Package jline.api.qsys

Class Qsys_geogeo1

java.lang.Object
jline.api.qsys.Qsys_geogeo1
public final class Qsys_geogeo1 extends Object
Geo/Geo/1 discrete-time queueing system analysis.

Time advances in slots of unit length. In every slot an arrival occurs with probability a and, if the server is engaged, a service completion occurs with probability s, both independently of everything else. The system content at slot boundaries is then a discrete birth-death chain whose stationary distribution is geometric, so all mean measures are available in closed form.

Under GeoGeo1Convention.LAS_DA the chain is 

   0 -> 1        with probability a
   n -> n-1      with probability s(1-a)          (n >= 1)
   n -> n+1      with probability a(1-s)          (n >= 1)
giving pi_0 = 1-rho and pi_n = (1-rho) rho/(1-a) r^(n-1) with rho = a/s and r = a(1-s)/(s(1-a)). Under GeoGeo1Convention.EAS an arrival may also depart within its own slot, so the upward transition has probability a(1-s) from every state including the empty one, giving the plain geometric pi_n = (1-r) r^n.

The two conventions describe the same queueing delay and differ only in whether the arrival slot is counted as part of the sojourn: the mean sojourn times differ by exactly one slot and the mean waiting times coincide. See GeoGeo1Convention for the exact observation epochs.

The LAS-DA branch reproduces H. Daduna, Queueing Networks with Discrete Time Scale, LNCS 2046, Springer 2001, corollary 2.7 term for term: his pi(n) = (1 - b/p) (bq/(cp))^n (1/q)^[n>0], with b = a, p = s, c = 1-a and q = 1-s, is the form below, and his tail ratio bq/(cp) is r. He likewise states the service time is geometric on {1,2,...}, which is the support jline's Geometric uses.

Since:
LINE 3.1.0

  • Method Details

    • qsys_geogeo1

      public static GeoGeo1Result qsys_geogeo1(double a, double s)
      Analyzes a Geo/Geo/1 queue under the late-arrival-delayed-access convention, which is the one realized by a slotted LDES simulation of a Geometric(a) source feeding a Geometric(s) FCFS queue.
      Parameters:
      a - per-slot arrival probability, 0 < a < s
      s - per-slot service completion probability, 0 < s <= 1
      Returns:
      the stationary metrics

    • qsys_geogeo1

      public static GeoGeo1Result qsys_geogeo1(double a, double s, GeoGeo1Convention convention)
      Analyzes a Geo/Geo/1 queue under the requested slot-boundary convention.
      Parameters:
      a - per-slot arrival probability, 0 < a < s
      s - per-slot service completion probability, 0 < s <= 1
      convention - slot-boundary convention
      Returns:
      the stationary metrics