  Consider a first-come, first-served single server queue with an initial
workload $x>0$ and customers who arrive according to an inhomogeneous Poisson
process with rate function $\lambda:[0,\infty)\rightarrow[0,\lambda_h ]$ for
some $\lambda_h>0$. For each $i\in\mathbb{N}$, let $S_i$ (resp., $Y_i$) be the
service (resp., patience) time of the $i$'th customer and assume that
$(S_1,Y_1),(S_2,Y_2),\ldots$ is an iid sequence of bivariate random vectors
with non-negative coordinates. A customer joins if and only if his patience
time is not less than his prospective waiting time (i.e., the left-limit of the
workload process at his arrival epoch). Let $\tau(x)$ be the first time when
the system becomes empty and let $N^*_\lambda(\cdot)$ be the arrival process of
those who join the queue. In the present work we suggest a novel coupling
technique which is applied to derive stochastic upper bounds for the
functionals: \begin{equation*}
  \int_0^{\tau(x)}g\circ W_x(t){\rm d}t\ \ \text{and}\ \ \int_0^{\tau(x)}g\circ
W_x(t){\rm d}N^*_\lambda(t)\,, \end{equation*} where $W_x(\cdot)$ is the
workload process in the queue and $g(\cdot)$ is any lower semi-continuous
function. We also demonstrate how to utilise these bounds via some examples
under the additional assumption that $\lambda(\cdot)$ is periodic.
