Теория вероятностей и математическая статистика. Билялов Р.Ф. - 123 стр.

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µ
k
µ
k
+λ
k
k k − 1.
τ k
n, τ
0
k, τ
00
n k.
m
k
= Mτ = M τ
0
+ Mτ
00
=
= Mτ
0
+P (k → k−1)M(τ
00
|k → k−1))+P (k → k+1)M(τ
00
|k → k+1) =
=
1
µ
k
+ λ
k
+
µ
k
µ
k
+ λ
k
m
k−1
+
λ
k
µ
k
+ λ
k
m
k+1
.
ξ
t
−
h → 0
P (ξ
h
= 2|ξ
0
= 1) = αh + o(h), P (ξ
h
= 1|ξ
0
= 2) = βh + o(h).
τ
k
(t), k = 1, 2,
k t
f
k
(t, s) = M (e
is(τ
1
(t)−τ
2
(t))
|ξ
0
= k).
f
1
(t + h) = M(e
is(τ
1
(t+h)−τ
2
(t+h))
|ξ
0
= 1) =
= M(e
is(τ
1
(h)−τ
2
(h)+τ
1
(t+h)−τ
1
(h)−(τ
2
(t+h)−τ
2
(h)))
|ξ
0
= 1) =
=
X
k
P (ξ
h
= k|ξ
0
= 1)
·M(e
is(τ
1
(h)−τ
2
(h))
· e
is(τ
1
(t+h)−τ
1
(h)−(τ
2
(t+h)−τ
2
(h))
|ξ
0
= 1, ξ
h
= k) =
X
k
P (ξ
h
= k|ξ
0
= 1) · M(e
is(τ
1
(h)−τ
2
(h))
|ξ
0
= 1, ξ
h
= k) · f
k
(t, s) =
(1 − αh + ···)e
ish
f
1
(t, s) + (αh + ···)(1 + ···)f
2
(t, s).
∂
∂t
f
1
(t, s) = (is − α)f
1
(t, s) + αf
2
(t, s).
Àíàëîãè÷íî, µkµ+λ
                k
                   k
                      âðåìÿ ïåðåõîäà èç k â k − 1.
   3) Ïóñòü τ âðåìÿ, çà êîòîðîå ñèñòåìà èç ñîñòîÿíèÿ k ïåðåõîäèò
â ñîñòîÿíèå n, τ 0  âðåìÿ äî ïåðâîãî âûõîäà èç ñîñòîÿíèÿ k, τ 00 
âðåìÿ ïðèõîäà âïåðâûå â ñîñòîÿíèå n ïîñëå âûõîäà èç ñîñòîÿíèÿ k.

                            mk = M τ = M τ 0 + M τ 00 =

= M τ 0 +P (k → k−1)M (τ 00 |k → k−1))+P (k → k+1)M (τ 00 |k → k+1) =
                       1        µk             λk
                =           +         mk−1 +         mk+1 .
                    µk + λk   µk + λk        µk + λk
   Çàäà÷à 10.9. Ïóñòü ξt − öåïü Ìàðêîâà ñ íåïðåðûâíûì âðåìåíåì
ñ äâóìÿ ñîñòîÿíèÿìè (1 è 2) è ïðè h → 0

    P (ξh = 2|ξ0 = 1) = αh + o(h), P (ξh = 1|ξ0 = 2) = βh + o(h).

Ïóñòü τk (t), k = 1, 2,  ñóììàðíàÿ äëèòåëüíîñòü ïðåáûâàíèÿ â ñî-
ñòîÿíèè k çà âðåìÿ t öåïè Ìàðêîâà. Ñîñòàâèòü äèôôåðåíöèàëüíîå
óðàâíåíèå äëÿ

                      fk (t, s) = M (eis(τ1 (t)−τ2 (t)) |ξ0 = k).

   Ðåøåíèå.
                f1 (t + h) = M (eis(τ1 (t+h)−τ2 (t+h)) |ξ0 = 1) =

        = M (eis(τ1 (h)−τ2 (h)+τ1 (t+h)−τ1 (h)−(τ2 (t+h)−τ2 (h))) |ξ0 = 1) =
                               X
                            =       P (ξh = k|ξ0 = 1)
                                  k
        is(τ1 (h)−τ2 (h))   is(τ1 (t+h)−τ1 (h)−(τ2 (t+h)−τ2 (h))
  ·M (e             ·e                                    |ξ0 = 1, ξh = k) =
   X
      P (ξh = k|ξ0 = 1) · M (eis(τ1 (h)−τ2 (h)) |ξ0 = 1, ξh = k) · fk (t, s) =
    k

         (1 − αh + · · · )eish f1 (t, s) + (αh + · · · )(1 + · · · )f2 (t, s).
                    ∂
                       f1 (t, s) = (is − α)f1 (t, s) + αf2 (t, s).
                    ∂t


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