. - , .
, , : -, ; -, , .
, (10.1)
N t,
.
,
N(0)=No, (10.2)
N(t)=N0eat, (10.3)
( ). , t a Δ t .
, Δ t λ μΔ t . i(t) , t i, i = 0, 1, 2,... pi(t + Δt). Δ t , , i, i 1, i i+1. pi(t + Δt) :
pi(t + Δt) = (i-1) λ pi-1 (t) Δt+(1-i(λ+μ) pi(t) Δt+(i+1) μi+1(t) Δt,
(i-1) λ pi-1 (t)- i(λ+μ) pi(t)+ (i+1) μi+1(t).
t → ∞,
(10.4)
(10.4) i= 2, 3, 4,.... i = 1 (10.4)
(10.5)
i = 0
(10.6)
( , p-1(t)≡0).
t =0 N0 , (10.4)(10.6) :
(10.7)
( [17]), (10.4)(10.7) (., , [47]). : , . . t
N(t)= (t) (10.8)
(), . . N(t)
. (10.9)
N(t) , (10.5) (10.4)
,
|
|
. .
N'(t)=(λ - μ)N(t) (10.10)
(10.7)
(10.11)
, ,
N(t)= , (10.12)
, λ > μ ( λ=μ+a (10.3)), λ < μ t → ∞. (. [17])
(10.13)
λ > μ
(10.14)
t → ∞ . , N0 N(t) , .
, . , (10.1),
(10.15)
N(0)=N0, (10.16)
y(t) , . ( (10.15) (10.16))
. (10.17)
y(t), . , y(t) : y(t)=yi, i-1≤t≤i, i =1,2,.... , [6]. ( ) . , , , , , , . , , , . , , , . , <, . . ={ <}, F(x), F'(x) = f(x) (. 10.1)
. (10.18)
σ ( ) X. ,
,
(10.19)
, (10.19) ,
, (10.20)
(10.19) , []=. X:
(10.21)
:
(10.22)
, σ (10.18) , . e. . ,
[i]=, D[yi]=var(yi)=σ2. (10.23)
, = , . ,
= ,
(10.24)
(10.17),
|
|
, (10.25)
N(t)
, (10.26)
D[N] = var(N)
(10.27)
(10.28)
,
(10.29)
t → ∞
. (10.30)
(10.26) (10.30) , , , N(t) , . , . , , , σ .
t p0(t):
, t, , , vaz(yt) = tσ2. ,
,
(10.31)
() = .
>0, . e. σ2>2a, → ∞ t → ∞, ,
, , , . , ; , .
1. , λ μ . N(t) t = 100, var (N(t)), , λ N0, t = 100 0,1%.
2. , , , = . N0 σ, t = 100 90%. (: (z), , , [6].)