Effects of Diversity and Procrastination in Priority Queuing Theory: the Different Power Law Regimes

TL;DR

This paper models the long-term decay of unpatched software and cyber threats using priority queuing theory, identifying a bifurcation parameter that explains different power law regimes and the impact of procrastination on persistence.

Contribution

It introduces a simple priority queuing model with a bifurcation parameter to explain power law decay regimes and incorporates diversity and procrastination effects.

Findings

Power law decay with exponent 1/2 for a homogeneous population.

Diversity of parameters leads to a range of decay exponents.

Procrastination causes even slower decay and longer persistence.

Abstract

Empirical analysis show that, after the update of a browser, the publication of the vulnerability of a software, or the discovery of a cyber worm, the fraction of computers still using the older version, or being not yet patched, or exhibiting worm activity decays as power laws $\sim 1/t^{\alpha}$ with $0 < \alpha \leq 1$ over time scales of years. We present a simple model for this persistence phenomenon framed within the standard priority queuing theory, of a target task which has the lowest priority compared with all other tasks that flow on the computer of an individual. We identify a "time deficit" control parameter $\beta$ and a bifurcation to a regime where there is a non-zero probability for the target task to never be completed. The distribution of waiting time ${\cal T}$ till the completion of the target task has the power law tail $\sim 1/t^{1/2}$, resulting from a first-passage solution of an equivalent Wiener process. Taking into account a diversity of time deficit parameters in a population of individuals, the power law tail is changed into $1/t^\alpha$ with $\alpha\in(0.5,\infty)$, including the well-known case $1/t$. We also study the effect of "procrastination", defined as the situation in which the target task may be postponed or delayed even after the individual has solved all other pending tasks. This new regime provides an explanation for even slower apparent decay and longer persistence.