An estimate of the probability that the queue length exceeds the maximum for a queue that is a generalized Ornstein–Uhlenbeck stochastic process

We consider the process \[ A(t)=mt+\sigma \int _0^t X(u) d u,\qquad t\geq 0, \] describing the queue length, where $m$ and $\sigma$ are positive constants, $X(u)$ is a $\varphi$-sub-Gaussian generalized Ornstein–Uhlenbeck stochastic process, and \[ \varphi (u)= \begin {cases} u^r, & |u| >1,\\ u^2, & |u|\le 1, \end {cases}\] $r\geq 2$. The classes of $\varphi$-sub-Gaussian and strictly $\varphi$-sub-Gaussian stochastic processes are wider than the class of Gaussian processes and are of interest for modeling stochastic processes appearing in queueing theory and in the mathematics of finance. We obtain an estimate of the probability that the queue length exceeds the maximum allowed for it, namely, \[ \mathsf {P}\left \{\sup _{t\geq 0}\left (A(t) -c t \right )>x \right \}\le L(\gamma ) x^{r/(r-1)}\exp \left \{-\kappa (\gamma )x^{r/(2(r-1))}\right \}, \] where $c>m$ is the service intensity, $x>0$ is the maximum queue length, and $L(\gamma )$ and $\kappa (\gamma )$ are some finite constants.