solver_mna_closed.m

function [Q,U,R,T,C,X,lG,totiter] = solver_mna_closed(sn, options)
 
config = options.config;
config.space_max = 16;
 
K = sn.nclasses;
rt = sn.rt;
S = 1./sn.rates;
scv = sn.scv; scv(isnan(scv))=0;
N = sn.njobs';
PH = sn.proc;
I = sn.nnodes;
M = sn.nstations;
C = sn.nchains;
V = cellsum(sn.visits);
Q = zeros(M,K);
 
 
U = zeros(M,K);
R = zeros(M,K);
T = zeros(M,K);
X = zeros(1,K);
 
lambda = zeros(1,C);
 
 
pie = {};
D0 = {};
% get service process
for ist=1:M
    switch sn.sched(ist)
        case {SchedStrategy.FCFS, SchedStrategy.INF,SchedStrategy.PS}
            for k=1:K
                pie{ist}{k} = map_pie(PH{ist}{k});
                D0{ist,k} = PH{ist}{k}{1};
                if any(isnan(D0{ist,k}))
                    D0{ist,k} = -GlobalConstants.Immediate;
                    pie{ist}{k} = 1;
                    PH{ist}{k} = map_exponential(GlobalConstants.Immediate);
                end
            end
    end
end
 
QNc = sn.njobs;
lambda_lb = zeros(1,K);
lambda_ub = zeros(1,K);
for k=1:K
    lambda_ub(k) = min(sn.rates(find(sn.nservers<Inf),k));
end
 
QN = zeros(1,K);
it_out = 0;
 
%% main iteration
while max(abs(QN-QNc))>options.iter_tol && it_out < options.iter_max
    it_out = it_out+1;
 
    if it_out ~=1
        for k=1:K
            if QN(k) < QNc(k)
                lambda_lb(k) = lambda(k);
            else
                lambda_ub(k) = lambda(k);
            end
            lambda(k) = (lambda_ub(k) + lambda_lb(k)) / 2;
        end
    else
        lambda = lambda_ub;
    end
    it =0;
    Q = zeros(M,K);
 
 
    U = zeros(M,K);
    R = zeros(M,K);
    T = zeros(M,K);
    X = zeros(1,K);
 
    a1 = zeros(M,K);
    a2 = zeros(M,K);
    a1_1 = a1+Inf;
    a2_1 = a2+Inf;
 
    d2 = zeros(M,1);
    f2 = zeros(M*K,M*K);
    for ist=1:M
        for jst=1:M
            if sn.nodetype(sn.stationToNode(jst)) ~= NodeType.Source
                for r=1:K
                    for s=1:K
                        if rt((ist-1)*K+r, (jst-1)*K+s)>0
                            f2((ist-1)*K+r, (jst-1)*K+s) = 1; % C^2ij,r
                        end
                    end
                end
            end
        end
    end
 
    while (max(max(abs(a1_1-a1))) > options.iter_tol || max(max(abs(a2_1-a2))) > options.iter_tol )&& it <= options.iter_max %#ok<max>
        it = it + 1;
        for c=1:C
            inchain = sn.inchain{c};
            Q(:,c) = sn.njobs(c) .* Q(:,c) / sum(Q(:,c));
        end
        Q_1 = Q;
 
        for k=1:K
            if sn.isslc(k)
                Q(:,k) = 0;
                Q(sn.refstat(k),k) = sn.njobs(k);
            end
        end
 
        a1_1 = a1;
        a2_1 = a2;
 
 
        % update throughputs at all stations
        if it==1
            for c=1:C
                inchain = sn.inchain{c};
                for m=1:M
                    T(m,inchain) = V(m,inchain) .* lambda(c);
                end
            end
        end
 
        % superposition
        for ist=1:M
            a1(ist,:) = 0;
            a2(ist,:) = 0;
            lambda_i = sum(T(ist,:));
            for jst=1:M
                for r=1:K
                    for s=1:K
                        a1(ist,r) = a1(ist,r) + T(jst,s)*rt((jst-1)*K+s, (ist-1)*K+r);
                        a2(ist,r) = a2(ist,r) + (1/lambda_i) * f2((jst-1)*K+s, (ist-1)*K+r)*T(jst,s)*rt((jst-1)*K+s, (ist-1)*K+r);
                    end
                end
            end
        end
 
        % update flow trhough queueing station
        for ind=1:I
            if sn.isstation(ind)
                ist = sn.nodeToStation(ind);
                switch sn.nodetype(ind)
                    case NodeType.Join
                        % no-op
 
                    otherwise
                        switch sn.sched(ist)
                            case SchedStrategy.INF
                                for r=1:K
                                    for s=1:K
                                        d2(ist,s) = a2(ist,s);
                                    end
                                end
                                for c=1:C
                                    inchain = sn.inchain{c};
                                    for k=inchain
                                        T(ist,k) = a1(ist,k);
                                        U(ist,k) = S(ist,k)*T(ist,k);
                                        Q(ist,k) = T(ist,k).*S(ist,k)*V(ist,k);
                                        R(ist,k) = Q(ist,k)/T(ist,k);
                                    end
                                end
                            case SchedStrategy.PS
                                for c=1:C
                                    inchain = sn.inchain{c};
                                    for k=inchain
                                        T(ist,k) = lambda(c)*V(ist,k);
                                        U(ist,k) = S(ist,k)*T(ist,k);
                                    end
                                    %Nc = sum(sn.njobs(inchain)); % closed population
                                    Uden = min([1-GlobalConstants.FineTol,sum(U(ist,:))]);
                                    for k=inchain
                                        Q(ist,k) = (U(ist,k)-U(ist,k)^(sum(sn.njobs(inchain))+1))/(1-Uden); % geometric bound type approximation
                                        %Q(ist,k) = UN(ist,k)/(1-Uden);
                                        R(ist,k) = Q(ist,k)/T(ist,k);
                                    end
                                end
                            case {SchedStrategy.FCFS}
                                mu_ist = sn.rates(ist,1:K);
                                mu_ist(isnan(mu_ist))=0;
                                rho_ist_class = a1(ist,1:K)./(GlobalConstants.FineTol+sn.rates(ist,1:K));
                                rho_ist_class(isnan(rho_ist_class))=0;
                                lambda_ist = sum(a1(ist,:));
                                mi = sn.nservers(ist);
                                rho_ist = sum(rho_ist_class) / mi;
                                if rho_ist < 1-options.tol
 
                                    for k=1:K
                                        mubar(ist) = lambda_ist ./ rho_ist;
                                        c2(ist) = -1;
                                        for r=1:K
                                            if mu_ist(r)>0
                                                c2(ist) = c2(ist) + a1(ist,r)/lambda_ist * (mubar(ist)/mi/mu_ist(r))^2 * (scv(ist,r)+1 );
                                            end
                                        end
                                    end
 
                                    d2(ist) = 1 + rho_ist^2*(c2(ist)-1)/sqrt(mi) + (1 - rho_ist^2) *(sum(a2(ist,:))-1);
                                else
                                    for k=1:K
                                        Q(ist,k) = sn.njobs(k);
                                    end
                                    d2(ist) = 1;
                                end
                                for k=1:K
                                    T(ist,k) = a1(ist,k);
                                    U(ist,k) = T(ist,k) * S(ist,k) /sn.nservers(ist);
                                    R(ist,k) = Q(ist,k) ./ T(ist,k);
                                end
 
                        end
                end
            else % not a station
                switch sn.nodetype(ind)
                    case NodeType.Fork
                        line_error(mfilename,'Fork nodes not supported yet by QNA solver.');
                end
            end
        end
 
 
        % splitting - update flow scvs
        for ist=1:M
            for jst=1:M
                if sn.nodetype(sn.stationToNode(jst)) ~= NodeType.Source
                    for r=1:K
                        for s=1:K
                            if rt((ist-1)*K+r, (jst-1)*K+s)>0
                                f2((ist-1)*K+r, (jst-1)*K+s) = 1 + rt((ist-1)*K+r, (jst-1)*K+s) * (d2(ist)-1);
                            end
                        end
                    end
                end
            end
        end
    end
 
    for ind=1:I
        if sn.isstation(ind)
            ist = sn.nodeToStation(ind);
            switch sn.sched(ist)
                case {SchedStrategy.FCFS}
                    mu_ist = sn.rates(ist,1:K);
                    mu_ist(isnan(mu_ist))=0;
                    rho_ist_class = a1(ist,1:K)./(GlobalConstants.FineTol+sn.rates(ist,1:K));
                    rho_ist_class(isnan(rho_ist_class))=0;
                    lambda_ist = sum(a1(ist,:));
                    mi = sn.nservers(ist);
                    rho_ist = sum(rho_ist_class) / mi;
                    if rho_ist < 1-options.tol
                        for k=1:K
 
                            if a1(ist,k)==0
                                arri_class = map_exponential(Inf);
                            else
                                arri_class = APH.fitMeanAndSCV(1/a1(ist,k),a2(ist,k)).getProcess; % MMAP repres of arrival process for class k at node ist
                                %arri_class = map_exponential(1/a1(ist,k));
                                arri_class = {arri_class{1},arri_class{2},arri_class{2}};
                            end
                            if k==1
                                arri_node = arri_class;
                            else
                                %arri_node = mmap_super(arri_node,arri_class,  'default');
 
                                arri_node = mmap_super_safe({arri_node,arri_class}, config.space_max, 'default'); % combine arrival process from different class
                            end
 
                        end
                        Qret = cell(1,K);
                        maxLevel = sum(N(isfinite(N)))+1;
                        D = {arri_node{[1,3:end]}};
                        pdistr = cell(1,K);
                        if map_lambda(D)< GlobalConstants.FineTol
                            for k=1:K
                                pdistr = [1-GlobalConstants.FineTol, GlobalConstants.FineTol];
                                Qret{k} = GlobalConstants.FineTol / sn.rates(ist);
                            end
                        else
                            [pdistr] = MMAPPH1FCFS(D, {pie{ist}{:}}, {D0{ist,:}}, 'ncDistr', maxLevel);
                            % rough approximation
                            for k=1:K
                                pdistr_k = abs(pdistr(1:(N(k)+1)));
                                pdistr_k(end) = abs(1-sum(pdistr(1:end-1)));
                                pdistr_k = pdistr_k / sum(pdistr_k(1:(N(k)+1)));
                                Qret{k} = max(0,min(N(k),(0:N(k))*pdistr_k(1:(N(k)+1))'));
                            end
 
                        end
                        Q(ist,:) = cell2mat(Qret);
                    else
                        for k=1:K
                            Q(ist,k) = sn.njobs(k);
                        end
                    end
 
                    for k=1:K
                        R(ist,k) = Q(ist,k) ./ T(ist,k);
                    end
            end
        end
    end
    QN = sum(Q,1);
end
 
for k=1:K
    if sn.isslc(k)
        Q(:,k) = 0;
        ist = sn.refstat(k);
        Q(ist,k) = sn.njobs(k);
        T(ist,k) = sn.njobs(k)*sn.rates(ist,k);
        R(ist,k) = Q(ist,k) ./ T(ist,k);
        U(ist,k) = S(ist,k)*T(ist,k);
    end
end
 
for c=1:C
    inchain = sn.inchain{c};
    if isfinite(sn.njobs(c))
        Q(:,c) = sn.njobs(c) .* Q(:,c) / sum(Q(:,c));
    end
end
for ist=1:sn.nstations
    switch sn.sched(ist)
        case SchedStrategy.INF
            U(ist,:) = Q(ist,:);
    end
end
 
%it_out
C = sum(R,1);
Q = abs(Q);
Q(isnan(Q))=0;
U(isnan(U))=0;
R(isnan(R))=0;
C(isnan(C))=0;
X(isnan(X))=0;
lG = 0;
totiter = it_out;
end