  Gene expression in individual cells is highly variable and sporadic, often
resulting in the synthesis of mRNAs and proteins in bursts. Bursting in gene
expression is known to impact cell-fate in diverse systems ranging from latency
in HIV-1 viral infections to cellular differentiation. It is generally assumed
that bursts are geometrically distributed and that they arrive according to a
Poisson process. On the other hand, recent single-cell experiments provide
evidence for complex burst arrival processes, highlighting the need for more
general stochastic models. To address this issue, we invoke a mapping between
general models of gene expression and systems studied in queueing theory to
derive exact analytical expressions for the moments associated with
mRNA/protein steady-state distributions. These moments are then used to derive
explicit conditions, based entirely on experimentally measurable quantities,
that determine if the burst distributions deviate from the geometric
distribution or if burst arrival deviates from a Poisson process. For
non-Poisson arrivals, we develop approaches for accurate estimation of burst
parameters.
