  Birth-death processes take place ubiquitously throughout the universe. In
general, birth and death rates depend on the system size (corresponding to the
number of products or customers undergoing the birth-death process) and thus
vary every time birth or death occurs, which makes fluctuations in the rates
inevitable. The differential-difference equation governing the time evolution
of such a birth-death process is well established, but it resists solving for a
non-asymptotic solution. In this work, we present the analytic solution of the
differential-difference equation for birth-death processes without
approximation. The time-dependent solution we obtain leads to an analytical
expression for counting statistics of products (or customers). We further
examine the relationship between the system size fluctuations and the birth and
death rates, and find that statistical properties (variance subtracted by mean)
of the system size are determined by the mean death rate as well as the
covariance of the system size and the net growth rate (i.e., the birth rate
minus the death rate). This work suggests a promising new direction for
quantitative investigations into birth-death processes.
