Structure of the big cone and Seshadri constant on Fano manifolds

I would like to know something about the following two questions.

1. Given $$X$$ Fano manifold and $$L$$ an ample line bundle on $$X$$, we define $$\begin{gather} \sigma(L,x):=\sup\{t>0\, :\, \mu^{*}L-tE \,\, \mbox{is big}\, \},\\ \end{gather}$$ for any $$x\in X$$ where $$\mu:Y\to X$$ is the blow-up at $$x$$ and $$E$$ is the exceptional divisor. Is there any known upper bound on $$\sigma(L):=\sup_{x\in X}\sigma(L,x)\, ?$$

2. In the same setting of point one and with the same notations, letting $$\begin{gather} \epsilon(L,x):=\sup\{t>0:\, \mu^{*}L-tE\, \,\mbox{is ample}\,\} \end{gather}$$ to be the one-point Seshadri constant of $$L$$ at $$x\in X$$, is there any known lower bound on $$\epsilon(L):=\inf_{x\in X} \epsilon(L,x)$$?

An uniform bound depending uniquely on $$n:=\dim X$$,$$(L^n)$$ and on $$(-K_{X}^{n})$$ would be great, but I believe it may be too optimistic. So any comment is very welcome, also regarding particular Fano manifolds.

An important remark. It is possible to prove that $$\sigma(L)\epsilon(L)^{n-1}\leq (L^{n})$$ so the second question is stronger than the first one.

As particular example, consider $$X=S_{r}$$ to be a surface given by the blow-up of $$P^{2}$$ at $$r$$-general point $$\{p_{1},\dots,p_{r}\}$$ for $$1\leq r\leq 9$$ and let $$L:=3H-E_{1}-\cdots-E_{r-1}-\delta E_{r}$$ for $$\delta$$ rational such that $$dL$$ is ample for $$d$$ natural number divisible enough. What is it known about $$\begin{gather} \sigma(dL),\epsilon(dL)? \end{gather}$$ Note that $$S_{r}$$ is a Del Pezzo surface for $$r\leq 8$$, while $$S_{9}$$ is not a Fano manifold but I would like to known more about the quantity described above also for this manifold.