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## Copyright (C) 2009 Dmitry Kolesnikov
## Copyright (C) 2012, 2016, 2022 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}, @var{p0}] =} qsmg1 (@var{lambda}, @var{xavg}, @var{x2nd})
##
## @cindex @math{M/G/1} system
##
## Compute utilization, response time, average number of requests and
## throughput for a @math{M/G/1} system. The service time distribution
## is described by its mean @var{xavg}, and by its second moment
## @var{x2nd}. The computations are based on results from L. Kleinrock,
## @cite{Queuing Systems}, Wiley, Vol 2, and Pollaczek-Khinchine formula.
##
## @strong{INPUTS}
##
## @table @code
##
## @item @var{lambda}
## Arrival rate
##
## @item @var{xavg}
## Average service time
##
## @item @var{x2nd}
## Second moment of service time distribution
##
## @end table
##
## @strong{OUTPUTS}
##
## @table @code
##
## @item @var{U}
## Service center utilization
##
## @item @var{R}
## Service center response time
##
## @item @var{Q}
## Average number of requests in the system
##
## @item @var{X}
## Service center throughput
##
## @item @var{p0}
## Probability that there is not any request at system
##
## @end table
##
## @var{lambda}, @var{xavg}, @var{t2nd} can be vectors of the
## same size. In this case, the results will be vectors as well.
##
## @seealso{qsmh1}
##
## @end deftypefn
## Author: Dmitry Kolesnikov
function [U R Q X p0] = qsmg1(lambda, xavg, x2nd)
if ( nargin != 3 )
print_usage();
endif
## bring the parameters to a common size
[ err lambda xavg x2nd ] = common_size( lambda, xavg, x2nd );
if ( err )
error( "parameters are of incompatible size" );
endif
mu = 1 ./ xavg;
rho = lambda ./ mu;
#coefficient of variation
Cx = (x2nd - xavg .* xavg) ./ (xavg .* xavg);
#PK mean formula(s)
Q = rho + rho .* rho .* (1 + Cx) ./ (2 .* (1 - rho));
R = xavg + xavg .* rho .* (1 + Cx) ./ (2 .* (1 - rho));
p0 = exp(-rho);
#General Results
#utilization
U = rho;
X = lambda;
endfunction
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