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## Copyright (C) 2008, 2009, 2010, 2011, 2012, 2016, 2018, 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} {pop_mix =} qncmpopmix (@var{k}, @var{N})
##
## @cindex population mix
## @cindex closed network, multiple classes
##
## Return the set of population mixes for a closed multiclass queueing
## network with exactly @var{k} customers. Specifically, given a
## closed multiclass QN with @math{C} customer classes, where there
## are @code{@var{N}(c)} class @math{c} requests, @math{c = 1, @dots{}, C}
## a @math{k}-mix @math{M} is a vector of length @math{C} with the following
## properties:
##
## @itemize
## @item @math{0 @leq{} M_c @leq{} @var{N}(c)} for all @math{c = 1, @dots{}, C};
## @item @math{\sum_{c=1}^C M_c = k}
## @end itemize
##
## In other words, a @math{k}-mix is an allocation of @math{k}
## requests to @math{C} classes such that the number of requests
## assigned to class @math{c} does not exceed the maximum value
## @code{@var{N}(c)}.
##
## @var{pop_mix} is a matrix with @math{C} columns, such
## that each row represents a valid mix.
##
## @strong{INPUTS}
##
## @table @code
##
## @item @var{k}
## Size of the requested mix (scalar, @code{@var{k} @geq{} 0}).
##
## @item @var{N}(c)
## number of class @math{c} requests (@code{@var{k} @leq{} sum(@var{N})}).
##
## @end table
##
## @strong{OUTPUTS}
##
## @table @code
##
## @item @var{pop_mix}(i,c)
## number of class @math{c} requests in the @math{i}-th population
## mix. The number of mixes is @code{rows(@var{pop_mix})}.
##
## @end table
##
## If you are interested in the number of @math{k}-mixes only, you can
## use the funcion @code{qnmvapop}.
##
## @strong{REFERENCES}
##
## @itemize
## @item
## Herb Schwetman, @cite{Implementing the Mean Value Algorithm for the
## Solution of Queueing Network Models}, Technical Report
## @uref{http://docs.lib.purdue.edu/cstech/286/, 80-355}, Department of Computer
## Sciences, Purdue University, revised February 15, 1982.
## @end itemize
##
## The slightly different problem of enumerating all tuples @math{k_1,
## @dots{}, k_N} such that @math{\sum_i k_i = k} and @math{k_i
## @geq{} 0}, for a given @math{k @geq{} 0} has been described in
## S. Santini, @cite{Computing the Indices for a Complex Summation},
## unpublished report, available at
## @url{http://arantxa.ii.uam.es/~ssantini/writing/notes/s668_summation.pdf}
##
## @seealso{qncmnpop}
##
## @end deftypefn
## Author: Moreno Marzolla <moreno.marzolla(at)unibo.it>
## Web: http://www.moreno.marzolla.name/
function pop_mix = qncmpopmix( k, population )
if ( nargin != 2 )
print_usage();
endif
isvector( population ) && all( population>=0 ) || ...
error( "N must be an array >=0" );
R = length(population); # number of classes
( isscalar(k) && k >= 0 && k <= sum(population) ) || ...
error( "valid range for k is [0, %d]", sum(population));
N = zeros(1, R);
const = min(k, population);
mp = 0;
pop_mix = []; # Init result
while ( N(R) <= const(R) )
x=k-mp;
## Fill the current configuration
i=1;
while ( x>0 && i<=R )
N(i) = min(x,const(i));
x = x-N(i);
mp = mp+N(i);
i = i+1;
endwhile
## here the configuration is filled. add it to the set of mixes
assert( sum(N), k );
pop_mix = [pop_mix; N];
## advance to the next feasible configuration
i = 1;
sw = true;
while sw
if ( ( mp==k || N(i)==const(i)) && ( i<R ) )
mp = mp-N(i);
N(i) = 0;
i=i+1;
else
N(i)=N(i)+1;
mp=mp+1;
sw = false;
endif
endwhile
endwhile
endfunction
%!demo
%! N = [2 3];
%! mix = qncmpopmix(3, N)
%!test
%! N = [-1 2 2];
%! fail( "qncmpopmix(1, N)" );
%!test
%! N = [2 2 2];
%! fail( "qncmpopmix(7, N)" );
%!test
%! N = [2 3 4];
%! f = qncmpopmix(0, N );
%! assert( f, [0 0 0] );
%! f = qncmpopmix( 1, N );
%! assert( f, [1 0 0; 0 1 0; 0 0 1] );
%! f = qncmpopmix( 2, N );
%! assert( f, [2 0 0; 1 1 0; 0 2 0; 1 0 1; 0 1 1; 0 0 2] );
%! f = qncmpopmix( 3, N );
%! assert( f, [2 1 0; 1 2 0; 0 3 0; 2 0 1; 1 1 1; 0 2 1; 1 0 2; 0 1 2; 0 0 3] );
%!test
%! N = [2 1];
%! f = qncmpopmix( 1, N );
%! assert( f, [1 0; 0 1] );
%! f = qncmpopmix( 2, N );
%! assert( f, [2 0; 1 1] );
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