snc_perc_backlog.h

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/*
 * Copyright (c) 2012-2026, QORE Lab, Imperial College London
 * All rights reserved.
 */
#ifndef LINE_API_SNC_PERC_BACKLOG_H
#define LINE_API_SNC_PERC_BACKLOG_H
 
/**
 * @file
 * @ingroup api_snc
 * Backlog quantile at a prescribed violation probability.
 *
 * Inverts `snc_bound_backlog` in b: at fixed theta the smallest level for which
 * the bound certifies P{B > b} <= eps is
 *
 *   b(theta) = sigmaA + sigmaS - log(eps*(1-exp(-theta*(rhoS-rhoA))))/theta,
 *
 * and the reported quantile is its minimum over the feasible thetas. The
 * minimizing theta differs from the one of the forward bound at a given level,
 * which is why the inversion is done in closed form and re-optimized.
 *
 * Port of matlab/src/api/snc/snc_perc_backlog.m.
 */
 
#include <cmath>
 
#include "line/api/snc/snc_thetaopt.h"
#include "line/api/snc/snc_types.h"
#include "line/util/error.h"
 
namespace line {
namespace snc {
 
/**
 * @brief Backlog quantile at a prescribed violation probability.
 *
 * @param arv      arrival envelope
 * @param srv      service envelope
 * @param eps      violation probability, 0 < eps < 1
 * @param thetamax upper end of the theta search
 */
inline SncResult snc_perc_backlog(const Envelope& arv, const Envelope& srv, double eps,
                                  double thetamax = 1e3) {
    if (!(eps > 0.0 && eps < 1.0))
        throw UnsupportedError("snc_perc_backlog: eps must lie in (0,1)");
    const SncResult r = snc_thetaopt(
        [&](double theta) {
            const detail::SncPair p = detail::snc_pair(arv, srv, theta);
            if (!p.ok) return std::numeric_limits<double>::infinity();
            return p.a.sigma + p.s.sigma -
                   std::log(eps * (1.0 - std::exp(-theta * (p.s.rho - p.a.rho)))) / theta;
        },
        thetamax);
    return SncResult{std::max(r.value, 0.0), r.theta};
}
 
}  // namespace snc
}  // namespace line
 
#endif  // LINE_API_SNC_PERC_BACKLOG_H