Remote Estimation Over a Packet-Drop Channel With Markovian State

Abstract

We investigate a remote estimation problem in which a transmitter observes a Markov source and chooses the power level to transmit it over a time-varying packet-drop channel. The channel is modeled as a channel with Markovian state where the packet drop probability depends on the channel state and the transmit power. A receiver observes the channel output and the channel state and estimates the source realization. The receiver also feeds back the channel state and an acknowledgment for successful reception to the transmitter. We consider two models for the source-finite state Markov chains and first-order autoregressive processes. For the first model, using ideas from team theory, we establish the structure of optimal transmission and estimation strategies and identify a dynamic program to determine optimal strategies with that structure. For the second model, we assume that the noise process has unimodal and symmetric distribution. Using ideas from majorization theory, we show that the optimal transmission strategy is symmetric and monotonic and the optimal estimation strategy is like Kalman filter. Consequently, when there are a finite number of power levels, the optimal transmission strategy may be described using thresholds that depend on the channel state. Finally, we propose a simulation based approach (Renewal Monte Carlo) to compute the optimal thresholds and optimal performance and elucidate the algorithm with an example.

Sampling of the Wiener Process for Remote Estimation Over a Channel With Random Delay

“…Some extensions of this research were reported in [24]- [26]. In [27], [28], [44], Chakravorty and Mahajan considered optimal communication scheduling and remote estimation over a few channel models, where it was proved that a threshold-based transmitter and a Kalman-like estimator are jointly optimal. In [45], Mahajan and Teneketzis provided a dynamic programming based numerical method to find the optimal transmission and estimation strategies over a channel with a constant transmission time but not over a queue with random service time.…”

“…One key difference between this paper and previous studies on remote estimation, e.g., [9]- [28], [44], [45], is that the channel between the sampler and estimator is modeled as a queue with random service times. As we will see later, this queueing model affects the structure of the optimal sampler.…”

Optimal Causal Rate-Constrained Sampling of the Wiener Process

“…In contrast to the scenarios in [1]- [9], where the communication channel is assumed to be perfect, [10]- [13] consider imperfect communication channels. Sun et al [10] proved that a symmetric threshold policy remains optimal even when the samples of the Wiener process experience an i.i.d.…”

“…For discrete-time first-order autoregressive Markov processes considered in [7]- [8], Ren et al [12] introduced a fading channel between the encoder and the decoder, where a successful transmission depends on both the channel gains and the transmission power, and found the optimal encoding and decoding policies that minimize an infinite horizon cost function combining the MSE and the power usage. For first-order autoregressive sources considered in [7][8] [12], Chakravorty and Mahajan [13] further proved that the optimal estimation policy is a Kalman-like filter and that the optimal sampling policy is symmetric threshold policy when the communication channel is a packet-drop channel with Markovian states.…”

“…Here we slightly abuse the notation: we have used f T in Definition 2(ii), and have shown in Proposition 1 that the compressing policy f T can be simplified to(13).…”