Sophie Chauvet

Veronique Fremeaux-Bacchi

Florent Petitprez

Alexandre Karras

Laurent Daniel

Stéphane Burtey

Gabriel Choukroun

Yahsou Delmas

Dominique Guerrot

Arnaud François

V´eronique Fr ´emeaux-Bacchi

St ´ephane Burtey

Arnaud Franc

Moglie Le Quintrec

Vincent Javaugue

David Ribes

Laurence Vrigneaud

Bertrand Arnulf

Jean Michel Goujon

Pierre Ronco

Guy Touchard

Frank Bridoux

Treatment of B-cell disorder improves renal outcome of patients with monoclonal gammopathy-associated C3 glomerulopathy

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

• Monoclonal gammopathy is associated with C3 glomerulopathy. • Specific treatment of the underlying B-cell clone improves renal survival.

Introduction

A wide spectrum of renal manifestations, including glomerular and tubular diseases, may occur in patients with monoclonal gammopathies. [START_REF] Bridoux | Diagnosis of monoclonal gammopathy of renal significance[END_REF] Except for myeloma cast nephropathy, invariably diagnosed in the setting of symptomatic multiple myeloma (MM), most renal disorders related to deposition or precipitation of a monoclonal immunoglobulin (MIg) occur in patients with indolent B-cell clone, most commonly of plasmocytic origin. The concept of monoclonal gammopathy of renal significance (MGRS) was introduced to depict the spectrum of renal diseases induced by MIg produced by indolent B-cell clones and to highlight the effect of chemotherapy on renal and patient outcomes. [START_REF] Bridoux | Diagnosis of monoclonal gammopathy of renal significance[END_REF] Trials on MGRS treatment are lacking, and therapeutic recommendations are currently based on consensus expert opinion. [START_REF] Fermand | International Kidney and Monoclonal Gammopathy Research Group. How I treat monoclonal gammopathy of renal significance (MGRS)[END_REF] It has been recently proposed to include C3 glomerulopathy (C3G) in the spectrum of MGRS. [START_REF] Bridoux | Diagnosis of monoclonal gammopathy of renal significance[END_REF] C3G is a heterogeneous group of rare glomerular diseases characterized by glomerular lesions with predominant complement component C3 fragment deposition without significant Ig deposits. [START_REF] Fakhouri | C3 glomerulopathy: a new classification[END_REF][START_REF] Servais | Primary glomerulonephritis with isolated C3 deposits: a new entity which shares common genetic risk factors with haemolytic uraemic syndrome[END_REF] According to the location and aspect of C3 deposits by immunofluorescence and electron microscopy, 2 different patterns of glomerular diseases, dense deposit disease (DDD), and glomerulonephritis with isolated C3 deposits (C3GN) are distinguished. [START_REF] Pickering | C3 glomerulopathy: consensus report[END_REF][START_REF] Hou | Toward a working definition of C3 glomerulopathy by immunofluorescence[END_REF] Both are associated with dysregulation of complement alternative pathway (AP) because of autoantibodies against complement proteins in most patients. [START_REF] Servais | Acquired and genetic complement abnormalities play a critical role in dense deposit disease and other C3 glomerulopathies[END_REF][START_REF] Sethi | Proliferative glomerulonephritis secondary to dysfunction of the alternative pathway of complement[END_REF][START_REF] Appel | Membranoproliferative glomerulonephritis type II (dense deposit disease): an update[END_REF][START_REF] Smith | Dense deposit disease[END_REF] The association between MIg and C3G has been recently described. [START_REF] Sethi | Dense deposit disease associated with monoclonal gammopathy of undetermined significance[END_REF][START_REF] Bridoux | Glomerulonephritis with isolated C3 deposits and monoclonal gammopathy: a fortuitous association?[END_REF][START_REF] Zand | C3 glomerulonephritis associated with monoclonal gammopathy: a case series[END_REF] In 2 series, MIg was found in 10/30 and 10/14 patients, indicating that the frequency of MIg in C3G patients older than 50 years largely exceeds that of the age-matched general population, and that the association is probably not fortuitous. [START_REF] Sethi | Dense deposit disease associated with monoclonal gammopathy of undetermined significance[END_REF][START_REF] Zand | C3 glomerulonephritis associated with monoclonal gammopathy: a case series[END_REF] Moreover, improvement of renal parameters after chemotherapy in a few patients suggests MIg could have a pathogenic role in AP activation and that chemotherapy targeting the underlying B-cell clone could favorably influence renal survival.

The aim of the present study was to evaluate renal outcomes after chemotherapy in a large cohort of patients with C3G and MIg.

Subjects and methods

Study population

Between 2000 and 2014, 201 patients older than 18 years were included in the French registry of C3G if complete renal pathological data were available. Patients with positive hepatitis B or C serology and antinuclear or anti-double-stranded DNA antibodies are excluded from the French registry.

Patients were included in this retrospective study on the basis of the following criteria [START_REF] Bridoux | Diagnosis of monoclonal gammopathy of renal significance[END_REF] : biopsy-proven C3G on native kidneys or renal allograft without evidence of associated Ig light chain (LC) or heavy chain glomerular deposits, except for trace IgM on glomerular sclerotic lesions by immunofluorescence [START_REF] Fermand | International Kidney and Monoclonal Gammopathy Research Group. How I treat monoclonal gammopathy of renal significance (MGRS)[END_REF] ; detectable serum and/or urine MIg [START_REF] Fakhouri | C3 glomerulopathy: a new classification[END_REF] ; and availability of clinical data. Patients with cryoglobulinemia were excluded (Figure 1).

For analysis, patients were divided into 3 groups according to treatment strategy, which consisted of chemotherapy adapted to the B-cell clone (n 5 29), conventional immunosuppressive therapy (n 5 8), or blockers of the renin-angiotensin system (n 5 13). Demographics and clinical and biological data were recorded at the time of the first kidney biopsy, at initiation of treatment, after completion of first-line treatment, and at last follow-up. Estimated glomerular filtration rate (eGFR) was calculated using the modified MDRD (Modification of Diet in Renal Disease) equation. [START_REF] Levey | Modification of Diet in Renal Disease Study Group. A more accurate method to estimate glomerular filtration rate from serum creatinine: a new prediction equation[END_REF] Impaired renal function was defined by an eGFR value of 60 mL/min/m 2 or less, and stages of chronic kidney disease (CKD) were assessed according to the US National Kidney Foundation. [START_REF] Kidney | K/DOQI clinical practice guidelines for chronic kidney disease: evaluation, classification, and stratification[END_REF] Nephrotic syndrome was defined by urinary protein excretion of 3 g/24 h or more or a urine protein-to-creatinine ratio of 3 g/g or higher and serum albumin levels of 30 g/L or lower.

Pathological studies

All kidney biopsy samples were processed for light and immunofluorescence microscopy. Immunofluorescence study was performed on sections directly frozen in liquid nitrogen (without transport in Michel's medium), using conjugates specific for a, g, and m Ig heavy chains; k and l Ig light chains (Dakopatts, Glostrup, Denmark); C3; C4; C1q; and albumin (Morphosys AbD, Düsseldorf, Germany). Electron microscopy was available in 25 cases.

The diagnosis of C3G was assessed by immunofluorescence, according to consensus recommendations, with bright diffuse predominant C3 glomerular staining ($21) of at least 2 orders of magnitude greater than any other immune reactant (ie, Ig). Patients with trace or weak amounts of IgM staining on glomerular sclerotic lesions were included, but those with weak staining for IgG, IgA, or Ig light chains were excluded. [START_REF] Pickering | C3 glomerulopathy: consensus report[END_REF] The diagnosis of DDD was confirmed by demonstration of diffuse, highly electron-dense osmiophilic deposits within the lamina densa by electron microscopy. In contrast, the appropriate, for comparison of continuous variables, and using chisquare or Fisher's exact tests for the comparison of categorical variables. Survival analysis was performed by the Kaplan-Meier method, using the log-rank test for comparison of groups. P values below .05 were considered significant. Results were analyzed using the Graph Pad Prism software.

Contribution to various factors to the probability of renal response was assessed using univariate and multivariate logistic regression model, using the R function glm. The contribution to end-stage renal disease (ESRD) was assessed using Cox proportional hazards models, both univariate and multivariate, computed using the R package survival.

Results

Epidemiological data

In the French cohort of 201 C3G patients, 60 (29.8%) had a monoclonal gammopathy. Among them, 50 patients were included in the study (including 6 patients described in Bridoux et al.) [START_REF] Bridoux | Glomerulonephritis with isolated C3 deposits and monoclonal gammopathy: a fortuitous association?[END_REF] (Figure 1). We excluded 10 patients because of positive cryoglobulinemia (n 5 4) and/ or unavailable clinical data (n 5 8). On the remaining 141 C3G patients without monoclonal gammopathy, only 29 were older than 50 years at the time of diagnosis. The prevalence of MIg in patients with C3G older than 50 years was 59% and increased at each decade, reaching 36%, 59%, and 94% of C3G patients within the age ranges of 50 to 59, 60 to 69, and 70 to 79 years, respectively (supplemental Figure 1, available on the Blood web site).

Clinical features and laboratory evaluation of patients at time of kidney biopsy and at onset of treatment

Median serum creatinine was 158 (68-990) mmol/L with eGFR of 37 (3-100) mL/min/1.73 m 2 . Forty-two patients (86%) presented with CKD, including 15/49 (31%) with CKD stage 3 and 27/49 (55%) with CKD stage 4 or 5. At diagnosis, 20/46 (43%) patients had nephroticrange proteinuria (Table 1). Renal parameters at diagnosis were similar among patients who received chemotherapy, immunosuppressive therapy, or conservative treatment (Table 1). Comparison of renal parameters in patients who received chemotherapy or immunosuppressive/conservative treatment are detailed in supplemental Table 1.

Median time from diagnosis to treatment initiation was 3 months. [START_REF] Bridoux | Diagnosis of monoclonal gammopathy of renal significance[END_REF][START_REF] Fermand | International Kidney and Monoclonal Gammopathy Research Group. How I treat monoclonal gammopathy of renal significance (MGRS)[END_REF][START_REF] Fakhouri | C3 glomerulopathy: a new classification[END_REF][START_REF] Servais | Primary glomerulonephritis with isolated C3 deposits: a new entity which shares common genetic risk factors with haemolytic uraemic syndrome[END_REF][START_REF] Pickering | C3 glomerulopathy: consensus report[END_REF][START_REF] Hou | Toward a working definition of C3 glomerulopathy by immunofluorescence[END_REF][START_REF] Servais | Acquired and genetic complement abnormalities play a critical role in dense deposit disease and other C3 glomerulopathies[END_REF][START_REF] Sethi | Proliferative glomerulonephritis secondary to dysfunction of the alternative pathway of complement[END_REF][START_REF] Appel | Membranoproliferative glomerulonephritis type II (dense deposit disease): an update[END_REF][START_REF] Smith | Dense deposit disease[END_REF][START_REF] Sethi | Dense deposit disease associated with monoclonal gammopathy of undetermined significance[END_REF][START_REF] Bridoux | Glomerulonephritis with isolated C3 deposits and monoclonal gammopathy: a fortuitous association?[END_REF][START_REF] Zand | C3 glomerulonephritis associated with monoclonal gammopathy: a case series[END_REF][START_REF] Levey | Modification of Diet in Renal Disease Study Group. A more accurate method to estimate glomerular filtration rate from serum creatinine: a new prediction equation[END_REF][START_REF] Kidney | K/DOQI clinical practice guidelines for chronic kidney disease: evaluation, classification, and stratification[END_REF][START_REF] Myeloma | Criteria for the classification of monoclonal gammopathies, multiple myeloma and related disorders: a report of the International Myeloma Working Group[END_REF][START_REF] Vardiman | The 2008 revision of the World Health Organization (WHO) classification of myeloid neoplasms and acute leukemia: rationale and important changes[END_REF][START_REF] Leung | Monoclonal gammopathy of renal significance: when MGUS is no longer undetermined or insignificant[END_REF][START_REF] Radhakrishnan | The KDIGO practice guideline on glomerulonephritis: reading between the (guide)lines-application to the individual patient[END_REF][START_REF] Durie | International uniform response criteria for multiple myeloma[END_REF] At onset of treatment, in the group of 29 patients treated with chemotherapy, median proteinuria was 3.55 g/day, with median serum creatinine of 202 mmol/L and eGFR of 28 mL/min/1.73 m 2 , including 6 (21%), 9 (31%), and 14 (48%) with CKD stage 1 to 2, CKD stage 3, and CKD stage 4 to 5, respectively (Table 1). In the 8 patients who received immunosuppressive therapy, at treatment initiation, median proteinuria was 2.9 g/day, with median serum creatinine of 198 mmol/L and eGFR of 29 mL/min/1.73 m 2 , including 1 (13%) and 7 (87%) patients with CKD stage 3 and CKD stage 4 to 5, respectively. From diagnosis to treatment initiation, 25% or more decrease in eGFR was observed in 8/29 and in 3/8 patients who received chemotherapy or immunosuppressive drugs, respectively.

Five patients had extra renal manifestations, including diffuse mucinosis (n 5 1), digital ischemia (n 5 2), skin purpuric lesions (n 5 1), and capillary leak syndrome (n 5 1; Table 1). Renal parameters at diagnosis were similar in C3G patients with MIg and in C3G patients older than 50 years without MIg (supplemental Table 2).

diagnosis of C3GN was established in patients showing deposits of lesser density without the characteristic distribution and "sausage shape" appearance of DDD.

Hematologic studies

Serum MIg were detected and quantified by electrophoresis and characterized by immunofixation. Serum free LCs (FLCs) were monitoredin32patients(BindingSite, Birmingham, United Kingdom). Abnormal FLC ratio was defined as a k/l ratio lower than 0.26 or higher than 1.65. Bone marrow smears and/or biopsy were performed in all cases, and clonality of plasma cells was tested by immunofluorescence (n 5 2) or flow cytometry (n 5 48). The diagnosis of MGRS, MM and chronic lymphocytic leukemia (CLL) was established according to international criteria. [START_REF] Myeloma | Criteria for the classification of monoclonal gammopathies, multiple myeloma and related disorders: a report of the International Myeloma Working Group[END_REF][START_REF] Vardiman | The 2008 revision of the World Health Organization (WHO) classification of myeloid neoplasms and acute leukemia: rationale and important changes[END_REF][START_REF] Leung | Monoclonal gammopathy of renal significance: when MGUS is no longer undetermined or insignificant[END_REF] Local ethics committees approved research, and all patients gave written informed consent.

Definition of renal and hematological responses

Renal response was assessed basedonKidneyDisease:Improving Global Outcomes practice guidelines on glomerulonephritis. [START_REF] Radhakrishnan | The KDIGO practice guideline on glomerulonephritis: reading between the (guide)lines-application to the individual patient[END_REF] Complete renal response was defined by proteinuria levels of 0.5 g/24 h or less, with albuminemia levels of 30 g/L or more and no more than 10% decrease in eGFR from baseline value. Partial renal response was defined by posttreatment proteinuria between 0.5 and 2.5 g/24 h or by a 50% or more reduction from baseline value, with albuminemia levels of 25 g/L or more and no more than a 10% decrease in eGFR from baseline value.

Hematological response was assessed according to the International Myeloma Working Group for patients with symptomatic MM. [START_REF] Durie | International uniform response criteria for multiple myeloma[END_REF] In patients with MGRS, response criteria were adapted from the 2012 International Society of Amyloidosis criteria 21 : complete hematological response was definedbynegativeimmunofixation of serum and urine with normal FLC ratio, very good partial response was defined by a 90% or more reduction of serum MIg and/or a 90% or more decrease in the difference between the concentration of the involved and uninvolved FLC (dFLC), and partial hematological response was defined by a 50% or more reduction of serum MIg and/or a 50% or more dFLC decrease.

Assays for complement component and genetic screening

All immunological and genetic analyses were performed at the reference laboratory for investigation of the complement system (European Hospital Georges Pompidou, France). EDTA plasma samples were obtained from all patients.PlasmaC3andC4levels were measured as previously described. [START_REF] Roumenina | Alternative complement pathway assessment in patients with atypical HUS[END_REF] Soluble C5b-9 (sC5b-9) level was determined using the MicroVue sC5b-9 Plus EIA Assay (Quidel, San Diego, CA), according to manufacturer's instructions. Normal values were evaluated after testing plasma from 100 healthy donors.

C3 nephritic factor activity was determined by assessing the ability of purified plasma IgG to stabilize membrane-bound C3bBb convertase. [START_REF] Fr Émeaux-Bacchi | Selective disappearance of C3NeF IgG autoantibody in the plasma of a patient with membranoproliferative glomerulonephritis following renal transplantation[END_REF] Titers of anti-Factor H (FH) IgG or IgA antibodies were determined by ELISA. Direct sequencing of all CFH, CFI, CD46,and C3 exons was undertaken in 28 patients with available DNA, as previously described. [START_REF] Roumenina | Alternative complement pathway assessment in patients with atypical HUS[END_REF]

Statistical analysis

Data are expressed as median for continuous variables and percentage for categorical variables. Statistical analysis was performed using the Student t test and Mann-Whitney and Kruskal-Wallis tests, as By light microscopy, 27 patients (54%) exhibited mesangial proliferative glomerulonephritis and 23 (46%) had a membranoproliferative pattern (Figure 2A-B). Glomerular crescents were found in 11 patients (22%), diffuse and cellular in 2 cases. Eight (16%) patients had thrombotic microangiopathy-like lesions, with occasional thrombi within glomerular capillaries, mesangiolysis, endothelial swelling and subendothelial lucent deposits. One patient with l LC MM had concomitant cast nephropathy with numerous fractured polychromatophilic l LC-positive casts in distal tubule lumens. By immunofluorescence, all patients displayed bright glomerular C3 deposits with prominent mesangial and subendothelial distribution (Figure 2C). Predominant subepithelial C3 deposits were observed in 14 patients (Figure 2D) (Table 1; supplemental Table 3).

Ultrastructural pattern was consistent with C3GN in 24/25 (96%) patients, with osmiophilic dense deposits located in mesangium (80%) and subendothelial space (76%). Humps were observed in 48% of patients (Figure 2E). Only 1 patient had DDD with typical dense "sausage-shaped" deposits within glomerular basement membranes and less abundant subendothelial osmiophilic deposits (Figure 2F).

Baseline hematological data

A serum monoclonal gammopathy (IgG, n 5 47; IgA, n 5 2; LC only, n ). The prevalence of MM was significantly higher in patients treated with chemotherapy or immunosuppressive drugs compared with untreated patients (P 5 .03; Table 1).

Complement studies

Plasma C3 and sC5b-9 levels in MIg-C3G patients were compared with those of the 29 C3G patients older than 50 years without MIg (supplemental Table 2). Median C3 level was similar (P 5 .56), but median sC5b-9 level was significantly higher in patients with MIg-C3G

(734 [164-2370] ng/mL vs 498 [215-1501] ng/mL, respectively; P 5 .03; supplemental Table 2).

Complement studies in patients with MIg-C3G are detailed in Table 1. Before initiation of hematological treatment, baseline complement C3 level was low in 22/50 (43%) patients, without any significant difference among patients treated with chemotherapy, immunosuppressive therapy, or conservative treatment (P 5 .7; Table 1). C4 level was normal in all patients. Twenty-seven of 34 tested patients had elevated sC5b-9 level without significant difference among the 3 groups of patients (P 5 .42; Table 1). C3NeF and anti-FH antibodies were found in 3 (6%) and 9 (19%) patients, respectively. Screening of variants in complement genes was performed in 28 patients. Two patients carried a rare variant of undetermined significance (p.D130N in CFH and p.E548Q in CFI).

Treatment

Thirteen patients were given conservative treatment alone with reninangiotensin system blockers. Eight patients received various immunosuppressive drugs after a median of 4 (1-54) months from diagnosis. Immunosuppressive regimens consisted of steroids alone (n 5 4), prednisone combined with rituximab (n 5 1), mycophenolate mofetil (n 5 1), azathioprine (n 5 1), or cyclophosphamide (n 5 1). Twentynine patients received chemotherapy targeting the associated B-cell clone after a median time from diagnosis of 3 (1-20) months. Chemotherapy was based on alkylating agents (cyclophosphamide, n 5 2; melphalan, n 5 3) with dexamethasone (DXM) in 5 patients, and on bortezomib in 22 patients (bortezomib-DXM: n 5 10; bortezomib-cyclophosphamide-DXM, n 5 10; bortezomib-thalidomide-DXM, n 5 2). Two patients with CLL received rituximab-based therapy. Patients received a median of 6 (3-9) cycles as first-line treatment (Table 2).

Tolerance to chemotherapy

Five of 29 patients who received chemotherapy experienced severe adverse events, including 6 episodes of infectious pneumonitis (bacterial, n 5 4; aspergillosis, n 5 1; H1N1 virus, n 5 1), leading to death in 3 cases. One patient treated with bortezomib developed peripheral neuropathy that required treatment discontinuation.

Hematological and renal response

Hematological response and renal responses were available in 37 treated patients (chemotherapy or immunosuppressive therapy) and 48 patients, respectively (Table 3). After first-line therapy, 18 of 37 treated patients achieved hematological response (assessed on serum monoclonal protein level by electrophoresis in 17 cases and/or FLC criteria in 6 cases). In 1 patient with l LC myeloma, hematological response was based on FLC criteria only. Of these patients, 15 (83%) had a renal response. In contrast, among 19 patients who did not achieve hematological response after treatment, 5/18 (28%) had a renal response (P 5 .002). By univariate analysis, predictive factors of renal response were proteinuria at onset of treatment (P 5 .005), use of chemotherapy (P 5 .001), use of bortezomib-based regimen (P 5 .0001), time to treatment initiation (P 5 .018), and hematological response (P 5 .0001; Table 4). By multivariate analysis, only hematological response was associated with renal response (odds ratio, 9; 95% confidence interval [CI], 7.98-10.02; P 5 .035).

Patient and renal survival

At last follow-up, 5 patients had died. Causes of death were cardiac (n 5 1), neurological (n 5 1), and infectious complications after chemotherapy (n 5 3). Overall survival of patients with MIg-associated C3G was not significantly different from that of patients older than 50 years without MIg (P 5 .16; Figure 3A). One patient treated with corticosteroids alone developed hepatocellular carcinoma 3 years after diagnosis. The median renal survival of MIg-C3G patients was significantly lower compared with that of C3G patients older than 50 years without monoclonal gammopathy (P , .003; hazard ratio [HR], 2.915; 95% CI, 1.41-6.01; Figure 3B). Among C3G patients under conservative treatment alone, median renal survival was also significantly higher in those without MIg (P 5 .02; supplemental Figure 1). After a median Median renal survival was higher in patients who had received chemotherapy compared with in those given conservative treatment (48 vs 31 months, respectively; P 5 .01; HR, 0.27; 95% CI, 0.09-0.77), but was not significantly different from that of patients receiving immunosuppressive therapy (28 months; P 5 .18; HR, 0.41; 95% CI, 0.11-1.51; Figure 4A). Median renal survival in patients who received chemotherapy was similar to that in C3G patients older than 50 years without MIg (data not shown). Renal survival was higher in patients who achieved a hematological response compared with those who did not (P 5 .009; HR, 0.17; 95% CI, 0.12-0.66; Figure 4B). Moreover, renal survival was higher in patients who reached complete hematological response or very good partial response compared with in patients who did not respond or achieved PR only (P 5 .05; Figure 4C). In patients who received chemotherapy, renal survival was significantly higher in patients with hematological response compared with in nonresponders (P 5 .04; supplemental Figure 3).

We performed Cox proportional hazards model analysis regarding progression to ESRD. By univariate analysis, renal response, hematological response and hematological treatment were predictive of renal survival (P 5 .048, P 5 .028, and P 5 .022, respectively; supplemental Figure 4). In 2 different models, by multivariate analysis, only the absence of renal response was independently associated with ESRD (P 5 .0006 and P 5 .008, respectively; supplemental Table 4).

Four patients, including 3 patients in the conservative treatment group and 1 treated with rituximab-prednisone, received a renal allograft. At the time of transplantation, none had achieved hematological response, and all showed disease recurrence after 3 to 12 months that resulted in graft loss in 1 case.

Complement assays after treatment

Posttreatment complement studies were available in 18 patients, including 9 patients with HR after chemotherapy. Plasma C3 level normalized in 6 patients, 5 of whom had achieved HR. C3 level in the 9 patients with HR was significantly higher after treatment compared with pretreatment C3 level (P 5 .007; Figure 5A). In contrast, C3 levels in 9 patients who did not achieve HR were unchanged after treatment (Figure 5B).

Discussion

In this study, we analyzed the effect of chemotherapy targeting the underlying B-cell clone in a large cohort of patients with MIg-C3G. We showed that efficient chemotherapy resulted in higher renal response rate and longer renal survival than conservative or immunosuppressive therapy.

C3G is a rare glomerular disease associated with dysregulation of the complement AP. The severity is variable, with one third of patients initially presenting with nephrotic range proteinuria and/or renal impairment. [START_REF] Servais | Acquired and genetic complement abnormalities play a critical role in dense deposit disease and other C3 glomerulopathies[END_REF] Renal prognosis of C3G remains poor, as 40% of patients progress to ESRD 8 to 10 years after diagnosis, [START_REF] Servais | Acquired and genetic complement abnormalities play a critical role in dense deposit disease and other C3 glomerulopathies[END_REF] and treatment is ill defined. In addition to nonspecific nephroprotective drugs such as renin-angiotensin system blockers, several observations suggest some efficacy of immunosuppressive therapies, patients with or without hematological response (complete response or very good partial response) in 37 patients who received treatment (immunosuppressive therapy or chemotherapy). Data regarding the hematological response were available in 36 of the 37 patients. CR, complete hematological response; HR, hematological response; IS, immunosuppressive; VGPR, very good partial response.

including eculizumab, a monoclonal anti-C5 antibody targeting the final step of complement pathway, and agents targeting T and/or B cells with a potential effect on glomerular inflammation and production of auto-antibodies activating complement AP. [START_REF] Bomback | Eculizumab for dense deposit disease and C3 glomerulonephritis[END_REF][START_REF] Nester | Treatment options for C3 glomerulopathy[END_REF][START_REF] Vivarelli | Eculizumab for the treatment of dense-deposit disease[END_REF][START_REF] Giaime | Remission of C3 glomerulopathy with rituximab as only immunosuppressive therapy[END_REF] In a recent study, it has been suggested that mycophenolate mofetil may improve renal survival in C3G. [START_REF] Rabasco | Spanish Group for the Study of Glomerular Diseases (GLOSEN). Effectiveness of mycophenolate mofetil in C3 glomerulonephritis[END_REF] C3G associated with monoclonal gammopathy remains poorly described. The high prevalence of monoclonal gammopathy in C3G patients older than 50 years, reaching 30% to 71% in 2 small series and 65% in the French C3G national cohort, strongly suggests a pathogenic link between the 2 conditions, even in the absence of MIg deposits on glomeruli. [START_REF] Sethi | Dense deposit disease associated with monoclonal gammopathy of undetermined significance[END_REF][START_REF] Zand | C3 glomerulonephritis associated with monoclonal gammopathy: a case series[END_REF] In the series from Zand et al, 56% of patients presented with nephrotic syndrome and 81% with renal failure at diagnosis. [START_REF] Zand | C3 glomerulonephritis associated with monoclonal gammopathy: a case series[END_REF] In our study, 43% of patients had nephrotic syndrome and 84% showed significant renal impairment at diagnosis. Of note, severe extra renal manifestations characterized by skin necrotic lesions were present in 3 patients, none of whom had evidence of cryoglobulinemia, despite repeated tests. Although dr üsen and lipodystrophy may occur in patients with C3G, distal ischemic symptoms have not been described outside the context of monoclonal gammopathy. [START_REF] Appel | Membranoproliferative glomerulonephritis type II (dense deposit disease): an update[END_REF] Whether overactivation of complement AP may be involved in severe skin involvement in MIg-C3G remains to be determined. MIg-C3G appears to be associated with shorter renal survival compared with C3G in patients older than 50 years without MIg. Moreover, in our study, median renal survival was 31 months in patients receiving immunosuppressive therapy, similar to that observed in patients receiving conservative therapy. This result suggests that targeting intraglomerular inflammation is not sufficient to improve renal prognosis in MIg-C3G. Renal response rate and renal survival were significantly higher in patients who rapidly achieved complete hematological response or very good partial response with chemotherapy, suggesting a pathogenic role for MIg in glomerular C3 deposition.

In our study, 90% of patients had an indolent clonal disorder, consistent with MGRS. As in other types of renal diseases in MGRS, it is likely that physicochemical peculiarities of MIg, rather than tumor mass, govern nephrotoxicity. The International Kidney and Monoclonal Gammopathy Research Group proposed to adapt treatment strategy to both the origin (plasmacytic or lymphocytic) of the causal clone and the severity of renal disease. [START_REF] Fermand | International Kidney and Monoclonal Gammopathy Research Group. How I treat monoclonal gammopathy of renal significance (MGRS)[END_REF] On the basis of these recommendations, 29 patients received chemotherapy, including bortezomib-based regimens in 22 patients. Our results suggest that the use of novel antimyeloma agents, such as bortezomib, represent a valuable therapeutic option in MIg-C3G. Because 5 patients treated with chemotherapy had severe adverse events, leading to death in 3 cases, therapeutic choices should take into account renal elimination of antineoplastic drugs to limit their adverse effects. The absence or insufficient suppression of the underlying clonal disorder resulted in early C3G recurrence after transplantation in 4 patients. Achieving sustained hematological remission before transplantation should be considered to limit recurrence and to avoid the use of chemotherapy in immunosuppressed patients.

Despite being the largest series to report renal outcomes after chemotherapy, this study has several limitations. It was a retrospective multicenter study, including different treatment regimens. Moreover, it only provides indirect arguments in favor of a pathophysiological link among MIg, AP dysregulation, and renal lesions. AP abnormalities identified in C3G patients are acquired in most cases. Indeed, C3NeF, an autoantibody that stabilizes the AP C3 convertase, or anti-FH antibodies are identified in more than 50% of patients. [START_REF] Servais | Acquired and genetic complement abnormalities play a critical role in dense deposit disease and other C3 glomerulopathies[END_REF][START_REF] Sethi | Proliferative glomerulonephritis secondary to dysfunction of the alternative pathway of complement[END_REF][START_REF] Appel | Membranoproliferative glomerulonephritis type II (dense deposit disease): an update[END_REF][START_REF] Blanc | Anti-factor H autoantibodies in C3 glomerulopathies and in atypical hemolytic uremic syndrome: one target, two diseases[END_REF] In our study, C3NeF and anti-FH antibodies were detected in only 6% and 19% of patients, respectively. In 1999, it was demonstrated that a circulating monoclonal l LC, identified in a patient with MPGN, was able to activate the AP in vitro in a manner different from C3NeF. Indeed, this dimeric LC behaved as a mini autoantibody against FH, leading to uncontrolled AP activation and C3 glomerular deposition. [START_REF] Jokiranta | Nephritogenic lambda light chain dimer: a unique human miniautoantibody against complement factor H[END_REF] As the anti-FH antibodies were not isotype-restricted to the monoclonal LC identified in our patients in most cases (data not shown), they were likely to be distinct from the circulating MIg. The main biomarker associated with C3G is plasma C3 level, which is decreased in 50% to 80% of patients. [START_REF] Servais | Acquired and genetic complement abnormalities play a critical role in dense deposit disease and other C3 glomerulopathies[END_REF] Elevated soluble C5b-9 has recently been described as a new biomarker of C5 convertase activity, mainly associated with C3GN pattern. [START_REF] Zhang | Defining the complement biomarker profile of C3 glomerulopathy[END_REF] In MIg-C3G, 40% of patients had low C3 level. Interestingly, hematological response was associated with a significant increase in C3 level, suggesting C3 level may serve as biomarker of treatment efficacy. This finding is of particular interest for the evaluation of response to treatment in patients with normal FLC ratio and a small monoclonal spike on serum electrophoresis. Elevated soluble C5b-9 was found in 80% of tested MIg-C3GN patients, with significantly higher levels compared with C3G patients without monoclonal gammopathy. AP dysregulation is associated with another rare renal disease, atypical hemolytic uremic syndrome, characterized by thrombotic microangiopathy. In our series, 16% of patients had lesions of thrombotic microangiopathy likely to result from local AP activation. Eculizumab is currently recommended for the treatment of atypical hemolytic uremic syndrome. Whether its use, combined with chemotherapy, might be beneficial in MIg-C3G patients, particularly those with concomitant renal thrombotic microangiopathy, remains to be evaluated.

In conclusion, to the best of our knowledge, this study is the first to report epidemiological data and renal outcomes after chemotherapy in patients with MIg-C3G. On the basis of our results, we recommend careful hematological workup in patients older than 50 years with a diagnosis of C3G. Our results indicate that achievement of clonal response may favorably influence renal outcomes. Chemotherapy adapted to the origin of the underlying B-cell clonal disorder and to renal function should be considered early in the disease course. The efficacy and tolerance of chemotherapy regimens, such as bortezomib combinations, remains to be established in patients with C3G and monoclonal gammopathy. Finally, experimental studies are required to unravel the link between monoclonal Ig and AP dysregulation in C3G.

French

Figure 1 .

 1 Figure 1. Patients' recruitment. Flowchart of inclusion of patients within the adult population of the French C3G registry (n 5 201). A total of 50 patients were eligible for enrollment in the study, based on the presence of monoclonal gammopathy by electrophoresis and immunofixation and the availability of clinical data.

  1) was detected by immunofixation in all cases. LC isotype was k in 37/50 patients. By electrophoresis, 47 patients had a detectable monoclonal spike, with a median level of 10 (2-38) g/L. Among 32 patients tested, 17 (53%) had an abnormal k/l ratio with median dFLC of 79 (2-20.800) mg/L. In 17 (34%) patients, monoclonal gammopathy had been diagnosed before renal disease, with a median interval of 44 (12-120) months. Bone marrow studies were performed on bone marrow smears (n 5 45) or biopsy (n 5 5). Seventeen patients (34%) had more than 10% infiltration by dystrophic and clonal plasma cells (as demonstrated by immunofluorescence or flow cytometry). Hematological diagnosis was MGRS (n 5 30), smoldering MM (n 5 15), symptomatic MM (n 5 2), and CLL (n 5 3

Figure 2 .

 2 Figure 2. Kidney biopsy findings of monoclonal gammopathy-associated C3 glomerulopathy. (A-B) Light microscopy. (A) Section of renal cortex showing mesangial proliferative glomerulonephritis (periodic acid-Schiff staining, 3400). (B) Diffuse membranoproliferative glomerulonephritis with nodular appearance of the mesangium and glomerular capillary wall deposits (light green trichrome, 3400). (C-D) Immunofluorescence microscopy. Diffuse granular capillary wall and mesangial bright deposits (anti-C3 fluorescein isothiocyanateconjugate, 3400).(E-F) Electron microscopy. (E) Nodular (pseudohumps) electron-dense mesangial deposits (asterisk) and intramembranous dense deposits (arrow). Scale bar, 1 mm (original magnification 38000). (F) Nodular electron-dense mesangial deposits (asterisk) and interrupted intramembranous dense deposits displaying a sausage-shaped (arrow). Scale bar, 1 mm (original magnification 315 000).

Figure 4 .

 4 Figure 4. Renal survival. (A) Kaplan-Meier renal survival analysis comparing MIg-C3G patients who received chemotherapy, immunosuppressive therapy, or conservative therapy. (B) Kaplan-Meier renal survival analysis comparing MIg-C3G patients who achieved or did not achieve hematological response. (C) Kaplan-Meier renal survival analysis comparing MIg-C3G patients with or without hematological response (complete response or very good partial response) in 37 patients who received treatment (immunosuppressive therapy or chemotherapy). Data regarding the hematological response were available in 36 of the 37 patients. CR, complete hematological response; HR, hematological response; IS, immunosuppressive; VGPR, very good partial response.

Figure 5 .

 5 Figure 5. C3 level before and after treatment. (A) In 9 patients who reached hematological response. (B) In 9 patients without evidence of hematological response.

Table 1 .

 1 Main demographic, clinical, and biological data at time of kidney biopsy and at onset of treatment All patients (N 5 50) Chemotherapy (N 5 29) IS therapy (N 5 8) Conservative treatment (N 5 13) P value

	Age, y	65 (38-82)	69 (36-83)	64 (57-70)	60 (40-77)	.58
	Male, n (%)	33 (66%)	18 (62%)	5 (75%)	10 (77%)	.62
	Renal manifestations at diagnosis
	Serum creatinine, mmol/L	158 (68-990)	158 (68-990)	168 (135-280)	(70-317)	.59
	eGFR, ml/min/1.73 m 2	37 (3-100)	37 (3-94)	32 (20-49)	39 (16-100)	.51
	CKD stage 1-2	8/49 (16%)	10/28 (36%)	0 (0%)	2 (15%)	.07
	CKD stage 3	15/49 (31%)	5 /28 (18%)	3 (37%)	7 (54%)	.06
	CKD stage 4-5	27/49 (55%)	13/28 (46%)	5 (63%)	4 (31%)	.35
	Nephrotic syndrome, n (%)	20/46 (43%)	14/22 (63%)	3 (38%)	3/11 (27%)	.11
	24-h proteinuria, g	3.15 (0.1-14)	4 (0.7-14)	1.7 (0.4-10)	1.8 (0.1-10)	.06
	Hematuria, n (%)	33/39 (84%)	19/24 (79%)	6/6 (100%)	8/9 (89%)	.41
	Renal manifestations at onset of treatment
	Serum creatinine, mmol/L	-	202 (68-990)	198 (70-450)	-	.47
	eGFR, ml/min/1.73 m 2	-	28 (4-100)	29 (11-100)	-	.32
	CKD stage 1-2	-	6 (21%)	0 (0%)	-	.71
	CKD stage 3	-	9 (31%)	1 (13%)	-	.76
	CKD stage 4-5	-	14 (48%)	7 (87%)	-	1
	Nephrotic syndrome, n (%)	-	17 (58%)	5 (65%)	-	.37
	24-h proteinuria, g	-	3.55 (0.7-13)	2.91 (0.6-10)	-	.06
	Hematological findings
	Serum monoclonal Ig, n (%)
	IgG k	36 (71%)	24 (80%)	5 (67%)	7 (54%)	.12
	IgG l	11 (22%)	3 (10%)	3 (38%)	5 (38%)	.06
	IgA k	1 (2%)	0 (0%)	0 (0%)	1 (8%)	1
	IgA l	1 (2%)	1 (3%)	0 (0%)	0 (0%)	1
	l LC only	1 (2%)	1 (3%)	0 (0%)	0 (0%)	1
	Abnormal FLC, n (%)	17/32 (53%)	12/22 (55%)	2/3 (67%)	3/7 (43%)	.76
	dFLC, mg/L	79 (2-20800)	79 (6-20800)	114 (15-959)	22 (2-820)	.55
	MIg concentration, g/L	10.2 (2-38)	11 (2-38)	6 (2-15)	9 (3-30)
	Underlying hematological disorder, n (%)
	MGRS	30 (60%)	12 (41%)	7 (88%)	11 (84%)	.006
	Smoldering MM	15 (30%)	13 (43%)	1 (14%)	1 (8%)	.02
	Symptomatic MM	2 (4%)	2 (7%)	0 (0%)	0 (0%)	.47
	CLL	3 (6%)	2 (7%)	0 (0%)	1 (8%)	1
	Extra renal manifestations, n (%)	5 (10%)	5 (17%)*	0 (0%)	0 (0%)	.13
	Biological thrombotic microangiopathy, n (%)	3 (6%)	3 (10%)	0 (0%)	0 (0%)	.31
	Immunological findings
	C3 level, mg/L	746 (78-1130)	689 (78-1220)	866 (410-1120)	(473-1130)	.95
	Low C3 level, n (%)	22 (43%)	14 (48%)	3 (38%)	7 (46%)	.76
	C4 level, mg/L	250 (104-575)	243 (109-575)	265 (193-321)	(104-376)	.82
	sC5b-9, ng/mL	734 (164-2370)	961 (164-2370)	705 (269-1035)	(266-960)	.17
	Elevated sC5b-9, upper 420 ng/mL	27/34 (79%)	15/18 (83%)	6/7 (86%)	6/9 (67%)	.42
	C3Nef, n (%)	3 (6%)	1 (3%)	1 (14%)	1 (8%)	.6
	Anti-FH antibody, n (%)	9 (19%)	4 (14%)	3 (38%)	2 (15%)	.29
	Pattern of glomerular disease by electron
	microscopy
	C3GN	24/25 (96%)	18/19 (95%)	-	6/6 (100%)	1
	DDD	1/25 (4%)	1/19 (5%)	-	0 (0%)	1
	Values for continuous variables are expressed as median [range], and values for categorical variables, as percentage. Statistical analysis: for contingency, chi2 test, for
	comparison, Kruskal-Wallis test.
	C3Nef, C3 nephritic factor; IS, immunosuppressive.
	*Diffuse mucinosis (n 5 1), digital ischemia (n 5 2), skin purpuric lesions (n 5 1), and capillary leak syndrome (n 5 1).

Table 2 .

 2 Treatment modalities

		Treated patients (N 5 37)
	Chemotherapy adapted to the clone, n (%)	29 (78%)
	Median time to treatment, mo	3 (1-20)
	Alkylating agent-dexamethasone*	5 (17%)
	Bortezomib based regimen-dexamethasone †	22 (76%)
	Prednisone-rituximab ‡	2 (7%)
	Immunosuppressive therapy, n	8 (22%)
	Median time to treatment, mo	4 (1-54)
	Prednisone-dexamethasone	4 (50%)
	Prednisone-cyclophosphamide	1 (12.5%)
	Prednisone-rituximab	1 (12.5%)
	Prednisone-mycophenolate mofetil	1 (12.5%)
	Prednisone-azathioprine	1 (12.5%)
	*Cyclophosphamide-dexamethasone (n 5 4), Melphalan-dexamethasone (n 5 2).
	†Bortezomib-dexamethasone (n 5 10), bortezomib-thalidomide-dexamethasone
	(n52), bortezomib-cyclophosphamide-dexamethasone (n 5 10).
	‡For treatment of chronic lymphoid leukemia.

Table 3 .

 3 Hematological and renal responses

		All patients (N 5 50)	Chemotherapy (N 5 29)	Others (IS/conservative) (N 5 21)	P value
	Hematological response, n (%)	18/49 (37%)	17/29 (59%)	1 /20 (5%)	.0002*
	Complete response	9 (18%)	8 (28%)	1 (5%)	.06 †
	Very good partial response	1 (2%)	1 (3%)	0 (0%)	1 ‡
	Partial response	8 (16%)	8 (28%)	0 (0%)	.01{
	No response	31 (63%)	12 (41%)	19 (95%)	.0002 §
	NA	1 (2%)		1 (%5)	1
	Severe adverse events	6 (12%)	5 (21%)	1 (5%)	.21
	Renal response, n (%)	20/48 (42%)	19/27 (74%)	1/21 (5%)	.0001
	Complete response	6 (13%)	6 (22%)	0 (0%)	.001
	Partial response	14 (29%)	13 (48%)	1 (5%)	.002
	No response	28 (58%)	9 (32%)	19 (90%)	.0001
	NA	2 (4%)	2 (6%)	0 (0%)	-
	Renal parameters after first-line therapy
	Serum creatinine, mmol/L	179 (70-450)	147 (70-450)	231 (132-400)	.17
	eGFR, mL/min/1.73m 2	34 (16-94)	37 (16-94)	24 (16-51)	.1
	24-h Proteinuria, g	2.4 (0.1-5)	2.4 (0.1-3)	2.6 (0.7-5)	.46
	ESRD, n (%)	12 (24%)	5 (17%)	7 (33%)	.31
	Data at last follow-up
	Follow-up, mo	24 (4-104)	24 (8-104)	25 (4-69)	.75
	Serum creatinine, mmol/L	144 (68-290)	135 (68-258)	249 (135-290)	.2
	eGFR, ml/min/1.73 m 2	38 (18-89)	38 (18-89)	21 (19-48)	.16
	24-h proteinuria, g	1.1 (0.08-6.5)	0.86 (0.08-17)	1.2 (0.4-7)	.59
	ESRD, n (%)	25 (50%)	9 (31%)	16 (76%)	.003
	Renal transplantation, n (%)	4 (8%)	0 (0%)	4 (13%)	-
	C3G recurrence, n (%)	4/4 (100%)	-	4/4 (100%)	-

Table 4 .

 4 Characteristics of renal respondersFigure 3. Renal and patient survival. (A) Kaplan-Meier patient survival analysis comparing C3G patients with or without MIg older than 50 years. (B) Kaplan-Meier renal survival analysis comparing patients with MIg-C3G and patients with C3G without MIg older than 50 years.

		Renal responders (n 5 20)	Nonrenal responders (n 5 28)		P value
	Age, y	62 (38-83)			67 (40-80)			.16
	Baseline eGFR, mL/min/1.73 m 2	54 (3-94)			34 (6-100)			.1
	CKD 1-2 at diagnosis, n (%)	10/20 (50%)			3/28 (11%)			.003
	24h-proteinuria at onset, g	4.4 (0.9-12.8)			2.5 (0.4-5.9)		.005
	M-spike g/L	8 (2-38)			11 (4-30)			.54
	Normal C3 level, n (%)	11 (52%)			19 (68%)			.33
	MGRS, n (%)	9 (45%)			20 (71%)			.12
	Specific chemotherapy, n (%)	19 (95%)			8 (29%)			.001
	Bortezomib -based regimen, n (%)	17 (85%)			5 (18%)			.0001
	Time from diagnosis to treatment, mo	3.5 (1-19)			14 (1-48)			.018
	Hematological response, n (%)	15 (75%)			2/26 (8%)			.0001
	M-spike, monoclonal spike.
		A
			100
		Percent patient survival	20 40 60 80					with MIg w/o MIg
					p=0,16
			0
				0	12	2 4	36	48	60
						Months
		Without MIg 26	23	15	11	9	3
		With MIg	50	39	25	11	9	7
		B
			100
							over 50 (w/o MIg)
		Percent renal survival	20 40 60 80				with MIg
					p=0,04
			0
				0	24	48	72	96	120
						Months
		Without MIg 26	14	8	4	4	2
		With MIg	50	2 7	8	2	2	0

Death, n (%) 5 (10%) 3 (10%) 2 (10%) 1 Specific comparison between the group chemotherapy and IS therapy: *P 5 .08; †P 5 .65; ‡P 5 1; {P 5 .3; and §P 5 .08. g, gramme; NA, not available.

Acknowledgments

The authors acknowledge M. Mignotet for technical support and all colleagues who participated in this study: R. Boudet (Department of Nephrology, Brives), P. Daugas (Department of Nephrology, Bichat, AP-HP), F. Fakhouri (Department of Nephrology, Nantes), F. Gallen Labbe (Department of Nephrology, Valence), P. Gobert (Department of Nephrology, Avignon), M. Hazzan (Department of Nephrology, Lille), L. Mercadal (Department of Nephrology, Pitié Salpétrière, AP-HP), M. Nouvier (Department of Nephrology, Lyon), N. Martin Silva (Department of Medicine, Caen), F. Merabet (Department of Hematology, Versailles), E. Renaudineau (Department of Nephrology, Saint Malo), J. B. Philit (Department of Nephrology, Chambery), D. Sarret (Department of Nephrology, Val de Grâce, Paris), A. Servais (Department of Nephrology, Necker, AP-HP), L. Taghipour (Department of Nephrology, Armentières), A. Tiple (Department of Nephrology, Clermont-Ferrand). This work was supported by grants from the Délégation Régionale à la Recherche Clinique, AP-HP (V.F.-B.), such as Programme Hospitalier de Recherche Clinique (AOM08198), by grants from the Fondation Pour La Recherche Medicale (FDM 20130727355) and the Association pour l'Information et la Recherche dans les maladies Rénales génétiques, by grants from European Community (FP7 Grant 2012-305608 EURenOmics), and by the Fondation du Rein (FRM, prix, 2012 Fondation pour la Recherche Médicale to V.F.-B.).

Authorship

Contribution: The study was conceived and designed by S.C., V.F.-B., and F.B.; S.C. conducted the analysis; V.F.-B., and F.B. reviewed the data analysis; S.C., V.F.-B., and F.B. were involved in the writing of the manuscript; and all other authors contributed to the conduct of the trial, recruited patients, and were involved in the review of results and final approval of the manuscript.

Conflict-of-interest disclosure: V.F.-B. received fees for participation in advisory boards, experts meetings, and/or teaching courses from Alexion Pharmaceutical. The remaining authors declare no competing financial interests.

Correspondence: Sophie Chauvet, Department of Nephrology, European Hospital Georges Pompidou, AP-HP, 20 Rue Leblanc, 75908 Paris Cedex 15, France; e-mail: sophie.chauvet@aphp.fr.


A Rybko
email: rybko@iitp.ru

Senya Shlosman
email: senya.shlosman@univ-amu.fr

A Vladimirov

Poisson Hypothesis for Open Networks at Low Load

We study large communication networks of the mean-field type. The input flows to the nodes of the network are supposed to be stationary and with low rate. We show that such a network is ergodic, i.e. it goes to the stationary state, which does not depend on the initial state of the network. This is in contrast with the high load regime, when the large time behavior of the network might depend on its initial state. Our technique is based on the coupling construction, which couples two Non-Linear Markov Processes.

Introduction and results

In this paper we study large information networks with low load. Our goal is to show that they obey a certain pattern, which is known under the name of Poisson Hypothesis.

The Poisson Hypothesis is a device to predict the behavior of large queuing networks. It was formulated first by L. Kleinrock in [K], and it was assumed to hold universally. It concerns the following situation. Suppose we have a large network of servers, through which many customers are traveling, being served at different nodes of the network. If the node is busy, the customers wait in the queue. Customers are entering into the network from the outside via some nodes, and these external flows of customers are Poissonian, with constant rates. The service time at each node is random, depending on the node, and the customer. The PH prediction about the (long-time, large-size) behavior of the network is the following:

• consider the total flow F of customers to a given node N . Then F is approximately equal to a Poisson flow, P, with a time dependent rate function λ N (T ) .

• The exit flow from Nnot Poissonian in general! -has a rate function γ N (T ) , which is smoother than λ N (T ) (due to averaging, taking place at the node N ).

• As a result, the flows λ N (T ) at various nodes N should go to a constant limits λN ≈ 1 T T 0 λ (t) dt, as T → ∞, the flows to different nodes being almost independent.

• The above convergence is uniform in the size of the network.

Note that the distributions of the service times at the nodes of the network can be arbitrary, so PH deals with quite a general situation. The range of validity of PH is supposed to be the class of networks where the internal flow to every node N is a union of flows from many other nodes, and each of these flows constitute only a small fraction of the total flow to N . If true, PH provides one with means to make easy computations of quantities of importance in network design.

The rationale behind this conjectured behavior is natural: since the inflow is a sum of many small inputs, it is approximately Poissonian. And due to the randomness of the service time the outflow from each node should be "smoother" than the total inflow to it. (This statement was proven in [START_REF] Rybko | Self-averaging property of queuing systems[END_REF] under quite general conditions.) In particular, the variation of the latter should be smaller than that of the former, and so all the flows should go with time to corresponding constant values.

In the paper [START_REF] Rybko | Poisson Hypothesis for Information Networks[END_REF] the Poisson Hypothesis is proven for simple networks in the infinite volume limit, under some natural conditions. The counterexamples to such universal behavior are also known by now. Some were constructed in [START_REF] Rybko | Phase transitions in the queuing networks and the violation of the Poisson hypothesis[END_REF], [START_REF] Rybko | Spontaneous Resonances and the Coherent States of the Queuing Networks[END_REF].

It was argued in [START_REF] Rybko | Spontaneous Resonances and the Coherent States of the Queuing Networks[END_REF], that the violation of the Poisson Hypothesis can happen only at high load, and that the load plays here the same role as the temperature in the statistical mechanics, so that the high (low) load corresponds to low (high) temperature. We have shown in [START_REF] Rybko | Absence of Breakdown of the Poisson Hypothesis. I. Closed Networks at Low Load[END_REF] that indeed the Poisson Hypothesis holds for the closed networks with low load. In the present paper we show that the Poisson Hypothesis behavior holds also for the open networks with low load. (The time gap between the paper [START_REF] Rybko | Absence of Breakdown of the Poisson Hypothesis. I. Closed Networks at Low Load[END_REF] and the present one is due to the fact that certain general results needed were not available earlier. Now they are established, see [BRS].)

The main technical innovation of our paper is that we prove our result by constructing a suitable coupling of two Non-Linear Markov Processes (in Sections 6.2, 6.3 below). The coupling constructed shows that the two NLMP merge together as time diverges, provided the load of the network is low. It is thus reminiscent of the FKG-uniqueness results of statistical mechanics, when it is obtained via coupling constructions for systems with attraction, [H].

In the next Section we describe the basic finite network, from which large networks of the mean-field type can be built. This mean-field construction is explained in the Section 3, and the corresponding Non-Linear Markov Process (NLMP) is defined in Section 4. The general properties of the networks with low input flows are established in the Section 5. The Section 6 contains the main result of the paper -that of ergodicity of the NLMP, corresponding to the low inflow. It is established by using the coupling of Sections 6.2, 6.3.

Basic network

The main object of our study are large queueing networks that are composed of many identical copies of a finite basic network G. In this section we will describe this basic network.

G : Markov chain of queues

The network G consists of a finite number of servers, s, also called nodes. At any given time t, each node is populated by a finite number of customers (also called particles) of different classes. The set J of all classes is finite, |J| < ∞. It is convenient from the very beginning to include into the class j of the customer the address of server s (j) ∈ G where the customer is currently located. The customers wait in the queue for their turn to be served by the node. The order of service is governed by the service discipline. After being served, the customer either jumps to some node of G (becoming there the last in the queue), or leaves the network.

The traffic of customers in G is governed by the routing matrix P . This is a |J| × |J| matrix, whose rows and columns correspond to the classes of customers in the network. The element p ij of P is the probability of a customer c of class i to change its class from i to j after the service. As soon as the service of c is over, the customer jumps with probability p ij to the node s (j) . With the complementary probability p i0 = 1 -j∈J p ij the customer leaves the network.

Queues

We restrict consideration to Markov networks where the service time of each customer is exponentially distributed. We also consider service disciplines that depend only on the order of arrival of customers to the queue (and not on the interarrival times or any other history). Namely, for each node n and each queue x = (j 1 , ..., j k ) of length k at this node, a collection γ n (x, j r ) : r = 1, . . . , k of service rates is defined by the service discipline. The formal definition will be given later.

For the alphabet J = {1, . . . , |J|} we define the configuration space

X = J * = k=0,1,... J k
of finite words in this alphabet, with J 0 = {∅} and J 1 = J. We will use the notation x = (x 1 , . . . , x k ) ≡ (j 1 , . . . , j k ) for the list of all customers in the network. Denote |x| = k.

The order of customers in the queue coincides with the order of their arrival, that is, x 1 is the class of the oldest customer in the queue and x k is that of the last arrived. (The customer x 1 is not necessarily the first to be served, since we consider a rather general class of service disciplines.)

Network events

The current state x ∈ X can be changed as a result of arrival events and service events. An arrival event of type j takes x to x ⊕ j = {j 1 , . . . , j |x| , j}.

Service event may only happen if x = ∅. If x = {j 1 , . . . , j k-1 , j k , j k+1 , . . . , j |x| }, i.e. k ≤ |x| , then the service event operation ⊖ * k is the jump from x to

x ⊖ * k = {j 1 , . . . , j k-1 , j k+1 , . . . , j |x| }.
With some abuse of notation we also denote this jump as x ⊖ j k . Service and arrival events may happen simultaneously, if a particle whose service is finished, joins the same queue again, see below.

Markov dynamics

The random evolution of the network G is a continuous-time Markov process G on the countable configuration space X. Let us write down the rates of all arrival and service events as functions of the current configuration x ∈ X.

The external arrivals of clients of type j ∈ J is a Poisson process with rate λ j (t). We will consider the time homogeneous case, when λ j (t) ≡ λ j , though for technical reasons we will also use sometime the general case of timedependent rates λ j (t). The external arrival events of customers of different types j ∈ J are independent of each other and of the state of the system.

A service event of type i ∈ x happens at appropriate location in x at rate γ(x, i). If the customer i stays in the network and turns into customer j, then the configuration of the system becomes x ⊖ i ⊕ j. That can happen with probability p ij . Another option is that the customer leaves the network, and the network state becomes x ⊖ i. That happens with probability p i0 = 1 -j∈J p ij .

Apart from the pairs of associated service and arrival events mentioned above, all the events in the system happen independently of each other. The non-zero transition rates r(x, y) of the process G are thus equal to

r(x, x ⊕ j) = λ j (t), j ∈ J, (1) r(x, x ⊖ i) = p i0 γ(x, i), i ∈ x, (2) r(x, x ⊖ i ⊕ j) = p ij γ(x, i), i ∈ x, j ∈ J. (3)
The state of the process G at the time t is given by a probability measure µ t on X. We denote by

P(X) = {µ ∈ R X : µ (x) ≥ 0, x ∈ X, x∈X µ (x) = 1}
the space of probability measures on X.

Mean-field networks

In order to construct the mean-field networks G M , we take the colection of M independent copies of G, and interconnect them as follows. The service event i at any given copy of G is followed by the corresponding arrival event j at any other copy of G with probability 1 M p ij . With the complementary probability, p i0 = 1 -j∈J p j i j , no arrival event happens, and the customer leaves the system.

G M : the mean-field type process

The resulting Markov process G M on the configuration product space

X M = {x = (x 1 , . . . , x M ) : x m ∈ X, m = 1, . . . , M}
can be described formally as follows. Let us introduce the notations:

x ⊕ m j = (x 1 , . . . , x m ⊕ j, . . . , x M ) and for i ∈ x m x ⊖ m i = (x 1 , . . . , x m ⊖ i, . . . , x M )
. Then the transition rates of the process G M are given by the relations

r(x, x ⊕ m j) = λ j (t), (4) r(x, x ⊖ m i) = (1 - j∈J p ij )γ(x m , i), (5) r(x, x ⊖ m i ⊕ m ′ j) = p ij γ(x m , i) M , (6)
for all m, m ′ = 1, . . . , M, i ∈ x m , j ∈ J.

Factor-process

The permutation group S M acts on X M , and the process G M is invariant under this action. The factor space can be identified with the atomic measures on X. So we will use the notation

x = 1 M M m=1 δ xm , (7)
where δ x is the unit measure at x ∈ X.

Denote by P M (X) ⊆ P(X) the subspace of all measures of the form (7). The factor-process, denoted by the same symbol, G M , becomes a continuoustime Markov process on the (countable) configuration space P M (X). Its state at time t is a probability measure

µ M t = {µ M t (x) :
x ∈ P M (X)} on P M (X) and thus on P(X). The expectation Eµ M t is an element of P(X) but, in general, not an element of P M (X).

Non-linear Markov process

Let us consider the limit of G M as M → ∞. To this end, let us choose one of M copies of G in the system G M -say, the first one, G 1 -and observe for t ≥ 0 the arrival and service events only at this copy. In other words, we take a projection H M of G M to G 1 , integrating out the remaining degrees of freedom.

If we want to know the evolution of the process H M , then, in addition to the initial configuration x 0 ∈ X we need, of course, to specify the initial state µ M t=0 ∈ P M (X) of the whole ensemble. Suppose that the initial measures µ M 0 converge weakly to some µ 0 ∈ P(X) as M → ∞, and consider the limiting process H = lim M →∞ H M . The existence and uniqueness of the limiting process is proven in [?]. This stochastic process will be referred to as the non-linear Markov process (NLMP) H. To specify its evolution we need to know the initial configuration x ∈ X and the initial measure µ 0 ∈ P(X). The NLMP H is not a Markov process on X because of its dependence on the whole measure µ 0 and not just on x(0).

The current configuration of this extended process is the pair (µ t , x(t)), and this configuration defines the distribution of the future configurations (µ s , x(s)), s ≥ t, uniquely. Moreover, the measure µ s is defined by µ t in a deterministic way, for s ≥ t. The measure µ t is a solution of a system of differential equations which we write down below. Once the family {µ t , t ≥ 0} is given, the process x(t) becomes an inhomogeneous Markov process on X, see below for more details.

The process equations

The evolution equations for the state µ t of the process H are conveniently written in terms of flow rates which we define now.

The outflow rate of class j customers from the system H at time t is given by

u j (t) = x∈X µ t (x) r:xr=j γ(x, x r ), j ∈ J (8)
The inflow rate of class j customers to the system is then equal to

v j (t) = λ j (t) + w j (t), j ∈ J, (9)
where

w j (t) = i∈J p ij u i (t), j ∈ J, (10)
is the internal inflow rate.

Thus the total inflow is a time-inhomogeneous (vector) Poisson process, whose rate vector is v(t) = λ(t) + w(t). The queueing process x(t) is then an inhomogeneous Markov process corresponding to the time-dependent arrival process with rates v j (t), j ∈ J, and the service rates γ(x, i), i ∈ x.

An arrival event of type j converts the configuration from x to x ⊕ j. The rate of this arrival equals to µ (x) v j . Analogously, a service event of type i converts x to x ⊖ i. Its rate is µ (x) γ(x, i).

We have described the dynamics of H in terms of arrival and service events by specifying their rates and the corresponding changes of the measure µ. The time evolution of µ t ∈ P(X) is given by the following countable system of ordinary differential equations:

μt = x∈X µ t (x)   |x| r=1 γ(x, x r )(δ x⊖xr -δ x ) + j∈J v j (t)(δ x⊕j -δ x )   . ( 11
)
Here we treat the measure µ t as a vector with coordinates µ t (x), and δ y -s denotes the δ-measures at various points y ∈ X. The first sum in brackets corresponds to service events that happen at configurations x ∈ X, |x| > 0, with (configuration-dependent) rates γ(x, j), and the second sum corresponds to arrival events that happen at all configurations x ∈ X with (configurationindependent) rates v j (t), j ∈ J.

Relations (8)-( 11) describe completely the evolution of µ t . The resulting dynamical system is non-linear since its right-hand side contains terms µ t (x)v j (t), which are quadratic in µ, since v j (t) depends on µ t linearly. The existence and uniqueness of the solution µ t on [0, +∞) of the system (11) for any initial state µ 0 is proven in [BRS] under minimal assumptions on the network. In particular, the existence and uniqueness hold under the following rate conditions: for some λ + < ∞ and 0 < γ -≤ γ + < ∞ we have λ j (t) ≤ λ + for all j ∈ J, t ≥ 0, (12)

γ -≤ γ(x) ≤ γ + for all x ∈ X, x = ∅, (13)
where γ(x) = |x| r=1 γ(x, x r ). Note that under the rate conditions the total inflow rates v j (t) in the system have a uniform upper bound V < ∞. So for each x ∈ J the function µ t (x) is Lipschitz continuous in t, with a Lipschitz constant which does not depend on x, t, or µ 0 .

Processes with low inflows

Bounds for a single queue

Now let us consider a server that receives independent Poisson inflows of particles of classes j ∈ J with rates λ j (t). As in the NLMP model, there is no feedback, that is, served customers never return to our server.

In what follows we assume that λ j (t) ≤ ε for all t ≥ 0, where ε > 0 is a small parameter. Let us derive some bounds on the length |x| of the queue at this server. Recall that our service discipline is rather general, but the total service rate γ(x) is neither too low nor too high for any x = 0, see rate conditions (13).

We will derive bounds on the distribution ν t of queue length, provided the queue is empty at t = 0. Denote by Q the random process of queue lengths. The trajectories of Q are step functions, taking values in Z + = {0, 1, 2, . . . }.

Denote by ν t (k) the probability that there are exactly k particles in the queue at time t. We have k=0,1,...

ν t (k) = 1, t ≥ 0.
Let us compare the process Q with the stationary process Q ′ , defined as follows:

• All the customers in Q ′ have the same exponentially distributed service time and the FIFO service discipline is used;

• The service rate in the process Q ′ is equal to the lower bound γ -on the service rates for the process Q;

• The inflow to Q ′ is a stationary Poisson flow with rate ε for each type j ∈ J.

For ε small enough such a process Q ′ exists and is unique. One easily sees that the queues in Q ′ dominates stochastically those in Q at any time t ≥ 0. Indeed, by coupling we may organize the pairs of evolutions q(t) and q ′ (t) in such a way that, apart from synchronous arrivals to both queues, there are additional arrivals to q ′ , and, apart from synchronous service attempts, there are additional service attempts at q. Then |q ′ (t)| ≥ |q(t)| for all t ≥ 0 since |q(0)| = 0.

For the stationary process Q ′ the explicit form of the queue length distribution ν ′ is known; it is the geometric distribution:

ν ′ (k) = (1 -κ)κ k , where κ = |J|ε γ - .
We have

ν ′ (0) = 1 -κ, k≥1 kν ′ (k) = κ (1 -κ) , k>1 kν ′ (k) = κ 2 (1 -κ) .
Hence, from the domination argument we have Lemma 1 Let the initial state ν t=0 has no customers. For κ

= |J|ε γ -, ν t (0) ≥ 1 -κ, t ≥ 0, ( 14
)
so the fraction of non-empty queues is O(ε);

k≥1 kν t (k) ≤ κ (1 -κ) ,
so the expected length of the queue is O(ε);

k>1 kν t (k) ≤ κ 2 (1 -κ) , ( 15
)
so the expected length of non-single queues is O(ε 2 ).

Let h(j) be the expected remaining number of services for a particle of class j. Clearly, h(j) depends only on the rooting matrix P and is finite since

ρ(P ) < 1 (the network is open). Let us put h = max j∈J h(j).
For a queue x ∈ X we denote

L(x) = |x| i=1 h(x i ),
where x = (j 1 , . . . , j |x| ). Finally,

L(µ) = X L(x)dµ(x).
As a consequence of Lemma 1, we get the following upper bound on L(ν t ):

Lemma 2 Under the hypothesis of Lemma 1, there exists a constant C 1 > 0, which depends only on the routing matrix P, such that

L(ν t ) ≤ hα(ν t ) + C 1 ε 2 , ( 16
)
where α(ν t ) = 1ν t (∅).

Proof. Note that L(x) ≤ h|x|. We estimate L(ν t ) separately over queues of length one and over longer queues. The first contribution is ≤ hα(ν t ), since ν t (1) ≤ α(ν t ), and for the second one we use (15) .

Bounds for the NLMP

Now we return to the NLMP H. Our goal is to prove that if ε > 0 is small, then any solution µ t converges weakly to the unique stationary solution µ * . We will do it, first, for the class of finite weight initial measures µ 0 , that is, for µ 0 with a finite expected length of the queue:

x∈X |x|µ 0 (x) < ∞.
Note that the state µ t is finite weight for all t > 0 iff µ 0 is finite weight. First, we will show that there exists a time moment T (µ 0 ) , such that the expected value of non-empty queues α(µ t ) = 1µ t (∅) in our system drops down below the value Cε for some constant C > 0. Second, we prove that for some C > 0 it remains below the value Cε at all later times. That implies that all the flow rates in the system will stay below Dε for some D > 0.

To make the first step we will construct a Lyapunov function. To implement the second step we 'split' our process H after the time moment T (µ 0 ) in two parts, such that the 'smaller' one has all the non-empty servers at the time T (µ 0 ) , and the 'larger' part starts from the empty state at T (µ 0 ) .

We begin by explaining the splitting construction. The splitting of the process H = {µ t , t ≥ 0} is a pair of processes µ 1 t and µ 2 t on X such that

ρµ 1 t + (1 -ρ)µ 2 t = µ t , t ≥ 0. ( 17
)
In words, we mark some part of queues at t = 0 and then consider the joint evolution of marked and unmarked queues within the NLMP H. The processes µ 1 t and µ 2 t are also NLMP-s, but with non-homogeneous external flows.

Formally, let µ 0 = ρµ 1 0 + (1ρ)µ 2 0 , where µ i 0 , i = 1, 2, are probability measures on X and 0 ≤ ρ ≤ 1. Let us describe the dynamics of µ i t , i = 1, 2, t ≥ 0 by means of internal flows. We define, in the same manner as above,

u k j (t) = x∈X µ k t (x) r:xr=j γ(x, x r ), j ∈ J, k = 1, 2, (18)
and

w k j (t) = i∈J p ij u k i (t), j ∈ J, k = 1, 2. ( 19
)
The total inflow will be defined in a different way:

v ′ j (t) = λ j (t) + ρw 1 j (t) + (1 -ρ)w 2 j (t), j ∈ J. (20)
Finally, the joint dynamics of µ k t , k = 1, 2, satisfies the following differential equations:

μk t = x∈X µ k t (x)   |x| r=1 γ(x, x r )(δ x⊖xr -δ x ) + j∈J v ′ j (t)(δ x⊕j -δ x )   , (21)
for k = 1, 2. Summing up equations (21) over k = 1, 2 with factors ρ and (1ρ), respectively, we conclude that (17) holds. It is easy to see that the evolution of each measure µ k t satisfies the equations of an NLMP with a modified routing matrices:

p 1 ij = ρp ij , p 2 ij = (1 -ρ) p ij (
fractions of outflows go to the other half of the system) and an additional time-dependent Poisson inflow (the customers that come from the other half of the system). Now we are prepared to prove the main statement of this section.

Lemma 3 There exists a constant C > 0 and, for each initial finite-weight measure µ 0 on X, a time T = T (µ 0 ) such that α(µ t ) < Cε for all t > T.

Proof. We use the mean number of remaining service events L(µ) as the Lyapunov function. The function L(µ) is finite for any finite-weight measure µ on X, since L(x) ≤ h|x| for any x ∈ X. Let us check that

d dt L(µ t ) = j∈J λ j (t)h(j) -S(µ t ), (22)
where

S(µ) = x∈X:x =∅ γ(x)µ (x)
is the mean service rate for a queue with the distribution µ (for empty queues this rate is zero). Indeed, the arrival event of type j adds the value h(j)

to L(µ). On the other hand, the expected number of services of a given customer decreases by 1 after its service is over, which explains the second term. Clearly, S(µ t ) ≥ γ -α(µ t ), where γ -is the minimal service rate (for nonempty queues). So from (22) we see that if γ -α(µ t ) > |J|εh, then the function L(µ t ) decays. Hence at a certain moment t = T we will have α(µ T ) ≤ |J|εh γ -. It remains to prove that for all t ≥ T the value of α(µ t ) will never exceed Cε for some constant C which does not depend on µ 0 and T . To do this we use the splitting construction expalined above, with ρ = α(µ T ), the initial states µ 1

T (•) = µ T (•|x = ∅) , µ 2 T (∅) = 1.
By definition, the first portionρµ 1 t (•) -of our process µ t remains small for all times t > T, since ρ is small. The total mass of the remaining measure (1ρ) µ 2

t is close to one, but the state µ 2 T is empty, and the rate of external flow is of the order of ε. Therefore one can expect that the state µ 2 t is sparcely populated at all times t > T. This is indeed the case:

Claim. Let λ(t) = j∈J λ j (t) and for all t ≥ 0 we have λ(t) ≤ ε. Let the initial state µ 0 (≡ µ 2 T ) has no customers, i.e. µ 0 (∅) = 1. Then there exists a constant C > 0 such that for all t ≥ 0 α(µ t ) ≤ Cε.

Proof of the Claim. Note first that for some C 3 > 0 we have

L(t) ≤ C 3 ε -γ -α(µ t ), t ≥ 0, (23)
as it follows from (22) . Note also, that if L(µ t ) ≥ 0, then α(µ t ) ≤ C 3 ε γ -≡ α + . We will show that for all t ≥ 0 we have α(µ t ) < ᾱ ≡ (1 + h) α + , provided ε is small. Suppose the opposite, and let T > 0 be the first occasion when α(µ t ) reaches the level ᾱ. Denote by T + the last occasion before T when α(µ T + ) = α + . Note that the total inflow rate |u(t)|, t ≤ T satisfies the inequality

|u(t)| ≤ ε + γ + ᾱ = C 4 ε.
Therefore, from (16) we have the inequality

L(µ T + ) ≤ hα + + C 5 ε 2 .
Since L(µ t ) ≤ 0 for all t ∈ T + , T , we have

L(µ T ) ≤ L(µ T + ) ≤ hα + + C 5 ε 2 . ( 24
)
On the other hand, we always have that L(µ) ≥ α(µ), so in particular

L(µ T ) ≥ α(µ T ) = hα + + α +
which contradicts (24) when ε > 0 is small enough, and the claim follows. The proof of Lemma 3 is thus also finished.

Stationary states of the NLMP

Let us assume that the external inflow is stationary, that is, λ(t) ≡ λ = (λ j ), t ≥ 0. If the corresponding stationary state µ λ of H exists, then the constant internal inflow rate w * is found from the equation

P (λ + w * ) = w * , (25)
that is, w * = (I -P ) -1 P λ.

Let us assume that λ j ≤ ε, j ∈ J, where ε > 0 is a small parameter.

Lemma 4 If ε is small enough, then H has a unique stationary state µ(t) ≡ µ * .

The uniqueness follows from (25) and (26). For the existence, it suffices to prove that the network without feedback is ergodic under the stationary Poisson inflow of rate λ + w * . The components of this rate vector are of order ε, hence, the assertion follows from the Foster criterion, see, for instance, [B], or directly from Lemma 1.

6 Ergodicity of the NLMP with low inflow

The main Theorem

Now we turn to the main result of the paper.

Theorem 5 For a given G, there exists an ε > 0 such that for all λ = (λ j ) with λ j < ε, j ∈ J the NLMP dynamics µ t with finite-weight initial measure µ 0 converges to the unique equilibrium µ λ : for every

x ∈ X lim t→∞ |µ t (x) -µ λ (x) | = 0.
service is over, the newly created pair either leaves the network, or else go to the same node, according to the routing matrix, i.e. the routing of the two customers is identical. The rule of staying white pair of customers or becoming red customers are then the same as for the arriving external white pairs.

Each red customer c is served individually. If c was participating in the process H ′ (resp., H ′′ ), then after the service he either leaves the network or jumps to the another node, still being in the process H ′ (resp., H ′′ ). If c was belonging to the red pair of queues (q ′ ⊕ c, q ′′ ) , and if q ′ = ∅ or q ′′ = ∅, then the resulting pair (q ′ , q ′′ ) stays red, while in the case when both q ′ = ∅ and q ′′ = ∅, the red pair (c, ∅) turns into the pair (∅, ∅) , which is declared white. If c stays in the network and enters into the pair (q ′ , q ′′ ) of queues, then the resulting pair (q ′ ⊕ c, q ′′ ) is declared red, independently of the colour of the pair (q ′ , q ′′ ) .

Summarizing, we see that a white particle turns red in two cases: if it joins a red queue, or if a red particle arrives to a white pair of queues (then all the white customers in both queues turn red). The queues in the white pair are identical, while the queues in the red pair may be identical or different. The pair (∅, ∅) is always white.

Our goal will be to prove that the fraction of red pairs r t vanishes as t → ∞. It will imply that the limits of µ ′ t and µ ′′ t , as t → ∞, exist and coinside with µ λ . In the end we will drop the assumption of µ ′ 0 to be finiteweight.

6.3

Equations for the coupled NLMP As was explained above, the state of our coupled process is given by the two measures -white and red:

{W t (x) , R t (y, z) : x, y, z ∈ X} ,
where we think of x as a pair (x, x) ∈ X × X on the diagonal. Sometime we will omit the subscript t.

Given such a state, let us define the corresponding -white and redflows. The outflows are The corresponding internal inflows are

v j (W ) = k p kj u k (W ), v ′ j (R) = k p k,j u ′ j (R), v ′′ j (R) = k p k,j u ′′ j (R).
Let us define v j (W, R) = v j (W ) + v ′ j (R) + v ′′ j (R). Now we write the differential equations for the white and red measures W and R, which describe formally the evolution explained in the previous section.

Infinite expected initial length

Here we say few words on extension of Theorem 5 to the case of infinite expected initial queue length. We give a sketch of the proof.

In order to reduce the situation to that of a finite initial length, we separate a small fraction β > 0 of queues so that the remaining ones are of finite expected length at t = 0. The value of β can be chosen arbitrarily small.

The flows to and from the separated part are of order β. We need to prove that, for finite initial weights and for different inflows that differ by order β, the solutions approach each other up to a distance of order β as well.

First, we prove that the fraction of red queues must eventually drop down to the order β. This follows from the differential equations for the dynamics of L(t) and r t . So we can suppose that r T ∼ β at some time T.

At this time T we separate, additionally, all the red queues (their fraction is r T ) and consider the remaining process that starts with no red queues. Then we can use the argument similar to that in the proof of Lemma 6 and show that r t remains of order β for all t ≥ T . The convergence to the equilibrium follows since β can be chosen arbitrarily small.

+

  + v j (W t , R t )) + W t (x : x ⊕ j = x)(λ j + v j (W t )) + |x|+1 i=1 x:x⊖x i =x W t (x)γ(x, xi ) + I {x=∅} j [R t (j, ∅) + R t (∅, j)] γ(j) ; d dt R t (y, z) = -R t (y, z)   j (λ j + v j (W t , R t )) + +R t (ȳ : ȳ ⊕ j = y, z : z ⊕ j = z)(λ j + v j (W t )) +R t (ȳ : ȳ ⊕ j = y, z)v ′ j (R) + R t (y, z : z ⊕ j = z)v ′′ j (R) + I {(y,z)=(j,∅)} W t (∅)v ′ j (R) + I {(y,z)=(∅,j)} W t (∅)v ′′ j (R) + I {(y,z) =(∅,∅)}   |y|+1 i=1 ȳ:ȳ⊖ȳ i =y R(ȳ, z)γ(ȳ, ȳi ) +|z|+1 i=1 z:z⊖z i =z R(y, z)γ(z, zi )   .

Part of this work has been carried out in the framework of the Labex Archimede (ANR-11-LABX-0033) and of the A*MIDEX project (ANR-11-IDEX-0001-02), funded by the "Investissements d'Avenir" French Government programme managed by the French National Research Agency (ANR). Part of this work has been carried out at IITP RAS. The support of Russian Foundation for Sciences (project No. 14-50-00150) is gratefully acknowledged.

1

We begin the proof with a technical lemma.

Lemma 6 There exist C, ε 0 > 0 such that, for any ε < ε 0 and any initial finite-weight measure µ 0 , there exists a time T = T (C, ε, µ 0 ) such that E µt |x| < Cε for all t > T.

Proof. This result follows from Lemma 2 and Lemma 3.

The coupling of Non-Linear Markov Processes

We are going to construct a coupling H ′ ⊗H ′′ of two NLMP H ′ and H ′′ , which live on the same graph, have the same inflows, and differ only by their initial states, µ ′ 0 and µ ′′ 0 . We remind the reader that the coupling of the processes µ ′ t and µ ′′ t is a process M t on X × X with marginals µ ′ t and µ ′′ t . In our case the process M t will be again a NLMP. Its initial state can be any coupling of µ ′ 0 and µ ′′ 0 ; for example the product µ ′ 0 × µ ′′ 0 will go. The coupling we construct has a form of a sum: M t = R t + W t ; the two (positive) processes R t and W t are interacting NLMP-s, except that the masses r t ≡ R t (X × X) , w t ≡ W t (X × X) of each of them can be less than one, while of course r t + w t = 1. The processes R t and W t are processes on pairs of queues, and we will call them the red and white pairs; the customers in these queues will be likewise called red and white customers. The choice of the initial states M r 0 and M w 0 of the processes R t and W t are not very important; one option can be R 0 = M 0 , W 0 = 0. The key property for us will be that the process W t has its support on the diagonal X ⊂ X × X, so we will be done if we will show that w t → 1 as t → ∞.

We start with the verbal Definition of the evolution of the (interacting) processes R t and W t . We then write down the differential equation, equivalent to that definition.

As usual, we couple the (equal) inflows to H ′ and H ′′ in such a way that the customers arrive in pairs (of identical customers) simulataneously to H ′ and H ′′ . The arriving external pairs of customers have white colour. The fraction w t of arriving identical pairs goes into the process W t . The remaining amount, r t , goes into red process, R t , and the corresponding white arriving customers become red instantly.

The service of white customers also happens in pairs, so the two paired (identical) customers start and end their service sinchronously. After the

Proof of stability

Now the colored dynamics is explicitely defined; it is a non-linear Markov process on the space of colored pairs of queues. Any pair is either white (and then the queues are identical and synchronized) or red (and then the queues evolve independently). The theorem 5 follows from the following Theorem 7 If ε is small enough, then r t → 0 as t → ∞.

Proof. First, we start the two NLMP H ′ and H ′′ and run them for the time T , given by Lemma 3. After that time we couple them as was explained above. The rates of all the flows in the coupled system do not exceed C 1 ε at any time t ≥ T . Denote by h(t) the expected number of remaining services of red particles only in both H ′ and H ′′ . By assumption, h(0) is finite and so h(t) remains finite for all t ≥ 0. The change of r t and h(t) in time is due to two factors. The first one is the termination of the service of red particles; it contributes to decline of h(t), which happens with the rate Cr t . The second one is the creation of new red particles, and it makes h (t) to increase, either when 1. white particles arrive to red queues, or when 2. red particles arrive to white queues.

In the case 1 the rate of emergence of the new red particles (and hence the rate of growth of h(t)) is at most of the order of εr t , since the flow of red particles has order r t , the fraction of non-empty white queues has order at most ε and the mean length of these queues is of order 1. In the case 2, the rate of growth of h(t) is also of order εr t since the flow of white particles has order ε and the fraction of red queues has order r t . Summarizing, we arrive at the following differential inequality:

Since h(t) is positive, the integral ∞ 0 r t dt is finite. Together with the fact that r t ≥ 0 and the derivative of r t is bounded, it implies that lim t→∞ r t = 0.