Real-Time Reconstruction of a Counting Process Through First-Come-First-Serve Queue Systems

For the emerging Internet of Things (IoT), one of the most critical problems is the real-time reconstruction of signals from a set of aged measurements. During the reconstruction, distortion occurs between the observed signal and the reconstructed signal due to sampling and queuing delay. We focus on minimizing the average distortion defined as the 1-norm of the difference of the two signals under the scenario that a Poisson counting process is reconstructed in real-time on a remote monitor. We consider the reconstruction under three special sampling policies. For each of the policy, we derive the closed-form expression of the average distortion by dividing the overall distortion area into polygons and analyzing their structures. It turns out that the polygons are built up by sub-polygons that account for distortions caused by sampling and queuing delay. The closed-form expressions of the average distortion help us find the optimal sampling parameters that achieve the minimum distortion. In addition, we propose an interpolation algorithm to further decrease the average distortion and give its lower-bound on distortion for one of the three sampling policies. Simulation results are provided to validate our conclusion.