Μελέτη κλειστών συστημάτων αναμονής διακριτού χρόνου

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IEEE 2015

Transcript of Μελέτη κλειστών συστημάτων αναμονής διακριτού χρόνου

  • 1 / 29-

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  • 1 ()

    2 pi

    3 /cp

    4

    2 / 29-

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  • 1 ()

    2 pi

    3 /cp

    4

    2 / 29-

    N

  • 1 ()

    2 pi

    3 /cp

    4

    2 / 29-

    N

  • 1 ()

    2 pi

    3 /cp

    4

    2 / 29-

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  • ()

    3 / 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

  • ()

    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

  • ()

    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

  • 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

  • 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

  • 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

  • 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 (/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

  • 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

  • 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

  • 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

  • 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

  • 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

  • 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

  • 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

  • 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