  We study a queueing system with Erlang arrivals with $k$ phases and Erlang
service with $m$ phases. Transition rates among phases vary periodically with
time. For these systems, we derive the asymptotic periodic distribution of the
level and phase as a function of time within the period. The asymptotic
periodic distribution is analogous to a steady-state distribution for a system
with constant rates. If the time within the period is considered part of the
state, then it is a steady-state distribution. We also obtain waiting time and
busy period distributions. These solutions are expressed as infinite series. We
provide bounds for the error of the estimate obtained by truncating the series.
Examples are provided comparing the solution of the system of ordinary
differential equation with a truncated state space to these asymptotic
solutions involving remarkably few terms of the infinite series.
