Suppose that $B_1, B_2,\ldots, B_n$ are a series of positive independent and identically distributed random variables. The moment generating function (MGF) of $B_i$'s is known, denoted as $M_{B}(\theta)$. I'm looking for an upper bound on the probability that the maximum of the $B_i$'s is above a threshold when $n\rightarrow\infty$, possibly in the following form: $$Pr\{\lim_{n\rightarrow\infty}\max_{i=1,\ldots,n}(B_1,\ldots,B_n)\geq x\}\leq\alpha(\theta,x)\cdot e^{\gamma(\theta,x)}$$ where $\alpha(\theta,x)$ and $\gamma(\theta,x)$ are (decreasing) functions of $x$.
If a general result is difficult to obtain, what if we know that $B_i$'s denote the duration of the busy period of a queueing system? The question is to bound the probability that the busy period exceeds a given threshold $x$ when time goes to infinity. For example, suppose that $B_i$ is the busy period of an M/M/1 queue. The Laplace transform for $B_i$'s is known to be [Eqn. (5.144) Kleinrock1975 "Queueing Systems"]: $$G^*(s)=\frac{\mu+\lambda+s-\sqrt{(\mu+\lambda+s)^2-4\mu\lambda}}{2\lambda}$$ where $\lambda$ and $\mu$ are the arrival and service rates, respectively. $\lambda<\mu$ such that the queue is stable, and the busy period is a renewal process. Then we have $M_B(\theta)=G^*(-\theta).$