Package jline.api.lossn

Class Lossn_revsens

java.lang.Object
jline.api.lossn.Lossn_revsens
public final class Lossn_revsens extends Object
Derivative of the long-run revenue rate of a loss network with respect to the offered loads, and the implied costs.

  • Method Details

    • lossn_revsens

      public static Lossn_revsens.LossnRevsensResult lossn_revsens(Matrix nu, Matrix A, Matrix C, Matrix w)
      Revenue sensitivities and implied costs by the exact method.
      Parameters:
      nu - offered load of route k (1xR)
      A - capacity requirement of link j for route k (JxR)
      C - capacity of link j (Jx1)
      w - revenue per unit time of a route-k call in progress (1xR)
      Returns:
      dW, c, W, clink and ci

    • lossn_revsens

      public static Lossn_revsens.LossnRevsensResult lossn_revsens(Matrix nu, Matrix A, Matrix C, Matrix w, String method)
      Revenue sensitivities and implied costs by the given method.
      Parameters:
      nu - offered load of route k (1xR)
      A - capacity requirement of link j for route k (JxR)
      C - capacity of link j (Jx1)
      w - revenue per unit time of a route-k call in progress (1xR)
      method - 'exact' (default), 'rec', 'erlangfp' or 'mci'
      Returns:
      dW, c, W, clink and ci

    • lossn_revsens

      public static Lossn_revsens.LossnRevsensResult lossn_revsens(Matrix nuVec, Matrix Amat, Matrix cVec, Matrix wVec, String method, Lossn_mci.Options options)
      Derivative of the long-run revenue rate with respect to offered loads.

      A loss network with offered loads nu (unit mean holding times), requirement matrix A and capacities C earns w_k per unit time for each route-k call in progress, so its revenue rate is W = sum_k w_k QLen_k. This function returns the sensitivity dW/dnu_k and the implied costs: the expected revenue lost over the lifetime of an admitted route-k call through the extra blocking it causes to later calls. They satisfy (Ross 1995, Theorem 5.11) 

            dW/dnu_k = (1 - B_k) (w_k - c_k),   c_k = W(C) - W(C - A e_k),
      so admitting a route-k call raises revenue only if w_k exceeds c_k. The link implied cost (shadow price) of route k on link j is W(C) - W(C - A(j,k) e_j), the revenue lost when link j alone loses the units of one route-k call.

      Supported methods:

      • 'exact' (or 'manjunath', 'ms'), 'rec': Theorem 5.11 with exact normalization constants
      • 'erlangfp': reduced-load approximation of the implied costs: the link costs c(j,k) solve Ross (1995) eq. 5.21, c(j,k) = sum_l D(j,l,k) eta(j,l) [w_l - sum_{i~=j} c(i,l)], with eta the Erlang fixed point reduced loads and D the change of the Kaufman-Roberts blocking of route l on link j when its capacity drops by A(j,k); then dW/dnu_k = prod_j (1 - E(j,k)) (w_k - sum_j c(j,k))
      • 'mci': Monte Carlo summation: dW/dnu_l = Cov(X_l, r(X))/nu_l from the importance samples, with delta-method confidence intervals

      References: K. W. Ross, "Multiservice Loss Models for Broadband Telecommunication Networks", Springer, 1995, Sec. 5.7 and 6.3. F. P. Kelly, "Routing in circuit-switched networks: optimization, shadow prices and decentralization", Adv. Appl. Probab. 20, 1988.

      Parameters:
      nuVec - offered load of route k (1xR)
      Amat - capacity requirement of link j for route k (JxR)
      cVec - capacity of link j (Jx1)
      wVec - revenue per unit time of a route-k call in progress (1xR)
      method - 'exact' (default when null or empty), 'rec', 'erlangfp' or 'mci'
      options - Lossn_mci options (samples, seed, alpha, gamma), used by 'mci'; may be null
      Returns:
      dW, c, W, clink and ci