Package jline.api.qsys

Class Qsys_mmapgk1_lcfspr

java.lang.Object
jline.api.qsys.Qsys_mmapgk1_lcfspr
public final class Qsys_mmapgk1_lcfspr extends Object
The MMAP[K]/G[K]/1 queue with preemptive-resume last-come-first-served service: K customer types with class-dependent GENERAL service, fed by a marked Markovian arrival process. THE METHOD, Takine (2002). With C = D0, D_k = D^(k) and F_k the service law of type k, let Q be the minimal solution of Q = C + sum_k D_k F_k(Q), F_k(Q) = int_0^inf exp(Qx) dF_k(x), the generator of the arrival phase with the busy periods excised (the same matrix as He's U in Qsys_mmapgk1), and kappa its stationary vector. With beta_k = D_k e E[S_k], Takine's Theorem 1 gives the rate matrices R_k = (beta_k kappa + D_k (I - F_k(Q))) (e kappa - Q)^-1, R = sum_k R_k, and the joint distribution of the per-class numbers in system has the matrix product form y(n) = sum_k y(n - e_k) R_k, y(0) = (1 - rho) kappa. From it: P(N = n) = (1 - rho) kappa R^n e, P(Y_k = m) = (1 - rho) kappa (A_k R_k)^m A_k e, A_k = (I - R + R_k)^-1, E[Y_k] = pi R_k (I - R)^-1 e, with pi the stationary vector of C + D. A type-k sojourn under LCFS-PR is the sub-busy period its own service generates, so its transform is E[exp(-s W_k)] = pi D_k F_k(Q*(s)) e / lambda_k, Q*(s) = C - sI + sum_j D_j F_j(Q*(s)), evaluated at complex s for the CDF by Euler inversion. A complex matrix A + iB is carried through the real matrix transform by its real embedding [A -B; B A], which the transform of a real law maps to [ReF -ImF; ImF ReF], since g(conj z) = conj g(z) for such a law. Reference: T. Takine, "Matrix product-form solution for an LCFS-PR single-server queue with multiple arrival streams governed by a Markov chain", Queueing Systems 42:131-151, 2002.

  • Method Details

    • qsys_mmapgk1_lcfspr

      public static QsysMmapGk1LcfsprResult qsys_mmapgk1_lcfspr(MatrixCell MMAP, List<?> svc)

    • qsys_mmapgk1_lcfspr

      public static QsysMmapGk1LcfsprResult qsys_mmapgk1_lcfspr(MatrixCell MMAP, List<?> svc, int nMax, Matrix wPoints, int quadNodes, double tol)
      Parameters:
      MMAP - LINE convention {D0, D1, D^(1), ..., D^(K)} with D1 = sum_k D^(k)
      svc - K service laws, each a Distribution, a MatrixCell {D0s, D1s} or a Qsys_mmapgk1.LstLaw
      nMax - if nonnegative, also return P(N = n) and P(Y_k = m) on 0..nMax
      wPoints - times at which to evaluate the per-type sojourn time CDF, or null
      quadNodes - quadrature order for a law with no closed-form matrix transform
      tol - fixed point tolerance on Q and Q*(s)
      Returns:
      the per-type solution