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
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
|
## Copyright (C) 2012, 2016, 2018, 2020, 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{Xl}, @var{Xu}, @var{Rl}, @var{Ru}] =} qncmcb (@var{N}, @var{D})
## @deftypefnx {Function File} {[@var{Xl}, @var{Xu}, @var{Rl}, @var{Ru}] =} qncmcb (@var{N}, @var{S}, @var{V})
##
## @cindex multiclass network, closed
## @cindex closed multiclass network
## @cindex bounds, composite
## @cindex composite bounds
##
## Compute Composite Bounds (CB) on system throughput and response time for closed multiclass networks.
##
## @strong{INPUTS}
##
## @table @code
##
## @item @var{N}(c)
## number of class @math{c} requests in the system.
##
## @item @var{D}(c, k)
## class @math{c} service demand
## at center @math{k} (@code{@var{S}(c,k) @geq{} 0}).
##
## @item @var{S}(c, k)
## mean service time of class @math{c}
## requests at center @math{k} (@code{@var{S}(c,k) @geq{} 0}).
##
## @item @var{V}(c,k)
## average number of visits of class @math{c}
## requests to center @math{k} (@code{@var{V}(c,k) @geq{} 0}).
##
## @end table
##
## @strong{OUTPUTS}
##
## @table @code
##
## @item @var{Xl}(c)
## @itemx @var{Xu}(c)
## Lower and upper bounds on class @math{c} throughput.
##
## @item @var{Rl}(c)
## @itemx @var{Ru}(c)
## Lower and upper bounds on class @math{c} response time.
##
## @end table
##
## @strong{REFERENCES}
##
## @itemize
## @item
## Teemu Kerola, @cite{The Composite Bound Method (CBM) for Computing
## Throughput Bounds in Multiple Class Environments}, Performance
## Evaluation Vol. 6, Issue 1, March 1986, DOI
## @uref{http://dx.doi.org/10.1016/0166-5316(86)90002-7,
## 10.1016/0166-5316(86)90002-7}. Also available as
## @uref{http://docs.lib.purdue.edu/cstech/395/, Technical Report
## CSD-TR-475}, Department of Computer Sciences, Purdue University, mar
## 13, 1984 (Revised Aug 27, 1984).
## @end itemize
##
## @end deftypefn
## Author: Moreno Marzolla <moreno.marzolla(at)unibo.it>
## Web: http://www.moreno.marzolla.name/
function [Xl Xu Rl Ru] = qncmcb( varargin )
if ( nargin < 2 || nargin > 3 )
print_usage();
endif
[err N S V m Z] = qncmchkparam( varargin{:} );
isempty(err) || error(err);
all(m == 1) || ...
error("this function only supports single-server FCFS centers");
all(Z == 0) || ...
error("this function does not support think time");
[C K] = size(S);
D = S .* V;
[Xl] = qncmbsb(N, D);
Xu = zeros(1,C);
D_max = max(D,[],2)';
for r=1:C
## This is equation (13) from T. Kerola, The Composite Bound Method
## (CBM) for Computing Throughput Bounds in Multiple Class
## Environments, Technical Report CSD-TR-475, Purdue University,
## march 13, 1984 (revised august 27, 1984)
## http://docs.lib.purdue.edu/cstech/395/
## The only modification here is to apply also the upper bound
## 1/D_max(r).
s = (1:C != r); # boolean array
tmp = (1 - Xl(s)*D(s,:)) ./ D(r,:);
Xu(r) = min([tmp 1/D_max(r)]);
endfor
Rl = N ./ Xu;
Ru = N ./ Xl;
endfunction
%!test
%! S = [10 7 5 4; ...
%! 5 2 4 6];
%! NN=20;
%! Xl = Xu = Rl = Ru = Xmva = Rmva = zeros(NN,2);
%! for n=1:NN
%! N=[n,10];
%! [a b c d] = qncmcb(N,S);
%! Xl(n,:) = a; Xu(n,:) = b; Rl(n,:) = c; Ru(n,:) = d;
%! [U R Q X] = qncmmva(N,S,ones(size(S)));
%! Xmva(n,:) = X(:,1)'; Rmva(n,:) = sum(R,2)';
%! endfor
%! assert( all(Xl <= Xmva) );
%! assert( all(Xu >= Xmva) );
%! assert( all(Rl <= Rmva) );
%! assert( all(Xu >= Xmva) );
%!demo
%! S = [10 7 5 4; ...
%! 5 2 4 6];
%! NN=20;
%! Xl = Xu = Rl = Ru = Xmva = Rmva = zeros(NN,2);
%! for n=1:NN
%! N=[n,10];
%! [a b c d] = qncmcb(N,S);
%! Xl(n,:) = a; Xu(n,:) = b; Rl(n,:) = c; Ru(n,:) = d;
%! [U R Q X] = qncmmva(N,S,ones(size(S)));
%! Xmva(n,:) = X(:,1)'; Rmva(n,:) = sum(R,2)';
%! endfor
%! subplot(2,2,1);
%! plot(1:NN,Xl(:,1), 1:NN,Xu(:,1), 1:NN,Xmva(:,1), ";MVA;", "linewidth", 2);
%! ylim([0, 0.2]);
%! title("Class 1 throughput"); legend("boxoff");
%! subplot(2,2,2);
%! plot(1:NN,Xl(:,2), 1:NN,Xu(:,2), 1:NN,Xmva(:,2), ";MVA;", "linewidth", 2);
%! ylim([0, 0.2]);
%! title("Class 2 throughput"); legend("boxoff");
%! subplot(2,2,3);
%! plot(1:NN,Rl(:,1), 1:NN,Ru(:,1), 1:NN,Rmva(:,1), ";MVA;", "linewidth", 2);
%! ylim([0, 700]);
%! title("Class 1 response time"); legend("location", "northwest"); legend("boxoff");
%! subplot(2,2,4);
%! plot(1:NN,Rl(:,2), 1:NN,Ru(:,2), 1:NN,Rmva(:,2), ";MVA;", "linewidth", 2);
%! ylim([0, 700]);
%! title("Class 2 response time"); legend("location", "northwest"); legend("boxoff");
|
