save some space, we report only the estimated coefficients associated to wife's participation. 10Employment as dependent variable In our baseline specification, the variable of main interest is the labor supply of the husband and the latter is explained, among others, by his wife's activity instrumented by her cohort LFPR at age 40. Although, this choice appears to be quite natural, it could be interesting to focus on another indicator of labor market attachment: the employment status. Indeed, as only a fraction of the participating wives are employed, we could imagine that for their husbands the shared leisure effect, the dominant effect in the baseline model, has more chances to be effective. In the current exercise, we replace labor market participation by employment. Corresponding estimates for the second stage are reported in row 2 of table 4.

Shared leisure and husband's age Another important robustness check is to establish if the results are of the same order with different age groups. In particular, as our baseline model suggests that the shared leisure effect is strong for married men aged between 50-64 years old, we could expect that the latter changes for younger men. For instance, we could imagine that preferences evolve with the age so that younger married men confer relatively less importance to the shared leisure value. Moreover, the income effect due to wife's participation could be more important for younger married men because on average their wives have more chances to be active. To address this potential issue, we run regressions by considering married men aged between 25-49 years old. Results from these new estimations are reported in row 2 of table 4.

Changing the cohort-specific LFPR In the baseline specification, we use as instrument the cohort participation rate at 40, a cohort being "strictly" defined by the exact women year of birth. As shown in figure 2, such a measure is a bit "noisy" and we are able to observe some weird variations around the clear upward trend. A potential explanation for such movements could be the relatively small number of observations belonging to each cohort. On average, there are 1045 available observations to compute the LFPR at 40. In order to ensure that the results are insensitive to such a noise, we compute a second measure of LFPR by changing the cohort definition. Specifically, a cohort is now defined as women year of birth ± 2 years. Doing so allows us to substantially increase the cohort size since the average cohort now contains 5200 observations. Figure 3 displays the alternative measure of cohort LFPR. As the first instrument, the second one displays a strong upward trend. However and as expected, the latter is smoother than the former. Row 4 of table 4 reports estimation results with this "extended" measure of cohort LFPR at 40 as instrument for wife's activity.

Sub-periods analysis Results of the causal effect presented in table 3 could be seen as an average over the 2003-2016 period. Implicitly, the model of equations ( 2) and (3) assumes that the impact of wife's labor supply on her husband's participation does not evolve over time. To investigate if the magnitude of this causal effect is the same or not throughout the sample period, we re-estimate two models by considering two sub-periods: i) the 2003-2006 period and ii) the 2013-2016 period. In the event that there are some changes in the influence of spousal participation on the participation of their husband, we are likely to observe a significant difference between the two γ H estimated. The post-2008 period As mentioned previously, older men participation rate followed a downward trend over the 2003-2007 periods before increasing again. Given that the increase in older men LFPR coincide with the beginning of the Great Recession in France, one could argue that such a pattern may disturb our general results. To address this issue, we run regressions by considering the post-2008 period. Results are displayed in the last row of table 4.

Comments As shown in table 4, the magnitude of the marginal effect of interest is quite insensitive to our set of robustness check. For each specification, the estimated marginal effect is highly significant. Overall, this complementary findings reinforce the idea that, on French data, the willingness to share leisure dominates over any income effects. However and in some case, the sensitivity analysis provides a refinement of the strength of the causal impact especially in regards to husband's age and sub-periods.

Changing the variable indicating labor market attachment confirms our initial intuition. Thus, the likelihood of being employed for married men increases of about 30 points when his wife is herself employed (Bivariate Probit estimates). Again, this finding clearly supports the shared leisure hypothesis. The definition of the cohort has no incidence on the marginal 1945 1950 1955 1960 1965 1970 1975 effect of interest. Indeed, estimated coefficients between this model and the baseline one are not significantly different. This suggests that the "noise" observed in figure 2 is not a matter of concern for the problem in hand. Now let me focus on the strength of the effect with the age of the married men. The third row of table 4 suggests that having a participating wife increases labor force participation of younger married men of 3 points. Even if the estimated marginal effects are significant, they are sharply lower compared to what we see in the baseline case. Such a finding indicates that preferences of prime-age men are different from those of their older counterparts. Thus, it is possible to conjecture that either the shared leisure effect is weaker or the income effect is stronger (or a combination of the two effects). When estimating the model for two different sub-periods, it seems that the magnitude of the causal effect changes over time. However, the estimates do not provide sufficient statistical power to strongly affirm this statement. Finally, estimations of the model by considering only the post-2008 lead to nearly the same results.

Concluding remarks

Working with data from the quarterly and the annual French Labour Force Survey, we investigate the impact of having an active partner on older men labor market behaviors in France over the 2003-2016 period. To provide a causal interpretation between the two phenomena, we estimate both Two-Stage Least Squares and Bivariate Probit using the cohort-specific participation rate of women at 40 as an exogenous variation in wives' labor supply. The construction of the instrument confirms the general increase in female's participation over the sample period. Estimates of the wife's equation indicate that a 1 point increase in cohort participation at 40 pushes current women likelihood of participation of about 2.5 points. Then, the marginal effect of interest, estimated with the second stage husband equation, is shown to be quite strong. More specifically, having an active wife induces an increase in male participation of 28 points (resp. 22) when Two-Stage Least Squares (resp. Bivariate Probit) are used. Then, we test for the sensitivity of the results by changing the dependent variable, the age group of interest and the measure of cohort-specific participation rate. Each time, our finding is confirmed and the causal effect of having a participating wife on her husband is estimated to be important.

All in all, this study should be seen as a further step in understanding the relationship between labor market decisions within couples. In particular, it suggests that, at least, husbands adjust their labor market behaviors to the decisions of their wife. In this respect, our finding is important from the policy maker viewpoint. Indeed, it suggest that, when implementing reforms changing the economic environment of elderly, policy designers should focus on the joint modeling of labor force participation decision in the household, especially because such modifications have both direct and indirect effects on the second partners.

Figure 1 :

 1 Figure 1: Labor force participation and employment rate of men and women aged 50-64 y.o. -France.

Figure 2 :

 2 Figure 2: Labor force participation rate at 40 of women born between 1943 and 1976. -France. Sources: French Labour Force Survey (1983-2016), author's own calculations.

Figure 3 :

 3 Figure 3: Labor force participation rate between 38-42 years old of women born between 1943 and 1976. -France. Sources: French Labour Force Survey (1983-2016), author's own calculations.

Table 1 :

 1 Summary statistics of older married men (50-64 y.o.)

		Married men	Wife in LFP	Wife not in LFP
		2003	2016	2003	2016	2003	2016
	Activity	0.68	0.69	0.76	0.79	0.53	0.49
	Wife's activity	0.64	0.67	1.00	1.00	0.00	0.00
	Age	55.4	56.9	54.7	56.0	56.6	58.8
	Wife's age	51.6	54.3	50.8	52.7	53.1	57.7
	Children in household	0.46	0.44	0.49	0.49	0.42	0.32
	Nb. children	0.81	0.74	0.80	0.82	0.81	0.57
	No degree	0.36	0.23	0.34	0.20	0.39	0.30
	< High school	0.37	0.43	0.39	0.44	0.35	0.40
	High school	0.10	0.12	0.11	0.13	0.09	0.11
	High school +2	0.05	0.08	0.06	0.09	0.04	0.07
	> High school +2	0.11	0.13	0.10	0.14	0.12	0.12

Sources

: French Labour Force Survey (2003[START_REF] Stancanelli | Partners' leisure time truly together upon retirement[END_REF]

, author's own calculations.

Table 2 :

 2 Wife's equation -"first stage" estimation. Standard errors are reported in parenthesis. Significant levels: * 10%; ** 5%; *** 1%. Samples contain wifes of married men aged between 50 and 64 years old. Marginal effects are mean marginal effects. Annual fixed effects are integrated in each regressions but they are not reported in the table.

	Sources: French Labour Force Survey (2003-2016), author's own calculations
	Notes:

Table 3 :

 3 Husband's equation -"second stage" estimation.

	Sources: French Labour Force Survey (2003-2016), author's own calculations.
	Notes: Standard errors are reported in parenthesis. Significant levels: * 10%; ** 5%; *** 1%. Samples contain married men aged between
	50 and 64 years old. Marginal effects are mean marginal effects. Annual fixed effects are integrated in each regressions but they are not
	reported in the table.

Table 4 :

 4 Robustness estimation -"second stage" estimation.

Sources: French Labour Force Survey

(2003)(2004)(2005)(2006)(2007)(2008)(2009)(2010)(2011)(2012)(2013)(2014)(2015)(2016)

, author's own calculations. Notes: Only parameters associated to wife's participation are reported. Standard errors are reported in parenthesis. Significant levels: * 10%; ** 5%; *** 1%. Except in row 3 and 4, samples contain married men aged between 50 and 64 years old. Marginal effects are mean marginal effects. Annual fixed effects are integrated in each regressions but they are not reported in the table.

Strictly speaking, the LFPR of older men falls between

and 2007 before increasing again. The sensitivity analysis conducted in this paper indicates that such a break is not a matter of concern for the question under investigation in this paper.2 The same pattern could be observed for the employment rate. In 1983, the gender gap was approximately equal to 28 points. It fell to 5 points in 2016.

It should be observed that we focus on the effect of wives' participation on the labor market behaviors of their husbands. Given that we do not have any instruments affecting exogenously male labor market status, the reversed effect, namely the causal effect of husbands' activity on their wives behaviors, can not be investigated.

Strictly speaking, the quarterly FLFS is also employed to construct measure of cohort participation between 2003 and 2016.

Depending on the year of birth, the reforms requires that people need to work more months to retire with maximum pension benefit.

In a more complete model, dummies for wife's education level will be added.

In particular, they are equal to 763 and 696 in each IV specification.

Detailed results are available upon request.

However, they remain available upon request.

Complete tables are available upon request.


Salah Zrigui

Raphael Y De Camargo

Denis Trystram

Arnaud Legrand

Improving the Performance of Batch Schedulers Using Online Job Size Classification

Job scheduling in high-performance computing platforms is a hard problem that involves uncertainties on both the job arrival process and their execution time. Users typically provide a loose upper bound estimate for job execution times that are hardly useful. Previous studies attempted to improve these estimates using regression techniques. Although these attempts provide reasonable predictions, they require a long period of training data. Furthermore, aiming for perfect prediction may be of limited use for scheduling purposes.

In this work, we propose a simpler approach by classifying jobs as small or large and prioritizing the execution of small jobs over large ones. Indeed, small jobs are the most impacted by queuing delays but they typically represent a light load and incur a small burden on the other jobs. The classifier operates online and learns by using data collected over the previous weeks, facilitating its deployment and enabling fast adaptations to changes in workload characteristics.

We evaluate our approach using four scheduling policies on six HPC platform workload traces. We show that: (i) incorporating such classification reduces the average bounded slowdown of jobs in all scenarios, and (ii) the obtained improvements are comparable, in most scenarios, to the ideal hypothetical situation where the scheduler would know the exact running time of jobs in advance.

I. INTRODUCTION

With the ever-increasing demand for computational resources, HPC platforms keep evolving and getting larger and more complex [START_REF]TOP500 Supercomputer Sites[END_REF], which instigates the need for more adaptive and elaborate scheduling schemes.

One way to deal with such complexity is to develop sophisticated ad-hoc scheduling algorithms that are often too situational, non-generalizable, and hard to reason about. Another, more appealing alternative, is to use a generic scheduling scheme like EASY-backfilling [START_REF] Mu | Utilization, predictability, workloads, and user runtime estimates in scheduling the IBM SP2 with backfilling[END_REF] coupled with some index policy. An index policy is a function that considers one or more job characteristics (such as arrival time, processing time, and requested resources), and performs an ordering based on a function of these characteristics. Two notable examples are First Come First Served (FCFS) and Shortest Processing time First (SPF) [START_REF] Srinivasan | Characterization of backfilling strategies for parallel job scheduling[END_REF].

One major problem with the aforementioned approach is that many crucial job characteristics, such as job runtimes, are unknown a priori. HPC platforms usually require their users to supply an upper bound to how long a job will execute on the platform. The scheduler also uses this estimate to define which job to include in the backfilling queue and, with policies such as SPF, job ordering in the main execution queue.

Job runtimes estimates are known to be inaccurate at best. Years of practice and experience have shown that users tend to overestimate their job execution times by a large margin, which reduces the effectiveness of backfilling. Considerable research has been conducted to improve the accuracy of these estimates.

Several studies have shown that it is possible to use learning schemes to generate better estimates of job execution times [START_REF] Tsafrir | Backfilling using systemgenerated predictions rather than user runtime estimates[END_REF]- [START_REF] Carastan-Santos | Obtaining dynamic scheduling policies with simulation and machine learning[END_REF]. However, predicting job execution times using historical data present in workload logs is difficult [START_REF] Michael | This is why ml-driven cluster scheduling remains widely impractical[END_REF]. These logs usually do not contain important information, such as application name and parameters, data input information, and job structure. Moreover, runtime information such as job placement and machine utilization are available only a posteriori. Finally, existing predictions frequently underestimate job execution times, causing their premature killing after they occupied the platform for a considerable time.

In this work, we propose a different approach. Instead of trying to predict the exact value, we propose a simple classification of the jobs into two classes: small and large. The small class contains all the jobs whose actual runtimes are short, regardless of user-supplied estimates. The large class encompasses all other jobs. We use an online machine learning classification algorithm that exploits user-specific information to learn whether jobs are small or large. We propose a modified version of the EASY algorithm that considers this classification by treating small jobs as high priority jobs.

We perform a thorough evaluation with six full workload traces from HPC platforms and four scheduling policies. We show that:

• Simple binary classification of jobs as small or large is sufficient to improve the performance of all evaluated scheduling policies in all evaluated workload traces; • The use of a simple safeguard mechanism that kills large jobs misclassified as small enables scheduling improvements similar to those obtainable with a perfect job size classifier; • Our scheme, combining job size classification and safeguard mechanism, provides improvements close to that obtainable using fully clairvoyant schedulers, with perfect knowledge of actual job execution times. The remainder of this paper is organized as follows. In Section II, we present related studies. In Sections III and IV, we discuss the reasoning and the method used for the classi-fication. Section V shows the experimental protocol, followed by the experimental evaluation (Section VI). Finally, in Section VII, we present the paper conclusions.

II. RELATED WORK

Researchers have used machine learning using two major approaches, which are: (i) improving runtime estimates, and (ii) designing or selecting scheduling policies.

The first approach consists in using machine learning techniques to improve runtime estimates. Feitelson et al. introduced EASY++, a variation of the classical EASY, which replaces user-provided runtimes estimates by the average runtime of the two previous jobs from the same user [START_REF] Tsafrir | Backfilling using systemgenerated predictions rather than user runtime estimates[END_REF]. Despite its simplicity, it allowed for improvement of around %25 over the classical EASY algorithm. Gaussier et al. used historical data from different traces and linear regression to predict runtimes with improved accuracy [START_REF] Gaussier | Improving backfilling by using machine learning to predict running times[END_REF]. They also showed that prediction could be used more effectively if coupled with a more aggressive backfilling heuristic (namely SPF). But prediction based approaches frequently suffer from the problem of underestimation of running times. Guo et al. proposed a framework that can be used to detect runtime underestimates [START_REF] Guo | Machine learning predictions for underestimation of job runtime on hpc system[END_REF], allowing it to adjust job running times.

An interesting phenomenon is that, increasing the inaccuracy (e.g., doubling the user-provided estimates) sometimes improves performance [START_REF] Zotkin | Job-length estimation and performance in backfilling schedulers[END_REF]. Such surprising behavior is related to Graham's scheduling anomalies and stems from the fact that index policies generally produce suboptimal scheduling. The policy used for scheduling had a major impact on the effectiveness of accurate predictions, with policies that favor shorter jobs benefiting more. More recent results [START_REF] Gaussier | Improving backfilling by using machine learning to predict running times[END_REF] show that, in some cases, predictions (which always have some inaccuracy) outperform their clairvoyant counterparts despite the latter's perfect knowledge of runtimes. During our own studies, we also encountered similar situations (especially when using workload resampling) but this remained an overall statistically insignificant effect.

In a recent study [START_REF] Michael | This is why ml-driven cluster scheduling remains widely impractical[END_REF], the authors explored the effectiveness and limitations of using machine learning to improve the performance of computing clusters. They show that the workload is highly variable among periods, with large user churn and changes in machine utilization levels, and that a few users generate most workload. Consequently, model performance can vary strongly on a day-to-day basis. Moreover, more accurate runtimes do not systematically lead to better scheduling performance, and with the few datasets available today, it is difficult to assess the model performances. Finally, they argue that training can take many months (or years) before it reaches a stable level when using a few features, which would prevent practical deployments.

Using the second approach, Carastan-Santos and Camargo used synthetic workloads and simulations to determine nonlinear regression functions that improve the slowdown metric [START_REF] Carastan-Santos | Obtaining dynamic scheduling policies with simulation and machine learning[END_REF], generating functions that resemble the Smallest Area First (SAF) policy. Zrigui et al. showed that using a linear combination of job characteristics allowed to build index policies that can significantly improve systems performance [START_REF] Legrand | Adapting Batch Scheduling to Workload Characteristics: What can we expect From Online Learning?[END_REF]. Yet, the authors show that the continuously changing nature of the data makes it very hard to learn online optimal weights for this linear combination and prevents any static policy to be fully effective. Sant'Ana et al. addressed the evolving nature of the workload by using machine learning techniques to select, in real time, the best scheduling policy to apply for the next day on a given cluster, based on the current cluster state [START_REF] Sant'ana | Real-time scheduling policy selection from queue and machine states[END_REF]. These attempts generated promising results but require system administrators to change fundamentally the scheduling policies in existing clusters.

In this work, we propose a simpler approach, which consists in classifying jobs into two classes: small and large. The objective is to allow faster training and adaptations to changes in the workload characteristics, while avoiding other issues of runtime predictions, such as underestimations of runtimes.

III. CHARACTERIZATION OF SMALL AND LARGE JOBS

A. Preliminary definitions

We use a data-driven approach, which relies on the characterization and identification of workload patterns from execution logs (traces) of HPC platforms. To ensure that our approach can be generalized and is not specific to a particular cluster or machine, we systematically studied datasets from six HPC platforms available from the Parallel Workloads Archive [START_REF] Feitelson | Experience with using the parallel workloads archive[END_REF], and whose main characteristics are shown in Table I. In this work, we adopt the simplest model of a HPC job as a rectangle, representing the runtime (width) and the number of requested resources (height). For each job j, we consider the following characteristics:

• The actual runtime p j , which is known only after job completion; • The estimated runtime pj , provided by the user at job submission and which is an upper bound of p j ≤ pj ; • The number of requested processors q j , which is static and provided by the user at job submission; • The submission time r j , also known as release date. Job runtime distributions change from one system to another, and building a unified runtime distribution model has proven to be a challenging task [START_REF] Lublin | The workload on parallel supercomputers: Modeling the characteristics of rigid jobs[END_REF]. Yet, the density of estimated runtimes for the six traces shows one or two peaks at small values, showing that most jobs have small processing time estimates (Figure 1, upper row). Other peaks also appear in some traces, with some containing a peak near the maximum allowed processing time. However, when comparing to the actual runtimes (Figure 1, bottom row), we can easily see the well-known mismatch between the estimated and actual runtimes.

But evaluating the actual runtimes of the six traces, we can also notice that they share an interesting similarity, with all distributions having a sharp peak at the small values and a large tail towards longer execution times. This distribution indicates that we could divide jobs into two classes: (i) small, encompassing the jobs at the peaks of the distributions, and (ii) large, comprising jobs at the tails of the distribution.

B. Job classes and their impact

We first examine the characteristics and impact of small jobs in HPC platform usage. To perform this examination, we must first define where the small job class ends, and the large begins. We applied two clustering algorithms, DBSN [START_REF] Ester | A density-based algorithm for discovering clusters a density-based algorithm for discovering clusters in large spatial databases with noise[END_REF] and EM [START_REF] Dempster | Maximum likelihood from incomplete data via the em algorithm[END_REF], to divide the classes into two groups. Both generated similar division, shown as a dashed black line in Figure 1. We also considered a simpler scheme, by using the median of the actual runtimes, represented by the solid green line on Figure 1. Although the divisions are not the same, we aimed for simplicity and considered that the median is sufficient to separate the initial peak from the rest of the distribution.

We considered a job as small if its runtime was smaller than the median (divider), and consider the job as large otherwise. We can further divide the small job class into two subclasses: (i) non-premature small jobs: short jobs that also had estimated runtimes smaller than divider, and (ii) premature small jobs: short jobs that had estimated runtimes larger than divider.

Table II shows the percentage of jobs from each class in the various platforms we tested. We can see that, there is always III shows that the total summed runtime and area1 of these premature jobs occupy a negligible portion of the total runtime and area (less than 0.5%).

Premature small jobs have a wildly over-estimated processing time, causing them to wait longer for execution, which results in large slowdown values. If one could correctly detect these premature small jobs, we would obtain a significant reduction in the overall average slowdown in the platform. In the next section, we propose a method for performing this classification.

IV. JOB SIZE CLASSIFICATION

The objective of the job size classifier is to map a set of job features into a class: small or large. We defined the features to use for classification based on two observations: (i) the runtime of a job is highly correlated with the user submission history, and (ii) users often submit more than one job type [START_REF] Michael | This is why ml-driven cluster scheduling remains widely impractical[END_REF]. One can consider that two jobs are in the same category when they have either the same requested processing time, requested number of resources, or submission date. In particular, the runtime of any given job is highly correlated with the runtime of the previous jobs that belong to the same category, as shown in Figure 2, with jobs closer in time having a higher correlation. Consequently, the class (small or large) of the previous jobs that belong to the same user and the same category are an important factor to predict the class of a submitted job.

Based on these observations, we decided to use the following features, shown in Table IV, for classification:

• Lag features: contains the class of the previous three jobs from the same category submitted by the user. • Aggregation feature: contains the percentage of jobs of the same category and submitted by the same user that belongs to the small class. This feature aggregates information from older jobs. • Temporal features: The hour of the day, the day of the week, the month, the week, and the quarter in which a job was submitted; • Job features: The estimated execution time and requested number of resources of the job. We used Random Forests [START_REF] Breiman | Random forests[END_REF] to perform the classifications as they allow to gracefully combine numerical and categorical features. Random forests work by creating multiple decision trees on randomly selected data samples, getting a prediction from each tree, and selecting the best solution by majority voting. Moreover, random forest

We measure the quality of our classifications using the three following indicators:

• Accuracy is the ratio of correctly predicted observation -TL (True Large): jobs correctly predicted as large -TS (True Small): jobs correctly predicted as small -FL (False Large): jobs incorrectly predicted as large -FS (False Small): jobs incorrectly predicted as small

• Precision is the ratio of correctly predicted small jobs to the total number of jobs that are predicted to be small: precision = T S T S+F S

• Recall is the ratio of correctly predicted small jobs to all observations in the small class: recall = T S T S+F N

Table V shows the results of applying Random Forest with the features presented in Table IV. These are the mean values obtained over all classifications performed during the experimental evaluation (Section VI). For all evaluated traces the value of three indicators was always above 80% The two types of prediction errors are false large (FL) and false small (FS). False large errors decrease the effectiveness of our approach since it causes small jobs to wait in the queue with large jobs, but causes no harmful effects. False small classifications are a more significant problem since a large job would be executed earlier, causing delay to true small jobs. We prevent this situation by killing false small jobs, i.e., jobs that execute for more than a specified time limit for small jobs.

V. EVALUATION FRAMEWORK

We tried to be as transparent as possible and to make our work as reproducible as possible [START_REF] Stodden | Implementing Reproducible Research[END_REF]. We provide a snapshot of the workflow we used throughout this work as a link to a git repository 2 , which includes a nix [START_REF] Dolstra | Imposing a memory management discipline on software deployment[END_REF] file that describes all the software dependencies and an R notebook that allows regenerating all the figures.

We consider HPC platforms as a collection of homogeneous resources, with jobs arriving at a centralized waiting queue, following the submission described in the workload logs. We implemented all simulations using Batsim [START_REF] Dutot | Batsim: A realistic language-independent resources and jobs management systems simulator[END_REF], a simulator based on SimGrid [START_REF] Casanova | Versatile, scalable, and accurate simulation of distributed applications and platforms[END_REF] that allows evaluating the behavior of scheduling algorithms under different conditions. We evaluate our method using the 6 traces presented in Table I.

A. Scheduling policies

We considered four scheduling policies:

• FCFS: First Come First Served [START_REF] Mu | Utilization, predictability, workloads, and user runtime estimates in scheduling the IBM SP2 with backfilling[END_REF] orders the jobs by their arrival time r j . FCFS is the most commonly used scheduling policy. • WFP: is a scheduling policy adopted by the Argonne National Labs [START_REF] Tang | Analyzing and adjusting user runtime estimates to improve job scheduling on the blue gene/p[END_REF] and is given by: W F P j = ( waitj pj ) 3 * q j . This policy attempts to strike a balance between the number requested resources, the estimated runtime and the waiting time of jobs. It puts emphasis on the number of requested resources while preventing small jobs from waiting too long in the queue.

• SPF: Shortest estimated Processing time First [START_REF] Srinivasan | Characterization of backfilling strategies for parallel job scheduling[END_REF] orders the jobs by the estimated processing time ( pj ) given by the user. • SAF: Smallest estimated Area First [START_REF] Carastan-Santos | One can only gain by replacing EASY Backfilling: A simple scheduling policies case study[END_REF] orders jobs by their estimated area ãj = pj * q j .

We chose FCFS and WFP because several existing HPC systems use them. SAF and SPF are less common, mostly because they are perceived as too risky since they could potentially induce job starvation. Starvation occurs when a job waits for an indefinite or a very long time in the queue. Yet, some studies [START_REF] Legrand | Adapting Batch Scheduling to Workload Characteristics: What can we expect From Online Learning?[END_REF], [START_REF] Carastan-Santos | One can only gain by replacing EASY Backfilling: A simple scheduling policies case study[END_REF] show that SAF and SPF provide better results on performance metrics in almost all cases. Furthermore, one can prevent starvation by putting a threshold on the waiting time [START_REF] Lelong | Tuning EASY-Backfilling Queues[END_REF]. When the waiting time of a job surpasses the threshold, the scheduler transfers the job to the head of the queue.

We implemented the four aforementioned policies in conjunction with the EASY [START_REF] Mu | Utilization, predictability, workloads, and user runtime estimates in scheduling the IBM SP2 with backfilling[END_REF] backfilling heuristic. When requested, the scheduler selects for execution the next job in the queue, ordered by the scheduling policy. The first time, the scheduler reaches a job that cannot start immediately, due to the immediate lack of resources, it makes a reservation for that job. Then, it schedules the next jobs in the queue that can execute to completion without delaying the job with the reservation.

2 https://gitlab.inria.fr/szrigui/job classification/

B. Learning and scheduling algorithm

When a user submits a job for execution, the classifier uses the job features to assign it to the small or large classes, represented by queues Q small and Q, respectively. In the first week, since the classifier does not have prior data to learn the classification task, it classifies all jobs as large and does not behave differently from a classical policy. After that, we update the classifier at the beginning of every week, with data from all previous weeks. The training has three steps. First, we extract job information from the execution logs (line 1 from Algorithm 1). Then we determine the divider (line 2) using the median of the past job true execution times. Finally, we use the extracted features and job classes to train the classifier (line 3). The resource manager calls the scheduler whenever a job finishes its execution, and computational resources become available. The scheduler then sorts the two queues Q and Q small independently, according to a chosen policy (FCFS, SAF, SPF, or SQF), and merge them in a single queue Q total , with the jobs belonging to the small class first. Finally, resource allocation is done using the EASY heuristic, as shown in Algorithm 2.

Algorithm 2: Scheduling

Input : Queue of large jobs Q Queue of small jobs Q small Scheduling policy P olicy (FCFS|WFP| . . . ) 1 Order Q according to P olicy 2 Order Q small according to P olicy 3 Q total = concat(Q small , Q) # small jobs are always put in the head of the final queue 4 EASY(Q total ) # Schedule the jobs in the final queue using the EASY heuristic

The only extra relevant overhead compared to the EASY scheduling policies are the job classification into classes small or large, which takes a few hundred milliseconds, and the classifier updating, which takes longer. The update includes finding the median execution time over the workload log of the previous week and training the classifier using the pairs f eatures, jobclass. The full execution of this procedure (Algorithm 1) takes only a few seconds and occurs only at the end of every week. Moreover, it runs independently from the scheduler, without blocking it.

False Small Jobs: Some large jobs can be wrongly classified as small by the classifier (false small jobs). Although the resource manager may allow these jobs to execute until completion, this would delay the execution of true small jobs. We employ the policy of killing these jobs when the execution time reaches the divider value. We consider that jobs are idempotent, which means they can be killed and restarted without changing the final execution outcome. The scheduler periodically goes through the list of running jobs (Algorithm 3). If it detects a job classified as small and has been executing for a period longer than the divider value, it kills the job and classifies it as large.

If jobs are not idempotent, the resource manager can allow them to execute until completion. The difference in the performance of the two approaches is discussed in detail in Section VI-D.

C. Evaluation metric

There exist several cost metrics, and each evaluates the performance of specific aspects of HPC platforms [START_REF] Feitelson | Metrics and benchmarking for parallel job scheduling[END_REF]. We use the bounded slowdown (bsld) metric, which represents the ratio between the time a job spent in the system and its running time. The reasoning behind the slowdown metric is that the response time of a job should be proportional to its runtime. Indeed, it would not seem fair to delay equally short and long jobs. Formally, it is defined as:

bsld j = max wait j + p j max(p j , τ ) , 1
The value wait j is the time the job spent in the queue, p j is the actual execution time, and τ is a constant that prevents short job times from generating large slowdown values. We set τ to 60 seconds. In this work, we use focus on the cumulative bounded slowdown, which is computed as the sum (resp. mean) of bsld of all the job from the beginning of the execution until the current time and is updated every time a job arrives.

VI. EXPERIMENTAL RESULTS

In Section IV, we presented the job size classifier and showed its accuracy from a pure learning perspective. However, achieving a high-quality classification is not our final goal. In the scheduling context, the effectiveness of an approach is measured by how much it improves end-to-end performance metrics, such as the average bounded slowdown.

A. Overall impact on scheduling performance

We evaluated the evolution of the cumulative bounded slowdown when applying the EASY-backfilling with the 4 scheduling policies (FCFS, WFP, SAF, and SPF). Figure 3 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q MetaCentrum-zegox SDSC-BLUE SDSC-SP2 shows the results for the scenarios with the job size classification and job-killing mechanism (in cyan) and without them (in black).

Comparing the curves for the four basic scheduling policies, we note that different SPF and SAF generated the lowest cumulative slowdown in all platforms. WFP had cumulative values close to SPF and SAF, while FSCS had the worst values by a large margin in all cases. These results are consistent with previous comparisons of scheduling policies [START_REF] Legrand | Adapting Batch Scheduling to Workload Characteristics: What can we expect From Online Learning?[END_REF], [START_REF] Lelong | Tuning EASY-Backfilling Queues[END_REF].

Applying the job size classification reduced the cumulative slowdown values in all scenarios, with the improvement depending on the trace and scheduling policy. For FCFS, we observed substantial improvements for all six traces, ranging between 33% to 79%. For the other policies, we observed smaller, albeit consistent, improvements in performance, ranging from 3% to 32% for SPF and 10% to 51% for SAF. We explore these results further in Section VI-E.

The cumulative slowdown increases most of the time smoothly, with some sharp rises. The slower increments occur during lightly or moderately loaded periods, in which we see steady increments in the gap between the scenarios with and without job size classification. The jumps are the result of high load periods and seem unavoidable, as they occur with all policies. But regardless of the policy and the trace used, our method always results in smaller cumulative slowdowns.

Since FCFS performed poorly compared to other policies, we decided to exclude it from the subsequent analysis. However, we note that the observations in the next sections also apply to FCFS.

B. Individual month improvement

The evolution of the cumulative bounded slowdown over long periods, although informative, can mask important details about the behavior of a scheduler at a smaller time scale, such as individual weeks or months. Ideally, a good scheduler should provide improvements somewhat equitably distributed throughout the evaluation period.

We investigate the effects of our approach on individual months in Figure 4. Each pair of connected points represents . Impact of using the classification-idempotent scheduler on the average bounded slowdown of small and large jobs, when compared to the base scheduler.

q q q q q q q q q q q q q q q q q q MetaCentrum-zegox the average bounded slowdown of a single month from the full workload execution, for the base and classification-idempotent cases. We note a reduction in the slowdown in most cases, with a decrease proportional to the base value. There are a few months where our approach seems to substantially degrade performance, such as in MetaCentrum-zegox/WFP. These are artifacts that emerge from splitting the results into one month periods, where workloads "spills" from one month to the other during periods of very high load. Overall, the results show that improvements are fairly distributed between months, even for the clusters that had large jumps in the cumulative slowdown, such as MetaCentrum-zegox and HPC2N.

C. Impact of prioritizing small jobs over large jobs

Our algorithm reduces the overall bounded slowdown by prioritizing small jobs. One crucial question is: what is the impact of favoring small jobs over the jobs classified as large?

We compute the average bounded slowdown for the jobs from each of the two classes (Figure 5). As expected, the small jobs had the most substantial reductions in the average slowdowns. The extent of the reduction is dependent on the platform and policy and is mostly proportional to the improvements in the cumulative bounded slowdown, shown in Figure 3. More importantly, there is only a small increase in the average slowdown of large jobs.

The use of job size classifier results in extensive improvements for small jobs, with little or no impact on large jobs. Consequently, we argue that there are no perceivable hidden costs for large jobs when prioritizing small jobs.

D. Impact of removing the job-killing mechanism

Assigning a large job to the small class can cause an overall increase in the average bounded slowdown of other jobs since it occupies resources for an extended period. We prevent this by killing the job when it reaches the job size divider value. But we cannot apply it for non-idempotent jobs. A question that arises is: can we still improve performance if we allow miss-classified jobs to run until completion?

We compared the cumulative bounded slowdown values at the end of the full workload trace simulations, for the six platforms, for three scenarios: (i) base, (ii) classificationidempotent, where we kill false small jobs, and (iii) classification, where we use classification without job-killing.

Preventing job-killing reduces the effectiveness of the classification in almost all scenarios (Figure 6). Scheduling large jobs ahead of others risk delaying all the jobs that come after in the waiting queue, especially true small jobs whose slowdown value can increase very quickly.

We note, however, that classification without job-killing still managed to improve the total slowdown for most cases, but to a lesser extent than classification-idempotent. The exceptions are the combinations where the classification-idempotent only managed to improve results by a small margin. In these cases, the classification without job-killing did not improve or caused very small degradations in performance. We conclude that removing the safeguard mechanism significantly reduces the effectiveness of our method without rendering it completely useless.

E. Comparison with clairvoyant

Finally, we evaluate what would be the improvements obtained by schedulers that would know in advance the actual execution time of each job. We compared three strategies that build the base policies (SPF, SAF, and WFP): (i) runtimesclairvoyant, where the scheduling heuristic is provided with the actual p j , instead of the requested (p j , processing times, (ii) class-clairvoyant, where the scheduler is indicated which class the jobs belong to (i.e., as if a perfect job class classification was achieved), and (iii) classification-idempotent, the method we propose and which only uses estimated execution times. Although the clairvoyant versions cannot occur in practice, they provide us with an upper bound on the achievable improvements.

Using the classification-idempotent results in improvements comparable to the class-clairvoyant (Figure 7), except for MetaCentrum-zegox. This result indicates that the job-killing q q q q q q q q q q q q q q q q q q MetaCentrum-zegox SDSC-BLUE SDSC-SP2 mechanism is effective in counteracting the misclassifications and that the overhead of job-killing has a small impact on performance. Moreover, it shows that our strategy of combining classification with job-killing is already very efficient and has little room for further improvements. The two clairvoyant strategies, class-clairvoyant and runtimes-clairvoyant, also had comparable performance, with slightly better results when using runtimes-clairvoyant. This result shows that a simple classification in two categories is, in most cases, sufficient to obtain important improvements in the bounded slowdown metric. It indicates that trying to accurately predict job execution time with elaborate regression techniques will not bring large improvements over the use of a simpler binary job size classification.

The most notable exception to the conclusions above is the MetaCentrum-zegox trace, where there are consistent improvements when moving from base to classification-idempotent, class-clairvoyant, and runtimes-clairvoyant. For this particular trace, there were several jumps in the cumulative bounded slowdown (Figure 3), caused by abnormally high loads. In these situations, a perfect knowledge of execution times appears to have a larger impact on scheduling performance.

Finally, we look at the cases were class-clairvoyant provided minor improvement: SDSC-SP2/SPF and KTH-SP2/SPF. In Figure 7 we can see that even with full knowledge no significant improvements where made. class-clairvoyant only improved over base SPF by 10% and 13% for SDSC-SP2 and KTH-SP2 respectively indicating that, for these two traces, SPF was already a very good policy.

VII. CONCLUSIONS

Scheduling parallel jobs is a hard problem in general, especially in online contexts, where important information, such as job execution time, is imprecise or missing. Predicting job execution time from the limited information provided by the platform is challenging and often generates only imprecise estimates. In this work, we show that a more straightforward classification of jobs into small and large classes is sufficient for improving scheduling performance. A simple safeguard mechanism that kills large jobs misclassified as small is important to prevent these jobs from delaying the others. Since the misclassification is detected very early, when the job execution time reaches the divider value between classes, it has a small overhead over the average slowdown metrics. We obtained improvements in scheduling performance for all combinations of six workload traces and four scheduling policies evaluated. Moreover, in most scenarios, we managed to obtain improvements in scheduling performance similar to that of clairvoyant schedulers with perfect knowledge of job execution times.

Furthermore, we can compare the improvements obtained by our approach with two regression-based approaches: EASY++ [START_REF] Tsafrir | Backfilling using systemgenerated predictions rather than user runtime estimates[END_REF] the one proposed by Gaussier et al. [START_REF] Gaussier | Improving backfilling by using machine learning to predict running times[END_REF]. Indeed, they also used the workload traces from SDSC-BLUE, SDSC-SP2, KTH-SP2, and CTC-SP2 and they report the improvement over the base EASY-backfilling with FCFS ordering policy (see Table VI). Although there are a few methodological differences between our evaluations, our classification approach combined with FCFS reduces the cumulative bounded slowdown by 50-79%, compared to 29-47% from EASY++, and 05-59% from Gaussier et al.. Relying on SPF instead of FCFS allows to decrease the cumulative bounded slowdown even further but most of the gain is provided by the classification mechanism.

These results indicate that a classification approach can be more effective than using regression for improving scheduling performance. Compared to regression-based techniques, our approach has two major advantages: (i) a two-class classification task is easier to learn than regression, requiring less training data, and (ii) misclassification of large jobs as small is detected very quickly during execution, opposed to regression, where underestimates are evident only after the job executed for the entire estimated period. Consequently, we believe that using the proposed scheme of job size classification is more appropriate for deployment in real HPC platforms than regression-based approaches.

Although distinguishing small jobs from large ones proved effective, we believe this classification is too rough. As a future work, we intend to identify other classes of jobs (e.g., long and thin jobs or series of jobs from a given group and whose duration is multi-modal) that could benefit from a specific treatment by the batch scheduler. We also intend to study how the (lack of) confidence of the classification could be exploited by the scheduling algorithm, similarly to what is done with Bayesian bandits in online learning.

Figure

  Figure Average auto-correlation between a job and its predecessors regarding the number of requested resources q. The x-axis shows the distance between the job and its predecessor (in number of jobs) and y-axis show the degree of correlation between them.

Algorithm 1 :

 1 Update classifier1 dataset = extract execution logs() 2 divider = cluster(dataset) 3 TrainRandomForestClassifier(dataset, divider)

Figure 3 .

 3 Figure 3. Evolution of the Cumulative Bounded Slowdown for the six platforms, using the base policies (black) and the same policies augmented with job size classification and idempotence (cyan).

Algorithm 3 : 5 # 7 kill(job j ) 8 Q

 3578 Killing False Small jobs 1 Q = {} # queue of large jobs 2 Q small = {} # queue of small jobs 3 job counter = 0 # number of submitted jobs 4 while Running do go through all jobs currently running 6 if job j .class == "small" & job j .runtime > divider then small .remove(job j ) Q.add(job j )

Figure 4 .

 4 Figure 4. Monthly average bounded slowdown. Each line links the values from the same month when using the base and classification-idempotent schedulers.

Figure 5

 5 Figure 5. Impact of using the classification-idempotent scheduler on the average bounded slowdown of small and large jobs, when compared to the base scheduler.

Figure 6 .

 6 Figure 6. Impact of disabling the job-killing mechanism for large jobs misclassified as small.

Figure 7 .

 7 Figure 7. Comparison with a perfect job size classifier and with a fully clairvoyant scheduler.

TABLE IV FEATURES

 IV USED FOR JOB CLASSIFICATION.

	Type	Feature	Description
	job features	pi	user supplied runtime estimate
		qi	user supplied number of resources
	Temporal features	h	hour of the day
		D week	day of the week
		d month	day of the month
		m	Month
		w	Week
		Q	Quarter
	Lag features	pi-1	class of the previous job that was submitted by the same user i and requested equal runtime
		pi-2	class of the second to previous previous job that was submitted by the same user i and requested equal runtime
		pi-3	class of the third to previous job that was submitted by the same user i and requested equal runtime
		qi-1	class of the previous job that was submitted by the same user i and requested equal number of resources
		qi-2	class of the second to previous job that was submitted by the same user i and requested equal number of resources
		qi-3	class of the third to previous job that was submitted by the same user i and requested equal number of resources
		di-1	class of the previous job that was submitted by the same user i on the same day
		di-2	class of the second to previous job that was submitted by the same user i on the same day
		di-3	class of the third to previous job that was submitted by the same user i on the same day
			percentage of jobs that where submitted by the same
	Aggregation features meaniq	user i and requested equal number of resources and belong
			to the class small

meanip

percentage of jobs that where submitted by the same user i and requested equal runtime and belong to the class small mean id percentage of jobs that where submitted by the same user i on the same day and belong to the class small

TABLE VI IMPROVEMENT

 VI OVER EASY-FCFS ACCORDING TO[START_REF] Tsafrir | Backfilling using systemgenerated predictions rather than user runtime estimates[END_REF] AND[START_REF] Gaussier | Improving backfilling by using machine learning to predict running times[END_REF].

				Classification-Idempotent
	EASY++ [4] Gaussier et. al. [5] FCFS-CI	SPF-CI
	KTH-SP2	36%	44 %	50 %	59%
	CTC-SP2	37%	59 %	79 %	85%
	SDSC-BLUE	47%	05 %	63 %	74%
	SDSC-SP2	29 %	15 %	66 %	75%

The area a j of a job j is defined as its runtime multiplied by the number of resources it requested: a j = p j * q j .

ACKNOWLEDGMENT

This research was financed in part by the French research agency (Energumen ANR-18-CE25-0008). The authors would like to thank Millian Poquet for his help and comments and Michael Mercier for his help in the simulation. We thank the contributors of the Parallel Workloads Archive, Victor Hazlewood (SDSC SP2), Travis Earheart and Nancy Wilkins-Diehr (SDSC Blue), Lars Malinowsky (KTH SP2), Dan Dwyer, Steve Hotovy (CTC SP2), Ake Sandgren (HPC2N), and The Czech National Grid Infrastructure( MetaCentrum).