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## Copyright (C) 2011, 2012, 2016, 2018 Moreno Marzolla
##
## This file is part of the queueing toolbox.
##
## The queueing toolbox is free software: you can redistribute it and/or
## modify it under the terms of the GNU General Public License as
## published by the Free Software Foundation, either version 3 of the
## License, or (at your option) any later version.
##
## The queueing toolbox is distributed in the hope that it will be
## useful, but WITHOUT ANY WARRANTY; without even the implied warranty
## of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
## General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with the queueing toolbox. If not, see <http://www.gnu.org/licenses/>.
## -*- texinfo -*-
##
## @deftypefn {Function File} {@var{M} =} ctmcfpt (@var{Q})
## @deftypefnx {Function File} {@var{m} =} ctmcfpt (@var{Q}, @var{i}, @var{j})
##
## @cindex first passage times, CTMC
## @cindex CTMC
## @cindex continuous time Markov chain
## @cindex Markov chain, continuous time
##
## Compute mean first passage times for an irreducible continuous-time
## Markov chain.
##
## @strong{INPUTS}
##
## @table @code
##
## @item @var{Q}(i,j)
## Infinitesimal generator matrix. @var{Q} is a @math{N \times N}
## square matrix where @code{@var{Q}(i,j)} is the transition rate from
## state @math{i} to state @math{j}, for @math{1 @leq{} i, j @leq{} N},
## @math{i \neq j}. Transition rates must be nonnegative, and
## @math{\sum_{j=1}^N Q_{i,j} = 0}
##
## @item @var{i}
## Initial state.
##
## @item @var{j}
## Destination state.
##
## @end table
##
## @strong{OUTPUTS}
##
## @table @code
##
## @item @var{M}(i,j)
## average time before state
## @var{j} is visited for the first time, starting from state @var{i}.
## We let @code{@var{M}(i,i) = 0}.
##
## @item m
## @var{m} is the average time before state @var{j} is visited for the first
## time, starting from state @var{i}.
##
## @end table
##
## @seealso{ctmcmtta}
##
## @end deftypefn
## Author: Moreno Marzolla <moreno.marzolla(at)unibo.it>
## Web: http://www.moreno.marzolla.name/
function result = ctmcfpt( Q, i, j )
persistent epsilon = 10*eps;
if ( nargin != 1 && nargin != 3 )
print_usage();
endif
[N err] = ctmcchkQ(Q);
(N>0) || ...
error(err);
if ( nargin == 1 )
result = zeros(N,N);
for j=1:N
QQ = Q;
QQ(j,:) = 0; # make state j absorbing
for i=1:N
p0 = zeros(1,N); p0(i) = 1;
result(i,j) = ctmcmtta(QQ,p0);
endfor
endfor
else
(isscalar(i) && i>=1 && j<=N) || error("i must be an integer in the range 1..%d", N);
(isvector(j) && all(j>=1) && all(j<=N)) || error("j must be an integer or vector with elements in 1..%d", N);
j = j(:)'; # make j a row vector
Q(j,:) = 0; # make state(s) j absorbing
p0 = zeros(1,N); p0(i) = 1;
result = ctmcmtta(Q,p0);
endif
endfunction
%!demo
%! Q = [ -1.0 0.9 0.1; ...
%! 0.1 -1.0 0.9; ...
%! 0.9 0.1 -1.0 ];
%! M = ctmcfpt(Q)
%! m = ctmcfpt(Q,1,3)
%!test
%! N = 10;
%! Q = reshape(1:N^2,N,N);
%! Q(1:N+1:end) = 0;
%! Q -= diag(sum(Q,2));
%! M = ctmcfpt(Q);
%! assert( all(diag(M) < 10*eps) );
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