In multi-source status update systems, sources need to be scheduled appropriately to maintain timely communication between each of the sources and the monitor. A cyclic schedule is an age-agnostic schedule in which the sources are served according to a fixed finite transmission pattern, which upon completion, repeats itself. Such a scheme has a low $O(1)$ runtime complexity, which is desirable in large networks. This paper's focus is on designing transmission patterns so as to be used in massive scale networking scenarios involving a very large number of sources, e.g., up to thousands of IoT sources, with service time requirements and weights being heterogeneous in nature. The goal is to minimize the weighted sum age of information (AoI), called weighted AoI, when transmitting users' packets over a channel susceptible to heterogeneous packet errors. The main tool we use is a stochastic modeling framework using either Markov chains (MC) or moment generating functions (MGF), by which we obtain the weighted AoI for a given transmission pattern, which is not straightforward in the presence of packet drops. Using this framework, we provide a lower bound on the weighted AoI for the particular case of two sources, and also an algorithm to attain this lower bound. Then, by using the same framework, we design a cyclic scheduler for general number of sources with reasonable complexity using convex optimization and well-established packet spreading algorithms, and comparatively evaluate the proposed algorithm and existing age-agnostic scheduling schemes for general number of sources (resp.~two sources) when the lower bound is not available (resp.~when it is available). We present extensive numerical results to validate the effectiveness of the proposed approach.

翻译：在多源状态更新系统中，需要合理调度各源节点以维持其与监控器间的及时通信。循环调度是一种与信息年龄无关的调度方案，其按照固定的有限传输模式为源节点提供服务，该模式在完成后将循环重复。此类方案具有较低的$O(1)$运行时复杂度，在大规模网络中极具应用价值。本文重点研究传输模式的设计，以适用于涉及海量源节点（例如多达数千个物联网源）的大规模网络场景，其中服务时间要求与权重具有异构特性。目标是在易受异构数据包错误影响的信道上传输用户数据包时，最小化加权信息年龄总和（称为加权AoI）。我们采用的主要工具是基于马尔可夫链（MC）或矩生成函数（MGF）的随机建模框架，通过该框架可获得给定传输模式下的加权AoI——这在存在丢包的情况下并非易事。利用此框架，我们针对双源特定情况给出了加权AoI的下界，并提供了达到该下界的算法。随后，基于同一框架，我们通过凸优化与成熟的包扩散算法，为任意数量源节点设计了具有合理复杂度的循环调度器，并分别在无法获得下界（任意数量源节点）与可获得下界（双源）的情况下，对所提算法与现有年龄无关调度方案进行了对比评估。我们提供了大量数值结果以验证所提方法的有效性。