Optimal Sampling for Data Freshness: Unreliable Transmissions with Random Two-way Delay

Abstract

In this paper, we study a sampling problem, in which freshly sampled data is sent to a remote destination via an unreliable channel, and acknowledgments are sent back on a feedback channel. Both the forward and feedback channels are subject to random transmission times. We optimize the sampling strategy at the source (e.g., a sensor), aiming to enhance the freshness of data samples delivered to the destination (e.g., an estimator). This sampling problem is motivated by a distributed sensing system, where an estimator estimates a signal by combining noisy signal observations collected from a local sensor and accurate signal samples received from a remote sensor. We show that the optimal estimation error is an increasing function of the age of received signal samples. The optimal sampling problem for general non-decreasing age functions is formulated as an MDP with an uncountable state space. An exact solution to this problem is derived, which has a simple thresholdtype structure. The threshold can be calculated by low-complexity bisection search and fixed-point iterations. We find that, after a successful packet delivery, the optimal sampler may wait before taking the next sample and sending it out, whereas no waiting time should be added if the previous transmission failed. 1 This assumption arises from the stop-and-wait Automatic Repeat Request (ARQ) mechanism. When the backward delay X i,j = 0, the policy that samples ahead of receiving feedback is always suboptimal. Because such a policy takes a new sample when the channel is busy and can be replaced by a policy that samples at exact time of receiving feedback [5]. When X i,j = 0, however, it may be optimal to transmit before receiving acknowledgement, which is out of the scope of this paper.

Query-Age-Optimal Scheduling Under Sampling and Transmission Constraints

“…] be a solution of the LP (11). By results of [5], an optimal stationary randomized policy for each source i, π * i ≜ {f i (s, a)}, is given by…”

“…To capture such relative freshness, we have used the AoI difference between the monitor and the transmitter for Source 1, and the AoI at the monitor itself for Source 2. Complexity analysis: The main complexity of finding the heuristic policy comes from solving the LP (11), which is (only) linear in the number of sources. Thus, the proposed policy has low complexity, and as numerically shown in the next section, obtains near-optimal performance.…”

“…Related work: Recently, the information freshness has been studied from the sampling and/or scheduling optimization perspectives by using the AoI metric in, e.g., [6]- [15], and using the query/request-based AoI metric in, e.g., [4], [16]- [18]. Assuming the generate-at-will model, the sampling times in [4], [6]- [8], [10], [11], [13]- [15], [18], and the sampling rate in [7] were optimized. The work [11] considered a limit on the sampling frequency.…”

“…Assuming the generate-at-will model, the sampling times in [4], [6]- [8], [10], [11], [13]- [15], [18], and the sampling rate in [7] were optimized. The work [11] considered a limit on the sampling frequency. The works [4], [6], [10], [12] considered the transmission cost, which limits the average number of transmissions [10], [12].…”

Query-Age-Optimal Scheduling under Sampling and Transmission Constraints

“…Assuming the generate-at-will model, the sampling times in [4], [6]- [8], [10], [11], [13]- [15], [18], and the sampling rate in [7] were optimized. The work [11] considered a limit on the sampling frequency. The works [4], [6], [10], [12] considered the transmission cost, which limits the average number of transmissions [10], [12].…”