Package jline.api.qsys

Class Qsys_mmcc_retrial_fp

java.lang.Object

jline.api.qsys.Qsys_mmcc_retrial_fp

public final class Qsys_mmcc_retrial_fp
extends Object

Fixed-point approximation for the M/M/c/c retrial queue.

Port of matlab/src/api/qsys/qsys_mmcc_retrial_fp.m. Customers arrive at rate lambda to a system of c servers each of rate mu, with no waiting room; a blocked customer joins an orbit and retries. Under the assumption that the retrial rate is small relative to the service rate, the total arrival flow (fresh plus retrial) is approximated by a Poisson process of rate lambda + r, with r the solution of

   r = (lambda + r) B(lambda/mu + r/mu, c),
 

B(a, c) being the Erlang-B blocking probability at offered load a.

With non-persistent customers (Phung-Duc 2019, Sec. 3.2), a blocked fresh customer joins the orbit with probability p and a blocked retrial customer returns to it with probability q. The orbit then sends retrials at the rate

   r = lambda p B / (1 - q B),   B = B((lambda + r)/mu, c),
 

which is the equation above when p = q = 1. A fresh customer is lost with probability lossProb = B(1-p) + B p (1-q) B/(1-qB), zero when p = q = 1.

References: Cohen (1957); Phung-Duc, "Retrial Queueing Models: A Survey on Theory and Applications", 2019, Eq. (1).

Nested Class Summary

Nested Classes

Modifier and Type	Class	Description
static final class	Qsys_mmcc_retrial_fp.Result	Blocking probability, orbit-induced rate and iteration count.

Method Summary

Modifier and Type	Method	Description
static double	erlangB(double a, int c)	Erlang-B by the recursion B_i = a B_{i-1}/(i + a B_{i-1}), numerically stable.
static Qsys_mmcc_retrial_fp.Result	qsys_mmcc_retrial_fp(double lambda, double mu, int c)	The fixed point at the reference defaults tol = 1e-10, maxiter = 10000.
static Qsys_mmcc_retrial_fp.Result	qsys_mmcc_retrial_fp(double lambda, double mu, int c, double tol, int maxiter)	The retrial fixed point.
static Qsys_mmcc_retrial_fp.Result	qsys_mmcc_retrial_fp(double lambda, double mu, int c, double tol, int maxiter, double p, double q)	The retrial fixed point with non-persistent customers.

Methods inherited from class java.lang.Object

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

Method Details

qsys_mmcc_retrial_fp

public static Qsys_mmcc_retrial_fp.Result qsys_mmcc_retrial_fp(double lambda,
 double mu,
 int c)

The fixed point at the reference defaults tol = 1e-10, maxiter = 10000.

qsys_mmcc_retrial_fp

public static Qsys_mmcc_retrial_fp.Result qsys_mmcc_retrial_fp(double lambda,
 double mu,
 int c,
 double tol,
 int maxiter)

The retrial fixed point.

Parameters:

lambda - fresh arrival rate

mu - service rate per server

c - number of servers, equal to the capacity

tol - convergence tolerance on r

maxiter - iteration cap

qsys_mmcc_retrial_fp

public static Qsys_mmcc_retrial_fp.Result qsys_mmcc_retrial_fp(double lambda,
 double mu,
 int c,
 double tol,
 int maxiter,
 double p,
 double q)

The retrial fixed point with non-persistent customers.

Parameters:

lambda - fresh arrival rate

mu - service rate per server

c - number of servers, equal to the capacity

tol - convergence tolerance on r

maxiter - iteration cap

p - probability that a blocked fresh customer joins the orbit

q - probability that a blocked retrial customer stays in the orbit

erlangB

public static double erlangB(double a,
 int c)

Erlang-B by the recursion B_i = a B_{i-1}/(i + a B_{i-1}), numerically stable.