Counting statistics based on the analytic solutions of the differential-difference equation for birth-death processes

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