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Procesosestocsticos. ProcesosdePoisson Procesosderenovacin

UnprocesoestocsticooprocesoaleatorioX(t) esunconceptomatemticoqueseutilizaparausarmagnitudesaleatoriasquevaranconeltiempooparacaracterizarunasucesindevariablesaleatoriasoestocsticasquevaranenfuncindeotravariable,generalmenteeltiempo.

n 1 n 1 n n n 1 n 1 2 2 1 1 n 1 n 1 n nP X x X x , X x , , X x , X x P X x X x+ + + + = = = = = = = =

paracualesquieran 0 y{ }n 1 1 2 nx , x , x , , x S+ Si S < sedicequelacadenadeMarkovesfinita.Encasocontrariosedicequeesinfinita.

Cadenahomognea: n n 1 1 0P X j X i P X j X i = = = = = paratodonodony i , j

ij 1 0 1 0j S j S

p P(X j X 1) P(X S X i) 1

= = = = = =

11 12 1k

21 22 2kij n 1 j n i

k1 k2 kk

p p p

p p pSi p P X s X s P

p p p

+

= = = =

Solucin:

0,2 0,8 0

P 0,

4 0 0,6

0,5 0,

2

35 0,15

3

1

2

3

=

0

E

1

st

0

P 1 /

2 0 1 / 2

1

1 2 3

1

0

2

3 0

=

1

2

3

0 1 0 0

0 0 1 0P

1 0 0 0

14 0 0 0

=

MATRIZDETRANSICINENPLAZOS

n

i ii 1

v 0 parai 1, 2, ...,n v 1=

Distribucindespusdel1paso:vP

Distribucindespusdel2paso: 2(vP)P vP=Distribucindespusdel3paso: 2 3(vP )P vP=Distribucindespusdenpasos: nvP

(n) (n) (n)11 12 1k(n) (n) (n)

(n) 21 22 2k

(n) (n) (n)k1 k2 kk

p p p

p p pP

p p p

=

(n)ij n 0 n m mp P(X j X i) P(X j X i)+= = = = = = ,portanto,

(1)ij ijp p=

Solucin:

Losposiblesestadosson { }S 1, 2, 3, 4, 5, 6=

Probabilidadesdetransicin ij n 1 np P X j X i+ = = = sonestacionariasporqueno

dependendelinstanteenqueseencuentreelproceso.

Matrizdetransicin:

11 12 13 14 15 16

21 22 23 24 25 26

31 32 33 34 35 36

41 42 43 44 45 46

51 52 53 54 55 56

61 62 63 64 65 66

p p p p p p

p p p p p p

p p p p p pP

p p p p p p

p p

p p p p

p p p p p p

6

1

2

3

4

5

6

=

11 n 1 n 12 n 1 n

13 n 1 n 14 n 1 n

15 n 1 n 16 n 1 n

p P X 1 X 1 p P X 2 X 1

p P X 3 X 1 0 p P X 4 X 1

p P X 5 X 1 0 p P X 6 X 1 0

+ +

+ +

+ +

= = = = = = = = = = = = = = = = = = = = =

21 n 1 n 22 n 1 n

23 n 1 n 24 n 1 n

25 n 1 n 26 n 1 n

p P X 1 X 2 p P X 2 X 2

p P X 3 X 2 0 p P X 4 X 2 0

p P X 5 X 2 0 p P X 6 X 2 0

+ +

+ +

+ +

= = = = = = = = = = = = = = = = = = = = = =

31 n 1 n 32 n 1 n

33 n 1 n 34 n 1 n

35 n 1 n 36 n 1 n

p P X 1 X 3 0 p P X 2 X 3 0

p P X 3 X 3 p P X 4 X 3 0

p P X 5 X 3 0 p P X 6 X 3

+ +

+ +

+ +

= = = = = = = = = = = = = = = = = = = = = =

41 n 1 n 42 n 1 n

43 n 1 n 44 n 1 n

45 n 1 n 46 n 1 n

p P X 1 X 4 p P X 2 X 4 0

p P X 3 X 4 0 p P X 4 X 4

p P X 5 X 4 0 p P X 6 X 4 0

+ +

+ +

+ +

= = = = = = = = = = = = = = = = = = = = = =

51 n 1 n 52 n 1 n

53 n 1 n 54 n 1 n

55 n 1 n 56 n 1 n

p P X 1 X 5 0 p P X 2 X 5 0

p P X 3 X 5 0 p P X 4 X 5 0

p P X 5 X 5 p P X 6 X 5

+ +

+ +

+ +

= = = = = = = = = = = = = = = = = = = = = =

61 n 1 n 62 n 1 n

63 n 1 n 64 n 1 n

65 n 1 n 66 n 1 n

p P X 1 X 6 0 p P X 2 X 6 0

p P X 3 X 6 p P X 4 X 6 0

p P X 5 X 6 p P X 6 X 6

+ +

+ +

+ +

= = = = = = = = = = = = = = = = = = = = =

Elvectordeprobabilidadinicial1 1 1 1 1 1

v , , , , ,6 6 6 6 6 6

=

ciertaceldaenelinstantedetiempo1.

11 12 14

21 22

33 36

1 / 3 1 / 3 0 1 / 3 0 0 p p p 1 / 3

1 / 2 1 / 2 0 0 0 0 p p 1 / 2

0 0 1 / 2 0 0 1 / 2 p pP

1 / 2 0 0 1