# Product number counting statistics from stochastic bursting birth-death processes

## TL;DR

This paper develops a theoretical model to analyze product number fluctuations in non-renewal birth processes with bursting, revealing how burst dynamics and product lifetime variability influence fluctuations over time.

## Contribution

The model quantitatively describes product number statistics in bursting and non-renewal birth-death processes, extending previous work to non-stationary conditions.

## Key findings

- Product number fluctuation decreases with increased product lifetime fluctuation in stationary processes.
- Bursting increases product number fluctuation compared to non-bursting processes.
- Product number fluctuation is finite at time zero in bursting processes, unlike in non-bursting ones.

## Abstract

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.

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Source: https://tomesphere.com/paper/1903.06370