Дронов С.В. Конспект лекций по теории случайных процессов
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b d (b + d ≤ 1)
b = λ∆t + o(∆t), d = µ∆t + o(∆t).
∆t
o(∆t)
b
n
(∆t), d
n
(∆t)
P(b
n
(∆t) = 1) = nλ∆t + o(∆t), P(d
n
(∆t) = 1) = nµ∆t + o(∆t),
P
n,n+1
(∆t) = P(b
n
(∆t) = 1, d
n
(∆t) = 0) + o(∆t) =
= (nλ∆t + o(∆t))(1 − nµ∆t + o(∆t)) + o(∆t) = nλ∆t + o(∆t);
P
n,n−1
(∆t) = nµ∆t + o(∆t);
P
n,n
(∆t) = P(b
n
(∆t) = 0, d
n
(∆t) = 0) + P(b
n
(∆t) = 1, d
n
(∆t) = 1)+
+o(∆t) = = 1 −n(λ + µ)∆t + o(∆t).
λ
n
= nλ, µ
n
= nµ
p
n
(t + ∆t) = p
n
(t)(1 − nλ∆t + o(∆t))(1 − nµ∆t + o(∆t))+
+p
n−1
(t)((n − 1)λ∆t + o(∆t))(1 − (n − 1)µ∆t + o(∆t))+
+p
n+1
(t)((n + 1)µ∆t + o(∆t))(1 −(n + 1)λ∆t + o(∆t)) + o(∆t).
p
n
(t + ∆t) − p
n
(t)
∆t
= −n(λ + µ)p
n
(t) + (n − 1)λp
n−1
(t)+
+(n + 1)µp
n+1
(t) + o(1).
∆t → 0
p
0
n
(t) = −n(λ + µ)p
n
(t) + (n − 1)λp
n−1
(t) + (n + 1)µp
n+1
(t).
n ≥ 1 n = 0
p
0
0
(t) = µp
1
(t).
n x
n
∞
X
n=0
p
0
n
(t)x
n
= µ
∞
X
n=1
p
n
(t)x
n−1
n + λx
2
∞
X
n=1
p
n
(t)x
n−1
n−
−(λ + µ)
∞
X
n=0
p
n
(t)x
n
n.
F (x, t) =
∞
X
n=0
p
n
(t)x
n
,
∂F
∂t
= (µ − (λ + µ)x + λx
2
)
∂F
∂x
.
p
1
(0) = 1
p
n
(0) = 0 n 6= 1
dt =
−dx
µ − (λ + µ)x + λx
2
,
t − c =
1
µ − λ
Z
1
1 − x
−
λ
µ − λx
!
dx =
=
1
µ − λ
ln
µ − λx
1 − x
.
F (t, x) = R
1
µ − λ
ln
µ − λx
1 − x
+ t
!
,
R
t = 0
F (x, t) = x,
R
1
µ − λ
ln
µ − λx
1 − x
!
= x,
R u =
1
µ−λ
ln
µ−λx
1−x
.
R(u) =
µ − e
(µ−λ)u
λ − e
(µ−λ)u
.
u R
F (x, t) =
µ(1 − e
(µ−λ)t
) + x(λe
(µ−λ)t
− µ)
λ − µe
(µ−λ)t
+ λx(e
(µ−λ)t
− 1)
.
F x
p
n
(t) =
(µ − λ)
2
e
(µ−λ)t
1 − e
(µ−λ)t
n−1
λ
n−1
(λ − µe
(µ−λ)t
)
n+1
, n ≥ 1,
p
0
(t) =
µ
1 − e
(µ−λ)t
λ − µe
(µ−λ)t
, p
≥1
(λ) =
λ − µ
λ − µe
(µ−λ)t
.
p
≥1
(t) =
P
∞
k=1
p
k
(t).
µ < λ t → ∞
p
0
(t) −→
µ
λ
, p
≥1
(t) −→ 1 −
µ
λ
,
µ > λ
p
0
(t) → 1, p
≥1
(t) → 0.
λ
n
=
λn + a, µ
n
= µn,
a
P
0
n,0
(t) = −aP
n,0
+ µP
n,1
(t),
P
0
n,j
(t) = (a + λ(j − 1))P
n,j−1
(t)−
−((λ + µ)j + a)P
n,j
(t) + µ(j + 1)P
n,j+1
(t), j ≥ 1.
j j M(t)
ξ(0) = n
M
0
(t) = a + (λ − µ)M(t).
λ 6= µ
M(t) = Ce
(λ−µ)t
−
a
λ − µ
.
t = 0
c = n +
a
λ − µ
.
M(t) =
a
λ − µ
e
(λ−µ)t
− 1
+ ne
(λ−µ)t
.
λ > µ M(t)
λ < µ
a/(µ − λ)
λ = µ
M(t) = at + n, M(t) −→ ∞.
ξ(t)
t
(∀n) λ
n
= λ, µ
n
= µ. τ
j
j
τ
j
(∀t ≥ 0) P(τ
j
≥ t) = e
−µt
.
n
τ τ
1
, ..., τ
n
P(τ ≥ t) =
n
Y
j=1
P(τ
j
≥ t) = e
−µnt
.
n
µ
j
= jµ j ≤ n µ
j
= nµ j > n
π
j
=
λ
µ
!
j
,
∞
X
j=1
π
j
=
µ
µ − λ
µ > λ
p
n
=
µ − λ
µ
λ
µ
!
n
, n ≥ 0.
n − 1
λ
n
= λ, µ
n
= nµ,
π
j
=
λ
j
j!µ
j
,
∞
X
j=1
π
j
= e
λ/µ
,
p
n
=
1
n!
λ
µ
!
n
e
−
λ
µ
, n ≥ 0.
n
t ∈ [a, b]
ξ(t), η(t)
(∀t) P(ξ(t) = η(t)) = 1.
ξ(t)
ξ(t)
r, α > 0
(∀s, t) M|x(t) − x(s)|
r
≤ c|t − s|
1+α
,
ξ(t) x(t)
g, q
(∀t, h > 0) P(|x(t + h) − x(t)| ≥ g(h)) ≤ q(h),
∞
X
n=0
2
n
q
b − a
2
n
!
< ∞,
∞
X
n=0
g
b − a
2
n
!
< ∞.
ξ(t) x(t)
γ > 0, α − rγ > 0
g(h) = h
γ
, q(h) = ch
1+α−rγ
,
c
r
P(|x(t + h) − x(t)| ≥ g(h)) ≤
ch
1+α
g
r
(h)
= q(h).
A
1
, A
2
, ...
∞
X
n=1
P(A
n
) < ∞.
A = ∩
∞
k=1
∪
∞
n=k
A
n
.
B
k
= ∪
n≥k
A
n
.
P(B
k
) ≤
∞
X
n=k
P(A
n
) −→ 0, k → ∞.
P(A) = lim
k→∞
P(B
k
) = 0.
X
(n)
(t) = 2
n
(x(t
n,k
)(t
n,k+1
− t) + x(t
n,k+1
)(t − t
n,k
)) ,
t ∈ [t
n,k
, t
n,k+1
],
t
n,k
=
k
2
n
, k = 0, ..., 2
n
(∀n, k) X
(n)
(t
n,k
) = x(t
n,k
)
|X
(n)
(t) −
X
(n+1)
(t)|
2k+1
2
n+1
P
n
≡ P
max |X
(n)
− X
(n+1)
| ≥ g(2
−(n+1)
)
≤ 2
n+1
q(2
−(n+1)
).
P
∞
n=1
P
n
{max |X
(n)
−
X
(n+1)
| ≥ g(2
−(n+1)
)}
N