Nontrivial Finite-Size Effects in Stochastic Residence Times: Lessons from Queues and Bacteria

Abstract

Queuing models, though originating in operations research, have become powerful metaphors for understanding stochastic processes in physics and biology. In particular, the dynamics of bacterial twitching on surfaces reveal residence times that span a surprisingly wide range, suggesting underlying mechanisms of delay and congestion. To capture this, we analyze the canonical $M/M/1/K$ queue—a single-server system with Poisson arrivals, exponential service times, and finite buffer size $K$. While the infinite-capacity version of this model is known to generate a scale-free distribution of waiting times when arrival and service rates balance, the consequences of finite system size have remained unclear. Our analysis shows that truncating the buffer leads to a sharp modification of the statistics: the power-law tail is preserved but terminated by an exponential cutoff, whose characteristic scale grows as $\sim K^{1.9}$. This nontrivial finite-size scaling highlights how microscopic constraints reshape macroscopic distributions, offering a general framework for interpreting residence-time spectra in stochastic transport—from bacterial surface motility to broader classes of nonequilibrium systems.