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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{U}, @var{R}, @var{Q}, @var{X}] =} qsammm (@var{lambda}, @var{mu})
##
## @cindex asymmetric @math{M/M/m} system
##
## Compute @emph{approximate} utilization, response time, average
## number of requests in service and throughput for an asymmetric
## @math{M/M/m} queue. In this type of system there are @math{m} different
## servers connected to a single queue. Each server has its own
## (possibly different) service rate. If there is more than one server
## available, requests are routed to a randomly-chosen one.
##
## @strong{INPUTS}
##
## @table @code
##
## @item @var{lambda}
## Arrival rate (@code{@var{lambda}>0})
##
## @item @var{mu}
## @code{@var{mu}(i)} is the service rate of server
## @math{i}, @math{1 @leq{} i @leq{} m}.
## The system must be ergodic (@code{@var{lambda} < sum(@var{mu})}).
##
## @end table
##
## @strong{OUTPUTS}
##
## @table @code
##
## @item @var{U}
## Approximate service center utilization,
## @math{U = \lambda / ( \sum_{i=1}^m \mu_i )}.
##
## @item @var{R}
## Approximate service center response time
##
## @item @var{Q}
## Approximate number of requests in the system
##
## @item @var{X}
## Approximate system throughput. If the system is ergodic,
## @code{@var{X} = @var{lambda}}
##
## @end table
##
## @strong{REFERENCES}
##
## @itemize
## @item
## G. Bolch, S. Greiner, H. de Meer and K. Trivedi, @cite{Queueing Networks
## and Markov Chains: Modeling and Performance Evaluation with Computer
## Science Applications}, Wiley, 1998
## @end itemize
##
## @seealso{qsmmm}
##
## @end deftypefn
## Author: Moreno Marzolla <moreno.marzolla(at)unibo.it>
## Web: http://www.moreno.marzolla.name/
function [U R Q X p0 pm] = qsammm( lambda, mu )
if ( nargin != 2 )
print_usage();
endif
isscalar(lambda) || ...
error( "lambda must be scalar" );
isvector(mu) || ...
error( "mu must be a vector" );
m = length(mu); # number of servers
all( lambda < sum(mu) ) || ...
error( "Processing capacity exceeded" );
X = lambda;
U = rho = lambda / sum(mu);
Q = p0 = 0;
#{
k=[0:m-1];
p0 = 1 / ( ...
sum( (m*rho).^k ./ factorial(k)) + ...
(m*rho)^m / (factorial(m)*(1-rho)) ...
);
pm = (m*rho)^m/(factorial(m)*(1-rho))*p0;
#}
p0 = 1 / ( ...
sumexpn( m*rho, m-1 ) + ...
expn(m*rho, m) / (1-rho) ...
);
pm = expn(m*rho, m)*p0/(1-rho);
Q = m*rho+rho / (1-rho) * pm;
R = Q / X;
endfunction
%!test
%! [U R Q X] = qsammm( 73,[10,15,20,20,25] );
%! assert( U, 0.81, 1e-2 );
%! assert( Q, 6.5278, 1e-4 );
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