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 $\sim 1/t^{\alpha }$ with $0<\alpha \le 1$ 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 $\mathcal{T}$ until 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 $\alpha ∊\left(0.5,\infty \right)$, 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.

Figure 1

Schematic illustration of the chain of events ${\cap }_{k=0}^{n\left(t\right)}{\mathcal{A}}_k$, leading to the occurrence of $\mathcal{T}>t$, where $\mathcal{T}$ is the “busy time duration” until the target task is performed.

Figure 2

(Color online) Typical realization of the stochastic process $V\left(k\right)$ defined by expression (10) as a function of the discrete argument $k$ indexing the successive tasks appearing after $t=0$ at which the target task has been initiated. The realization shown here obeys the inequality $V\left(k\right)>-{\eta }_0^{\prime }$ defining $Q\left(t\right)$ in Eq. (14) over the whole time interval shown.

Figure 3

(Color online) Dependence of $\kappa \left(\rho ;\beta \right)$ given by Eq. (35) as a function of $\rho$ for three values of the normalized drift parameter $\beta =0.01;0.05;0.1$. Dashed straight line corresponds to the pure power law $\sim {\rho }^{-3/2}$. The larger the value of $\beta$, the narrower the interval in $\rho$ for which the intermediate power asymptotic regime holds.

Figure 4

(Color online) Complementary cumulative distribution $\mathcal{Q}\left(\rho ;\beta \right)$ for $\beta =\pm {10}^{-3}$ as a function of the normalized waiting time $\rho$ for completing the target task, demonstrating the qualitatively different asymptotic behavior of $\mathcal{Q}\left(\rho ;\beta \right)$ for $\beta >0$ and for $\beta <0$. Dashed straight line shows the asymptotic power law (41) corresponding to the balanced case $\beta =0$.

Figure 5

(Color online) Dependence of the normalized pdf $\overline{\kappa }\left(\rho ;{\beta }_0\right)$ given by expression (48) of the waiting times until the completion of the target task as a function of the normalized time $\rho$ for ${\beta }_0=0.01;0.1;1$. For the smaller values of $\beta$, one can observe the intermediate asymptotic law $\sim {\rho }^{-3/2}$ progressively crossing over to the asymptotic power law tail $\sim {\rho }^{-2}$.

Figure 6

(Color online) Dependence of the averaged cumulative distribution $\overline{\mathcal{Q}}\left(\rho ;{\beta }_0\right)$ as a function of the normalized time $\rho$. Solid lines correspond to ${\beta }_0=0.1;0.05;0.01;0.005;0.001$ (top to bottom). Dashed straight line is the asymptotic power law (41).

Figure 7

(Color online) Plot of the limiting probability ${\overline{\mathcal{Q}}}_{\infty }\left({\beta }_0\right)$ preventing shaping of the power law of the average complementary distribution $\overline{\mathcal{Q}}\left(\rho ;{\beta }_0\right)$ for $\rho \to \infty$

Figure 8

(Color online) Dependence of the complementary cumulative distribution ${\overline{\mathcal{Q}}}_{-}\left(\rho ;{\beta }_0\right)$ given by Eq. (55) as a function of the normalized waiting time $\rho$ until the completion of the target task. Top to bottom: ${\beta }_0=0.001;0.01;0.1$. One can observe the crossover from $\sim {\rho }^{-1/2}$ to $\sim {\rho }^{-1}$ for ${\beta }_0=0.01$ and the unique power law $\sim {\rho }^{-1}$ for ${\beta }_0=0.1$

Figure 9

(Color online) Plot of the pdf $\kappa \left(y,\alpha \right)$ given by Eq. (82) as a function of the reduced variable $y=t/{\chi }^{1/\alpha }$ for $\alpha =0.7$. The two dashed lines correspond to the intermediate asymptotic regime and to the tail asymptotic law given by Eq. (85).

Figure 10

(Color online) Dependence of the normalized complementary cumulative distribution $\mathcal{K}\left(y,\alpha \right)$ given by Eq. (84), for $\alpha =0.7$, as a function of the normalized time $y=t/{\chi }^{1/\alpha }$. Dashed straight line corresponds to the asymptotic power law $\sim y^{-\alpha }$. The two dotted straight lines are the power laws $\sim y^{-1/3}$ and $\sim y^{-1/4}$ to help the eyes suggest the presence of an intermediate apparent slow power law decay over some limited time range.