-/* $Id$ */
-
-/* Copyright (c) 2007 Arnaud Legrand, Pedro Velho. All rights reserved. */
+/* Copyright (c) 2007, 2008, 2009, 2010. The SimGrid Team.
+ * All rights reserved. */
/* This program is free software; you can redistribute it and/or modify it
* under the terms of the license (GNU LGPL) which comes with this package. */
*/
#include "xbt/log.h"
#include "xbt/sysdep.h"
-#include "xbt/mallocator.h"
#include "maxmin_private.h"
#include <stdlib.h>
#include <math.h>
#endif
-
XBT_LOG_NEW_DEFAULT_SUBCATEGORY(surf_lagrange, surf,
- "Logging specific to SURF (lagrange)");
+ "Logging specific to SURF (lagrange)");
+XBT_LOG_NEW_SUBCATEGORY(surf_lagrange_dichotomy, surf_lagrange,
+ "Logging specific to SURF (lagrange dichotomy)");
-void lagrange_solve(lmm_system_t sys)
+#define SHOW_EXPR(expr) XBT_CDEBUG(surf_lagrange,#expr " = %g",expr);
+
+double (*func_f_def) (lmm_variable_t, double);
+double (*func_fp_def) (lmm_variable_t, double);
+double (*func_fpi_def) (lmm_variable_t, double);
+
+/*
+ * Local prototypes to implement the lagrangian optimization with optimal step, also called dichotomy.
+ */
+//solves the proportional fairness using a lagrange optimizition with dichotomy step
+void lagrange_solve(lmm_system_t sys);
+//computes the value of the dichotomy using a initial values, init, with a specific variable or constraint
+static double dichotomy(double init, double diff(double, void *),
+ void *var_cnst, double min_error);
+//computes the value of the differential of variable param_var applied to mu
+static double partial_diff_mu(double mu, void *param_var);
+//computes the value of the differential of constraint param_cnst applied to lambda
+static double partial_diff_lambda(double lambda, void *param_cnst);
+
+static int __check_feasible(xbt_swag_t cnst_list, xbt_swag_t var_list,
+ int warn)
{
+ xbt_swag_t elem_list = NULL;
+ lmm_element_t elem = NULL;
+ lmm_constraint_t cnst = NULL;
+ lmm_variable_t var = NULL;
+
+ double tmp;
+
+ xbt_swag_foreach(cnst, cnst_list) {
+ tmp = 0;
+ elem_list = &(cnst->element_set);
+ xbt_swag_foreach(elem, elem_list) {
+ var = elem->variable;
+ if (var->weight <= 0)
+ continue;
+ tmp += var->value;
+ }
+
+ if (double_positive(tmp - cnst->bound)) {
+ if (warn)
+ XBT_WARN
+ ("The link (%p) is over-used. Expected less than %f and got %f",
+ cnst, cnst->bound, tmp);
+ return 0;
+ }
+ XBT_DEBUG
+ ("Checking feasability for constraint (%p): sat = %f, lambda = %f ",
+ cnst, tmp - cnst->bound, cnst->lambda);
+ }
+
+ xbt_swag_foreach(var, var_list) {
+ if (!var->weight)
+ break;
+ if (var->bound < 0)
+ continue;
+ XBT_DEBUG("Checking feasability for variable (%p): sat = %f mu = %f", var,
+ var->value - var->bound, var->mu);
+
+ if (double_positive(var->value - var->bound)) {
+ if (warn)
+ XBT_WARN
+ ("The variable (%p) is too large. Expected less than %f and got %f",
+ var, var->bound, var->value);
+ return 0;
+ }
+ }
+ return 1;
+}
+
+static double new_value(lmm_variable_t var)
+{
+ double tmp = 0;
+ int i;
+
+ for (i = 0; i < var->cnsts_number; i++) {
+ tmp += (var->cnsts[i].constraint)->lambda;
+ }
+ if (var->bound > 0)
+ tmp += var->mu;
+ XBT_DEBUG("\t Working on var (%p). cost = %e; Weight = %e", var, tmp,
+ var->weight);
+ //uses the partial differential inverse function
+ return var->func_fpi(var, tmp);
+}
+
+static double new_mu(lmm_variable_t var)
+{
+ double mu_i = 0.0;
+ double sigma_i = 0.0;
+ int j;
+
+ for (j = 0; j < var->cnsts_number; j++) {
+ sigma_i += (var->cnsts[j].constraint)->lambda;
+ }
+ mu_i = var->func_fp(var, var->bound) - sigma_i;
+ if (mu_i < 0.0)
+ return 0.0;
+ return mu_i;
+}
+
+static double dual_objective(xbt_swag_t var_list, xbt_swag_t cnst_list)
+{
+ lmm_constraint_t cnst = NULL;
+ lmm_variable_t var = NULL;
+
+ double obj = 0.0;
+
+ xbt_swag_foreach(var, var_list) {
+ double sigma_i = 0.0;
+ int j;
+
+ if (!var->weight)
+ break;
+
+ for (j = 0; j < var->cnsts_number; j++)
+ sigma_i += (var->cnsts[j].constraint)->lambda;
+
+ if (var->bound > 0)
+ sigma_i += var->mu;
+
+ XBT_DEBUG("var %p : sigma_i = %1.20f", var, sigma_i);
+
+ obj += var->func_f(var, var->func_fpi(var, sigma_i)) -
+ sigma_i * var->func_fpi(var, sigma_i);
+
+ if (var->bound > 0)
+ obj += var->mu * var->bound;
+ }
+
+ xbt_swag_foreach(cnst, cnst_list)
+ obj += cnst->lambda * cnst->bound;
+ return obj;
+}
+
+void lagrange_solve(lmm_system_t sys)
+{
/*
* Lagrange Variables.
*/
- double epsilon_min_error = 1e-6;
- double overall_error = 1;
- double sigma_step = 0.5e-3;
- double capacity_error, bound_error;
- double sum_capacity = 0;
- double sum_bound = 0;
-
+ int max_iterations = 100;
+ double epsilon_min_error = MAXMIN_PRECISION;
+ double dichotomy_min_error = 1e-14;
+ double overall_modification = 1;
/*
* Variables to manipulate the data structure proposed to model the maxmin
* fairness. See docummentation for more details.
*/
- lmm_element_t elem = NULL;
xbt_swag_t cnst_list = NULL;
- lmm_constraint_t cnst1 = NULL;
- lmm_constraint_t cnst2 = NULL;
- xbt_swag_t var_list = NULL;
- xbt_swag_t elem_list = NULL;
- lmm_variable_t var1 = NULL;
- lmm_variable_t var2 = NULL;
+ lmm_constraint_t cnst = NULL;
+ xbt_swag_t var_list = NULL;
+ lmm_variable_t var = NULL;
/*
* Auxiliar variables.
*/
- int iteration=0;
- int max_iterations=100000;
- double mu_partial=0;
- double lambda_partial=0;
+ int iteration = 0;
+ double tmp = 0;
+ int i;
+ double obj, new_obj;
+ XBT_DEBUG("Iterative method configuration snapshot =====>");
+ XBT_DEBUG("#### Maximum number of iterations : %d", max_iterations);
+ XBT_DEBUG("#### Minimum error tolerated : %e",
+ epsilon_min_error);
+ XBT_DEBUG("#### Minimum error tolerated (dichotomy) : %e",
+ dichotomy_min_error);
- if ( !(sys->modified))
+ if (XBT_LOG_ISENABLED(surf_lagrange, xbt_log_priority_debug)) {
+ lmm_print(sys);
+ }
+
+ if (!(sys->modified))
return;
-
+
+
+ /*
+ * Initialize lambda.
+ */
+ cnst_list = &(sys->active_constraint_set);
+ xbt_swag_foreach(cnst, cnst_list) {
+ cnst->lambda = 1.0;
+ cnst->new_lambda = 2.0;
+ XBT_DEBUG("#### cnst(%p)->lambda : %e", cnst, cnst->lambda);
+ }
+
/*
* Initialize the var list variable with only the active variables.
- * Associate an index in the swag variables and compute the sum
- * of all round trip time constraints. May change depending on the
- * function f(x).
+ * Associate an index in the swag variables. Initialize mu.
*/
- var_list = &(sys->active_variable_set);
- i=0;
- xbt_swag_foreach(var1, var_list) {
- if(var1->weight != 0.0){
- i++;
- sum_bound += var1->bound;
+ var_list = &(sys->variable_set);
+ i = 0;
+ xbt_swag_foreach(var, var_list) {
+ if (!var->weight)
+ var->value = 0.0;
+ else {
+ int nb = 0;
+ if (var->bound < 0.0) {
+ XBT_DEBUG("#### NOTE var(%d) is a boundless variable", i);
+ var->mu = -1.0;
+ var->value = new_value(var);
+ } else {
+ var->mu = 1.0;
+ var->new_mu = 2.0;
+ var->value = new_value(var);
+ }
+ XBT_DEBUG("#### var(%p) ->weight : %e", var, var->weight);
+ XBT_DEBUG("#### var(%p) ->mu : %e", var, var->mu);
+ XBT_DEBUG("#### var(%p) ->weight: %e", var, var->weight);
+ XBT_DEBUG("#### var(%p) ->bound: %e", var, var->bound);
+ for (i = 0; i < var->cnsts_number; i++) {
+ if (var->cnsts[i].value == 0.0)
+ nb++;
+ }
+ if (nb == var->cnsts_number)
+ var->value = 1.0;
}
}
/*
- * Compute the sum of all capacities constraints. May change depending
- * on the function f(x).
+ * Compute dual objective.
*/
- cnst_list=&(sys->active_constraint_set);
- xbt_swag_foreach(cnst1, cnst_list) {
- sum_capacity += cnst1->value;
- }
+ obj = dual_objective(var_list, cnst_list);
-
/*
* While doesn't reach a minimun error or a number maximum of iterations.
*/
- while(overall_error > epsilon_min_error && iteration < max_iterations){
+ while (overall_modification > epsilon_min_error
+ && iteration < max_iterations) {
+/* int dual_updated=0; */
iteration++;
+ XBT_DEBUG("************** ITERATION %d **************", iteration);
+ XBT_DEBUG("-------------- Gradient Descent ----------");
-
-
- /* d Dual
- * Compute the value of ----------- (\lambda^k, \mu^k) this portion
- * d \mu_i^k
- * of code depends on function f(x).
+ /*
+ * Improve the value of mu_i
*/
- bound_error = 0;
- xbt_swag_foreach(var1, var_list) {
-
- mu_partial = 0;
-
- //for each link elem1 that uses flow of variable var1 do
- //mu_partial += elem1->weight + var1->bound;
+ xbt_swag_foreach(var, var_list) {
+ if (!var->weight)
+ break;
+ if (var->bound >= 0) {
+ XBT_DEBUG("Working on var (%p)", var);
+ var->new_mu = new_mu(var);
+/* dual_updated += (fabs(var->new_mu-var->mu)>dichotomy_min_error); */
+/* XBT_DEBUG("dual_updated (%d) : %1.20f",dual_updated,fabs(var->new_mu-var->mu)); */
+ XBT_DEBUG("Updating mu : var->mu (%p) : %1.20f -> %1.20f", var,
+ var->mu, var->new_mu);
+ var->mu = var->new_mu;
- mu_partial = - (1 / mu_partial) + sum_bound;
-
- var1->bound = var1->bound + sigma_step * mu_partial;
+ new_obj = dual_objective(var_list, cnst_list);
+ XBT_DEBUG("Improvement for Objective (%g -> %g) : %g", obj, new_obj,
+ obj - new_obj);
+ xbt_assert1(obj - new_obj >= -epsilon_min_error,
+ "Our gradient sucks! (%1.20f)", obj - new_obj);
+ obj = new_obj;
+ }
}
-
-
/*
- * Verify for each capacity constraint (lambda) the error associated.
+ * Improve the value of lambda_i
*/
- xbt_swag_foreach(cnst1, cnst_list) {
- cnst2 = xbt_swag_getNext(cnst1,(var_list)->offset);
- if(cnst2 != NULL){
- capacity_error += fabs(cnst1->value - cnsts2->value);
- }
+ xbt_swag_foreach(cnst, cnst_list) {
+ XBT_DEBUG("Working on cnst (%p)", cnst);
+ cnst->new_lambda =
+ dichotomy(cnst->lambda, partial_diff_lambda, cnst,
+ dichotomy_min_error);
+/* dual_updated += (fabs(cnst->new_lambda-cnst->lambda)>dichotomy_min_error); */
+/* XBT_DEBUG("dual_updated (%d) : %1.20f",dual_updated,fabs(cnst->new_lambda-cnst->lambda)); */
+ XBT_DEBUG("Updating lambda : cnst->lambda (%p) : %1.20f -> %1.20f",
+ cnst, cnst->lambda, cnst->new_lambda);
+ cnst->lambda = cnst->new_lambda;
+
+ new_obj = dual_objective(var_list, cnst_list);
+ XBT_DEBUG("Improvement for Objective (%g -> %g) : %g", obj, new_obj,
+ obj - new_obj);
+ xbt_assert1(obj - new_obj >= -epsilon_min_error,
+ "Our gradient sucks! (%1.20f)", obj - new_obj);
+ obj = new_obj;
}
/*
- * Verify for each variable the error of round trip time constraint (mu).
+ * Now computes the values of each variable (\rho) based on
+ * the values of \lambda and \mu.
*/
- bound_error = 0;
- xbt_swag_foreach(var1, var_list) {
- var2 = xbt_swag_getNext(var1,(var_list)->offset);
- if(var2 != NULL){
- bound_error += fabs( var2->weight - var1->weight);
+ XBT_DEBUG("-------------- Check convergence ----------");
+ overall_modification = 0;
+ xbt_swag_foreach(var, var_list) {
+ if (var->weight <= 0)
+ var->value = 0.0;
+ else {
+ tmp = new_value(var);
+
+ overall_modification =
+ MAX(overall_modification, fabs(var->value - tmp));
+
+ var->value = tmp;
+ XBT_DEBUG("New value of var (%p) = %e, overall_modification = %e",
+ var, var->value, overall_modification);
}
}
- overall_error = capacity_error + bound_error;
+ XBT_DEBUG("-------------- Check feasability ----------");
+ if (!__check_feasible(cnst_list, var_list, 0))
+ overall_modification = 1.0;
+ XBT_DEBUG("Iteration %d: overall_modification : %f", iteration,
+ overall_modification);
+/* if(!dual_updated) { */
+/* XBT_WARN("Could not improve the convergence at iteration %d. Drop it!",iteration); */
+/* break; */
+/* } */
+ }
+
+ __check_feasible(cnst_list, var_list, 1);
+
+ if (overall_modification <= epsilon_min_error) {
+ XBT_DEBUG("The method converges in %d iterations.", iteration);
+ }
+ if (iteration >= max_iterations) {
+ XBT_DEBUG
+ ("Method reach %d iterations, which is the maximum number of iterations allowed.",
+ iteration);
+ }
+/* XBT_INFO("Method converged after %d iterations", iteration); */
+
+ if (XBT_LOG_ISENABLED(surf_lagrange, xbt_log_priority_debug)) {
+ lmm_print(sys);
+ }
+}
+
+/*
+ * Returns a double value corresponding to the result of a dichotomy proccess with
+ * respect to a given variable/constraint (\mu in the case of a variable or \lambda in
+ * case of a constraint) and a initial value init.
+ *
+ * @param init initial value for \mu or \lambda
+ * @param diff a function that computes the differential of with respect a \mu or \lambda
+ * @param var_cnst a pointer to a variable or constraint
+ * @param min_erro a minimun error tolerated
+ *
+ * @return a double correponding to the result of the dichotomyal process
+ */
+static double dichotomy(double init, double diff(double, void *),
+ void *var_cnst, double min_error)
+{
+ double min, max;
+ double overall_error;
+ double middle;
+ double min_diff, max_diff, middle_diff;
+ double diff_0 = 0.0;
+ min = max = init;
+
+ XBT_IN("");
+
+ if (init == 0.0) {
+ min = max = 0.5;
+ }
+
+ min_diff = max_diff = middle_diff = 0.0;
+ overall_error = 1;
+
+ if ((diff_0 = diff(1e-16, var_cnst)) >= 0) {
+ XBT_CDEBUG(surf_lagrange_dichotomy, "returning 0.0 (diff = %e)", diff_0);
+ XBT_OUT();
+ return 0.0;
}
+ min_diff = diff(min, var_cnst);
+ max_diff = diff(max, var_cnst);
+
+ while (overall_error > min_error) {
+ XBT_CDEBUG(surf_lagrange_dichotomy,
+ "[min, max] = [%1.20f, %1.20f] || diffmin, diffmax = %1.20f, %1.20f",
+ min, max, min_diff, max_diff);
+
+ if (min_diff > 0 && max_diff > 0) {
+ if (min == max) {
+ XBT_CDEBUG(surf_lagrange_dichotomy, "Decreasing min");
+ min = min / 2.0;
+ min_diff = diff(min, var_cnst);
+ } else {
+ XBT_CDEBUG(surf_lagrange_dichotomy, "Decreasing max");
+ max = min;
+ max_diff = min_diff;
+ }
+ } else if (min_diff < 0 && max_diff < 0) {
+ if (min == max) {
+ XBT_CDEBUG(surf_lagrange_dichotomy, "Increasing max");
+ max = max * 2.0;
+ max_diff = diff(max, var_cnst);
+ } else {
+ XBT_CDEBUG(surf_lagrange_dichotomy, "Increasing min");
+ min = max;
+ min_diff = max_diff;
+ }
+ } else if (min_diff < 0 && max_diff > 0) {
+ middle = (max + min) / 2.0;
+ XBT_CDEBUG(surf_lagrange_dichotomy, "Trying (max+min)/2 : %1.20f",
+ middle);
+
+ if ((min == middle) || (max == middle)) {
+ XBT_CWARN(surf_lagrange_dichotomy,
+ "Cannot improve the convergence! min=max=middle=%1.20f, diff = %1.20f."
+ " Reaching the 'double' limits. Maybe scaling your function would help ([%1.20f,%1.20f]).",
+ min, max - min, min_diff, max_diff);
+ break;
+ }
+ middle_diff = diff(middle, var_cnst);
+
+ if (middle_diff < 0) {
+ XBT_CDEBUG(surf_lagrange_dichotomy, "Increasing min");
+ min = middle;
+ overall_error = max_diff - middle_diff;
+ min_diff = middle_diff;
+/* SHOW_EXPR(overall_error); */
+ } else if (middle_diff > 0) {
+ XBT_CDEBUG(surf_lagrange_dichotomy, "Decreasing max");
+ max = middle;
+ overall_error = max_diff - middle_diff;
+ max_diff = middle_diff;
+/* SHOW_EXPR(overall_error); */
+ } else {
+ overall_error = 0;
+/* SHOW_EXPR(overall_error); */
+ }
+ } else if (min_diff == 0) {
+ max = min;
+ overall_error = 0;
+/* SHOW_EXPR(overall_error); */
+ } else if (max_diff == 0) {
+ min = max;
+ overall_error = 0;
+/* SHOW_EXPR(overall_error); */
+ } else if (min_diff > 0 && max_diff < 0) {
+ XBT_CWARN(surf_lagrange_dichotomy,
+ "The impossible happened, partial_diff(min) > 0 && partial_diff(max) < 0");
+ abort();
+ } else {
+ XBT_CWARN(surf_lagrange_dichotomy,
+ "diffmin (%1.20f) or diffmax (%1.20f) are something I don't know, taking no action.",
+ min_diff, max_diff);
+ abort();
+ }
+ }
+
+ XBT_CDEBUG(surf_lagrange_dichotomy, "returning %e", (min + max) / 2.0);
+ XBT_OUT();
+ return ((min + max) / 2.0);
+}
+
+static double partial_diff_lambda(double lambda, void *param_cnst)
+{
+
+ int j;
+ xbt_swag_t elem_list = NULL;
+ lmm_element_t elem = NULL;
+ lmm_variable_t var = NULL;
+ lmm_constraint_t cnst = (lmm_constraint_t) param_cnst;
+ double diff = 0.0;
+ double sigma_i = 0.0;
+
+ XBT_IN("");
+ elem_list = &(cnst->element_set);
+
+ XBT_CDEBUG(surf_lagrange_dichotomy, "Computing diff of cnst (%p)", cnst);
+
+ xbt_swag_foreach(elem, elem_list) {
+ var = elem->variable;
+ if (var->weight <= 0)
+ continue;
+
+ XBT_CDEBUG(surf_lagrange_dichotomy, "Computing sigma_i for var (%p)",
+ var);
+ // Initialize the summation variable
+ sigma_i = 0.0;
+
+ // Compute sigma_i
+ for (j = 0; j < var->cnsts_number; j++) {
+ sigma_i += (var->cnsts[j].constraint)->lambda;
+ }
+
+ //add mu_i if this flow has a RTT constraint associated
+ if (var->bound > 0)
+ sigma_i += var->mu;
+
+ //replace value of cnst->lambda by the value of parameter lambda
+ sigma_i = (sigma_i - cnst->lambda) + lambda;
+
+ diff += -var->func_fpi(var, sigma_i);
+ }
+
+
+ diff += cnst->bound;
+
+ XBT_CDEBUG(surf_lagrange_dichotomy,
+ "d D/d lambda for cnst (%p) at %1.20f = %1.20f", cnst, lambda,
+ diff);
+ XBT_OUT();
+ return diff;
+}
+
+/** \brief Attribute the value bound to var->bound.
+ *
+ * \param func_fpi inverse of the partial differential of f (f prime inverse, (f')^{-1})
+ *
+ * Set default functions to the ones passed as parameters. This is a polimorfism in C pure, enjoy the roots of programming.
+ *
+ */
+void lmm_set_default_protocol_function(double (*func_f)
+
+
+
+
+
+
+ (lmm_variable_t var, double x),
+ double (*func_fp) (lmm_variable_t
+ var, double x),
+ double (*func_fpi) (lmm_variable_t
+ var, double x))
+{
+ func_f_def = func_f;
+ func_fp_def = func_fp;
+ func_fpi_def = func_fpi;
+}
+
+
+/**************** Vegas and Reno functions *************************/
+/*
+ * NOTE for Reno: all functions consider the network
+ * coeficient (alpha) equal to 1.
+ */
+
+/*
+ * For Vegas: $f(x) = \alpha D_f\ln(x)$
+ * Therefore: $fp(x) = \frac{\alpha D_f}{x}$
+ * Therefore: $fpi(x) = \frac{\alpha D_f}{x}$
+ */
+#define VEGAS_SCALING 1000.0
+
+double func_vegas_f(lmm_variable_t var, double x)
+{
+ xbt_assert1(x > 0.0, "Don't call me with stupid values! (%1.20f)", x);
+ return VEGAS_SCALING * var->weight * log(x);
+}
+
+double func_vegas_fp(lmm_variable_t var, double x)
+{
+ xbt_assert1(x > 0.0, "Don't call me with stupid values! (%1.20f)", x);
+ return VEGAS_SCALING * var->weight / x;
+}
+
+double func_vegas_fpi(lmm_variable_t var, double x)
+{
+ xbt_assert1(x > 0.0, "Don't call me with stupid values! (%1.20f)", x);
+ return var->weight / (x / VEGAS_SCALING);
+}
+
+/*
+ * For Reno: $f(x) = \frac{\sqrt{3/2}}{D_f} atan(\sqrt{3/2}D_f x)$
+ * Therefore: $fp(x) = \frac{3}{3 D_f^2 x^2+2}$
+ * Therefore: $fpi(x) = \sqrt{\frac{1}{{D_f}^2 x} - \frac{2}{3{D_f}^2}}$
+ */
+#define RENO_SCALING 1.0
+double func_reno_f(lmm_variable_t var, double x)
+{
+ xbt_assert0(var->weight > 0.0, "Don't call me with stupid values!");
+
+ return RENO_SCALING * sqrt(3.0 / 2.0) / var->weight *
+ atan(sqrt(3.0 / 2.0) * var->weight * x);
+}
+
+double func_reno_fp(lmm_variable_t var, double x)
+{
+ return RENO_SCALING * 3.0 / (3.0 * var->weight * var->weight * x * x +
+ 2.0);
+}
+
+double func_reno_fpi(lmm_variable_t var, double x)
+{
+ double res_fpi;
+
+ xbt_assert0(var->weight > 0.0, "Don't call me with stupid values!");
+ xbt_assert0(x > 0.0, "Don't call me with stupid values!");
+
+ res_fpi =
+ 1.0 / (var->weight * var->weight * (x / RENO_SCALING)) -
+ 2.0 / (3.0 * var->weight * var->weight);
+ if (res_fpi <= 0.0)
+ return 0.0;
+/* xbt_assert0(res_fpi>0.0,"Don't call me with stupid values!"); */
+ return sqrt(res_fpi);
+}
+
+
+/* Implementing new Reno-2
+ * For Reno-2: $f(x) = U_f(x_f) = \frac{{2}{D_f}}*ln(2+x*D_f)$
+ * Therefore: $fp(x) = 2/(Weight*x + 2)
+ * Therefore: $fpi(x) = (2*Weight)/x - 4
+ */
+#define RENO2_SCALING 1.0
+double func_reno2_f(lmm_variable_t var, double x)
+{
+ xbt_assert0(var->weight > 0.0, "Don't call me with stupid values!");
+ return RENO2_SCALING * (1.0 / var->weight) * log((x * var->weight) /
+ (2.0 * x * var->weight +
+ 3.0));
+}
+
+double func_reno2_fp(lmm_variable_t var, double x)
+{
+ return RENO2_SCALING * 3.0 / (var->weight * x *
+ (2.0 * var->weight * x + 3.0));
+}
+
+double func_reno2_fpi(lmm_variable_t var, double x)
+{
+ double res_fpi;
+ double tmp;
+
+ xbt_assert0(x > 0.0, "Don't call me with stupid values!");
+ tmp = x * var->weight * var->weight;
+ res_fpi = tmp * (9.0 * x + 24.0);
+
+ if (res_fpi <= 0.0)
+ return 0.0;
+
+ res_fpi = RENO2_SCALING * (-3.0 * tmp + sqrt(res_fpi)) / (4.0 * tmp);
+ return res_fpi;
}
