1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
|
## 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{Xl}, @var{Xu}, @var{Rl}, @var{Ru}] =} qncsbsb (@var{N}, @var{D})
## @deftypefnx {Function File} {[@var{Xl}, @var{Xu}, @var{Rl}, @var{Ru}] =} qncsbsb (@var{N}, @var{S}, @var{V})
## @deftypefnx {Function File} {[@var{Xl}, @var{Xu}, @var{Rl}, @var{Ru}] =} qncsbsb (@var{N}, @var{S}, @var{V}, @var{m})
## @deftypefnx {Function File} {[@var{Xl}, @var{Xu}, @var{Rl}, @var{Ru}] =} qncsbsb (@var{N}, @var{S}, @var{V}, @var{m}, @var{Z})
##
## @cindex bounds, balanced system
## @cindex closed network, single class
## @cindex balanced system bounds
##
## Compute Balanced System Bounds on system throughput and response time for closed, single-class networks with @math{K} service centers.
##
## @strong{INPUTS}
##
## @table @code
##
## @item @var{N}
## number of requests in the system (scalar, @code{@var{N} @geq{} 0}).
##
## @item @var{D}(k)
## service demand at center @math{k} (@code{@var{D}(k) @geq{} 0}).
##
## @item @var{S}(k)
## mean service time at center @math{k} (@code{@var{S}(k) @geq{} 0}).
##
## @item @var{V}(k)
## average number of visits to center @math{k} (@code{@var{V}(k)
## @geq{} 0}). Default is 1.
##
## @item @var{m}(k)
## number of servers at center @math{k}. This function supports
## @code{@var{m}(k) = 1} only (single-eserver FCFS nodes); this
## parameter is only for compatibility with @code{qncsaba}. Default is
## 1.
##
## @item @var{Z}
## External delay (@code{@var{Z} @geq{} 0}). Default is 0.
##
## @end table
##
## @strong{OUTPUTS}
##
## @table @code
##
## @item @var{Xl}
## @itemx @var{Xu}
## Lower and upper bound on the system throughput.
##
## @item @var{Rl}
## @itemx @var{Ru}
## Lower and upper bound on the system response time.
##
## @end table
##
## @strong{REFERENCES}
##
## @itemize
## @item
## Edward D. Lazowska, John Zahorjan, G. Scott Graham, and Kenneth
## C. Sevcik, @cite{Quantitative System Performance: Computer System
## Analysis Using Queueing Network Models}, Prentice Hall,
## 1984. @url{http://www.cs.washington.edu/homes/lazowska/qsp/}. In
## particular, see section 5.4 ("Balanced Systems Bounds").
## @end itemize
##
## @seealso{qncmbsb}
##
## @end deftypefn
## Author: Moreno Marzolla <moreno.marzolla(at)unibo.it>
## Web: http://www.moreno.marzolla.name/
function [Xl Xu Rl Ru] = qncsbsb( varargin )
if (nargin<2 || nargin>5)
print_usage();
endif
[err N S V m Z] = qncschkparam( varargin{:} );
isempty(err) || error(err);
all(m==1) || ...
error( "this function supports M/M/1 servers only" );
D = S .* V;
D_max = max(D);
D_tot = sum(D);
D_ave = mean(D);
Xl = N/(D_tot+Z+( (N-1)*D_max )/( 1+Z/(N*D_tot) ) );
Xu = min( 1/D_max, N/( D_tot+Z+( (N-1)*D_ave )/(1+Z/D_tot) ) );
Rl = max( N*D_max-Z, D_tot+( (N-1)*D_ave )/( 1+Z/D_tot) );
Ru = D_tot + ( (N-1)*D_max )/( 1+Z/(N*D_tot) );
endfunction
%!test
%! fail("qncsbsb(-1,0)", "N must be");
%! fail("qncsbsb(1,[])", "nonempty");
%! fail("qncsbsb(1,[-1 2])", "nonnegative");
%! fail("qncsbsb(1,[1 2],[1 2 3])", "incompatible size");
%! fail("qncsbsb(1,[1 2 3],[1 2 3],[1 2])", "incompatible size");
%! fail("qncsbsb(1,[1 2 3],[1 2 3],[1 2 1])", "M/M/1 servers");
%! fail("qncsbsb(1,[1 2 3],[1 2 3],[1 1 1],-1)", "nonnegative");
%! fail("qncsbsb(1,[1 2 3],[1 2 3],[1 1 1],[0 0])", "scalar");
%!test
%! S = [1 0.8 1.2 0.5];
%! V = [1 2 2 1];
%! D = S .* V;
%! N = 50;
%! tol = 1e-7; # compensate for numerical inaccuracies
%! for n=1:N
%! [U R Q X] = qncsmva(n, S, V);
%! Xs = X(1)/V(1);
%! Rs = dot(R,V);
%! [Xl Xu Rl Ru] = qncsbsb( n, D );
%! assert( Xl <= Xs+tol );
%! assert( Xu >= Xs-tol );
%! assert( Rl <= Rs+tol );
%! assert( Ru >= Rs-tol );
%! endfor
|
