P(x<S1≤x+h,y<S2≤y+h,N(t)=1 lim P(N()=0, N(+h)-N(r)=l N(t)-N(x+h)=0,N(y)-N(t)=0,N(y+h)-N(y=1)/h2 (1)当n=0时 若x≤t,P(S1≤x,N(t)=0)=0 P(SI t)=0) (S1≤x|N(t)=0)P(N(t)=0) P(W(t)<I-tIN(t)=OP(N(t)=0) P(W(t)<I-tP(N(t)=0) (1 当n≥1时 若x<0.P(S1≤x,N(t)=n)=0 若x≥0 P(S1≤x,N(t)=m) P(S1 <aN(t=n)P(N(t)=n P(U(1)≤x)P(N(t)=m)l(x≤)+P(N(t)=n)la (At)e (2)当n=0时 若x≤t,P(S2≤x,N(t)=0) (S2≤x,N()=0) P(N(t)=0,N(x)-N(t)≥2) 当n=1时 若x≤t,P(S2≤x,N(t)=1)=0 若x>t P(S2≤x,N(t)=1) P(S2≤|N(t)=1)P(N(t)=1) P(W(t)<I-tIN(t)=IP(N(t)=1) P(W(t)<I-t)P(N()=1)(3) f(x, y) = lim h→0 P(x < S1 ≤ x + h, y < S2 ≤ y + h, N(t) = 1 h 2 = lim h→0 P(N(x) = 0, N(x + h) − N(x) = 1, N(t) − N(x + h) = 0, N(y) − N(t) = 0, N(y + h) − N(y = 1))/h2 = lim h→0 e −λxλhe−λhe −λ(t−x−h) e −λ(y−h−t)λhe−λh h 2 = λ t e −λ(y−t) 2.29 )µ (1)  n = 0 ž§ e x ≤ t, P(S1 ≤ x, N(t) = 0) = 0 " e x > t P(S1 ≤ x, N(t) = 0) = P(S1 ≤ x|N(t) = 0)P(N(t) = 0) = P(W(t) ≤ x − t|N(t) = 0)P(N(t) = 0) = P(W(t) ≤ x − t)P(N(t) = 0) = (1 − e −λ(x−t) )e −λt = e −λt − e −λx  n ≥ 1 ž e x < 0, P(S1 ≤ x, N(t) = n) = 0 " e x ≥ 0 P(S1 ≤ x, N(t) = n) = P(S1 ≤ x|N(t) = n)P(N(t) = n) = P(U(1) ≤ x)P(N(t) = n)I(x≤t) + P(N(t) = n)I(x>t) = [1 − µ t − x t ¶n ] (λt) ne −λt n! I(x≤t) + (λt) ne −λt n! I(x>t) (2)  n = 0 ž§ e x ≤ t, P(S2 ≤ x, N(t) = 0) = 0 " e x > t P(S2 ≤ x, N(t) = 0) = P(N(t) = 0, N(x) − N(t) ≥ 2) = P(N(t) = 0)P(N(x) − N(t) ≥ 2) = e −λt − e −λx − λ(x − t)e −λx  n = 1 ž e x ≤ t, P(S2 ≤ x, N(t) = 1) = 0 " e x > t P(S2 ≤ x, N(t) = 1) = P(S2 ≤ x|N(t) = 1)P(N(t) = 1) = P(W(t) ≤ x − t|N(t) = 1)P(N(t) = 1) = P(W(t) ≤ x − t)P(N(t) = 1) = (1 − e −λ(x−t) )λte−λt 9