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## Copyright (C) 2012, 2016, 2020 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{V} =} qncsvisits (@var{P})
## @deftypefnx {Function File} {@var{V} =} qncsvisits (@var{P}, @var{r})
##
## Compute the mean number of visits to the service centers of a
## single class, closed network with @math{K} service centers.
##
## @strong{INPUTS}
##
## @table @code
##
## @item @var{P}(i,j)
## probability that a request which completed service at center
## @math{i} is routed to center @math{j} (@math{K \times K} matrix).
## For closed networks it must hold that @code{sum(@var{P},2)==1}. The
## routing graph must be strongly connected, meaning that each node
## must be reachable from every other node.
##
## @item @var{r}
## Index of the reference station, @math{r \in @{1, @dots{}, K@}};
## Default @code{@var{r}=1}. The traffic equations are solved by
## imposing the condition @code{@var{V}(r) = 1}. A request returning to
## the reference station completes its activity cycle.
##
## @end table
##
## @strong{OUTPUTS}
##
## @table @code
##
## @item @var{V}(k)
## average number of visits to service center @math{k}, assuming
## @math{r} as the reference station.
##
## @end table
##
## @end deftypefn
## Author: Moreno Marzolla <moreno.marzolla(at)unibo.it>
## Web: http://www.moreno.marzolla.name/
function V = qncsvisits( P, r )
if ( nargin < 1 || nargin > 2 )
print_usage();
endif
issquare(P) || ...
error("P must be a square matrix");
[res err] = dtmcchkP(P);
(res>0) || ...
error( "invalid transition probability matrix P" );
K = rows(P);
if ( nargin < 2 )
r = 1;
else
isscalar(r) || ...
error("r must be a scalar");
(r>=1 && r<=K) || ...
error("r must be an integer in the range [1, %d]",K);
endif
V = zeros(size(P));
A = P-eye(K);
b = zeros(1,K);
A(:,r) = 0; A(r,r) = 1;
b(r) = 1;
V = b/A;
## Make sure that no negative values appear (sometimes, numerical
## errors produce tiny negative values instead of zeros)
V = max(0,V);
endfunction
%!test
%! P = [-1 0; 0 0];
%! fail( "qncsvisits(P)", "invalid" );
%! P = [1 0; 0.5 0];
%! fail( "qncsvisits(P)", "invalid" );
%! P = [1 2 3; 1 2 3];
%! fail( "qncsvisits(P)", "square" );
%! P = [0 1; 1 0];
%! fail( "qncsvisits(P,0)", "range" );
%! fail( "qncsvisits(P,3)", "range" );
%!test
%!
%! ## Closed, single class network
%!
%! P = [0 0.3 0.7; 1 0 0; 1 0 0];
%! V = qncsvisits(P);
%! assert( V*P,V,1e-5 );
%! assert( V, [1 0.3 0.7], 1e-5 );
%!test
%!
%! ## Test tolerance of the qncsvisits() function.
%! ## This test builds transition probability matrices and tries
%! ## to compute the visit counts on them.
%!
%! for k=[5, 10, 20, 50]
%! P = reshape(1:k^2, k, k);
%! P = P ./ repmat(sum(P,2),1,k);
%! V = qncsvisits(P);
%! assert( V*P, V, 1e-5 );
%! endfor
%!demo
%! P = [0 0.3 0.7; ...
%! 1 0 0 ; ...
%! 1 0 0 ];
%! V = qncsvisits(P)
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