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
|
## Copyright (C) 2012, 2016, 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} {[@var{Xl}, @var{Xu}, @var{Rl}, @var{Ru}] =} qncsaba (@var{N}, @var{D})
## @deftypefnx {Function File} {[@var{Xl}, @var{Xu}, @var{Rl}, @var{Ru}] =} qncsaba (@var{N}, @var{S}, @var{V})
## @deftypefnx {Function File} {[@var{Xl}, @var{Xu}, @var{Rl}, @var{Ru}] =} qncsaba (@var{N}, @var{S}, @var{V}, @var{m})
## @deftypefnx {Function File} {[@var{Xl}, @var{Xu}, @var{Rl}, @var{Ru}] =} qncsaba (@var{N}, @var{S}, @var{V}, @var{m}, @var{Z})
##
## @cindex bounds, asymptotic
## @cindex asymptotic bounds
## @cindex closed network, single class
##
## Compute Asymptotic Bounds for the system throughput and response
## time of closed, single-class networks with @math{K} service
## centers.
##
## Single-server and infinite-server nodes are supported.
## Multiple-server nodes and general load-dependent servers are not
## supported.
##
## @strong{INPUTS}
##
## @table @code
##
## @item @var{N}
## number of requests in the system (scalar, @code{@var{N}>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}).
##
## @item @var{m}(k)
## number of servers at center @math{k}
## (if @var{m} is a scalar, all centers have that number of servers). If
## @code{@var{m}(k) < 1}, center @math{k} is a delay center (IS);
## if @code{@var{m}(k) = 1}, center @math{k} is a M/M/1-FCFS server.
## This function does not support multiple-server nodes. Default
## is 1.
##
## @item @var{Z}
## External delay (scalar, @code{@var{Z} @geq{} 0}). Default is 0.
##
## @end table
##
## @strong{OUTPUTS}
##
## @table @code
##
## @item @var{Xl}
## @itemx @var{Xu}
## Lower and upper bounds on the system throughput.
##
## @item @var{Rl}
## @itemx @var{Ru}
## Lower and upper bounds on the system response time.
##
## @end table
##
## @seealso{qncmaba}
##
## @end deftypefn
## Author: Moreno Marzolla <moreno.marzolla(at)unibo.it>
## Web: http://www.moreno.marzolla.name/
function [Xl Xu Rl Ru] = qncsaba( varargin ) #N, S, V, m, Z )
if (nargin<2 || nargin>5)
print_usage();
endif
[err N S V m Z] = qncschkparam( varargin{:} );
isempty(err) || error(err);
all(m<=1) || ...
error( "multiple server nodes are not supported" );
D = S.*V;
Dtot_single = sum(D(m==1)); # total demand at single-server nodes
Dtot_delay = sum(D(m<1)); # total demand at IS nodes
Dtot = sum(D); # total demand
Dmax = max(D); # max demand
Xl = N/(N*Dtot_single + Dtot_delay + Z);
Xu = min( N/(Dtot+Z), 1/Dmax );
Rl = max( Dtot, N*Dmax - Z );
Ru = N*Dtot_single + Dtot_delay;
endfunction
%!test
%! fail("qncsaba(-1,0)", "N must be");
%! fail("qncsaba(1,[])", "nonempty");
%! fail("qncsaba(1,[-1 2])", "nonnegative");
%! fail("qncsaba(1,[1 2],[1 2 3])", "incompatible size");
%! fail("qncsaba(1,[1 2 3],[1 2 -1])", "nonnegative");
%! fail("qncsaba(1,[1 2 3],[1 2 3],[1 2])", "incompatible size");
%! fail("qncsaba(1,[1 2 3],[1 2 3],[1 2 1])", "not supported");
%! fail("qncsaba(1,[1 2 3],[1 2 3],[1 1 1],-1)", "nonnegative");
%! fail("qncsaba(1,[1 2 3],[1 2 3],1,[0 0])", "scalar");
## Example 9.6 p. 913 Bolch et al.
%!test
%! N = 20;
%! S = [ 4.6*2 8 ];
%! Z = 120;
%! [X_l X_u R_l R_u] = qncsaba(N, S, ones(size(S)), ones(size(S)), Z);
%! assert( [X_u R_l], [0.109 64], 1e-3 );
%!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] = qncsaba( n, D );
%! assert( Xl <= Xs+tol );
%! assert( Xu >= Xs-tol );
%! assert( Rl <= Rs+tol );
%! assert( Ru >= Rs-tol );
%! endfor
|
