  Bursting and non-renewal processes are common phenomena in birth-death
process, yet no theory can quantitatively describe a non-renewal birth process
with bursting. Here, we present a theoretical model that yields the product
number counting statistics of product creation occurring in bursts and of a
non-renewal creation process. When product creation is a stationary process,
our model confirms that product number fluctuation decreases with an increase
in the product lifetime fluctuation, originating from the non-Poisson
degradation dynamics, a result obtained in previous work. Our model
additionally demonstrates that the dependence of product number fluctuation on
product lifetime fluctuation varies with time, when product creation is a
non-stationary process. We find that bursting increases product number
fluctuation, compared to birth-processes without bursting. At time zero, in a
burst-less birth process, product number fluctuation is unsurprisingly found to
be zero, but we discover that, in a bulk creation process characterized by
bursting, product number fluctuation is a finite value at time zero. The
analytic expressions we obtain are applicable to many fields related to the
study system population, such as queueing models and gene expression.
