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Semi-Markov Models by Yuriy E Obzherin, Elena G Boyko

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ψ()=0f(x+y)h(y)dy+0π(x,y)dy0f(y+t)h(t)dt,ϕ1(x1,x2)=ρ00f(x1+y)f(x2+y)hr(y)dy++0ψ(x1+t)f(x2+t)hr(t)dt+0ψ(x2+t)f(x1+t)hr(t)dt,ϕ2(x,z)=ϕ3(x,z)=ρ00f(y)f(x+y)νr(y,z)dy++0ψ(t)f(x+t)νr(t,z)dt+0ψ(x+t)f(t)νr(t,z)dt,ϕ6(z)=ϕ7(z)=ρ00dt0f(y)f(t+y)νr(y,t+z)dy++0dt0ψ(y)f(t+y)νr(y,t+z)dy+0dt0ψ(t+y)f(y)νr(y,t+z)dy,

image
where hr(t)=n=1r*(n)(t)image is the density of renewal function Hr(t) of the renewal process generated by RV δ; νr(z,x)=r(z+x)+0zr(z+xs)hr(s)ds is the distribution density of the direct residual time for the renewal process ...

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