  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.
