jmc

First, we note that $x$ must be positive, since otherwise $\lceil x\rceil \cdot x$ is nonpositive. Now, knowing that $\lceil x\rceil -1<x\le \lceil x\rceil ,$ we see that $\lceil x\rceil$ must be $12,$ since $11\cdot 11<135\le 12\cdot 12.$

Now we see that $\lceil x\rceil \cdot x=12x=135,$ so $x=\frac{135}{12}=\boxed{11.25}.$