renewal equation for $\sim U(0,1)$ interarrivals

Question

renewal equation for $\sim U(0,1)$ interarrivals should be

$m(t)=t+\int_0^t{m(t-s)f(s)ds}$

how can this be solved?

can I make substitution $y=t-s$ to get

$m(t)=t+\int_0^t{m(y)f(y)dy}$ if all I know is that t>1 ?

What will make this substitution valid?

Answer 1

You need to correctly do the substitution $y=t-s$ to get

$$m(t)=t+\int_{-\infty}^{t}m(y)dy$$

And now differentiate in $t$:

$$m'=1+m$$

Which you can solve easily. You'll need an initial value of $m(t)$ which you can deduce from the limit condition $m(t)/t$ for renewal processes.

By the way you should check to see if you've written things down right. There should be a density function inside the integral: $m(t-s)f(s)$ and the density is 0 when $s>1$.