Package jline.api.dqsys

Class Dqsys_bernoulli1

java.lang.Object
jline.api.dqsys.Dqsys_bernoulli1
public final class Dqsys_bernoulli1 extends Object
Exact analysis of a state dependent Bernoulli server on a discrete time scale.

Port of matlab/src/api/dqsys/dqsys_bernoulli1.m. Time advances in slots. In the slot starting at t with n jobs present the job in service departs with probability p(n) and an arrival occurs with probability b(n), independently; both are recorded at the end of the slot with the departure resolved first (Daduna's LA rule and D/A rule). The queue length at slot boundaries is a discrete birth-death chain with

   pi(n) = [prod_{m=0}^{n-1} b(m) / prod_{m=0}^{n} c(m)]
         * [prod_{m=1}^{n-1} q(m) / prod_{m=1}^{n} p(m)] / H

c = 1-b and q = 1-p, which is theorem 2.3, and corollary 2.8 once b(n) = 0 above the capacity. For constant b and p it collapses to the Geo/Geo/1 law of Dqsys_geogeo1 under the LAS_DA convention.

The law seen by an arriving customer, with himself not counted, is theorem 2.11 and is returned in arrivalPmf. It is not the time-stationary law: discrete time has no PASTA analogue, and the two differ even when the arrival stream is a state independent Bernoulli process. In that state independent case pi_1 is exactly the EAS-convention queue length law of Dqsys_geogeo1, geometric with ratio r = b(1-p)/(p(1-b)).

  • Method Details

    • dqsys_bernoulli1

      public static Bernoulli1Result dqsys_bernoulli1(double b, double p)
      Unbounded buffer with constant arrival and service probabilities.

    • dqsys_bernoulli1

      public static Bernoulli1Result dqsys_bernoulli1(double[] b, double[] p, int L)
      Finite buffer, state dependent probabilities. An arrival in a slot that finds L jobs present is lost, which is the loss system of corollary 2.8.
      Parameters:
      b - offered arrival probabilities, b[n] for n = 0..L, or a single-entry array for a state independent stream
      p - service probabilities, p[n-1] = p(n) for n = 1..L, or a single-entry array for a state independent server
      L - buffer capacity in jobs