Package jline.api.mam

Class Dmap_batch

java.lang.Object
jline.api.mam.Dmap_batch
public final class Dmap_batch extends Object
Discrete-time batch arrival streams: event rate, superposition, Bernoulli splitting and order reduction.

A batch stream is a MatrixCell {A_0, A_1, ...} where A_k carries the slots delivering k events. A plain DMAP is the two-entry case. These four operations are what a slotted decomposition needs between stations.

MATLAB twins: dmap_lambda.m, dmap_super.m, dmap_thin.m, dmap_compress.m, dmap_compress_batch.m

  • Method Details

    • dmap_lambda

      public static double dmap_lambda(MatrixCell A)
      Mean number of EVENTS per slot, pi * sum_k k*A_k * e. A slot carrying a batch of two counts twice, which is what Little's law consumes downstream.

    • dmap_super

      public static MatrixCell dmap_super(MatrixCell A, MatrixCell B)
      Superposition, E_k = sum_{i+j=k} kron(A_i, B_j).

      NOT closed on DMAPs: two slotted streams fire in the same slot with positive probability, so the merged stream carries batches. Folding E_2 into E_1 would conserve neither the arrival rate nor the slot in which the work appears, so the batch dimension is kept and the downstream station is solved as an M/G/1-type chain instead of a QBD.

    • dmap_thin

      public static MatrixCell dmap_thin(MatrixCell A, double p)
      Bernoulli thinning, B_k = sum_{n>=k} C(n,k) p^k (1-p)^(n-k) A_n. The phase process is untouched, so this is exact for PROB/RAND routing.

    • dmap_compress

      public static MatrixCell dmap_compress(MatrixCell DMAP, int maxOrder)
      Order reduction of a DMAP by matching three interevent moments through BuTools DPH2From3Moments, leaving it untouched while it stays within maxOrder. Correlation is NOT preserved, mirroring the continuous-time 'mixture.order1' compression, and that is where the multi-station discrete-time path becomes approximate. Outside the DPH(2) region the fallback keeps the exact mean with a Geometric, so the arrival rate is conserved in every branch.

    • dmap_compress_batch

      public static MatrixCell dmap_compress_batch(MatrixCell B, int maxOrder)
      Order reduction of a batch stream. Keeps the two features the downstream M/G/1-type solve consumes: the law of the time between NONEMPTY slots, matched to three moments, and the stationary batch-size distribution conditional on a nonempty slot, kept exactly. The event rate of the reduced stream equals the original one by construction.