aoi_lcfspr_gim1.h

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/*
 * Copyright (c) 2012-2026, QORE Lab, Imperial College London
 * All rights reserved.
 */
#ifndef LINE_API_AOI_LCFSPR_GIM1_H
#define LINE_API_AOI_LCFSPR_GIM1_H
 
/**
 * @file
 * @ingroup api_aoi
 * Mean Age of Information, its transform and the peak age of a GI/M/1
 * preemptive LCFS queue.
 *
 * Templated port of matlab/src/api/aoi/aoi_lcfspr_gim1.m, cross-checked
 * against jar/src/main/java/jline/api/aoi/Aoi_lcfspr_gim1.java (identical).
 *
 *   E[A]      = lambda E[Y^2]/2 + 1/mu
 *   q         = P(S < Y) = 1 - Y*(mu)
 *   E[S|succ] = (1/mu + Y*'(mu) - Y*(mu)/mu) / q
 *   E[Apeak]  = E[S|succ] + 1/(lambda q)
 *   A*(s)     = mu/(s + mu) * lambda (1 - Y*(s))/s
 *
 * With memoryless service the age is the backward recurrence time of the
 * arrivals plus an independent Exp(mu); the old E[Y] + 1/mu and Y*(s) mu/(s+mu)
 * hold only for Poisson arrivals (general age formula of Inoue et al. 2019).
 *
 * static_assert(num_traits<T>::has_transcendental) -- MATLAB's
 * finite-difference derivative of Y* at mu. The mean is rational in E_Y, E_Y2
 * and mu and is exact; the transform is exact wherever Y* is.
 */
 
#include "line/api/aoi/aoi_types.h"
#include "line/num/number.h"
#include "line/util/error.h"
 
namespace line {
namespace aoi {
 
/**
 * @brief Mean Age of Information, its transform and the peak age of a GI/M/1
 *        preemptive LCFS queue.
 *
 * @param Y_lst LST of the interarrival time
 * @param mu    service rate, > 0
 * @param E_Y   mean interarrival time, > 0
 * @param E_Y2  second moment of the interarrival time, >= E_Y^2
 * @return      [meanAoI, peakAoI, A*(s)]
 */
template <class T>
AoiLstResult<T> aoi_lcfspr_gim1(const Lst<T>& Y_lst, const T& mu, const T& E_Y, const T& E_Y2) {
    static_assert(num_traits<T>::has_transcendental,
                  "aoi_lcfspr_gim1 requires transcendental arithmetic");
    detail::require_positive(mu, "aoi_lcfspr_gim1", "the service rate mu");
    detail::require_positive(E_Y, "aoi_lcfspr_gim1", "the mean interarrival time E_Y");
    if (!Y_lst) throw InputError("aoi_lcfspr_gim1: the interarrival LST must be callable");
    if (E_Y2 < E_Y * E_Y) throw InputError("aoi_lcfspr_gim1: E_Y2 must be at least E_Y^2");
    const T one = num_traits<T>::from_int(1), two = num_traits<T>::from_int(2);
    const T lambda = one / E_Y;
    const T rho = lambda / mu;
    detail::require_stable(rho, "aoi_lcfspr_gim1");
 
    const T meanAoI = lambda * E_Y2 / two + one / mu;
    const T dYstar = detail::lst_derivative<T>(Y_lst, mu);
    const T q = one - Y_lst(mu);
    if (q == num_traits<T>::from_int(0))
        throw NumericError("aoi_lcfspr_gim1: Y*(mu) = 1, no update ever completes");
    const T ES_succ = (one / mu + dYstar - Y_lst(mu) / mu) / q;
    const T peakAoI = ES_succ + one / (lambda * q);
 
    Lst<T> lstAoI = [Y_lst, mu, lambda, one](const T& s) {
        if (num_traits<T>::to_double(s < T(0) ? T(-s) : s) < 1e-12)
            return one;  // removable singularity: lambda (1 - Y*(s))/s -> 1
        return T((mu / (s + mu)) * lambda * (one - Y_lst(s)) / s);
    };
    return {meanAoI, peakAoI, lstAoI, true};
}
 
}  // namespace aoi
}  // namespace line
 
#endif  // LINE_API_AOI_LCFSPR_GIM1_H