On the Timeliness of the Stalest Stream Among Multiple Status Updating Streams.

In practical status updating systems, most decisions made at monitors are based on diverse data streams. Due to the cask effect, it can be cognised that the effectiveness of decisions is often constrained by the stalest one, i.e., the straggler among all the streams. This work studies the statistical characteristics of age of the stalest information (AoSI) which describes the timeliness of the stalest stream. The AoSI is defined as the time elapsed since the latest successfully received update of the currently stalest stream among all different streams at the monitor was generated. Peak age of the stalest information (PAoSI) is also studied for evaluating the worst cases, i.e., the peaks of AoSI process. We develop an analytical approach to derive the AoSI and PAoSI based on the per-stream age of information (AoI) and peak age of information (PAoI), for the multi-stream single-monitor system with separate status updating. In particular, to comprehensively characterize the timeliness of the stalest stream, the distributions of AoSI and PAoSI are derived in closed-form for the general multi-stream system. Moreover, we concisely derive the explicit expressions of the distributions and averages of AoSI and PAoSI, upon a typical two-stream case with the classical automatic repeat-request protocol. Finally, the accuracy of the theoretical analyses is validated by the numerical results. Appropriateness and advantages of the AoSI (PAoSI) are elaborated by comparing with the maximum average AoI (PAoI), i.e., the maximal one among the averages of all the per-stream AoIs (PAoIs).