Package jline.api.pfqn.sens

Class Pfqn_sens_respt

java.lang.Object
jline.api.pfqn.sens.Pfqn_sens_respt
public final class Pfqn_sens_respt extends Object

  • Method Details

    • pfqn_sens_respt

      public static Ret.pfqnSensRespt pfqn_sens_respt(Matrix S, Matrix V, Matrix N, Matrix Z, Matrix b, int tmax)
      Exact raw moments E[W_(i,l)^t], t = 1..tmax, of the sojourn time of a class-l job at an FCFS b-server center i of a closed product-form queueing network, together with the variance of that sojourn time.

      This is Theorem 4.1 of the reference. Its mechanism is the arrival theorem of Lavenberg-Reiser and Sevcik-Mitrani: a class-l job arriving at center i finds j jobs already there with probability p_i(j, N - 1_l). Conditioning the sojourn time on j and inverting the Laplace transform of the conditional density gives

         E[W_(i,l)^t] = t!/mu^t + sum_{tau=0..t} a_(t,tau)(0) E[Qt_i^tau]
                  - sum_{j=0..b-1} p_i(j,N-1_l) sum_{tau=0..t} a_(t,tau)(0) j^tau

      where mu = 1/S(i) is the rate of each of the b servers, Qt_i is the total queue length at center i at population N - 1_l (so its moments are those of Pfqn_sens_mom evaluated one job down in class l), and the coefficients a_(t,tau)(0) depend only on b and mu, not on the network (Remark 4.3 of the reference). The moments E[Qt_i^tau] up to tau = 3 need the second derivative of the MVA recursion, so this routine carries a second-order forward-mode pass exactly as Pfqn_sens_mom does, but over the b-server recursion (4.1)-(4.2) rather than the single-server one.

      For b = 1 the coefficients a_(t,0)(0) vanish identically and the double-sum correction disappears, so no marginal probabilities are needed (Remark 4.2 of the reference); the routine still evaluates the general expression, which reduces to that case on its own.

      Only FCFS centers are covered. The reference is explicit that the sojourn-time distribution at PS and LCFS centers is in general not known, so no analogue exists there. FCFS in a BCMP network further requires the service time to be exponential and class-independent, which is why this routine takes a per-station service time S(i) and a separate visit-ratio matrix V rather than a demand matrix: the sojourn time is per visit, so the per-visit rate mu = 1/S(i) must be known and cannot be recovered from the demand L(i,l) = S(i)*V(i,l) alone.

      Reference: J. C. Strelen, "Moment Analysis for Closed Queuing Networks and its Linearizer", Performance Evaluation 11:127-142, 1990, Theorem 4.1 with equations (4.1)-(4.5) and Remarks 4.2-4.3.

      Restricted to closed populations. Load-dependent rates are not covered here; the b-server dependence is the only state dependence, and it is carried exactly by (4.1)-(4.2).

      Parameters:
      S - service time at each station (M x 1), common to all classes
      V - visit ratio matrix (M x R); the demand is L(i,r) = S(i)*V(i,r)
      N - population vector (1 x R)
      Z - think time vector (1 x R), null or empty for zeros
      b - number of servers at each station (M x 1), null for ones
      tmax - highest sojourn-time moment to return, 1..3; the coefficients a_(t,tau)(0) are tabulated in the reference up to t = 3
      Returns:
      the base measures together with the exact sojourn-time moments

    • pfqn_sens_respt

      public static Ret.pfqnSensRespt pfqn_sens_respt(Matrix S, Matrix V, Matrix N, Matrix Z, Matrix b)

    • pfqn_sens_respt

      public static Ret.pfqnSensRespt pfqn_sens_respt(Matrix S, Matrix V, Matrix N, Matrix Z)

    • pfqn_sens_respt

      public static Ret.pfqnSensRespt pfqn_sens_respt(Matrix S, Matrix V, Matrix N)