We consider a multiclass production system processed by a single machine with cyclic order. Each part belongs to one of several classes and arrives at each queue in a Poisson process. The processing discipline at each queue is either exhaustive or gated. We introduce a new approach to analysis of mean waiting times for each queue. We formulate mean waiting times as cost functions on state of the system. The mean waiting times are obtained by solving a set of O($J^2$) linear equations, where J is the number of queues in the system. We also obtain the mean number of parts of each queue in the system at an arbitrary time. Based on this result, we plot the numerical values of the mean waiting times for several parameter settings.