Suppose that a stationary random process $S(t)$ can be characterized as the figure below, which for most of the time is a straight line $S(t)=c\cdot t$, but occasionally would "stall" for a period of time $B_i$. The stalling time $B_i$'s are i.i.d. random variables, with the first and second moments known as $\mathrm{E}[B]$ and $\mathrm{E}[B^2]$, respectively. The appearance of $B_i$'s can be modeled as a renewal process. In a fraction of $\rho$ time the system would see a $B_i$. Let $N(t)$ be the number of $B_i$'s in a duration of time $t$. $\lim_{t\rightarrow \infty}\frac{N(t)}{t}=\frac{\rho}{\mathrm{E}[B]}$.

Given the above conditions, is it possible to (approximately) determine an affine envelop for the negative moment generating function of $S(t)$? That is, for $\theta>0$, determine a function $\sigma(\theta)$ such that:

$\mathrm{E}\left[e^{-\theta S(t)}\right]\leq e^{-\theta(c\cdot t-\sigma(\theta))}.$

For some background, if think of $S(t)$ as the service process of a server, this kind of envelop (read dash line) corresponds to a lower bound of $S(t)$ known as the serivce curve of the server in the language of stochastic network calculus. $\sigma(\theta)$ may correspond to the latency incurred by the service and $\theta$ is some decaying rate.

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