Package jline.api.mam

Class Map_m1ps

java.lang.Object

jline.api.mam.Map_m1ps

public final class Map_m1ps
extends Object

Sojourn time distribution of the MAP/M/1 processor-sharing queue.

Port of matlab/src/api/mam/map_compute_R.m, map_m1ps_h_recursive.m and map_m1ps_sojourn.m, twin of the native Python api.mam.mapm1ps. The processor-sharing discipline shares the server equally, so with n customers present each is served at rate mu/n; the sojourn law is therefore not the FCFS waiting law and is not available in closed form.

Implements Theorem 1 of Masuyama, H. and Takine, T., "Sojourn time distribution in a MAP/M/1 processor-sharing queue", Operations Research Letters 31(6), 2003, 406-412:

   Wbar(x) = (1/lambda) sum_n pi_0 R^n D sum_k p_k(x) h_{n,k}
 

with p_k(x) the Poisson(( theta + mu) x) mass, R the minimal nonnegative solution of D + R(C - mu I) + mu R^2 = 0, and h_{n,k} the uniformized coefficients of map_m1ps_h_recursive(jline.util.matrix.Matrix, jline.util.matrix.Matrix, double, int, int).

Nested Class Summary

Nested Classes

Modifier and Type	Class	Description
static final class	Map_m1ps.SojournResult	Result of map_m1ps_sojourn(jline.util.matrix.Matrix, jline.util.matrix.Matrix, double, double[]).

Method Summary

Modifier and Type	Method	Description
static Matrix	map_compute_R(Matrix C, Matrix D, double mu)	Minimal nonnegative solution R of D + R(C - mu I) + mu R^2 = 0, by the fixed point R <- -D (C - mu I + mu R)^-1 started at R = 0.
static double[][][]	map_m1ps_h_recursive(Matrix C, Matrix D, double mu, int N, int K)	The h_{n,k} vectors of Theorem 1, returned as h[n][k], each M x 1:
static Map_m1ps.SojournResult	map_m1ps_sojourn(Matrix C, Matrix D, double mu, double[] x)	map_m1ps_sojourn at the reference defaults epsilon = 1e-11, epsilon' = 1e-10.
static Map_m1ps.SojournResult	map_m1ps_sojourn(Matrix C, Matrix D, double mu, double[] x, double epsilon, double epsilonPrime)

Methods inherited from class java.lang.Object

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

Method Details

map_compute_R

public static Matrix map_compute_R(Matrix C,
 Matrix D,
 double mu)

Minimal nonnegative solution R of D + R(C - mu I) + mu R^2 = 0, by the fixed point R <- -D (C - mu I + mu R)^-1 started at R = 0.

map_m1ps_h_recursive

public static double[][][] map_m1ps_h_recursive(Matrix C,
 Matrix D,
 double mu,
 int N,
 int K)

The h_{n,k} vectors of Theorem 1, returned as h[n][k], each M x 1:

   h_{n,0}   = e
   h_{n,k+1} = [ n mu/(n+1) h_{n-1,k} + (theta I + C) h_{n,k} + D h_{n+1,k} ]
               / (theta + mu)
 

with h_{-1,k} = 0 and the level dimension truncated at N, i.e. h_{N+1,k} = 0.

map_m1ps_sojourn

public static Map_m1ps.SojournResult map_m1ps_sojourn(Matrix C,
 Matrix D,
 double mu,
 double[] x)

map_m1ps_sojourn at the reference defaults epsilon = 1e-11, epsilon' = 1e-10.

map_m1ps_sojourn

public static Map_m1ps.SojournResult map_m1ps_sojourn(Matrix C,
 Matrix D,
 double mu,
 double[] x,
 double epsilon,
 double epsilonPrime)

Parameters:

C - M x M MAP transitions without an arrival

D - M x M MAP transitions with an arrival

mu - service rate, mu > 0

x - time points at which Pr[W > x] is evaluated

epsilon - queue-length truncation tolerance

epsilonPrime - uniformization truncation tolerance