Μελέτη κλειστών συστημάτων αναμονής διακριτού χρόνου
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Transcript of Μελέτη κλειστών συστημάτων αναμονής διακριτού χρόνου
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1 / 29-
N
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1 ()
2 pi
3 /cp
4
2 / 29-
N
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1 ()
2 pi
3 /cp
4
2 / 29-
N
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1 ()
2 pi
3 /cp
4
2 / 29-
N
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1 ()
2 pi
3 /cp
4
2 / 29-
N
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()
3 / 29-
N
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()
t = 0, 1, 2, . . ., pi pi
S = {1, 2, . . . , k}, N,
P = (pij) , i, j = 1, 2, . . . , k , pi
nij(t), i, j = 1, 2, . . . , k , pi pi pi i j [t, t + 1)
ni(t), i = 1, 2, . . . , k , pi pi i t
n(t) = (n1(t), n2(t), . . . , nk(t))T ,
t
4 / 29-
N
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()
t = 0, 1, 2, . . ., pi pi
S = {1, 2, . . . , k}, N,
P = (pij) , i, j = 1, 2, . . . , k , pi
nij(t), i, j = 1, 2, . . . , k , pi pi pi i j [t, t + 1)
ni(t), i = 1, 2, . . . , k , pi pi i t
n(t) = (n1(t), n2(t), . . . , nk(t))T ,
t
4 / 29-
N
-
()
t = 0, 1, 2, . . ., pi pi
S = {1, 2, . . . , k}, N,
P = (pij) , i, j = 1, 2, . . . , k , pi
nij(t), i, j = 1, 2, . . . , k , pi pi pi i j [t, t + 1)
ni(t), i = 1, 2, . . . , k , pi pi i t
n(t) = (n1(t), n2(t), . . . , nk(t))T ,
t
4 / 29-
N
-
()
t = 0, 1, 2, . . ., pi pi
S = {1, 2, . . . , k}, N,
P = (pij) , i, j = 1, 2, . . . , k , pi
nij(t), i, j = 1, 2, . . . , k , pi pi pi i j [t, t + 1)
ni(t), i = 1, 2, . . . , k , pi pi i t
n(t) = (n1(t), n2(t), . . . , nk(t))T ,
t
4 / 29-
N
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()
t = 0, 1, 2, . . ., pi pi
S = {1, 2, . . . , k}, N,
P = (pij) , i, j = 1, 2, . . . , k , pi
nij(t), i, j = 1, 2, . . . , k , pi pi pi i j [t, t + 1)
ni(t), i = 1, 2, . . . , k , pi pi i t
n(t) = (n1(t), n2(t), . . . , nk(t))T ,
t
4 / 29-
N
-
()
t = 0, 1, 2, . . ., pi pi
S = {1, 2, . . . , k}, N,
P = (pij) , i, j = 1, 2, . . . , k , pi
nij(t), i, j = 1, 2, . . . , k , pi pi pi i j [t, t + 1)
ni(t), i = 1, 2, . . . , k , pi pi i t
n(t) = (n1(t), n2(t), . . . , nk(t))T ,
t
4 / 29-
N
-
()
t = 0, 1, 2, . . ., pi pi
S = {1, 2, . . . , k}, N,
P = (pij) , i, j = 1, 2, . . . , k , pi
nij(t), i, j = 1, 2, . . . , k , pi pi pi i j [t, t + 1)
ni(t), i = 1, 2, . . . , k , pi pi i t
n(t) = (n1(t), n2(t), . . . , nk(t))T ,
t
4 / 29-
N
-
()
t = 0, 1, 2, . . ., pi pi
S = {1, 2, . . . , k}, N,
P = (pij) , i, j = 1, 2, . . . , k , pi
nij(t), i, j = 1, 2, . . . , k , pi pi pi i j [t, t + 1)
ni(t), i = 1, 2, . . . , k , pi pi i t
n(t) = (n1(t), n2(t), . . . , nk(t))T ,
t
4 / 29-
N
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tn1(t)
n2(t)
ni(t)
nk(t)
n1(t + 1)
n2(t + 1)
nj (t + 1)
nk (t + 1)
pi1
pi2ni1(t)
ni2(t)
nij(t) pij
nik(t)
pik
t + 1
5 / 29-
N
-
tn1(t)
n2(t)
ni(t)
nk(t)
n1(t + 1)
n2(t + 1)
nj (t + 1)
nk (t + 1)
pi1
pi2ni1(t)
ni2(t)
nij(t) pij
nik(t)
pik
t + 1
5 / 29-
N
-
tn1(t)
n2(t)
ni(t)
nk(t)
n1(t + 1)
n2(t + 1)
nj (t + 1)
nk (t + 1)
pi1
pi2ni1(t)
ni2(t)
nij(t) pij
nik(t)
pik
t + 1
5 / 29-
N
-
tn1(t)
n2(t)
ni(t)
nk(t)
n1(t + 1)
n2(t + 1)
nj (t + 1)
nk (t + 1)
pi1
pi2
ni1(t)
ni2(t)
nij(t) pij
nik(t)
pik
t + 1
5 / 29-
N
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tn1(t)
n2(t)
ni(t)
nk(t)
n1(t + 1)
n2(t + 1)
nj (t + 1)
nk (t + 1)
pi1
pi2ni1(t)
ni2(t)
nij(t) pij
nik(t)
pik
t + 1
5 / 29-
N
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pi
S = {1, 2, . . . , k}, pi P = (pij), i, j = 1, 2, . . . , k, n(t) = (n1(t), . . . , nk(t)). pi i, i = 1, 2, . . . , k , pipi ci N. t
ni(t) ci , i = 1, 2, . . . , k.
pi: pi pi () pi k + 1. nk+1(t) t .
6 / 29-
N
-
pi
S = {1, 2, . . . , k}, pi P = (pij), i, j = 1, 2, . . . , k, n(t) = (n1(t), . . . , nk(t)). pi i, i = 1, 2, . . . , k , pipi ci N. t
ni(t) ci , i = 1, 2, . . . , k.
pi: pi pi () pi k + 1. nk+1(t) t .
6 / 29-
N
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pi
S = {1, 2, . . . , k}, pi P = (pij), i, j = 1, 2, . . . , k, n(t) = (n1(t), . . . , nk(t)). pi i, i = 1, 2, . . . , k , pipi ci N. t
ni(t) ci , i = 1, 2, . . . , k.
pi: pi pi () pi k + 1. nk+1(t) t .
6 / 29-
N
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pi (/cp) :
S = {1, 2, . . . , k + 1}, ,
P =
(P 0pT 0
), pi pi,
n(t) = (n1(t), n2(t), . . . , nk(t), nk+1(t))T ,
t .
7 / 29-
N
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pi (/cp) :
S = {1, 2, . . . , k + 1}, ,
P =
(P 0pT 0
), pi pi,
n(t) = (n1(t), n2(t), . . . , nk(t), nk+1(t))T ,
t .
7 / 29-
N
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pi (/cp) :
S = {1, 2, . . . , k + 1}, ,
P =
(P 0pT 0
), pi pi,
n(t) = (n1(t), n2(t), . . . , nk(t), nk+1(t))T ,
t .
7 / 29-
N
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pi (/cp) :
S = {1, 2, . . . , k + 1}, ,
P =
(P 0pT 0
), pi pi,
n(t) = (n1(t), n2(t), . . . , nk(t), nk+1(t))T ,
t .
7 / 29-
N
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pi t pipi pi :
pi t pi . pi pi P (pi pi).
pi pi pi pi pi pi . pipi pi pi pi pi pi pi, pi pi.
8 / 29-
N
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pi t pipi pi :
pi t pi . pi pi P (pi pi).
pi pi pi pi pi pi . pipi pi pi pi pi pi pi, pi pi.
8 / 29-
N
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pi t pipi pi :
pi t pi . pi pi P (pi pi).
pi pi pi pi pi pi . pipi pi pi pi pi pi pi, pi pi.
8 / 29-
N
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t = 1, 2, . . ., pi pi ,
N, N N, pi ,k , k N, pi pi, ( pi pi) pi p pipi pi pi,
pi pipi pi q.
9 / 29-
N
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t = 1, 2, . . ., pi pi ,
N, N N, pi ,k , k N, pi pi, ( pi pi) pi p pipi pi pi,
pi pipi pi q.
9 / 29-
N
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t = 1, 2, . . ., pi pi ,
N, N N, pi ,k , k N, pi pi, ( pi pi) pi p pipi pi pi,
pi pipi pi q.
9 / 29-
N
-
t = 1, 2, . . ., pi pi ,
N, N N, pi ,k , k N, pi pi, ( pi pi) pi p pipi pi pi,
pi pipi pi q.
9 / 29-
N
-
t = 1, 2, . . ., pi pi ,
N, N N, pi ,k , k N, pi pi, ( pi pi) pi p pipi pi pi,
pi pipi pi q.
9 / 29-
N
-
t = 1, 2, . . ., pi pi ,
N, N N, pi ,k , k N, pi pi, ( pi pi) pi p pipi pi pi,
pi pipi pi q.
9 / 29-
N
-
t = 1, 2, . . ., pi pi ,
N, N N, pi ,k , k N, pi pi, ( pi pi) pi p pipi pi pi,
pi pipi pi q.
9 / 29-
N
-
.
p
p
..
q
q
10 / 29-
N
-
/cp
pi:
P =
1 p p 0q 1 q 00 1 0
.
11 / 29-
N
-
/cp
t+1t
n1(t)
n2(t)
n3(t)
1-p
n1(t+1)
n2(t+1)
n3(t+1)
c1=N
c2=k
pi:
P =
1 p p 0q 1 q 00 1 0
.
11 / 29-
N
-
/cp
t+1t
n1(t)
n2(t)
n3(t)
1-p
n1(t+1)
n2(t+1)
n3(t+1)
c1=N
c2=k
pi:
P =
1 p p 0q 1 q 00 1 0
.
11 / 29-
N
-
/cp
12 / 29-
N
-
/cp
p(bij ; n(t)) = P( j t
i
t pi j),
p(bij ; n(t)) = P( pi pi j
t i
t pi j),
i = 1, 2, 3, j = 1, 2.
13 / 29-
N
-
/cp
p(bij ; n(t)) = P( j t
i
t pi j),
p(bij ; n(t)) = P( pi pi j
t i
t pi j),
i = 1, 2, 3, j = 1, 2.
13 / 29-
N
-
pi n(t) pi (Vasiliadis and Tsaklidis, 2011)
P(n(t)) =
(P 4 H(n(t)) P 4 H(n(t))e
pTp(b32; n(t)) p(b32; n(t))
),
pi
H(n(t)), 2 2 pi pi p(bij ; n(t)), i, j = 1, 2,H(n(t)), 2 2 pi pip(bij ; n(t)), i, j = 1, 2,e ,
P =
(1 p pq 1 q
),
p = (0, 1)T ,
4 Hadamard .
14 / 29-
N
-
pi n(t) pi (Vasiliadis and Tsaklidis, 2011)
P(n(t)) =
(P 4 H(n(t)) P 4 H(n(t))e
pTp(b32; n(t)) p(b32; n(t))
),
pi
H(n(t)), 2 2 pi pi p(bij ; n(t)), i, j = 1, 2,H(n(t)), 2 2 pi pip(bij ; n(t)), i, j = 1, 2,e ,
P =
(1 p pq 1 q
),
p = (0, 1)T ,
4 Hadamard .
14 / 29-
N
-
pi n(t) pi (Vasiliadis and Tsaklidis, 2011)
P(n(t)) =
(P 4 H(n(t)) P 4 H(n(t))e
pTp(b32; n(t)) p(b32; n(t))
),
pi
H(n(t)), 2 2 pi pi p(bij ; n(t)), i, j = 1, 2,H(n(t)), 2 2 pi pip(bij ; n(t)), i, j = 1, 2,e ,
P =
(1 p pq 1 q
),
p = (0, 1)T ,
4 Hadamard .
14 / 29-
N
-
pi n(t) pi (Vasiliadis and Tsaklidis, 2011)
P(n(t)) =
(P 4 H(n(t)) P 4 H(n(t))e
pTp(b32; n(t)) p(b32; n(t))
),
pi
H(n(t)), 2 2 pi pi p(bij ; n(t)), i, j = 1, 2,H(n(t)), 2 2 pi pip(bij ; n(t)), i, j = 1, 2,e ,
P =
(1 p pq 1 q
),
p = (0, 1)T ,
4 Hadamard .
14 / 29-
N
-
pi n(t) pi (Vasiliadis and Tsaklidis, 2011)
P(n(t)) =
(P 4 H(n(t)) P 4 H(n(t))e
pTp(b32; n(t)) p(b32; n(t))
),
pi
H(n(t)), 2 2 pi pi p(bij ; n(t)), i, j = 1, 2,H(n(t)), 2 2 pi pip(bij ; n(t)), i, j = 1, 2,e ,
P =
(1 p pq 1 q
),
p = (0, 1)T ,
4 Hadamard .
14 / 29-
N
-
pi n(t) pi (Vasiliadis and Tsaklidis, 2011)
P(n(t)) =
(P 4 H(n(t)) P 4 H(n(t))e
pTp(b32; n(t)) p(b32; n(t))
),
pi
H(n(t)), 2 2 pi pi p(bij ; n(t)), i, j = 1, 2,H(n(t)), 2 2 pi pip(bij ; n(t)), i, j = 1, 2,e ,
P =
(1 p pq 1 q
),
p = (0, 1)T ,
4 Hadamard .
14 / 29-
N
-
pi n(t) pi (Vasiliadis and Tsaklidis, 2011)
P(n(t)) =
(P 4 H(n(t)) P 4 H(n(t))e
pTp(b32; n(t)) p(b32; n(t))
),
pi
H(n(t)), 2 2 pi pi p(bij ; n(t)), i, j = 1, 2,H(n(t)), 2 2 pi pip(bij ; n(t)), i, j = 1, 2,e ,
P =
(1 p pq 1 q
),
p = (0, 1)T ,
4 Hadamard .
14 / 29-
N
-
pipipi pi P(n(t)).
E[nT (t)
]=
s
sT P(s)P (n(t 1) = s) ,
pi s
s n(t) = s t .
15 / 29-
N
-
pipipi pi P(n(t)).
E[nT (t)
]=
s
sT P(s)P (n(t 1) = s) ,
pi s
s n(t) = s t .
15 / 29-
N
-
pi
16 / 29-
N
-
pi
pi pi :
pi, xT = (x1, x2),
xT xT = (x21 , x1x2, x2x1, x22),
xT xT = (x(2)1 , x(1)1 x(1)2 , x(1)2 x(1)1 , x(2)2 ),
xT xT = (x1(x1 1), x1x2, x2x1, x2(x2 1)).
17 / 29-
N
-
pi
pi pi : pi, xT = (x1, x2),
xT xT = (x21 , x1x2, x2x1, x22),
xT xT = (x(2)1 , x(1)1 x(1)2 , x(1)2 x(1)1 , x(2)2 ),
xT xT = (x1(x1 1), x1x2, x2x1, x2(x2 1)).
17 / 29-
N
-
pi pi pi pi . : P(n(t)) 3 3 pi (), P(n(t)) P(n(t)) 32 32 pi, pi pipi pi Kronecker P(n(t)) P(n(t)),
p(bij ; n(t))p(brs; n(t)) p(bijbrs; n(t)),p(bij ; n(t))p(brs; n(t)) p(bijbrs; n(t)),p(bij ; n(t))p(brs; n(t)) p(bijbrs; n(t)),p(bij ; n(t))p(brs; n(t)) p(bijbrs; n(t)),
i, j, r, s {1, 2, 3}.
18 / 29-
N
-
pipipi pi P(n(t)).
E
nT (t) . . . nT (t) r
=
E
(nT (t 1) . . . nT (t 1)) r
(P(n(t 1)) . . . P(n(t 1)) r
)
,
E
nT (t) . . . nT (t) r
= s
sT . . . sT r
(P(s) . . . P(s) r
)P(nT (t 1) = sT
),
pi s
s n(t) = s t .
19 / 29-
N
-
pipipi pi P(n(t)). pi ni(t), i = 1, 2, 3, t , pi
P [n1(t) = r1, n2(t) = r2, n3(t) = r3] =1
r1!r2!r3!E[n(r1)1 (t)n
(r2)2 (t)n
(r3)3 (t)
].
20 / 29-
N
-
21 / 29-
N
-
mij(z), i, j = 1, 2, 3, z = 1, 2, . . . , s, pi j i nz .
22 / 29-
N
-
pipipi pi pi P.
mij(z) = eTi (I Qj)1e, i, j = 1, 2, 3, z = 1, 2, . . . , s,
pi
ei 3s 1 pi i + 3(z 1) 1 pipi 0,
e 3s 1 1,I pi 3s 3s,Qj pi 3s 3s pi pipi pi pi Q j + 3(w 1),w = 1, 2, . . . , s, 0.
23 / 29- N
-
wz , z = 1, 2, . . . , s, nz .
wz = m32(z), z T ,pi T {1, 2, . . . , s} pi pi .
z T ,
wz =s
i=1
wiP[n(1) = ni
n(0) = nz]=
iT
wiP[n(1) = ni
n(0) = nz]+iT
wiP[n(1) = ni
n(0) = nz]=
iT
m32(i)P[n(1) = ni
n(0) = nz]+iT
wiP[n(1) = ni
n(0) = nz],
24 / 29-
N
-
wz , z = 1, 2, . . . , s, nz .
wz = m32(z), z T ,pi T {1, 2, . . . , s} pi pi . z T ,
wz =s
i=1
wiP[n(1) = ni
n(0) = nz]=
iT
wiP[n(1) = ni
n(0) = nz]+iT
wiP[n(1) = ni
n(0) = nz]=
iT
m32(i)P[n(1) = ni
n(0) = nz]+iT
wiP[n(1) = ni
n(0) = nz],
24 / 29-
N
-
25 / 29-
N
-
q=0.4
26 / 29-
N
-
pi pi P
P =
0.8 0.2 00.4 0.6 00 1 0
.
N = 4, t = 0
n(0) = (4, 0, 0)T .
pi, pi pi pipi t ,
n1(t) = (3, 1, 0)T , n2(t) = (2, 1, 1)
T , n3(t) = (1, 1, 2)T ,
n4(t) = (0, 1, 3)T , n5(t) = (4, 0, 0)
T .
27 / 29-
N
-
pi pi P
P =
0.8 0.2 00.4 0.6 00 1 0
. N = 4, t = 0
n(0) = (4, 0, 0)T .
pi, pi pi pipi t ,
n1(t) = (3, 1, 0)T , n2(t) = (2, 1, 1)
T , n3(t) = (1, 1, 2)T ,
n4(t) = (0, 1, 3)T , n5(t) = (4, 0, 0)
T .
27 / 29-
N
-
pi pi P
P =
0.8 0.2 00.4 0.6 00 1 0
. N = 4, t = 0
n(0) = (4, 0, 0)T .
pi, pi pi pipi t ,
n1(t) = (3, 1, 0)T , n2(t) = (2, 1, 1)
T , n3(t) = (1, 1, 2)T ,
n4(t) = (0, 1, 3)T , n5(t) = (4, 0, 0)
T .
27 / 29-
N
-
28 / 29-
N
-
pi Wq , pi w5.
w5 = w1P[n(1) = n1
n(0) = n5] +m32(2)P [n(1) = n2n(0) = n5]++m32(3)P
[n(1) = n3
n(0) = n5] +m32(4)P [n(1) = n4n(0) = n5] + w5P [n(1) = n5n(0) = n5],
w1 = w1P[n(1) = n1
n(0) = n1] +m32(2)P [n(1) = n2n(0) = n1]++m32(3)P
[n(1) = n3
n(0) = n1] +m32(4)P [n(1) = n4n(0) = n1] + w5P [n(1) = n5n(0) = n1].
pipi
w5 = 3.49954, w1 = 3.5162.
, pi
Wq = w5 = 3.49954.
29 / 29-
N
-
pi Wq , pi w5.
w5 = w1P[n(1) = n1
n(0) = n5] +m32(2)P [n(1) = n2n(0) = n5]++m32(3)P
[n(1) = n3
n(0) = n5] +m32(4)P [n(1) = n4n(0) = n5] + w5P [n(1) = n5n(0) = n5],
w1 = w1P[n(1) = n1
n(0) = n1] +m32(2)P [n(1) = n2n(0) = n1]++m32(3)P
[n(1) = n3
n(0) = n1] +m32(4)P [n(1) = n4n(0) = n1] + w5P [n(1) = n5n(0) = n1]. pipi
w5 = 3.49954, w1 = 3.5162.
, pi
Wq = w5 = 3.49954.
29 / 29-
N
To omogen'ec Markobian'o s'usthma (OMS) diakrito'u qr'onouTo OMS diakrito'u qr'onou me qwrhtik'othta stic katast'aseic kai proterai'othta stic metakin'hseicKleist'o s'usthma anamon'hc diakrito'u qr'onouMel'eth tou sust'hmatoc anamon'hc me th bo'hjeia en'oc OMS/cp
Efarmog'h