Masuda N, Kim J S, Kahng B
Graduate School of Information Science and Technology, The University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-8656, Japan.
Phys Rev E Stat Nonlin Soft Matter Phys. 2009 Mar;79(3 Pt 2):036106. doi: 10.1103/PhysRevE.79.036106. Epub 2009 Mar 16.
Recently increased accessibility of large-scale digital records enables one to monitor human activities such as the interevent time distributions between two consecutive visits to a web portal by a single user, two consecutive emails sent out by a user, two consecutive library loans made by a single individual, etc. Interestingly, those distributions exhibit a universal behavior, D(tau) approximately tau(-delta) , where tau is the interevent time, and delta approximately 1 or 32 . The universal behaviors have been modeled via the waiting-time distribution of a task in the queue operating based on priority; the waiting time follows a power-law distribution P(w)(tau) approximately tau(-alpha) with either alpha=1 or 32 depending on the detail of queuing dynamics. In these models, the number of incoming tasks in a unit time interval has been assumed to follow a Poisson-type distribution. For an email system, however, the number of emails delivered to a mail box in a unit time we measured follows a power-law distribution with general exponent gamma . For this case, we obtain analytically the exponent alpha , which is not necessarily 1 or 32 and takes nonuniversal values depending on gamma . We develop the generating function formalism to obtain the exponent alpha , which is distinct from the continuous time approximation used in the previous studies.
最近，大规模数字记录的可获取性不断提高，使得人们能够监测人类活动，比如单个用户两次连续访问门户网站的事件间隔时间分布、用户发出的两封连续电子邮件、单个个体的两次连续图书馆借阅等等。有趣的是，这些分布呈现出一种普遍行为，即(D(\tau)\approx\tau^{-\delta})，其中(\tau)是事件间隔时间，且(\delta\approx1)或(32)。这种普遍行为已通过基于优先级运行的队列中任务的等待时间分布进行建模；等待时间遵循幂律分布(P_w(\tau)\approx\tau^{-\alpha})，其中(\alpha = 1)或(32)取决于排队动态的细节。在这些模型中，假设单位时间间隔内传入任务的数量遵循泊松型分布。然而，对于一个电子邮件系统，我们测量到的单位时间内投递到邮箱的电子邮件数量遵循具有一般指数(\gamma)的幂律分布。对于这种情况，我们通过解析得到指数(\alpha)，它不一定是(1)或(32)，而是取决于(\gamma)取非通用值。我们开发了生成函数形式体系来获得指数(\alpha)，这与先前研究中使用的连续时间近似不同。
