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

Empirical analyses show that after the update of a browser, or the publication of the vulnerability of a software, or the discovery of a cyber worm, the fraction of computers still using the older browser or software version, or not yet patched, or exhibiting worm activity decays as a power law approximately 1/t(alpha) with 0<alpha<or=1 over a time scale 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 to 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 nonzero probability for the target task to never be completed. The distribution of waiting time T until the completion of the target task has the power law tail approximately 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 is an element of (0.5,infinity), 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 regime provides an explanation for even slower apparent decay and longer persistence.