Isabelle Carrière
email: isabelle.carriere@inserm.fr

PhD Laure Anne Gutierrez

MSc Karine Pérès

PhD Claudine Berr

PhD Pascale Barberger-Gateau

PhD Karen Ritchie

PhD Marie Laure Ancelin

Late life depression and incident activity limitations: influence of gender and symptom severity

Keywords: Aged, Longitudinal studies, Disability evaluation, Depression, Gender differences

Background

Mental disorders, especially depression, are one of the principal causes of disablement.

Previous research has been limited partly due to the failure to take into account subsyndromal states and the very large number of candidate mediating and confounding factors.

Methods

Longitudinal associations between baseline depressive symptomatology and activity limitations were examined in a community-dwelling elderly cohort of 3191 participants.

Mixed logistic models were used to determine associations between mild or severe depressive symptoms (Center for Epidemiologic Studies-Depression Scale, CES-D), and 7-year incident disability on four limitation scales assessing instrumental and basic activities of daily living, mobility using the Rosow and Breslau scale, and social restriction.

Results

In men, mild depressive symptomatology was associated with increased incident limitations on instrumental activities of daily living (IADL) (odds ratio (OR) (95%CI)=5.07(2.25-11.42)). In women, severe depressive symptomatology was related to social restriction (OR(95%CI)=2.36(1.31-4.25)), IADL (OR(95%CI)=1.89(1.13-3.15)) and activities of daily living (OR(95%CI)=11. 15(3.43-36.23)). Men and women with a 2-year increase in CES-D score were highly at risk for social restriction and limitations in mobility and IADL.

Conclusion

Depression is an independent predictor of disability in the elderly population; the relation is gender-dependent and varies with symptom load.

Introduction

Mental disorders are one of the principal causes of disablement; the Global Burden of Disease reporting that neuropsychiatric conditions account for up to a quarter of all Disability-Adjusted Life-Years (DALYs) [START_REF] Murray | The Global Burden of Disease: a comprehensive assessment of mortality and disability from diseases, injuries and risk factors in 1990 and projected to 2020[END_REF][START_REF] Prince | No health without mental health[END_REF]. Depression in particular is associated with high mortality and disability, accounting for 4.46% of DALYs and 12.1% Years of Life with Disability (YLDs) [START_REF] Ustun | Global burden of depressive disorders in the year 2000[END_REF]. It is above all late life depression which significantly reduces disability-free life expectancy [START_REF] Peres | Impact of late-life self-reported emotional problems on Disability-Free Life Expectancy: results from the MRC Cognitive Function and Ageing Study[END_REF].

Estimates of the life-time prevalence of major depression in elderly cohorts vary widely (3 to 19%) [START_REF] Lyness | Psychiatric disorders in older primary care patients[END_REF][START_REF] Ritchie | Prevalence of DSM-IV psychiatric disorder in the French elderly population[END_REF], this variability being largely attributable to assessment methods (clinical/standardised examination/self-evaluation) and the type of depression investigated (major, minor or depressive symptoms). Most elderly people who have clinically significant depressive symptoms do not meet the full diagnostic criteria for major depression [START_REF] Blazer | Depression in late life: review and commentary[END_REF][START_REF] Rollman | Minor and subsyndromal depression: functional disability worth treating[END_REF], however, sub-syndromal depression has both a high prevalence in later life and poor response to anti-depressant treatment [START_REF] Kirsch | Initial severity and antidepressant benefits: a meta-analysis of data submitted to the Food and Drug Administration[END_REF]. It may thus constitute a major cause of disability for which treatment methods are presently unsatisfactory.

A better understanding of the association between both major and sub-syndromal depression and disability constitutes an important first step in identifying alternative intervention points for therapeutic intervention. Previous research has been limited not only due to failure to take into account sub-syndromal states and the very large number of candidate mediating and confounding factors, but also the inadequacy of the disability measures used. Previous longitudinal studies have principally used partial dependency measures incorporating different items from activities of daily living (ADL) or instrumental activities of daily living (IADL) scales [START_REF] Carriere | Modelling complex pathways between late-life depression and disability: evidence for mediating and moderating factors[END_REF][START_REF] Schillerstrom | Depression, disability and intermediate pathways: a review of longitudinal studies in elders[END_REF]. Furthermore, although gender differences have been consistently observed in the aetiology and clinical presentation of depression, differential susceptibility to depression-related disability has surprisingly seldom been investigated. We recently reported that the association between depression and 4-year mortality is dependent both on gender and disease severity, the greatest risk being observed for men with severe depression [START_REF] Ryan | Late-life depression and mortality: influence of gender and antidepressant use[END_REF].

The purpose of this study is to examine the long-term association between depressive symptoms and incident activity limitations in a large elderly prospective community cohort, for which information on a large number of potential confounding factors was available. The analyses are based on four independent scales of activity limitation corresponding to different levels of activity and social restriction.

METHOD

Study sample

Subjects were recruited as part of a multi-site cohort study of community-dwelling persons aged 65 years and over from the electoral rolls of three French cities (Bordeaux, Dijon and Montpellier) between 1999and 2001(The 3C Study Group, 2003). The study protocol was approved by the Ethical Committee of the University-Hospital of Bicêtre (France) and written informed consent was obtained from each participant. A standardized evaluation with a face to face interview and a number of clinical examinations was undertaken in the three study centres at baseline, 2 and 4 years. In the centres of Bordeaux and Montpellier this follow-up was extended with an examination at 7 years. Of the 4215 dementia-free participants included in these two centres, 408 were missing all follow-up evaluations for activity limitations (of whom 170 died) and 616 had missing data for at least one baseline adjustment variable. The present analyses were thus conducted on 3191 subjects (1617 from Bordeaux and 1574 from Montpellier; 1311 men and 1880 women).

Compared with the analysed sample, those not included in this analysis were more likely to be older (p < 0.0001), have lower education (p<0.0001), be more depressed (p< 0.0001), use antidepressants (p=0.004), live alone (p=0.02) and have disabilities (p<0.0001), cognitive impairment (p<0.0001), comorbidity (p<0.0001), visual impairment (p=0.03) and hearing impairment (p<0.0001).

Disability measures

Three complementary measurements of activity limitations were investigated as outcomes: mobility, complex or instrumental activities of daily living (IADLs) and basic activities of daily living (ADLs). Mobility was assessed according to the Rosow and Breslau scale [START_REF] Rosow | A Guttman health scale for the aged[END_REF] which evaluates ability to do heavy housework, walk half a mile, and climb stairs. The Lawton-Brody IADL scale was used to assess individual ability to use the telephone, to manage medication and money, to use public or private transport, to shop, and for women only, to prepare meals and do housework and laundry [START_REF] Lawton | Assessment of older people: self-maintaining and instrumental activities of daily living[END_REF].

For ADLs, participants were asked whether they needed help for any task on the Katz ADL scale: bathing, dressing, transferring from bed to chair, toileting, and eating (incontinence was not considered since it reflects organ impairment rather than functional limitation) [START_REF] Katz | Progress in development of the index of ADL[END_REF]. For each domain of disability, participants indicating inability to perform one or more activities without help were considered as having mobility, IADL, or ADL limitations. A fourth outcome was social restriction (confinement to bed, home or outings restricted to neighbourhood) which can be considered as participation restriction according to the International Classification of Functioning, Disability, and Health (ICF, 2001 ).

Depression

The Mini International Neuropsychiatric Interview (MINI) [START_REF] Sheehan | The Mini-International Neuropsychiatric Interview (M.I.N.I.): the development and validation of a structured diagnostic psychiatric interview for DSM-IV and ICD-10[END_REF], a standardised psychiatric examination which has been validated in the general population, was used for the diagnosis of current and past major depressive episodes (MDE), according to DSM-IV criteria. Severity of depressive symptoms was assessed using the 20-item Center for Epidemiologic Studies-Depression scale (CES-D) [START_REF] Radloff | The CES-D scale: a self-report depression scale for research in the general population[END_REF]. For this analysis, participants were classified into one of three groups [START_REF] Ryan | Late-life depression and mortality: influence of gender and antidepressant use[END_REF]. 'Severe depressive symptomatology' (Severe DS group) included participants with a current MDE or a CES-D score of 23 or over (allowing for the fact that some participants with severe symptoms did not reach DSM classification criteria, principally because of the duration of symptoms). 'Mild depressive symptomatology' (Mild DS group) was defined as a CES-D score between 16 and 22 and 'no DS' included participants with a CES-D score lower than 16. Deterioration in depressive symptoms between inclusion and the first follow-up examination was defined as an increase of 3 points or more in the CES-D score. Current use of antidepressants was validated by presentation of the prescription or medication to the interviewer, and type of medication was noted according to the World Health Organization's Anatomical Therapeutic Chemical (ATC) classification.

Socio-demographic and clinical variables

The standardized interview included questions on demographic characteristics, education level (classified in four groups corresponding to 5, 9, 12, and 12+ years of education), marital status, income, mode of living (alone or not), height, and weight. Information was obtained on type and quantity of alcohol consumption (number of units of alcohol per day; 0, 1-36, >36 g/day), tobacco use (classified as past, present or never users) as well as use of hormonal therapy (HT) (never, past, current use) for women. Detailed medical questionnaires included history of stroke, angina pectoris, myocardial infarction, and cardio-vascular surgery established according to standardized questions, history of cancer and fracture with hospitalisation during the last two years, asthma (attacks during the last year), chronic bronchitis, dyspnoea, hypertension (>160/95 mm Hg or treated), diabetes (fasting glycemia> 7mmol/l or treated) and thyroid disease. Blood pressure was measured during the interview using a digital electronic tensiometer OMRON M4. The number of chronic diseases was calculated including: hypertension, diabetes, cardiovascular diseases, respiratory diseases, dyspnoea, thyroid disease, and cancer.

Cognitive impairment was defined as a Mini Mental State Examination (MMSE) [START_REF] Folstein | Mini-mental state". A practical method for grading the cognitive state of patients for the clinician[END_REF] score lower than 24. Visual impairment was defined as having a corrected near visual acuity (Parinaud scale) of less than 4 or difficulties recognizing a familiar face at 4 meters. Hearing impairment was defined deafness or only able to hear a conversation when a single person speaks loudly.

Statistical analyses

Separate analyses were conducted for men and women. The Chi2 test was used to identify gender-related differences. For each indicator of activity limitations longitudinal associations with the levels of DS were established from subjects free of that activity limitation at baseline. Thus the sample size was 951 men and 954 women for mobility, 1245 men, and 1723 women for IADL, 1303 men and 1864 women for ADL and 1276 men and 1738 women for social restriction.

In longitudinal studies, the within-subject responses (repeated evaluations of activity limitations) are correlated. This correlation was accounted for by using a mixed logistic model [START_REF] Carriere | Choosing marginal or random-effects models for longitudinal binary responses: application to self-reported disability among older persons[END_REF]. Briefly, this model has four basic characteristics: (i) the individual time evolutions of activity limitations are entirely taken into account including possible reversion to normal state (ii) subjects with incomplete responses across time are included in the analysis; (iii) subjects do not have to be evaluated at the same time points; (iv) the model allows within-subjects dependency to vary from one subject to another, via the random part of the covariable linear combination. The SAS procedure NLMIXED was used to estimate the parameters (version 9.1).

We used univariate and multivariate models to determine if baseline DS was associated with odds of activity limitations or social restriction during follow up. After gender stratification, univariate odds ratios were adjusted for centre, age, time and interaction time*age (Model 0). Multivariate adjusted logistic regression included covariates that were associated with the follow-up responses (p<0.15 in univariate model). Model 1 was adjusted for age, centre, time, time*age, alcohol, body mass index (BMI), smoking, number of chronic pathologies, cognitive impairment, visual impairment, hearing impairment and HT (women only). Model 2 was further adjusted for income, education level and living alone and Model 3 for antidepressant treatment and past MDE.

A sensitivity analysis was also performed. The non-respondents at 7 years were contacted by telephone and invited to answer a short questionnaire exploring medication use, activity restrictions (ADL, IADL, mobility), cognitive performances (MMSE, verbal fluency test) and depressive symptoms (CES-D). Information obtained by telephone were pooled with the study data base in order to examine potential bias due to non random loss to follow-up.

RESULTS

Within this elderly community-dwelling sample, the baseline prevalence of Severe DS (major depressive disorder or CES-D ≥ 23) was 11.8% and that of Mild DS (CES-D 16-22) was 9.2%. Past MDE was reported by 40.3% of subjects with Severe DS and 20.5% with Mild DS and the mean age (SD) of first episode onset was 43.9 (16.1) years. Women had a significantly higher prevalence of Severe and Mild DS than men, were more often treated with antidepressants, and declared more frequently a history of MDE (Table 1).

For every baseline disability scale except ADL, women reported activity limitations significantly more frequently than men (7.50% vs 2.52% for social restriction, 48.63% vs 26.45% for mobility, and 7.96% vs 4.59% for IADL). Compared to men, women were also less educated, more likely to live alone, have a lower incomes more visual and cognitive impairment but were less likely to have hearing impairment, comorbidity, to be overweight or obese, to smoke and take alcohol.

The analysis of the repeated activity limitation proportions was undertaken for each scale on the subjects without limitation on the considered scale at baseline. Table 2 shows the rates of activity limitations over 7 years in men and women according to the level of baseline DS and the odds ratios of having incident activity limitations adjusted for age, centre, time and the interaction between time and age (model 0). Over the follow-up period, the probability of becoming disabled tended to increase for all the indicators and whatever the level of DS. All mixed logistic models showed a significant effect of age and age*time interaction indicating that the effect of time on activity limitations was more pronounced in the oldest old. For ADL in men time and age were significant, but not the interaction.

Compared with men without DS at baseline, the risk of having IADL limitations during follow-up was highly significantly greater in men with Mild DS (OR(95%CI) =5.20 (2.37-11.38)) (Table 2). This association was also significant for social restriction (OR(95%CI)=2.71 (1.01-7.26)) and mobility (OR(95%CI)=1.87(1.00-3.50)). In women whatever the outcome scale, baseline Mild DS was not found to be significantly related to the odds of activity limitations during follow-up. Baseline Severe DS was only found to be significant in men for social restriction (OR(95%CI)=3.85(1.05-14.04)) while in women it was highly significant for social restriction (OR(95%CI)=3.57(1.93-6.61)), IADL (OR(95%CI)=2.47(1.48-4.11)) and ADL (OR(95%CI)=13.29(4.50-39.31)).

When the mixed logistic models were further adjusted for possible confounding factors (model 3) all the preceding associations persisted except in men for social restriction and mobility. In men (Table 3), Mild DS was highly significantly associated with the adjusted odds of IADL limitations (OR(95%CI)=5.07(2.25-11.42)) while in women (Table 4) Severe DS was related to social restriction (OR(95%CI)=2.36(1.31-4.25)), IADL (OR(95%CI)=1.89

(1.13-3.15)) and ADL limitations (OR(95%CI)=11. 15(3.43-36.23)). The multi-adjusted OR values tended to slightly decrease when covariates linked to medical burden were added to the model (model 1) as well as when past MDE and antidepressant use were further included (model 3) but not when socioeconomic status was added (model 2). These results were not modified when the data from the phone interview were added as a sensitivity analysis to test the potential bias due to non-random dropout (321 additional time points at 7 years). The associations found in model 3 also remained significant when excluding the case of incident dementia (130 women and 78 men) except for severe DS and IADL in women (OR(95%CI)=1.50(0.86-2.60)). In men, Mild DS remained significantly associated with IADL limitations (OR(95%CI)=4.31(1.85-10.06)) and in women Severe DS was still associated with social restriction (OR(95%CI)=1.99(1.04-3.80)) and ADL limitations (OR(95%CI)=7.47(1.81-30.84)).

No interactions were found between DS (mild or severe) and centre, past MDE or the number of chronic pathologies or cognitive, visual or hearing impairments, except in women for social restriction (p-value for interaction=0.04); depressed women with hearing impairment being at higher risk of confinement to home or neighbourhood (n=53, p=0.03) whereas the association with DS was not significant in those who were not hearing impaired (n=1685, p=0.42).

Among the 2377 subjects (1070 men and 1307 women) free of DS and of disability at baseline and with a 2-year depression evaluation, 27.1% of the men and 34.3% of the women had an increase in the CES-D score of at least three points over the first two years (χ 2 test, p=0.0002). The odds of having activity limitations associated with a 2-year increase in CES-D score was significantly higher in men and women for social restriction, mobility and IADL (Table 5). These associations were especially strong in men for social restriction and IADL with an OR (95%CI) equal to 3.43 (1.72-6.86) and 3.00 (1.72-5.24), respectively.

DISCUSSION

Gender differences in depression-related activity limitation risk

The most striking findings of this large study of community-dwelling elderly were that activity limitations risk varied according to DS severity and by sex. Women with Mild DS did not show increased risk in relation to any disability type, but those with severe DS were at higher risk of limitations in IADL, ADL and social restriction. On the other hand, men with Mild DS were at high risk of IADL limitations; an increased risk in social and mobility restriction was also found in an age-adjusted model, but this failed to reach significance in the multi-adjusted models. In men with Severe DS, we did not observe a significant association with any of the disability measures after 7 years of follow-up, probably due to lower number (4.9% compared to 12.2% for women) and a higher mortality rate. Indeed in the same cohort, an increase in depression-related 4-year mortality risk was observed in men with mild (OR=2.8) and treated severe DS (OR=5.3) whereas in women only non-treated severe DS which slightly increased risk (OR=1.8) [START_REF] Ryan | Late-life depression and mortality: influence of gender and antidepressant use[END_REF].

Several previous studies have reported an increase in activity limitations (most frequently severe ADL limitations) associated with DS in elderly community cohorts (see [START_REF] Carriere | Modelling complex pathways between late-life depression and disability: evidence for mediating and moderating factors[END_REF][START_REF] Schillerstrom | Depression, disability and intermediate pathways: a review of longitudinal studies in elders[END_REF], for reviews). Only two prospective studies have examined the impact of gender differences on severe level of disability [START_REF] Barry | Longitudinal association between depressive symptoms and disability burden among older persons[END_REF][START_REF] Dalle Carbonare | Physical disability and depressive symptomatology in an elderly population: a complex relationship. The Italian Longitudinal Study on Aging (ILSA)[END_REF]. Dalle Carbonare et al observed an association between DS and short-term limitations in ADL for both gender and physical function disability for men [START_REF] Dalle Carbonare | Physical disability and depressive symptomatology in an elderly population: a complex relationship. The Italian Longitudinal Study on Aging (ILSA)[END_REF]. Barry et al using a 4-item ADL scale found an increased risk of experiencing severe disability associated with moderate and high levels of DS in men, and only with severe DS in women [START_REF] Barry | Longitudinal association between depressive symptoms and disability burden among older persons[END_REF]. We also observed an association between ADL and Severe DS in women. In men with Severe DS, the association tended to be significant only in age-adjusted but not multi-adjusted model, probably due to a lack of power; few people being severely disabled in our sample.

Disability in relation to depression severity and gender differences

Our study is, we believe, the first to use several activity limitation scales in their validated forms. Mobility, IADL and ADL limitations correspond to an increased gradient of severity [START_REF] Barberger-Gateau | A hierarchical model of domains of disablement in the elderly: a longitudinal approach[END_REF]. Severe DS was predictive of multiple disabilities in women whereas the more likely consequence in men would be on mortality. Whereas ADL appears as an "ultimate" indicator of disability associated with Severe DS, IADL appears to be a more sensitive marker of disability as a function of depression severity level in our sample. Men with Mild DS were at very high risk of IADL disability (OR=5.07) whereas women even with Severe DS were at lower risk (OR=1.89). On the other hand, mobility limitations which correspond to a lower level of disability are probably not sufficiently sensitive to DS in this elderly sample (39.5% with such activity limitations at baseline). Mobility limitations were observed in people without DS at baseline but showing an early slight deterioration in CES-D score during follow-up. This was also the case for social restriction and IADL. We observed a slight gender difference in mobility and social restriction and a notable difference in IADL, men being at higher risk of disability. Our study thus gives an indication of the dynamics of the evolution of activity limitations at increasing levels of DS severity; different patterns of disability evolution possibly explaining gender differences. Some authors have distinguished subjects who rapidly develop severe disability (in less than one year -also called catastrophic disability) from those who develop severe disability for a longer period (progressive or insidious disability) [START_REF] Ferrucci | Progressive versus catastrophic disability: a longitudinal view of the disablement process[END_REF][START_REF] Gill | The development of insidious disability in activities of daily living among community-living older persons[END_REF][START_REF] Gill | Trajectories of disability in the last year of life[END_REF]. Catastrophic disability could be the consequence of precipitating events such as stroke, hip fracture, cancer while progressive disability is more likely to be related to the frailty syndrome defined as an aggregate of subclinical losses of reserve across multiple physiologic systems [START_REF] Bortz | A conceptual framework of frailty: a review[END_REF][START_REF] Fried | Frailty in older adults: evidence for a phenotype[END_REF] or insidious onset of cognitive deterioration. Both time spent in disability and time before death are shorter in men [START_REF] Deeg | Health profiles and profile-specific health expectancies of older women and men: The Netherlands[END_REF][START_REF] Peres | Disability-free life expentancy of older French people: gender and education differentials from PAQUID cohort[END_REF] indicating that men may be more vulnerable to catastrophic disability whereas women are at greater risk of an insidious progression with longer time spent in states of severe disability.

Our results suggest that moderate DS could contribute to more rapid disability progression in men while in women severe DS is associated with slower progression.

Mechanisms potentially involved in the association between depression and disability

The mechanisms by which depression may generate activity limitation are not fully understood and may involve a combination of factors including physical comorbidity, cognitive and sensorial impairment, health behaviours, and biological factors. Vascular comorbidity is of particular importance in this context as it has been reported to be associated with functional decline [START_REF] Wang | Predictors of functional change: a longitudinal study of nondemented people aged 65 and older[END_REF]. Depression could be an intermediate state between cerebral vascular disease and disability [START_REF] Alexopoulos | Vascular depression' hypothesis[END_REF][START_REF] Schillerstrom | Depression, disability and intermediate pathways: a review of longitudinal studies in elders[END_REF]. We however, controlled for hypertension, diabetes, stroke, and cardiovascular disease and the association remained significant, suggesting that this pathway does not fully explain the association.

The early symptoms of cognitive decline are often destabilising and may also induce a depressive state. Inversely depression is associated with apathy and impaired cognitive functioning. The relationships persisted even after adjusting for baseline cognitive level and there was no significant interaction between cognitive impairment and depression in our study. [START_REF] Mehta | Cognitive impairment, depressive symptoms, and functional decline in older people[END_REF] found that in elderly people without ADL limitations, but not in those with ADL limitations at baseline, cognitive impairment and DS were both risk factors for functional decline. In a sample of vulnerable elders eligible for nursing home care, Li et al [START_REF] Li | Effects of changes in depressive symptoms and cognitive functioning on physical disability in home care elders[END_REF] found that cognitive decline was associated with more elevated IADL scores when depression status worsened, but this was not observed for ADL scores. The strength of the interaction between DS and cognition probably depends on both cognitive level and degree of disability within the study population. A relatively small proportion of the participants in our study were at high risk for dementia (4.2% had baseline MMSE <24) which may explain the lack of interaction.

Sensory impairments, have been demonstrated to be a predictor of both onset and persistence of depression and functional disability [START_REF] Chou | Combined effect of vision and hearing impairment on depression in older adults: evidence from the English Longitudinal Study of Ageing[END_REF][START_REF] Crews | Vision impairment and hearing loss among communitydwelling older Americans: implications for health and functioning[END_REF][START_REF] Owsley | Effect of refractive error correction on health-related quality of life and depression in older nursing home residents[END_REF]. In our analysis, the relationship persisted after adjustment for both visual and hearing impairment. However, the assessment of hearing impairment was self-reported and the adjustment may not be complete considering that sensory impairments are often unrecognized by elderly people due to their slow progression.

Several behavioural mechanisms could also explain the disability risk associated with depression including impaired motivation and lower treatment compliance [START_REF] Ciechanowski | Depression and diabetes: impact of depressive symptoms on adherence, function, and costs[END_REF][START_REF] Vinkers | Does depression in old age increase only cardiovascular mortality? The Leiden 85-plus Study[END_REF]. Men in particular are likely to behave in ways which are detrimental to their overall health [START_REF] Ciechanowski | Depression and diabetes: impact of depressive symptoms on adherence, function, and costs[END_REF], however, the associations persisted even after controlling for comorbidity and "self-care" variables (i.e. smoking, alcohol, BMI, and level of visual and hearing impairment). More subjective factors such as impairment in affect regulation, social perception and greater tendency to amplify physical symptoms [START_REF] Ormel | Common mental disorders and disability across cultures. Results from the WHO Collaborative Study on Psychological Problems in General Health Care[END_REF] may also have played a role. Finally, dysfunctioning in the hypothalamic-pituitary-adrenal and sympathetic axes and immune system may also be involved [START_REF] Katon | Increased medical costs of a populationbased sample of depressed elderly patients[END_REF].

Limitations

The data concerning disability outcomes were self-reported which may have led to overreporting in depressed participants. Bias could also be introduced by the exclusion of participants with missing baseline data or lost to follow-up, who were more likely to be depressed, disabled or to have died. A difference in the attrition of our cohort was observed, men with DS being more often lost to follow-up (31.4% of depressed men and 14.9% of depressed women died during follow-up, 57.9% of depressed men and 48.5% of depressed women had less than 4 examinations). Men with DS may thus be censored before becoming disabled, so that associations between depression and disability were underestimated.

However, the same results were obtained when data from telephone interviews of nonrespondents were included (data not shown). Finally, in our study, we had no information regarding level of pain and its association with depression [START_REF] Lin | Effect of improving depression care on pain and functional outcomes among older adults with arthritis: a randomized controlled trial[END_REF] which may have increased the risk of over-adjustment. We did, however, adjust for chronic pathologies commonly associated with pain, and this did not modify the associations.

Strengths

Our prospective study based on a large community sample permitted a dynamic evaluation of activity limitation with four evaluations over 7 years. We also used a number of validated scales exploring the major components of disability corresponding to distinct levels of disability severity. Depression was assessed using two distinct measures validated in the general population, including a structured diagnostic interview [START_REF] Radloff | The CES-D scale: a self-report depression scale for research in the general population[END_REF][START_REF] Sheehan | The Mini-International Neuropsychiatric Interview (M.I.N.I.): the development and validation of a structured diagnostic psychiatric interview for DSM-IV and ICD-10[END_REF], thus minimizing exposure misclassification. We were able to cover a wide range of depression profiles, from sub-clinical symptoms to major depression, thus addressing the problem of the varying definitions of depression in previous research. We also evaluated the early impact of 2 year-DS deterioration on disability. We controlled for a large number of potentially confounding factors, particularly measures of physical health (comorbidity, sensorial impairments, health behaviors), cognitive impairment, past MDE and antidepressant use (with the risk of over-adjusting) which only slightly affected the results.

In conclusion, our findings suggest that the relationship between DS and incident activity limitations in the elderly is gender-dependent and also varies according to symptom load.

Findings from this study suggest an increased risk of disability in men, even at the level of sub-clinical depression. This also highlights the importance of early detection of depressive symptoms in general practice, associated with treatments adapted to severity level.

Table 1 . Characteristics of the study population

 1

	Men (n=1311)	Women (n=1880)	Chi2
	%	%	p

Table 2 . Incident cases of disability by baseline depressive symptomatology and risk of disability adjusted for age, centre, time and time*age (model 0)

 2

	Men

Table 3 : Multi-adjusted associations of baseline depressive symptomatology with disability incidence in men

 3 Model 1: adjusted for age, centre, time and time*age, alcohol, BMI, smoking, number of chronic pathologies, cognitive impairment, visual impairment, hearing impairment Model 2: adjusted for age, centre, time and time*age, alcohol, BMI, smoking, number of chronic pathologies, cognitive impairment, visual impairment, hearing impairment, income, study level and, living alone Model adjusted for age, centre, time and time*age, alcohol, BMI, smoking, number of chronic pathologies, cognitive impairment, visual impairment, hearing impairment, income, study level, living alone, past major depressive episode and, antidepressant treatment.

		Model 1		Model 2		Model 3
		OR (95%CI)	p-value	OR (95%CI)	p-value	OR (95%CI)	p-value
	Home or neighborhood confined, N=1276
	No DS	1		1		1
	Mild DS	2.16(0.77-6.08)	0.14	2.29 (0.81-6.46)	0.12	2.22(0.80-6.13)	0.13
	Severe DS	3.39(0.88-13.08)	0.08	3.57 (0.9-14.14)	0.07	2.17(0.51-9.2)	0.29
	Mobility: Rosow and Breslau, N=951
	No DS					1
	Mild DS	1.68(0.9-3.12)	0.10	1.69 (0.91-4.49)	0.10	1.68(0.90-3.16)	0.11
	Severe DS	0.6(0.26-1.39)	0.24	0.59 (0.25-1.36)	0.21	0.59(0.25-1.39)	0.23
	IADL, N=1245
	No DS	1		1		1
	Mild DS	4.85(2.13-11.03)	0.0002	5.4 (2.38-12.24)	<0.0001 5.07(2.25-11.42)	<0.0001
	Severe DS	1.78(0.54-5.87)	0.35	2.07 (0.62-6.94)	0.24	1.78(0.52-6.02)	0.36
	ADL, N=1303
	No DS	1		1		1

DS: Depressive Symptomatology

Table 4 : Multi-adjusted associations of baseline depressive symptomatology with disability incidence in women

 4

		Model 1		Model 2		Model 3
		OR (95%CI)	p-value	OR (95%CI)	p-value	OR (95%CI)	p-value
	Home or neighborhood confined, N=1738
	No DS					1
	Mild DS	0.87(0.49-1.56)	0.64	0.88 (0.5-1.56)	0.66	0.82(0.46-1.46)	0.50
	Severe DS	3.01(1.68-5.41)	0.0002	2.94 (1.65-5.24)	0.0003	2.36(1.31-4.25)	0.004
	Mobility: Rosow and Breslau, N=954
	No DS					1
	Mild DS	1.33(0.86-2.04)	0.20	1.30 (0.84-2.00)	0.23	1.27(0.82-1.97)	0.28
	Severe DS	1.48(0.9-2.43)	0.12	1.44 (0.87-2.38)	0.15	1.32(0.79-2.21)	0.29
	IADL, N=1723
	No DS					1
	Mild DS	1.26(0.79-2.00)	0.34	1.35 (0.84-2.17)	0.22	1.26(0.78-2.02)	0.34
	Severe DS	2.03(1.24-3.33)	0.005	2.22 (1.34-3.67)	0.002	1.89(1.13-3.15)	0.02
	ADL, N=1864

Table 5 : Adjusted associations of 2-year depressive deterioration (increase of 3 points or more at CES-D) with disability incidence in subjects free of depressive symptomatology at baseline

 5 adjusted for age, centre, time and time*age, alcohol, BMI, smoking, number of chronic pathologies, cognitive impairment, visual impairment, hearing impairment, hormonal treatment (women only), income, study level, living alone, past major depressive episode and, antidepressant treatment.

		Men		Women
		OR (95%CI)*	p-value	OR (95%CI)*	p-value
	Home or neighborhood confined	N=1049		N=1234
		3.43 (1.72-6.86)	0.0005	2.76 (1.78-4.27)	<0.0001
	Mobilty: Rosow and Breslau	N=794		N=725
		2.18 (1.5-3.18)	<0.0001	1.68 (1.2-2.35)	0.003
	IADL	N=1022		N=1214
		3.00 (1.72-5.24)	0.0001	1.55 (1.03-2.34)	0.04
	ADL	N=1065		N=1300
		1.64 (0.67-4.03)	0.28	2.67 (0.95-7.53)	0.06

*

Acknowledgement Funding: The 3C Study is conducted under a partnership agreement between Inserm, the Victor Segalen-Bordeaux II University and Sanofi-Synthélabo. The Fondation pour la Recherche Médicale funded the preparation and first phase of the study. The 3C-Study was also supported by the Caisse Nationale Maladie des Travailleurs Salariés, Direction Générale de la Santé, MGEN, the Institut de la Longévité, Agence Française de Sécurité Sanitaire des Produits de Santé, the Regional Governments of Aquitaine, Bourgogne and Languedoc-Roussillon, the Fondation de France, the Ministry of Research-Inserm Programme 'Cohorts and collection of biological material' and Novartis. The Lille Genopole was supported by an unconditional grant from Eisai. This work was carried out with the financial support of the "ANR -Agence Nationale de la Recherche -The French National Research Agency" under the "Programme National de Recherche en Alimentation et nutrition humaine", project "COGINUT ANR-06-PNRA-005" and under the "Programme Longévité et vieillissement", project 07-LVIE-004" and by The Institut de Recherche en Santé Publique (IReSP), Paris, France.

Role of funding source: the sponsors had no role in study design; in collection, analysis, and interpretation of data; in the writing of the report; and in the decision to submit the paper for publication.

Conflict of interest: None.

Contributors. IC: study concept, statistical analyses and drafting of the manuscript, LAG: statistical analyses, KP, CB, PBG, KR, MLA: study concept and design, management of the cohort. All authors contributed to and approved the final version of the manuscript.


B B Laszczyszyn
email: bartek.blaszczyszyn@ens.fr

M K Karray

Keywords: Radio communication, Opportunistic, Scheduling, Quality of Service, Information theory, Markov processes

Effect of Opportunistic Scheduling on the Quality of Service Perceived by the Users in OFDMA Cellular Networks

Bart lomiej B laszczyszyn • Mohamed Kadhem Karray

Received: April 18, 2011 Abstract Our objective is to analyze the impact of fading and opportunistic scheduling on the quality of service perceived by the users in an Orthogonal Frequency-Division Multiple Access (OFDMA) cellular network.

To this end, assuming Markovian arrivals and departures of customers that transmit some given datavolumes, as well as some temporal channel variability (fading), we study the mean throughput that the network offers to users in the long run of the system. Explicit formulas are obtained in the case of allocation policies, which may or may-not take advantage of the fading, called respectively opportunistic and nonopportunistic.

The main practical results of the present work are the following. Firstly we evaluate for the non-opportunistic allocation the degradation due to fading compared to Additive White Gaussian Noise (AWGN) (that is, a decrease of at least 13% of the throughput). Secondly, we evaluate the gain induced by the opportunistic allocation. In particular, when the traffic demand per cell exceeds some value (about 2 Mbits/s in our numerical example), the gain induced by opportunism compensates the degradation induced by fading compared to AWG N .

Introduction

Wireless cellular networks are in permanent and rapid evolution. They provide service for constant bit-rate (CBR) calls (such as voice calls, video conferences, etc.), and carry variable bit-rate (VBR) calls (such as mail, ftp), which accept fluctuations of the rates. Customers arrive to the network and require some service. Once they are admitted, they are allocated some resource (power, bandwidth, code) for the duration of their service, then they depart from the network. Besides the dynamics due to user arrivals and departures, the channel conditions vary due to fading.

Growing traffic requires better planning and/or dimensioning of the networks. This task can be substantially simplified by analytical study of the dynamic network performance.

The objective of the present paper is to account for the effect of fading in the dynamic performance evaluation of the downlink of an Orthogonal Frequency-Division Multiple Access (OFDMA) cellular network.

To this aim, the following two groups of elements are crucial. On one hand, the network geometry and the resource allocation, which can be chosen by the network designer. On the other hand, channel conditions and the user traffic (intensity of arrivals and requested service) can only be predicted by the designer. In order to optimize the network performance, the resource allocation should be adapted to the channel and traffic conditions. Besides better coding (up to the information theory's limit) one is looking for allocations Effect of Opportunistic Scheduling on the Quality of Service Perceived by the Users in OFDMA Cellular Networks

Bart lomiej B laszczyszyn and Mohamed Kadhem Karray that are opportunistic; i.e., take advantage of the actual channel state and user position, preserving some fairness of the mean service rates offered to users. More precisely, the opportunistic allocation assumes that at each time, each bandwidth portion is allocated to the user with the most favorable fading state. Once the resource allocation is given, the quality of service (QoS; i.e., blocking probability for CBR and throughput for VBR) perceived by the users may be evaluated using queueing theory.

Individual elements of the above puzzle (i.e. information theory, resource allocation and queueing theory) are often studied and optimized separately. The main contribution of this paper is a global approach that combines these elements to deduce the performance of the non-opportunistic and the opportunistic resource allocations. In doing so we systematically use a separation of the times scales of different elements of the network dynamics, described in Section 2.1.

The main practical results of the present work are the following. Firstly we evaluate for the non-opportunistic allocation the degradation due to fading compared to Additive White Gaussian Noise (AWGN). Secondly, we evaluate the gain induced by the opportunistic allocation.

The remaining part of this paper is organized as follows. In Section 1.1 we list these existing results in the literature that are used in our global approach. Our model of the OFDMA network is presented in Section 2. In Section 2.1 we describe the network dynamics and its decomposition into three time scales. They correspond respectively to information theory, resource allocation and queueing theory which are studied in Sections 3, 4 and 5 respectively. The numerical results are described in Section 6. Finally Section 8 describes some interesting perspectives.

Related Work

A growing interest in OFDMA has resulted in many publications on performance evaluation of such networks. It is not in the scope of this paper to make a thorough review of this literature. For some references on this field see for example those cited in [START_REF] Karray | Analytical evaluation of QoS in the downlink of OFDMA wireless cellular networks serving streaming and elastic traffic[END_REF]. Here we remark only those results that are directly related to our approach.

Our starting point is our work in [START_REF] Karray | Analytical evaluation of QoS in the downlink of OFDMA wireless cellular networks serving streaming and elastic traffic[END_REF] assuming AWGN channels between the users and the base stations. Coluccia et al. [START_REF] Coluccia | On the optimality of max-min fairness in resource allocation[END_REF] study optimality and fairness in resource allocation. As regard to the performance of the opportunistic allocation policy over fading, we extend the approach of [START_REF] Borst | User-level performance of channel-aware scheduling algorithms in wireless data networks[END_REF] and [START_REF] Kushner | Convergence of proportional-fair sharing algorithms under general conditions[END_REF] to OFDMA networks. Moreover we use the known results for the stability and stationary distribution of the multi-class extension of the processor sharing queue; see e.g. [START_REF] Cohen | On regenerative processes in queueing theory[END_REF][START_REF] Serfozo | Introduction to stochastic networks[END_REF].

The above results are used as building blocks to evaluate the dynamic performance of the non-opportunistic and the opportunistic resource allocations in OFDMA multi-cell networks. The present paper is an extended version of [START_REF] Karray | Fading effect on the dynamic performance evaluation of ofdma cellular networks[END_REF]. Besides more detailed proofs and analysis, the major extension is to show that the capacity lower bounds (which are the basis of our analytical approach) hold true even if there are a few number of interferers (Propositions 1 and 3 below). We carry this extension by building an argument based on the Jensen's inequality in place of the law of large numbers argument used in our previous work [START_REF] Karray | Fading effect on the dynamic performance evaluation of ofdma cellular networks[END_REF].

Some attempts to make other extensions of the present work are currently carried by other researchers; for example Combes et al. [START_REF] Combes | Scheduling gain for frequency-selective rayleigh-fading channels with application to self-organizing packet scheduling[END_REF] attempt to account for frequency selective fading channels.

Model assumptions

We will consider a wireless network composed of several base stations (BS). Each BS is equipped with a single antenna (no MIMO) and its total power is limited to some given maximal value. The same frequency spectrum is available to all BS (frequency reuse factor equal to one). Each BS allocates disjoint sub-carriers to its users without macrodiversity (each user is served by exactly one BS). Thus, any given user receives only other-BS interference that is the sum of powers emitted by other BS on the sub-carriers allocated to him by his BS.

When the radio signal propagates (from a BS to a given user) it is attenuated. We model the propagationloss as the product of three factors called distance-loss, shadowing and fading. The first factor is due to the distance between the transmitter and the receiver. The shadowing is due to the attenuation by the obstacles between the transmitter and the receiver. The fading is due to the reflection on the obstacles in the neighborhood of the receiver which generate multiple paths. We account only for the distance and the fading effects in the present study.

We assume that the bandwidth of each sub-carrier is smaller than the coherence frequency of the channel, so that we can consider that the fading in each sub-carrier is flat. That is, the output of the channel at a given time depends on the input only at the same instant of time. We don't make any assumption on the correlation of the fading processes corresponding to different subcarr i e r s( f o rag i v e nu s e ra n dag i v e nB S ) .H o w e v e r ,t h e fading processes for different users or base stations are assumed independent.

Time is divided into time-slots of length smaller than the coherence time of the channel, so that, for a given sub-carrier, the fading remains constant during each time-slot and the fading process in different timeslots may be assumed ergodic. (Such model for fading generalizes the so-called quasi-static model where the fading process at different time-slots is assumed to independent and identically distributed.)

The fading in each time-slot and each sub-carrier is a Rayleigh distributed random variable.

The codeword duration equals the time-slot, which is assumed sufficiently large so that the capacity within each time-slot may be defined in the asymptotic sense of the information theory.

Users perform single user detection;t h u st h ei n t e rference is considered as additive noise and added to the additive white Gaussian noise (AWGN). The statistical properties of the interference are not known a priory since they depend of the codings of the other users. Moreover the signals transmitted by different base stations are assumed independent.

For each sub-carrier, and each time-slot, efficient (e.g. turbo) codes are used to obtain bit-rates close to the capacity given by information theory (i.e. the maximal rate for which there exist coding schemes with error probability vanishing when the length of the code is sufficiently large). In particular, for the AWGN channel the capacity equals

C = w log 2 (1+SNR) ( 1
)
where w is the bandwidth and SNR designates the signal to noise power ratio.

Separation of time scales

We assume that there are a few time scales each associated to the evolution of some stochastic process.

Roughly speaking, we assume that the duration of each time scale is sufficiently large so that the corresponding process converges to its stationary state; and sufficiently small so that the processes of the larger time scales don't evolve within this duration.

We describe now the time scales from the fastest to the slowest one.

The fastest time scale corresponds to information theory. The users number and positions as well as powers and bandwidths allocated to them are assumed fixed. Then the bit-rate of each user can not exceed the capacity given by information theory (i.e. the maximal rate for which there exist coding schemes with error probability vanishing when the length of the code is sufficiently large). This information theory constraint (i.e. existence of a coding scheme corresponding to the bitrate) together with the maximal power and bandwidth constraints constitute the resource allocation constraints.

The intermediate time scale corresponds to the resource allocation problem. The users number and positions are assumed fixed. The network attempts to allocate powers, bandwidths and bit-rates (coding schemes) to all users respecting the resource allocation constraints.

This problem may be reformulated in terms of some condition on the bit-rates. More precisely, by feasibility condition we mean a condition regarding bit-rates of different users which guarantees the existence of powers and bandwidths satisfying the resource allocation constraints.

It is the role of the load control to guarantee this condition. The load control comprises the admission control (for CBR traffic) which checks at the arrival instant of each new user the feasibility condition and decides whether it can be admitted or not; and the congestion control (for VBR traffic) which may modify the user bit-rates to satisfy the feasibility condition.O n c e this is done, the network allocates powers and bandwidths to all users supporting the (feasible) bit-rates.

The slowest time scale corresponds to queueing theory, where we consider the stochastic process of users positions which is driven by call arrivals and durations or data volumes. This process is subject at each time the feasibility condition described above. The stationary state (or distribution) of this process permits to calculate the quality of service perceived by the users.

Power allocation

The network may operate a power adaptation. Such power adaptation may be either a power control to transmit just what is necessary to compensate the interference as described in [START_REF] Karray | Analytical evaluation of QoS in the downlink of OFDMA wireless cellular networks serving streaming and elastic traffic[END_REF] or a water filling attempting to maximize the cell capacity by allocating the powers among the users according to their fadings as described in [20, Theorem 2 p.119 ].

We shall assume that the signal transmitted by each BS has a given power constant over time (which equals to the maximal authorized value) and a constant power spectral density (i.e. the power is the same for all subcarriers). The performance obtained with this assumption gives a lower bound of the performance of a network which operates a power adaptation. We shall carry our analysis by considering further lower bounds when necessary. The dimensioning based on such lower bounds is conservative.

Resource allocation

We always assume that each user (receiver) knows its own fading state; a situation called CSIR (Channel State Information at Receiver) in literature.

We shall consider two resource allocation schemes:

1. Non-opportunistic: The BS doesn't take into account the fading when allocating the sub-carriers to the users. This may be due to the fact that the BS (transmitter) doesn't know the fading states of its users. 2. Opportunistic: The BS takes into account the fading when allocating the sub-carriers to the users. This requires that the BS (transmitter) knows the fading states of its users; a situation called CSIT (Channel State Information at Transmitter) in literature.

The gain obtained by the opportunistic allocation is called the multiuser diversity gain.

Possible model extensions

Even though our model seems to have many simplifying assumptions, we will show that it permits to capture and analyze the principal impacts of fading and multiuser diversity gain in cellular OFDMA networks. Moreover the following remarks show how to extend the model.

In [START_REF] Jang | Transmit power adaptation for multiuser OFDM systems[END_REF] it is observed that for OFDMA systems implementing a family of M -QAM modulations (as those described in [START_REF] Goldsmith | Variable-rate variablepower MQAM for fading channels[END_REF]) with some BER target, the AWGN capacity formula (1) should be replaced by The assumption that the fading is constant over a time slot may be replaced by the less restrictive assumption that the fading is ergodic and perfectly known by the receiver (see [START_REF] Caire | On the capacity of some channels with channel state information[END_REF]).

C = w log 2 1+ SNR Γ where Γ = -ln (5 × BER) /1.
In what follows we analyze the three time scales of our model described in Section 2.1.

Information theory

We assume that the users don't move at the considered time scale. Consider a given user served by a given BS u. For each sub-carrier, and each time-slot, say [0,T], the channel output Y (t) is related to the transmitted signal X(t)b y

Y (t)=S u × X(t)+Z(t)+I(t),t ∈ [0,T]
where S u represent the fading assumed constant, Z(t) designate the noise assumed AWGN with power spectral density N 0 and I(t) is the interference. (We skip for the moment the propagation loss induced by the distance.)

Lets denote by p be the power in a given sub-carrier, that is

E |X(t)| 2 = p
For each interfering BS v,l etX v (t) be its transmitted signal and S v be the fading assumed constant. Then the interference equals to

I(t)= v =u S v X v (t)
Since the signals transmitted by different base stations are assumed independent (and centred),

E |I(t)| 2 = v =u |S v | 2 E |X v (t)| 2 = p v =u |S v | 2
Using [22, Theorem 18], we may show that the worst noise process distribution (not necessarily white nor Gaussian) for capacity with given second moment, is the AWGN. We deduce that the capacity C within the considered time-slot is lower bounded by

C ≥ w log 2 1+ p |S u | 2 wN 0 + p v =u |S v | 2
where w is the bandwidth of a sub-carrier.

Remark 1 An alternative way to show the above inequality is to use [1, Lemma 1 p.30] giving the capacity of the interference channel. It is frequent to see in literature the equality sign in place of the inequality in the above display, as for example [20, p.118 ]. We don't see why the equality may be justified in this generality (see [2, §4 p.811] for a similar question).

We will now introduce propagation loss L v induced by the distance between BS v and the given user. In order to account for these losses, the above formula should be replaced by

C ≥ w log 2 1+ p |S u | 2 /L u wN 0 + p v =u |S v | 2 /L v (2)

Ergodic capacity

We will see soon that the expectation of the capacity with respect to the fading process (called ergodic capacity) plays a crucial role. We will not make any assumption of the distribution of the fading process except that

E |S u | 2 =1
Proposition 1 The ergodic capacity E [C] is lower bounded by

E [C] ≥ wE log 2 1+β |S u | 2
where the expectation is with respect to the fading random variables and

β = p/L u wN 0 + p v =u 1/L v (3)
which may be viewed as the signal to interference and noise ratio (SINR) in our model.

Proof Note that the expectations in the present proof are with respect to the fading random variables S u and (S v ) v =u .L e tE [•|S u ] designates the expectation conditionally to S u . By the properties of the conditional expectation we have

E [C]=E [E [C|S u ]]
Equation ( 2) implies

E [C|S u ] ≥ wE log 2 1+ P|S u | 2 /L u N + P v =u |S v | 2 /L v |S u
Since the function x → log(1+1/x)i sc o n v e xo nR * + (its second derivative is 2x+1 x 2 (x+1) 2 ), we deduce from Jensen's inequality that the right-hand side of the above equation is larger than

E log 2 1+β |S u | 2 |S u where β is
given by [START_REF] Baccelli | Elements of queueing theory[END_REF]. Thus

E [C]=E [E [C|S u ]] ≥ wE E log 2 1+β |S u | 2 |S u = wE log 2 1+β |S u | 2 4
Resource allocation

Non-opportunistic allocation

We consider a given sub-carrier and multiple time-slots.

Recall that the fading for different time-slots are independent and identically distributed. By the law of large number, the capacity averaged over a large number of time-slots would approach the so-called ergodic capacity E [C] where the expectation is with respect to the fading states. Proposition 1 gives a lower bound for E [C].

Averaging over a large number of time-slots corresponds to exploiting the so-called time-diversity which is suitable for the analysis of the performance of VBR traffic as observed in [8, §I].

Remark 2 Note that the formula

E log 2 1+β |S u | 2
is similar to the capacity of a channel with ergodic fading in [START_REF] Caire | On the capacity of some channels with channel state information[END_REF] (or i.i.d. fading in [14, Equation ( 2)]) and CSIR (Channel State Information known at Receiver; i.e. the receiver knows the fading). Nevertheless, the fading varies during a codeword there, whereas in the model considered in the present paper the fading is constant during a codeword.

Since |S u | is Rayleigh distributed, i.e. it has the probability density function

f |Su| (s)=2se -s 2 1 s≥0 then H = |S u | 2 has exponential distribution f H (h)=e -h 1 h≥0
In this case,

E log 2 1+β |S u | 2 = ∞ 0 log 2 (1 + βh) e -h dh = 1 ln 2 e β -1 Ei 1,β -1 where Ei (1,x)= ∞ 1 e -xt t dt
is the exponential-integral function.

Remark 3 Frequency diversity.W econsidernowagiven time-slot and large number n of sub-carriers. Assume that the fading variables for different sub-carriers are i.i.d. (or more generally the fading process is ergodic with respect to the number n of subcarriers). Then again, by the law of large number, the capacity of a large number n of sub-carriers is lower bounded by

nwE log 2 1+β |S u | 2
Thus the ergodic capacity is also appropriate for CBR traffic (where we can't always count on time-diversity) when the number of sub-carriers allocated to each user is large enough.

Feasible bit-rates

A base station u allocates to each user m in its cell (denoted with a slight abuse of notation by m ∈ u)a number of sub-carriers of total bandwidth w m (that is the bandwidth w of a given sub-carrier multiplied by the number of sub-carriers allocated to the user). The following proposition gives a condition on the users bit rates (r m ) m∈u assuring the feasibility of the bandwidths allocation.

Proposition 2 Let u be a given base station, W be the total system bandwidth and

β m = p/L u,m wN 0 + p v =u 1/L v,m (4)
be the SINR for user m ∈ u,wher eN and P are respectively the noise and signal powers in a given sub-carrier (of bandwidth w)a n dL v,m is the propagation loss between BS v and user m.I ft h eu s e r sb i tr a t e s(r m ) m∈u satisfy the following condition

m∈u r m E log 2 1+β m |S u | 2 ≤ W (5)
then the allocation of the bandwidths to the users is feasible.

Proof The feasibility of a bandwidth allocation may be formulated as follows. Do there exist bandwidths (w m ) m∈u whose sum doesn't not exceed the total bandwidth W ,t h a ti s m∈u w m ≤ W

We deduce from Proposition 1 that the following bitrates are feasible from the information theory point of view

r m = w m E log 2 1+β m |S u | 2 ,m ∈ u
where the SINR β m is given by ( 4). Since β m is independent of the number of sub-carriers allocated to the user, we may easily express the bandwidth as function of the bit rate as follows

w m = r m E log 2 1+β m |S u | 2
The condition (5) on the bit-rates assures that the sum of the bandwidths given by the above equation doesn't not exceed the total bandwidth W .

Note that (5) may be written as follows m∈u r m γ m ≤ 1 [START_REF] Borst | User-level performance of channel-aware scheduling algorithms in wireless data networks[END_REF] where

γ m = WE log 2 1+β m |S u | 2 -1 (7)
Each bit-rate allocation satisfying (6) will be called non-opportunistic allocation. Note that the constraint ( 6) on the bit-rates do not depend on the current values of the fadings but only on their statistics.

Opportunistic allocation

We assume that the BS knows the fading states of all the users in its cell and that each user knows his own fading state.

Analysis for a given subcarrier

We account now for the fading variations at the resource allocation time scale. In what follows, by opportunistic allocation we mean that at a given time-slot, each subcarrier is allocated to the user with the most favorable fading state;m o r ep r e c i s e l y ,e a c hB Su allocates the subcarrier to the user m such that

|S u,m | =max n∈u |S u,n | (8)
We aim now to calculate the ergodic throughput of each user per time-slot under this policy.

If we allocate a given subcarrier to a user m ∈ u, then we deduce from (2) that, for a given time-slot, we can find a coding scheme to support the following bit-rate

R m = w log 2 1+ p |S u,m | 2 /L u,m N + p v =u |S v,m | 2 /L v,m ,m ∈ u
where the S v,m are the fading states between the user m and BS v for the considered subcarrier and time-slot.

Let us denote by Rm the bit-rate effectively allocated by the opportunistic policy to a given user m averaged over a large number of time-slots. From the assumed temporal ergodicity, we deduce that

Rm = E R m × 1 |S u,m | =max n∈u |S u,n |
Proposition 3 Assume that for a given time-slot the bit-rate of user m is given by

R m = wφ p |S u,m | 2 /L u,m N + p v =u |S v,m | 2 /L v,m ,m ∈ u (9)
where φ is a function defined on R + having a well defined Laplace transform L φ (u)= +∞ 0 φ (x) e -ux dx; and such that the function x → φ (1/x) is convex on R + . Then for the opportunistic allocation ( 8), the bitrate effectively allocated to user m and averaged over a large number of time-slots is lower bounded by

Rm ≥ R ′ m (10) := wE φ β m |S u,m | 2 × 1 |S u,m | =max n∈u |S u,n | = w +∞ 0 φ (β m x) e -x 1 -e -x M-1 dx (11) = w β m M-1 k=0 M -1 k (-1) k L φ k +1 β m ( 12
)
where β m is given by ( 4) and M is the number of the users in the cell.

Proof Since the function x → log(1+1/x)i sc o n v e x , the inequality (10) follows from Jensen's inequality. Since the |S u,n | 2 are exponentially distributed, we get

R ′ m = w R M + φ (β m x m )1 x m =max n∈u x n n∈u e -xn dx n = w R+ φ (β m x) e -x R M -1 + n =m 1 {x n ≤ x} e -xn dx n dx = w R+ φ (β m x) e -x n =m R+ 1 {x n ≤ x} e -xn dx n dx = w R+ φ (β m x) e -x 1 -e -x M-1 dx = w +∞ 0 φ (β m x) e -x 1 -e -x M-1 dx
Using the binomial formula

1 -e -x M-1 = M-1 k=0 M -1 k (-1) k e -kx
we get

R ′ m = w M-1 k=0 M -1 k (-1) k +∞ 0 φ (β m x) e -(k+1)x dx = w M-1 k=0 M -1 k (-1) k +∞ 0 φ (y) e -k+1 βm y dy β m = w β m M-1 k=0 M -1 k (-1) k L φ k +1 β m
We prove now that the Shannon function log 2 (1 + x) satisfies the conditions of the above proposition.

Lemma 1 The function φ (x) = log 2 (1 + x) has a well defined Laplace transform given by

L φ (u)= e u u ln 2 Ei (1,u) (13)
and the function

x → φ (1/x) is convex on R + . Proof The Laplace transform of φ (x) = log 2 (1 + x)i s L φ (u)= +∞ 0 log 2 (1 + x) e -ux dx = 1 ln 2 +∞ 0 ln (1 + x) e -ux dx = 1 ln 2 +∞ 0 e -ux u (1 + x) dx = e u u ln 2 ∞ 1 e -ut t dt = e u u ln 2 Ei (1,u)
where for the third line we use an integration by parts. The convexity of the function x → φ (1/x)o nR + was already shown in the proof of Proposition 1.

Symmetric case

The expression of R ′ m (defined by ( 10)) in the case when φ is linear; that is φ (x)=x/ ln 2 was already given by Borst [START_REF] Borst | User-level performance of channel-aware scheduling algorithms in wireless data networks[END_REF]§II] who showed that

R ′ m = 1 M M k=1 1 k wβ m ln 2 (14)
This case is called symmetric in [6, §II] since the random variables φ β m |S u,m | 2 normalized by their expectations with respect to the fading states are identically distributed. On the other hand, L φ (u)= 1 u 2 ln 2 , thus we deduce from [START_REF] Combes | Scheduling gain for frequency-selective rayleigh-fading channels with application to self-organizing packet scheduling[END_REF] that

R ′ m = M-1 k=0 M -1 k (-1) k 1 (k +1) 2 wβ m ln 2
which is coherent with Equation ( 14) since it is straightforward to prove that

M-1 k=0 M -1 k (-1) k 1 (k +1) 2 = 1 M M k=1 1 k (Indeed, letting f (x)= M-1 k=0 M-1 k (-1) k e -(k+1)x (k+1) 2 and observing that f ′′ (x)=e -x (1 -e -x ) M-1 we get f (x)= 1 M M k=1 1-(1-e -x
) k k which applied to x =0g i v e st h e desired result.)

Thus the expression of R ′ m in our Proposition 3 may be seen as an extension of the result in [START_REF] Borst | User-level performance of channel-aware scheduling algorithms in wireless data networks[END_REF]§II] to the non-symmetric case; since it holds true even when the random variables φ β m |S u,m | 2 normalized by their expectations are not necessarily identically distributed.

Note finally that Equation ( 14) may be written as follows

R ′ m = 1 h(M ) w W 1 γ ′ m (15)
where h(M )=M/ M k=1 1/k and γ ′ m = ln 2 Wβm .T h u s , in the symmetric case, R ′ m is decomposed into some function the number M of the users multiplied by some function of the position of user m. This particular form allows an explicit calculus of the performance at the queueing theory time-scale as will be shown in Proposition 5 below.

Feasible bit-rates

Thebit-rateforuserm over all the bandwidth, say r ′ m , is related to the bit-rate R ′ m per sub-carrier by

r ′ m = W w R ′ m (16)
where R ′ m is given by [START_REF] Coluccia | On the optimality of max-min fairness in resource allocation[END_REF].

Example 1 In the particular case when R ′ m is given by ( 15), then

r ′ m = 1 h(M ) 1 γ ′ m (17)
5 Queueing theory

Traffic dynamics

Denote the geographic region covered by the a cell by D that is assumed to be a bounded subset of the plane R 2 . Consider only VBR calls whose inter-arrival times to D are independent and identically distributed (i.i.d.) exponential random variables with rate λ (mean 1/λ). The position of each arrival is picked at random in D according to the uniform distribution. We assume that users don't move during their calls. Each call requires to transmit a given volume of data (amount of bits that has to be sent or received), which is modeled by an independent of everything else, exponential random variable with parameter µ.T h eq u a n t i t yρ = λ/µ is called the traffic demand (expressed in Mbps 1 ) per cell. Users are served by the BS according to some bit-rate allocation policy.

The set of positions of all users served at a given time is called configuration of users.L e tM be the set 1 The abbreviation Mbps designates "Kilo-bit per second". of all possible configurations (this can be formalized e.g. on the basis of the theory of point processes). We denote by {N t } t≥0 the process describing the evolution in time of the user configurations in D (due to arrivals and departures). It takes its values in M. If the process {N t } t≥0 isn't ergodic, then the mean number of users in the system grows unboundedly in the long run of the system. This situation has to be avoided; in which case we say that the system is stable (or equivalently ergodic).

One distinguish two milestones of the analytical evaluation of the network performance: identification of its stability region, and the evaluation of the steady state characteristics (e.g. the mean throughput and delay).

Non-opportunistic allocation

Define the critical traffic demand ρ c by

ρ c = 1 γ
where γ is the expectation of γ m (which is given by ( 7)) with respect to the position of the user m uniformly distributed in the cell.

Proposition 4 Assume any non-opportunistic workconserving bit-rate allocation satisfying Condition [START_REF] Borst | User-level performance of channel-aware scheduling algorithms in wireless data networks[END_REF].

If the traffic demand per cell

ρ<ρ c ( 18
)
then the system is stable. Consider the particular allocation

r m = 1 Mγ m (19)
Then, at the steady state, the expected number of users in cell u,d e n o t e d N ,e q u a l s

N = ρ ρ c -ρ (20)
the mean delay per call, denoted T ,i s

T = 1 µr = 1 µ (ρ c -ρ) (21)
and the mean throughput, denoted r,e q u a l s

r = ρ c -ρ (22)
Proof A bit-rate allocation is said to be work-conserving when m∈u r m γ (m) = 1 as long as there is at least a user in the cell. The stability condition [START_REF] Kleinrock | Queueing systems[END_REF] and the expression [START_REF] Liu | OFDM-based broadband wireless networks: Design and optimization[END_REF] of the users number at the steady state follow from known results for multi-class processor sharing queues [START_REF] Cohen | The multiple phase service network with generalized processor sharing[END_REF][START_REF] Kleinrock | Queueing systems[END_REF]. Applying Little's formula [START_REF] Baccelli | Elements of queueing theory[END_REF] we get the expression (21) for the delay. Using the fact that the throughput r is the ratio of the date volume average , which equals 1/µ,a n dt h ed e l a y T proves Equation [START_REF] Shannon | A mathematical theory of communication[END_REF].

Opportunistic allocation

The following proposition gives the performance of the opportunistic allocation [START_REF] Karray | Analytical evaluation of QoS in the downlink of OFDMA wireless cellular networks serving streaming and elastic traffic[END_REF]. then the system is stable. Moreover, at the steady state, the mean throughput per user, denoted r′ ,e q u a l s

r′ = 1 γ′ 1 H(γ ′ ρ)
where the function H(s) is defined for s>0 by

H(s)= E[H(X +1)] E[H(X)] ,H (M )= M k=1 h(k) if M ≥ 1 1 if M =0
where X is a Poisson random variable with parameter s. Moreover, the mean delay is

Approximation

Note that Proposition 5 assumes a particular case when the decomposition (17) holds true as in the symmetric case described in Section 4.2.2.

Unfortunately, for the more general allocation (16) we do not know a closed form for the performance; especially because the geometric parameters β m depend of the location of the user m. The idea is then to replace these parameters by a constant β 0 calculated as follows.

β 0 is determined in such a way that the average over the cell of the factors γ m given by Equation ( 7) should remain unchanged when the β m are replaced by β 0 .(By Proposition 4, this is equivalent to say that the performance of the non-opportunistic allocation [START_REF] Kushner | Convergence of proportional-fair sharing algorithms under general conditions[END_REF] remains unchanged when the β m are replaced by β 0 .)

We will approximate the performance of our opportunistic scheme ( 16) by replacing β m with β 0 and applying Proposition 5.

6 Numerical results

Model specification

In order to obtain numerical values, we consider the most popular hexagonal network model,w h e ret h eb ase stations are placed on a regular infinite hexagonal grid. Let R be the radius of the disc whose area is equal to that of the hexagonal cell served by each base station, and call R the cell radius.W et a k eR =0 .525km. We assume a distance-loss L(r)=( Kr) η , with η =3 .38, K = 8667km -1 . This means that the distance-loss between BS u and user m is equal to L(|x u ,x m |)w h e r e x u ,x m denote, respectively, the geometric location of u and m and |•| is the Euclidean distance.

We consider VBR traffic with traffic demand assumed (spatially) uniform over the cell. We consider a traffic demand per cell ρ varying from 0 to 20Mbps.

The BS maximal total power is P = 52dBm (accounting for antenna gains and losses), the total bandwidth is W = 5MHz and the ambient noise power WN 0 = -103dBm.

Results

Figure 1 shows the throughput per user as function of the traffic demand per cell for the opportunistic allocation. We show the result of the simulations and of the approximation described in Section 5.3.1. The two curves are close which validates the considered approximation. Figure 2 shows the throughput per user as function of the traffic demand per cell for the AWGN, nonopportunistic and opportunistic allocation. Firstly, we observe that for the non-opportunistic allocation, the fading decreases the throughput per user of at least 13% compared to AWGN. Note moreover that the nonopportunistic allocation gives, like AWGN, the throughput that is a linear function of the demanded traffic in the bounded stability region. Secondly, the gain induced by opportunism increases with the traffic. When the traffic demand is very small, the gain induced by the opportunistic allocation is negligible. Nevertheless, when the traffic demand per cell exceeds some value (about 2 Mbps in our example) the gain induced by opportunism compensates the degradation induced by fading compared to AWGN.

7C o n c l u s i o n

We have given analytic expressions for the mean throughput and delay in OFDMA cellular networks with VBR traffic. These formulas take into account in a simple but not simplistic way all important elements of the network performance: the network geometry, the traffic dynamics, fading and bit-rate allocation policy. They allow, in particular, to compare in a systematic way the effect on performance of fading for either non-opportunistic or opportunistic allocation policies.

Perspectives

Even though we assumed Rayleigh distribution for the fading, the approach may be extended to other distributions such as Rice, Nakagami, etc.

It may be useful to study the effect of grouping the sub-carriers into the so-called chunks (i.e. a group of sub-carriers and a number of time-slots); see [20, §6.2].

As observed in [20, §6.2], the non-opportunistic allocation scheme is suitable for fast varying fading, due to high mobility of users for example, which makes difficult to track of the rapidly varying channel. On the other hand, the opportunistic allocation scheme is suitable for slow varying fading as in the case of fixed (or pedestrian) users for example. Thus, from the point of view of the present paper, mobility decreases performance. It is interesting to couple this with the account of the speed effect at the queueing theory time scale in [START_REF] Karray | Performance evaluation of scalable congestion control schemes for elastic traffic in cellular networks with power control[END_REF] which have shown that mobility ameliorates performance.

Proposition 5

 5 Consider the opportunistic bit-rate allocation (17) and denote by γ′ the expectation of γ ′ m with respect to the user's location m uniformly distributed in the cell. If the traffic demand per cell

  γ ′ ρ) µ and the mean number of users at the steady state equals γ′ ρH(γ ′ ρ) Proof See [6, Proposition 3.1] and [16, §11.1].

Fig. 1

 1 Fig.1Validation of the approximation for the opportunistic allocation.

Fig. 2

 2 Fig.2Comparison of the perfomance of the different allocations.

Acknowledgments

We thank F. Baccelli at INRIA for motivating discussions on resource allocation and queueing theory. We thank also M. Debbah from Alcatel-Lucent for instructive exchanges about information theory.


M A Sbai
email: a.sbai@brgm.fr

SCALABLE PARALLEL DOMAIN DECOMPOSITION FOR MIXED FINITE ELEMENTS: APPLICATION TO ANALYSIS OF CONVECTIVE MIXING IN CO 2 STORAGE

Accurate prediction of the long-term fate of carbon dioxide in saline aquifers is a numerical and computational challenge. High permeability rocks are ubiquitous in many sedimentary bassins targeted by this technology. In many of such cases, convective mixing of dissolved supercritical CO 2 rich waters with the initial saline waters is likely to occur involving a multiscale time-space dependent problem. Spatial resolutions needed to capture dissolution fingers for practical applications like the first commercial CO 2 injection project in Sleipner (North-Sea) are in the orders of few centimetres. Therefore, efficient and highly scalable parallel decomposition methods need to be developed, hopefully as a black-box simulator, to advance scientific knowledge for such applications in as much to demonstrate the long-term safety of future storage sites. To this end, we implemented a balancing domain decomposition mixed finite elements for second-order elliptic partial differential equations. Problems in this form arise in many other applications including single-phase and multiphase subsurface flow, electrostatics, implicit discretisations of parabolic and hyperbolic equations, and as a subproblem in the application of operator-splitting techniques to advection-dispersion-reaction transport equations. The Balancing Domain Decomposition method, is a substructuring method that involves at each iteration the solution of a local problem with Dirichlet boundary conditions, a local problem with Neumann boundary conditions, and a coarse grid problem to propagate information globally (over all domain processors) and to ensure the consistency of the Neumann problems. Our numerical experiments show that the condition number, consistently with theory, grows at worst like O 1 + log H h 2 in both two and three dimensions, where H is the characteristic subdomain size, and h the characteristic mesh size. Computational results from an MPI message passing parallel implementation on IBM CLX/1024 Linux cluster machine at CINECA demonstrate the scalability properties of the method and show almost optimal linear speed-up for up to 128 processors.

Acknowledgement: This research work is supported by the French Agency for Scientific Research (ANR), and BRGM Research Direction, under contract number ANR-06-CO2-006-01 in the framework of Heterogeneities-CO2 project. Support for a HPC-EUROPA visit program in 2007 at CINECA host in Bologna is gratefully acknowledged.