优先权理论中的多样性和拖延效应：不同的权力法律制度
Effects of Diversity and Procrastination in Priority Queuing Theory: the Different Power Law Regimes
摘要：
经验分析表明，在更新浏览器、发布软件漏洞或发现网络蠕虫后，仍使用旧版本或尚未修补或显示蠕虫活动的计算机的比例随着时间尺度的年数按幂律$\sim 1/t^α$与$0 < α\leq 1$衰减。我们提出了一个简单的模型框架内的标准优先级排队理论，目标任务，它具有最低的优先级相比，所有其他任务的计算机上的个人流动的这种持久性现象。我们确定了一个“时间赤字”的控制参数$β$和分叉的制度，有一个非零的概率，目标任务永远不会完成。直到目标任务完成的等待时间${\cal T}$的分布具有幂律尾部$\sim 1/t^{1/2}$，这是由等效Wiener过程的首次通过解得到的。考虑到在一个人口的个人的时间赤字参数的多样性，幂律尾部被改变为$1/t^α$与$α\in(0.5,\infty)$，包括著名的情况$1/t$。我们还研究了“拖延”的影响，定义为目标任务可能被推迟或延迟的情况下，即使个人已经解决了所有其他悬而未决的任务。这种新的机制为更慢的明显衰减和更长的持久性提供了解释。
英文摘要：
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 $\sim 1/t^α$ with $0 < α\leq 1$ 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 $β$ 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 $1/t^α$ with $α\in(0.5,\infty)$, 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.
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