We consider a situation that multiple monitoring applications (each with a different sensor-monitor pair) compete for a common service resource such as a communication link. Each sensor reports the latest state of its own time-varying information source to its corresponding monitor, incurring queueing and processing delays at the shared resource. The primary performance metric of interest is the age of information (AoI) of each sensor-monitor pair, which is defined as the elapsed time from the generation of the information currently displayed on the monitor. Although the multi-source first-come first-served (FCFS) M/GI/1 queue is one of the most fundamental model to describe such competing sensors, its exact analysis has been an open problem for years. In this paper, we show that the Laplace-Stieltjes transform (LST) of the stationary distribution of the AoI in this model, as well as the mean AoI, is given by a simple explicit formula, utilizing the double Laplace transform of the transient workload in the M/GI/1 queue.

翻译：我们考虑多个监控应用（每个应用包含一个不同的传感器-监视器对）竞争同一服务资源（如通信链路）的场景。每个传感器将其自身时变信息源的最新状态报告给对应的监视器，导致共享资源处产生排队和处理延迟。关注的性能指标是每个传感器-监视器对的信息年龄（AoI），其定义为监视器当前显示信息自生成以来所经过的时间。尽管多源先到先服务（FCFS）M/GI/1队列是描述此类竞争传感器的基础模型之一，但其精确分析多年来一直是一个未解决的问题。在本文中，我们利用M/GI/1队列中瞬时工作量的双拉普拉斯变换，证明了该模型中AoI平稳分布的拉普拉斯-斯蒂尔切斯变换（LST）以及平均AoI可由一个简单的显式公式给出。