Constructor Detail

Method Detail

• fj_xmax_exp

   final static Double fj_xmax_exp(Integer K, Double mu)

  Expected maximum of K i.i.d. exponential random variables.

  X_K^max = H_K / mu

  Parameters:

  K - Number of parallel servers (positive integer)

  mu - Service rate (mean service time is 1/mu)

  Returns:

  Expected maximum service time

• fj_xmax_2

   final static Double fj_xmax_2(Double lambda1, Double lambda2)

  Expected maximum of 2 exponential random variables.

  Y_2^max = 1/lambda1 + 1/lambda2 - 1/(lambda1 + lambda2)

  For identical rates: Y_2^max = 1.5/lambda = H_2/lambda

  Parameters:

  lambda1 - Rate of first exponential

  lambda2 - Rate of second exponential (default: same as lambda1)

  Returns:

  Expected maximum Y_2^max

• fj_xmax_2

   final static Double fj_xmax_2(Double lambda)

  Expected maximum of 2 i.i.d. exponential random variables (same rate).

  Parameters:

  lambda - Rate of both exponentials

  Returns:

  Expected maximum Y_2^max = 1.5/lambda

• fj_xmax_erlang

   final static Double fj_xmax_erlang(Integer K, Integer k, Double mu)

  Expected maximum of K i.i.d. Erlang-k random variables.

  For k=2 (Erlang-2), uses the closed-form formula: X_K^max = (1/mu) * sum_{n=1}^{K} C(K,n) * (-1)^{n-1} * sum_{m=1}^{n} C(n,m) * m! / (2*n^{m+1})

  For general k, uses numerical integration: X_K^max = integral_0^inf 1 - F_Erlang(t)^K dt

  Parameters:

  K - Number of parallel servers (positive integer)

  k - Number of Erlang stages (positive integer)

  mu - Rate parameter per stage (mean service time = k/mu)

  Returns:

  Expected maximum service time

• fj_xmax_hyperexp

   final static Double fj_xmax_hyperexp(Integer K, Double p1, Double mu1, Double mu2)

  Expected maximum of K i.i.d. Hyperexponential-2 random variables.

  X_K^max = sum_{n=1}^{K} (-1)^{n+1} * sum_{m=0}^{n} C(n,m) * p1^m * p2^{n-m} / (m*mu1 + (n-m)*mu2)

  Parameters:

  K - Number of parallel servers (positive integer)

  p1 - Probability of branch 1 (0 < p1 < 1)

  mu1 - Rate of branch 1

  mu2 - Rate of branch 2

  Returns:

  Expected maximum service time

• fj_xmax_normal

   final static FJXmaxNormalResult fj_xmax_normal(Integer K, Double mu, Double sigma, String method)

  Expected maximum for normal distribution.

  Johnson et al. approximation: EY_K ~ mu + sigma * sqrt(2*ln(K)) - (ln(ln(K)) - ln(4*pi) + 2*gamma) / (2*sqrt(2*ln(K)))

  Arnold approximation: EY_K ~ mu + sigma * sqrt(2*ln(K))

  Corrected version adds Petzold bias correction: delta(K) = 0.1727 * K^(-0.2750)

  Variance approximation: VarY_K ~ 1.64492 * sigma^2 / (2*ln(K))

  Parameters:

  K - Number of random variables (K >= 2)

  mu - Mean of the normal distribution

  sigma - Standard deviation

  method - "johnson" (default), "arnold", or "corrected"

  Returns:

  FJXmaxNormalResult containing expected maximum and variance

• fj_xmax_pareto

   final static Double fj_xmax_pareto(Integer K, Double beta, Double k)

  Expected maximum for Pareto distribution.

  Pareto CDF: F(x) = 1 - (k / (k + x))^beta, x >= 0, beta 2

  For the standardized Pareto with mean 1: k = beta - 1

  Parameters:

  K - Number of random variables (positive integer)

  beta - Shape parameter (beta 2 required for finite moments)

  k - Scale parameter (default: beta - 1 for mean = 1)

  Returns:

  FJXmaxParetoResult containing expected maximum and characteristic maximum

• fj_xmax_pareto

   final static Double fj_xmax_pareto(Integer K, Double beta)

  Expected maximum for Pareto distribution with default scale for mean = 1.

  Parameters:

  K - Number of random variables (positive integer)

  beta - Shape parameter (beta 2)

  Returns:

  Expected maximum

• fj_xmax_pareto_char_max

   final static Double fj_xmax_pareto_char_max(Integer K, Double beta, Double k)

  Characteristic maximum M_K for Pareto distribution.

  m_K = k * (K^(1/beta) - 1) M_K = m_K + K * integral_{m_K}^{inf} S(x) dx

  For Pareto: integral_{m_K}^{inf} (k/(k+x))^beta dx = k^beta * (k+m_K)^{1-beta} / (beta-1)

  Parameters:

  K - Number of random variables (positive integer)

  beta - Shape parameter (beta 2)

  k - Scale parameter (default: beta - 1)

  Returns:

  Characteristic maximum M_K

• fj_xmax_approx

   final static Double fj_xmax_approx(Integer K, Double muX, Double sigmaX, String distType)

  General approximation for expected maximum of K random variables.

  X_K^max ~ mu_X + sigma_X * G(K)

  G(K) depends on distribution type:

  • Exponential: G(K) = H_K - 1

  • Uniform: G(K) = sqrt(3) * (K-1) / (K+1)

  • EVD: G(K) = sqrt(6) * ln(K) / pi

  • General: G(K) <= (K-1) / sqrt(2K-1) (upper bound)

  Parameters:

  K - Number of random variables (positive integer)

  muX - Mean of the distribution

  sigmaX - Standard deviation of the distribution

  distType - Distribution type: "exp", "uniform", "evd", "bound"

  Returns:

  Expected maximum X_K^max

• fj_xmax_emma

   final static Double fj_xmax_emma(Integer K, Double mu)

  Expected maximum using EMMA (Expected Maximum from Marginal Approximation).

  Based on: [F_X(EY_K)]^K ~ phi = 0.570376 Therefore: EY_K ~ F^{-1}(phi^{1/K})

  For exponential with rate mu: EY_K = -(1/mu) * ln(1 - phi^{1/K})

  The constant phi = exp(-exp(-gamma)) ~ 0.570376.

  Parameters:

  K - Number of random variables (positive integer)

  mu - Rate parameter for exponential distribution

  Returns:

  Approximate expected maximum