Title: Steady-state analysis of single exponential vacation in a $PH/MSP/1/\infty$ queue using roots,
Abstract: We consider an infinite-buffer single-server queue where inter-arrival times
are phase-type ($PH$), the service is provided according to Markovian service
process $(MSP)$, and the server may take single, exponentially distributed
vacations when the queue is empty. The proposed analysis is based on roots of
the associated characteristic equation of the vector-generating function (VGF)
of system-length distribution at a pre-arrival epoch. Also, we obtain the
steady-state system-length distribution at an arbitrary epoch along with some
important performance measures such as the mean number of customers in the
system and the mean system sojourn time of a customer. Later, we have
established heavy- and light-traffic approximations as well as an approximation
for the tail probabilities at pre-arrival epoch based on one root of the
characteristic equation. At the end, we present numerical results in the form
of tables to show the effect of model parameters on the performance measures.