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## Copyright (C) 2008, 2009, 2010, 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} =} ctmctaexps (@var{Q}, @var{t}, @var{p0})
## @deftypefnx {Function File} {@var{M} =} ctmctaexps (@var{Q}, @var{p0})
##
## @cindex Markov chain, continuous time
## @cindex time-alveraged sojourn time, CTMC
## @cindex continuous time Markov chain
## @cindex CTMC
##
## Compute the @emph{time-averaged sojourn time} @code{@var{M}(i)},
## defined as the fraction of the time interval @math{[0,t]} (or until
## absorption) spent in state @math{i}, assuming that the state
## occupancy probabilities at time 0 are @var{p}.
##
## @strong{INPUTS}
##
## @table @code
##
## @item @var{Q}(i,j)
## Infinitesimal generator matrix. @code{@var{Q}(i,j)} is the transition
## rate from state @math{i} to state @math{j},
## @math{1 @leq{} i,j @leq{} N}, @math{i \neq j}. The
## matrix @var{Q} must also satisfy the condition @math{\sum_{j=1}^N Q_{i,j} = 0}
##
## @item @var{t}
## Time. If omitted, the results are computed until absorption.
##
## @item @var{p0}(i)
## initial state occupancy probabilities. @code{@var{p0}(i)} is the
## probability that the system is in state @math{i} at time 0, @math{i
## = 1, @dots{}, N}
##
## @end table
##
## @strong{OUTPUTS}
##
## @table @code
##
## @item @var{M}(i)
## When called with three arguments, @code{@var{M}(i)} is the expected
## fraction of the interval @math{[0,t]} spent in state @math{i}
## assuming that the state occupancy probability at time zero is
## @var{p}. When called with two arguments, @code{@var{M}(i)} is the
## expected fraction of time until absorption spent in state @math{i};
## in this case the mean time to absorption is @code{sum(@var{M})}.
##
## @end table
##
## @seealso{ctmcexps}
##
## @end deftypefn
## Author: Moreno Marzolla <moreno.marzolla(at)unibo.it>
## Web: http://www.moreno.marzolla.name/
function M = ctmctaexps( Q, varargin )
persistent epsilon = 10*eps;
if ( nargin < 2 || nargin > 3 )
print_usage();
endif
L = ctmcexps(Q,varargin{:});
M = L ./ repmat(sum(L,2),1,columns(L));
endfunction
%!test
%! Q = [ 0 0.1 0 0; ...
%! 0.9 0 0.1 0; ...
%! 0 0.9 0 0.1; ...
%! 0 0 0 0 ];
%! Q -= diag( sum(Q,2) );
%! M = ctmctaexps(Q, [1 0 0 0]);
%! assert( sum(M), 1, 10*eps );
%!demo
%! lambda = 0.5;
%! N = 4;
%! birth = lambda*linspace(1,N-1,N-1);
%! death = zeros(1,N-1);
%! Q = diag(birth,1)+diag(death,-1);
%! Q -= diag(sum(Q,2));
%! t = linspace(1e-5,30,100);
%! p = zeros(1,N); p(1)=1;
%! M = zeros(length(t),N);
%! for i=1:length(t)
%! M(i,:) = ctmctaexps(Q,t(i),p);
%! endfor
%! clf;
%! plot(t, M(:,1), ";State 1;", "linewidth", 2, ...
%! t, M(:,2), ";State 2;", "linewidth", 2, ...
%! t, M(:,3), ";State 3;", "linewidth", 2, ...
%! t, M(:,4), ";State 4 (absorbing);", "linewidth", 2 );
%! legend("location","east"); legend("boxoff");
%! xlabel("Time");
%! ylabel("Time-averaged Expected sojourn time");
## This example is from: David I. Heimann, Nitin Mittal, Kishor S. Trivedi,
## "Availability and Reliability Modeling for Computer Systems", sep 1989,
## section 2.5
%!demo
%! sec = 1;
%! min = sec*60;
%! hour = 60*min;
%! day = 24*hour;
%!
%! # state space enumeration {2, RC, RB, 1, 0}
%! a = 1/(10*min); # 1/a = duration of reboot (10 min)
%! b = 1/(30*sec); # 1/b = reconfiguration time (30 sec)
%! g = 1/(5000*hour); # 1/g = processor MTTF (5000 hours)
%! d = 1/(4*hour); # 1/d = processor MTTR (4 hours)
%! c = 0.9; # coverage
%! Q = [ -2*g 2*c*g 2*(1-c)*g 0 0; ...
%! 0 -b 0 b 0; ...
%! 0 0 -a a 0; ...
%! d 0 0 -(g+d) g; ...
%! 0 0 0 d -d];
%! p = ctmc(Q);
%! printf("System availability: %f\n",p(1)+p(4));
%! TT = linspace(0,1*day,101);
%! PP = ctmctaexps(Q,TT,[1 0 0 0 0]);
%! A = At = Abart = zeros(size(TT));
%! A(:) = p(1) + p(4); # steady-state availability
%! for n=1:length(TT)
%! t = TT(n);
%! p = ctmc(Q,t,[1 0 0 0 0]);
%! At(n) = p(1) + p(4); # instantaneous availability
%! Abart(n) = PP(n,1) + PP(n,4); # interval base availability
%! endfor
%! clf;
%! semilogy(TT,A,";Steady-state;", ...
%! TT,At,";Instantaneous;", ...
%! TT,Abart,";Interval base;");
%! ax = axis();
%! ax(3) = 1-1e-5;
%! axis(ax);
%! legend("boxoff");
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