Package jline.api.qsys

Class Qsys_gigk_rqt

java.lang.Object

jline.api.qsys.Qsys_gigk_rqt

public final class Qsys_gigk_rqt
extends Object

Method Summary

Modifier and Type	Method	Description
static double[]	qsys_gigk_rqt(double lambda, double mu, double Gamma_a, double Gamma_s, int k, double alpha_a, double alpha_s)	Robust Queueing Theory worst-case system time of a G/G/k FCFS queue.

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Method Details

qsys_gigk_rqt

public static double[] qsys_gigk_rqt(double lambda,
 double mu,
 double Gamma_a,
 double Gamma_s,
 int k,
 double alpha_a,
 double alpha_s)

Robust Queueing Theory worst-case system time of a G/G/k FCFS queue. The arrival and service processes are not described by distributions but by the polyhedral uncertainty sets U^a = { T : (sum_{i=k+1}^n T_i - (n-k)/lambda)/(n-k)^(1/alpha_a) >= -Gamma_a }, U^s = { X : (sum_{i=k}^n X_i - (n-k+1)/mu)/(n-k+1)^(1/alpha_s) <= Gamma_s }, whose shape follows the (generalized) central limit theorem: alpha=2 is the finite-variance regime and alpha in (1,2) the heavy-tailed one.

The first returned value is the closed-form bound of Theorem 3 (Theorem 8 when the two tail coefficients differ, with alphabar = min(alpha_a,alpha_s)), W <= (ab-1)/ab^(ab/(ab-1)) lambda^(1/(ab-1)) (Gamma_a+Gamma_s/k^(1/ab))^(ab/(ab-1)) / (1-rho)^(1/(ab-1)) + k/lambda, which for k=1 reduces to Theorem 2 and, at alphabar=2, to the Kingman-like form (lambda/4)(Gamma_a+Gamma_s)^2/(1-rho) + 1/lambda. The third is the exact worst case over the uncertainty sets, eq. (45), the supremum over the integer x >= 1 of x/mu + Gamma_s x^(1/alpha_s) - k(x-1)/lambda + Gamma_a (k(x-1))^(1/alpha_a). The arrival deviation ADDS to the worst case, since the adversary shortens the interarrival times.

W is a SYSTEM time (waiting plus service), and its additive term is k/lambda rather than the mean service time 1/mu.

Parameters:

lambda - arrival rate

mu - service rate of each server

Gamma_a - variability parameter of the arrival uncertainty set

Gamma_s - variability parameter of the service uncertainty set

k - number of servers

alpha_a - arrival tail coefficient in (1,2]

alpha_s - service tail coefficient in (1,2]

Returns:

array {W, rhohat, Sworst}: closed-form bound on the system time, modified utilization, and exact worst-case system time