In this paper we consider the asymptotic time average variance of the actual waiting time of the PH/PH/1 queue. To this end Poisson's equation plays a major role, since the solution to Poisson's equation is shown to be intimately connected with the desired variance. We derive explicit solutions to Poisson's equation using phase-type methodology, and present some numerical examples.
"The variance constant for the actual waiting time of the $PH/PH/1$ queue." Ann. Appl. Probab. 6 (3) 766 - 777, August 1996. https://doi.org/10.1214/aoap/1034968227