Package jline.api.qsys

Class Qsys_geoxgeo1

java.lang.Object
jline.api.qsys.Qsys_geoxgeo1
public final class Qsys_geoxgeo1 extends Object
Geo^X/Geo/1 discrete-time queueing system analysis.

Generalizes Qsys_geogeo1 from single to batch arrivals. In every slot a batch arrives with probability a; the batch size X is supported on {1,2,...} and drawn independently. Service is geometric with per-slot completion probability s and a single server, so the arrival rate is lambda = a E[X] and stability requires lambda < s.

With A(z) = 1 - a + a X(z) the pgf of the number of jobs arriving in one slot, the slot-boundary queue length obeys X(t+1) = X(t) - D(t) + A(t) with the departure resolved first (Daduna's LA- and D/A-rules), giving 

   P(z) = p0 s (z-1) A(z) / ( z - A(z)(s + (1-s)z) ),   p0 = 1 - lambda/s

Differentiating at z = 1 yields the mean, which needs only the first two factorial moments of the batch: 

   E[N] = lambda + ( a E[X(X-1)]/2 + lambda(1-s) ) / (s - lambda)

At X == 1 this collapses to a(1-a)/(s-a), the Geo/Geo/1 result, so Qsys_geogeo1 is the degenerate-batch special case. The batch enters only through its first two factorial moments, so the primary entry point takes those directly and any batch law is supported; the three-argument overload is the geometric-batch convenience form that gives the model its name.

Reference: H. Daduna, Queueing Networks with Discrete Time Scale, LNCS 2046, Springer 2001, chapter 6 (networks with batch movements) for the discrete-time batch framework; the single-node pgf above is the standard discrete-time M/G/1-type derivation.

Since:
LINE 3.1.0

  • Method Details

    • qsys_geoxgeo1

      public static GeoXGeo1Result qsys_geoxgeo1(double a, double beta, double s)
      Analyzes a Geo^X/Geo/1 queue with geometrically distributed batch sizes under the LAS-DA convention.
      Parameters:
      a - per-slot probability that a batch arrives, 0 < a <= 1
      beta - batch-size geometric parameter; the batch is supported on {1,2,...} with mean 1/beta, 0 < beta <= 1
      s - per-slot service completion probability, 0 < s <= 1
      Returns:
      the stationary metrics

    • qsys_geoxgeo1

      public static GeoXGeo1Result qsys_geoxgeo1(double a, double beta, double s, GeoGeo1Convention convention)
      Analyzes a Geo^X/Geo/1 queue with geometrically distributed batch sizes.

      For a geometric batch on {1,2,...}, E[X] = 1/beta and E[X(X-1)] = 2(1-beta)/beta^2. At beta = 1 the batch is always a single job and the result equals Qsys_geogeo1.

      Parameters:
      a - per-slot probability that a batch arrives
      beta - batch-size geometric parameter
      s - per-slot service completion probability
      convention - observation epoch
      Returns:
      the stationary metrics

    • qsys_geoxgeo1_moments

      public static GeoXGeo1Result qsys_geoxgeo1_moments(double a, double batchMean, double batchSecondFactorial, double s, GeoGeo1Convention convention)
      Analyzes a Geo^X/Geo/1 queue for an arbitrary batch-size law, specified by its first two factorial moments.
      Parameters:
      a - per-slot probability that a batch arrives, 0 < a <= 1
      batchMean - E[X], must be at least 1 since a batch that arrives carries at least one job
      batchSecondFactorial - E[X(X-1)], must be non-negative and at least batchMean^2 - batchMean by Cauchy-Schwarz
      s - per-slot service completion probability, 0 < s <= 1
      convention - observation epoch
      Returns:
      the stationary metrics