In this paper, we derive the mean message waiting times in a local area network that uses the Demand-Priority Access Method. We model the system as a two-priority M/G/1 queue with switchover time between service periods. This switchover time accounts for the polling and port selection performed by the repeater after each message transmission. The service discipline is non-preemptive and the length of the switchover time is dependent upon the priority class of the preceding message served as well as that of the message to be served next. The dependency in the switchover times is motivated by the polling and port selection operation of the protocol and it makes the analysis much more involved. In order to avoid the complexities of an exact analysis, we make some independence assumptions and thus obtain an approximate solution. Laplace-Stieltjes transforms of the stationary probability distribution functions for the waiting time of high- and normal-priority messages are derived, and subsequently, the expressions for the mean message waiting times. Numerical results computed using these expressions are verified using simulations which model the actual protocol. These numerical results which are shown to be accurate can be easily computed with widely available mathematical software.