lossn_erlangfp.m

%{ @file lossn_erlangfp.m
 %  @brief Erlang fixed point approximation for loss networks
 %
 %  @author LINE Development Team
%}
 
%{
 % @brief Computes performance metrics for loss networks using Erlang FP
 %
 % @details
 % This function uses the Erlang fixed point approximation to compute
 % performance metrics for loss networks. Calls on route r arrive according
 % to Poisson rate nu_r with unit mean service times. Link capacity
 % requirements are: sum_r A(j,r) n(j,r) <= C(j) for all links j.
 %
 % @par Syntax:
 % @code
 % [QLen, Loss, E, niter] = lossn_erlangfp(nu, A, C)
 % @endcode
 %
 % @par Parameters:
 % <table>
 % <tr><th>Name<th>Description
 % <tr><td>nu<td>Arrival rate of route (class) r (1xR vector)
 % <tr><td>A<td>Capacity requirement of link j for route r (JxR matrix)
 % <tr><td>C<td>Available capacity of link j (Jx1 vector)
 % </table>
 %
 % @par Returns:
 % <table>
 % <tr><th>Name<th>Description
 % <tr><td>QLen<td>Mean queue-length for route r
 % <tr><td>Loss<td>Loss probability for route r
 % <tr><td>E<td>Blocking probability for link j
 % <tr><td>niter<td>Number of iterations required for convergence
 % </table>
%}
function [QLen,Loss,E,niter] = lossn_erlangfp(nu,A,C)
% [QLen,Loss,E] = LOSS_ERLANGFP(rho,A,C)
% Erlang fixed point approximation for loss networks
%
% Calls (i.e., jobs) on route (i.e., class) r arrive according to Poisson
% rate nu_r, r=1..R. Call service times on route r have unit mean.
%
% Link capacity requirements are:
%                        \sum_r A(j,r) n(j,r) <= C(j)
% for all links j=1..J, where n(j,r) counts the calls on route r on link j.
%
% INPUT:
% nu  (1xR): arv. rate of route (class) r = 1..R
% A   (J,R): capacity requirement of link j for route r = 1..R
% C   (J,1): available capacity of link j
%
% OUTPUT:
% Q   (1xR): mean queue-length for route r = 1..R calls
% L   (1xR): loss probability for route r = 1..R calls
% E   (Jx1): blocking probability of for link j = 1..J
%
% NOTE: nu_r may be replaced by a utilization rho_r=nu_r/mu_r, where mu_r
% is the service rate for route r.
 
R = length(nu);
J = length(C);
 
% Erlang fixed point on the link blocking probabilities, driven by the
% generic DA successive-substitution driver
fpopts = struct('iter_max', 10000, 'iter_tol', 1e-8);
fpopts.config.da_norm = @(d) norm(d,1);
fpopts.config.da_nanstop = true; % legacy while-loop exited on NaN measure
[E, niter] = da_fpi(@efp_sweep, 0.5*ones(J,1), fpopts);
 
    function [Enew, Eref] = efp_sweep(E_1, ~)
        Enew = E_1;
        for j=1:J
            rhoj_1 = 0;
            for r=1:R
                if A(j,r)>0
                    termj=nu(r)*A(j,r);
                    for i=1:J
                        if A(i,r)>0
                            termj=termj*(1-E_1(i))^A(i,r);
                        end
                    end
                    rhoj_1 = rhoj_1 + termj;
                end
            end
            rhoj_1 = rhoj_1 / (1-E_1(j));
            Enew(j) = ErlangB(rhoj_1,C(j));
        end
        Eref = E_1;
    end
QLen = nu;
for r=1:R
    for j=1:J
        QLen(r) = QLen(r)*(1-E(j))^A(j,r);
    end
end
QLen = real(QLen);
Loss = 1 - QLen./nu;   % blocking probability = 1 - carried/offered
end
 
function blockProb = ErlangB(nu,C)
% Erlang B formula
% blockProb = E(nu,C) = (nu^C/C!) / (sum_{i=1}^C nu^i/i!)
den = 0;
for i=0:C
    d = i*log(nu)-factln(i);
    den = den + exp(d);
end
blockProb = C*log(nu) -factln(C) -log(den);
blockProb = exp(blockProb);
end