Estimating tail probabilities of delays and queue lengths for telecommunication applications

在大多数服务行业，特别是电信系统中，服务提供商通常希望确定与性能度量相关的单个统计数据，例如队列长度或等待时间，以用作用户服务质量（QoS）的指标。通常用作单一度量的统计量是平均值。然而，就用户而言，平均值被视为确定QoS目的的性能度量的良好指标。尾部概率更有意义，被认为是更好的衡量标准。例如，告诉用户他们的等待时间不会以一定的概率（尾部概率）超过给定值，比告诉用户他们的平均等待时间是一个特定的值更有意义。估计性能度量的尾部概率对于服务提供商规划其系统至关重要。他们需要这样的统计数据来评估他们的系统被用户认为有多成功，这最终决定了哪些用户订阅了系统，从而转化为服务提供商的收入（和利润）。然而，挑战在于如何在不计算整个分布的情况下确定度量的尾部概率，因为在实际的现实生活系统中获得整个分布通常是非常麻烦的。本提案的目的是开发精确和近似的技术来估计具有轻尾的系统中测量的尾部概率，即尾部具有几何衰减特性。对于这种情况，我们只需要计算两个参数；衰减速率和衰减（渐近）常数。有几种计算衰变率的方法，但几乎没有计算衰变常数的方法。我们计划改进现有的计算衰减率的技术，并开发计算单节点队列衰减常数的近似技术，然后将结果扩展到用于电信系统应用的串联队列。

In most service industries, especially the telecommunication systems, service providers usually want to determine a single statistic associated with a performance measure, such as queue length or waiting times,to be used as an indicator of users' quality of service (QoS).  The statistic often used as a single measure is the mean value.  Mean values are however viewed as not good indicator of performance measure for purpose of determining QoS as far as a user is concerned.  Tail probabilities are more meaningful and accepted as better measures.  For example, telling a user that their waiting time will not exceed a given value with a certain probability (tail probability) is more meaningful than been told that their average (mean) waiting time is of a particular value. Estimating the tail probabilities of performance measures is critical for service providers in planning their systems.  They need such statistic in order to assess how successful their system is perceived to be by users and this is what in the end determines what users subscribe to the system which translates to revenue (and profit) to service providers. The challenge, however, is how to determine the tail probabilities of a measure without computing the whole distribution, since it is usually very cumbersome to obtain a whole distribution in a practical real life system.  The aim of this proposal is to develop exact and approximate techniques for estimating the tail probabilities for measures in systems that have light tails, i.e for which the tail has a geometric decay property.  For such cases we only need to compute two parameters; the decay rate and the decay (asymptotic) constant.  There are several methods for computing decay rates, but hardly any for computing decay constants. We plan to improve existing techniques for computing decay rates and develop approximate techniques for computing decay constants for single node queues, and then extend the results to tandem queues for applications in telecommunication systems.