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
|
## Copyright (C) 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{L} =} dtmcexps (@var{P}, @var{n}, @var{p0})
## @deftypefnx {Function File} {@var{L} =} dtmcexps (@var{P}, @var{p0})
##
## @cindex expected sojourn times, DTMC
## @cindex DTMC
## @cindex discrete time Markov chain
## @cindex Markov chain, discrete time
##
## Compute the expected number of visits to each state during the first
## @var{n} transitions, or until abrosption.
##
## @strong{INPUTS}
##
## @table @code
##
## @item @var{P}(i,j)
## @math{N \times N} transition matrix. @code{@var{P}(i,j)} is the
## transition probability from state @math{i} to state @math{j}.
##
## @item @var{n}
## Number of steps during which the expected number of visits are
## computed (@math{@var{n} @geq{} 0}). If @code{@var{n}=0}, returns
## @var{p0}. If @code{@var{n} > 0}, returns the expected number of
## visits after exactly @var{n} transitions.
##
## @item @var{p0}(i)
## Initial state occupancy probabilities; @code{@var{p0}(i)} is
## the probability that the system is in state @math{i} at step 0.
##
## @end table
##
## @strong{OUTPUTS}
##
## @table @code
##
## @item @var{L}(i)
## When called with two arguments, @code{@var{L}(i)} is the expected
## number of visits to state @math{i} before absorption. When
## called with three arguments, @code{@var{L}(i)} is the expected number
## of visits to state @math{i} during the first @var{n} transitions.
##
## @end table
##
## @strong{REFERENCES}
##
## @itemize
## @item Grinstead, Charles M.; Snell, J. Laurie (July
## 1997). @cite{Introduction to Probability}, Ch. 11: Markov
## Chains. American Mathematical Society. ISBN 978-0821807491.
## @end itemize
##
## @seealso{ctmcexps}
##
## @end deftypefn
## Author: Moreno Marzolla <moreno.marzolla(at)unibo.it>
## Web: http://www.moreno.marzolla.name/
function L = dtmcexps ( P, varargin )
persistent epsilon = 10*eps;
if ( nargin < 2 || nargin > 3 )
print_usage();
endif
[K err] = dtmcchkP(P);
(K>0) || ...
error(err);
if ( nargin == 2 )
p0 = varargin{1};
else
n = varargin{1};
p0 = varargin{2};
endif
( isvector(p0) && length(p0) == K && all(p0>=0) && abs(sum(p0)-1.0)<epsilon ) || ...
error( "p0 must be a state occupancy probability vector" );
p0 = p0(:)'; # make p0 a row vector
if ( nargin == 3 )
isscalar(n) && n>=0 || ...
error("n must be >= 0");
n = fix(n);
L = zeros(sizeof(p0));
## It is know that
##
## I + P + P^2 + P^3 + ... + P^n = (I-P)^-1 * (I-P^(n+1))
##
## and therefore we could succintly write
##
## L = p0*inv(eye(K)-P)*(eye(K)-P^(n+1));
##
## Unfortunatly, the method above is numerically unstable (at least
## for small values of n), so we use the crude approach below.
PP = p0;
L = zeros(1,K);
for p=0:n
L += PP;
PP *= P;
endfor
else
## identify transient states
tr = find(diag(P) < 1);
k = length(tr); # number of transient states
if ( k == K )
error("There are no absorbing states");
endif
N = zeros(size(P));
tmpN = inv(eye(k) - P(tr,tr)); # matrix N = (I-Q)^-1
N(tr,tr) = tmpN;
L = p0*N;
endif
endfunction
%!test
%! P = dtmcbd([1 1 1 1], [0 0 0 0]);
%! L = dtmcexps(P,[1 0 0 0 0]);
%! t = dtmcmtta(P,[1 0 0 0 0]);
%! assert( L, [1 1 1 1 0] );
%! assert( sum(L), t );
%!test
%! P = dtmcbd(linspace(0.1,0.4,5),linspace(0.4,0.1,5));
%! p0 = [1 0 0 0 0 0];
%! L = dtmcexps(P,0,p0);
%! assert( L, p0 );
|
