  The scaling properties of the roughness of surfaces grown by two different
processes randomly alternating in time, are addressed. The duration of each
application of the two primary processes is assumed to be independently drawn
from given distribution functions. We analytically address processes in which
the two primary processes are linear and extend the conclusions to non-linear
processes as well. The growth scaling exponent of the average roughness with
the number of applications is found to be determined by the long time tail of
the distribution functions. For processes in which both mean application times
are finite, the scaling behavior follows that of the corresponding cyclical
process in which the uniform application time of each primary process is given
by its mean. If the distribution functions decay with a small enough power law
for the mean application times to diverge, the growth exponent is found to
depend continuously on this power law exponent. In contrast, the roughness
exponent does not depend on the timing of the applications. The analytical
results are supported by numerical simulations of various pairs of primary
processes and with different distribution functions. Self-affine surfaces grown
by two randomly alternating processes are common in nature (e.g., due to
randomly changing weather conditions) and in man-made devices such as
rechargeable batteries.
