The first time independent, parallel processes consisting of two alternating renewal functions, say function zero and one, will be in phase zero together under equilibrium and non-equilibrium starting conditions is found. With function one an exponential distribution and function zero a non-lattice type, the distribution of the aforementioned time is found by taking advantage of a point of renewal. Further results are derived when both functions are exponential with the same parameter value by likening the situation to an Ehrenfest model