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Bursting and non-renewal processes are common phenomena in birth-deathprocess, yet no theory can quantitatively describe a non-renewal birth processwith bursting. Here, we present a theoretical model that yields the productnumber counting statistics of product creation occurring in bursts and of anon-renewal creation process. When product creation is a stationary process,our model confirms that product number fluctuation decreases with an increasein the product lifetime fluctuation, originating from the non-Poissondegradation dynamics, a result obtained in previous work. Our modeladditionally demonstrates that the dependence of product number fluctuation onproduct lifetime fluctuation varies with time, when product creation is anon-stationary process. We find that bursting increases product numberfluctuation, compared to birth-processes without bursting. At time zero, in aburst-less birth process, product number fluctuation is unsurprisingly found tobe zero, but we discover that, in a bulk creation process characterized bybursting, product number fluctuation is a finite value at time zero. Theanalytic expressions we obtain are applicable to many fields related to thestudy system population, such as queueing models and gene expression.

Product number counting statistics from stochastic bursting birth-death processes