Bounds for the distribution of some functionals of processes with $\varphi$-sub-Gaussian increments

Bounds for the distribution of some functionals of a stochastic process $\{X(t),t\in T\}$ belonging to the class $V(\varphi ,\psi )$ are obtained. An example of the functionals studied in the paper is given by \[ \mathsf {F}\left \{\sup _{s\le t; s,t \in B}(X(t)-X(s)-(f(t)-f(s)))>x\right \}, \] where $f(t)$ is a continuous function that can be viewed as a service output rate of a queue formed by the process $X(t)$. For the latter interpretation, the bounds can be viewed as upper estimates for the buffer overflow probabilities with buffer size $x>0$. The results obtained in the paper apply to Gaussian stochastic processes. As an example, we show an application for the generalized fractional Brownian motion defined on a finite interval.