We use the techniques of Karchmer and Widgerson KW90] to derive strong lower bounds on the expected parallel time to compute boolean functions by circuits. By average time, we mean the time needed on a self-timed circuit, a model introduced recently by Jakoby, Reischuk, and Schindelhauer, JRS94] in which gates compute their output as soon as it is determined (possibly by a subset of the inputs ...