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## Copyright (C) 2012, 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{P} =} dtmcbd (@var{b}, @var{d})
##
## @cindex Markov chain, discrete time
## @cindex DTMC
## @cindex discrete time Markov chain
## @cindex birth-death process, DTMC
##
## Returns the transition probability matrix @math{P} for a discrete
## birth-death process over state space @math{@{1, @dots{}, N@}}.
## For each @math{i=1, @dots{}, (N-1)},
## @code{@var{b}(i)} is the transition probability from state
## @math{i} to @math{(i+1)}, and @code{@var{d}(i)} is the transition
## probability from state @math{(i+1)} to @math{i}.
##
## Matrix @math{\bf P} is defined as:
##
## @tex
## $$ \pmatrix{ (1-\lambda_1) & \lambda_1 & & & & \cr
## \mu_1 & (1 - \mu_1 - \lambda_2) & \lambda_2 & & \cr
## & \mu_2 & (1 - \mu_2 - \lambda_3) & \lambda_3 & & \cr
## \cr
## & & \ddots & \ddots & \ddots & & \cr
## \cr
## & & & \mu_{N-2} & (1 - \mu_{N-2}-\lambda_{N-1}) & \lambda_{N-1} \cr
## & & & & \mu_{N-1} & (1-\mu_{N-1}) }
## $$
## @end tex
## @ifnottex
## @example
## @group
## / \
## | 1-b(1) b(1) |
## | d(1) (1-d(1)-b(2)) b(2) |
## | d(2) (1-d(2)-b(3)) b(3) |
## | |
## | ... ... ... |
## | |
## | d(N-2) (1-d(N-2)-b(N-1)) b(N-1) |
## | d(N-1) 1-d(N-1) |
## \ /
## @end group
## @end example
## @end ifnottex
##
## @noindent where @math{\lambda_i} and @math{\mu_i} are the birth and
## death probabilities, respectively.
##
## @seealso{ctmcbd}
##
## @end deftypefn
## Author: Moreno Marzolla <moreno.marzolla(at)unibo.it>
## Web: http://www.moreno.marzolla.name/
function P = dtmcbd( b, d )
if ( nargin != 2 )
print_usage();
endif
( isvector( b ) && isvector( d ) ) || ...
error( "birht and death must be vectors" );
b = b(:); # make b a column vector
d = d(:); # make d a column vector
size_equal( b, d ) || ...
error( "birth and death vectors must have the same length" );
all( b >= 0 ) || ...
error( "birth probabilities must be >= 0" );
all( d >= 0 ) || ...
error( "death probabilities must be >= 0" );
all( ([b; 0] + [0; d]) <= 1 ) || ...
error( "d(i)+b(i+1) must be <= 1");
P = diag( b, 1 ) + diag( d, -1 );
P += diag( 1-sum(P,2) );
endfunction
%!test
%! birth = [.5 .5 .3];
%! death = [.6 .2 .3];
%! fail("dtmcbd(birth,death)","must be");
%!demo
%! birth = [ .2 .3 .4 ];
%! death = [ .1 .2 .3 ];
%! P = dtmcbd( birth, death );
%! disp(P)
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