For monitoring applications, the Age of Information (AoI) metric has been the primary focus of recent research, but closely related to monitoring is the problem of real-time or remote estimation. Age of Information has been shown to be insufficient for minimizing remote estimation error, but recently a metric known as Age of Incorrect Information (AoII) was proposed that characterizes the cost of a monitor being in an erroneous state over time. In this work, we study the AoII metric in the simple context of monitoring a symmetric binary information source over a delay system with feedback. We compare three different performance metrics: probability of error, AoI, and AoII. For each metric, we formulate the optimal sampling problem as a Markov decision process and apply a dynamic programming algorithm to compute the optimal performance and policy. We also simulate the system for two sampling policies: sample-at-change and zero-wait, and we observe which policy coincides with the optimal policy for each metric. For a variety of delay distributions and AoII penalty functions, we observe that the optimal policy for the probability of error and for AoII are equal to the sample-at-change policy, whereas the optimal policy for AoI is a threshold policy.

We consider a finite-state Discrete-Time Markov
Chain (DTMC) source that can be sampled for detecting the
events when the DTMC transits to a new state. Our goal is to
study the trade-off between sampling frequency and staleness in
detecting the events. We argue that, for the problem at hand,
using Age of Information (AoI) for quantifying the staleness of
a sample is conservative and therefore, introduce age penalty
for this purpose. We study two optimization problems: minimize
average age penalty subject to an average sampling frequency
constraint, and minimize average sampling frequency subject to
an average age penalty constraint; both are Constrained Markov
Decision Problems. We solve them using linear programming
approach and compute Markov policies that are optimal among
all causal policies. Our numerical results demonstrate that the
computed Markov policies not only outperform optimal periodic
sampling policies, but also achieve sampling frequencies close
to or lower than that of an optimal clairvoyant (non-causal)
sampling policy, if a small age penalty is allowed.

Real-time status updating plays a pivotal role in automated control, situational awareness, and networked monitoring. However, how to balance data freshness and its distortion due to lossy compression remains open. In this paper, we are interested in the optimal cross-layer policy that minimizes the Age-of-Information (AoI) and distortion simultaneously over an ON/OFF channel. Specifically, we formulate a hierarchical framework to jointly schedule the lossy compression in the application layer and the packet scheduling in the physical layer. In the application layer, we characterize the compression loss using age or forgetting factor weighted distortion. The optimal tradeoff between the age reduction and compression loss is then revealed via convex optimizations. It allows us to further determine how many packets to send in the physical layer based on a probabilistic scheduling policy. To this end, a Constrained Markov Decision Process (CMDP) problem is formulated and solved by Linear Programming (LP), which gives the optimal tradeoff between the data freshness and distortion, as well as, the optimal cross-layer strategy to strike the AoI-distortion tradeoff.

In this paper, transmission scheduling for multi-loop wireless networked control systems sharing the wireless channel is considered. A linear quadratic cost offset has been proposed to evaluate the performance gap induced by the non-ideal communication. A functional relationship between linear quadratic offset metric and Age of Information has been built up. Based on the offset metric, we come up with an age-based scheduling policy and numerical simulations show that there is a significant improvement compared to the former work.

In this paper, we introduce a generalized definition of age of information (AoI) for actuation update in real-time wireless control systems. In such a system, a general queueing model, i.e., M/M/1/1 queueing model, is used to describe the
actuation update, in which the sampling packets arrive at the remote controller following the Poisson process, the process from the controller to the actuator follows the exponential distribution, and the actuation intends to update at the actuator at the predictive time. Then, the initial time of the AoI for the new
actuation update is the predictive time for the latest update, which is significantly different from the traditional calculation in status update. By the relationship between communication time from the controller to the actuator and predictive time, the AoI calculation falls into two cases, where the conventional AoI in status update is a specific case in this paper. Simulation results show the performance of our method.

This paper is motivated by emerging edge computing systems which consist of sensor nodes that acquire and process information and then transmit status updates to an edge receiver for possible further processing. As power is a scarce resource at the sensor nodes, the system is modeled as a tandem computation-transmission queue with power-efficient computing. Jobs arrive at the computation server with rate \lambda as a Poisson process with no available data buffer. The computation server can be in one of three states: (i) OFF: the server is turned off and no jobs are observed or processed, (ii) ON-Idle: the server is turned on but there is no job in the server, (iii) ON-Busy: the server is turned on and a job is processed in the server. These states cost zero, one and p_c units of power, respectively. Under a long-term power constraint, the computation server switches from one state to another in sequence: first a deterministic T_o time units in OFF state, then waiting for a job arrival in ON-Idle state and then in ON-Busy state for an independent identically distributed compute time duration. The transmission server has a single unit data buffer to save incoming packets and applies last come first serve with discarding as well as a packet deadline to discard a sitting packet for maintaining information freshness, which is measured by the Age of Information (AoI). Additionally, there is a monotonic functional relation between the mean time spent in ON-Busy state and the mean transmission time. We obtain closed-form expressions for average AoI and average peak AoI. Our numerical results illustrate various regimes of operation for best AoI performances optimized over packet deadlines with relation to power efficiency.

In this paper, we derive an explicit expression for the probability distribution of the age of information (AoI) in the GI/GI/\(\infty\) queue with loss. Two special cases M/GI/\(\infty\) and D/GI/\(\infty\) are discussed, where the distribution function of the AoI is shown to take a simple closed-form. In addition, a comparison result between the M/GI/\(\infty\) and D/GI/\(\infty\) queues in terms of the AoI distribution is presented.

In this paper, we consider an M/M/1 status update system consisting of two independent sources, one server, and one sink. We consider the following last-come first-served (LCFS) prioritized packet management policy. When the system is empty, any arriving packet immediately enters the server; when the server is busy, a packet of a source waiting in the queue is replaced if a new packet of the same source arrives and the fresh packet goes at the head of the queue. We derive the average age of information (AoI) of the considered M/M/1 queueing model by using the stochastic hybrid systems (SHS) technique. Numerical results illustrate the effectiveness of the proposed packet management policy compared to several baseline policies.

The Age-of-Information (AoI) is a new performance metric recently proposed for measuring the freshness of information in information-update systems. In this work, we conduct a systematic and comparative study to investigate the impact of scheduling policies on the AoI performance in single-server queues and provide useful guidelines for the design of AoI-efficient scheduling policies. Specifically, we first perform extensive simulations to demonstrate that the update-size information can be leveraged for achieving a substantially AoI improvement compared to non-size-based (or arrival-time-based) policies. Then, by utilizing both the update-size and arrival-time information, we propose three AoI-based policies. Observing improved AoI performance of policies that allow service preemption and that prioritize informative updates, we further propose preemptive, informative, AoI-based scheduling policies. Our simulation results show that such policies empirically achieve the best AoI performance among all the considered policies. Interestingly, we also prove sample-path equivalence between some size-based policies and AoI-based policies. This provides an intuitive explanation for why some size-based policies, such as Shortest-Remaining-Processing-Time (SRPT), achieve a very good AoI performance.

For many time-critical Internet of Things applications, the performance depends heavily on the freshness of information. A fundamental problem in this scenario is how to support heterogeneous information traffic whose freshness may be measured by different metrics. This paper studies a single-link wireless communication system where a sender supports two types of traffic---status update and delay-constrained traffic. The sender decides whether to serve the delay-constrained traffic or sample an underlying status process and update the receiver (central controller) on the status. The optimal scheduling policy is designed to minimize the long-term average \emph{age of information} (AoI) of the status at the central controller while guaranteeing minimum timely-throughput of the delay-constrained traffic. This problem is first formulated as a Constrained Markov Decision Process (CMDP) and then converted into unconstrained MDP by Lagrangian relaxation. Structure property of the optimal policy for the CMDP is derived and an optimal policy is developed. Moreover, considering the computation overhead of MDP, we develop a rather simple scheduling policy based on the Lyapunov-drift method. The performance is analyzed theoretically and verified by simulations.

The optimal transmission strategy to minimize the weighted combination of age of information (AoI) and total energy consumption is studied in this paper. It is assumed that the status update information is obtained and transmitted at fixed rate over a Rayleigh fading channel in a packet-based wireless communication system. A maximum transmission round on each packet is enforced to guarantee certain reliability of the update packets. Given fixed average transmission power, the age-energy tradeoff can be formulated as a constrained Markov decision process (CMDP) problem considering the sensing power consumption as well. Employing the Lagrangian relaxation, the CMDP problem is transformed into a Markov decision process (MDP) problem. An algorithm is proposed to obtain the optimal power allocation policy. Through simulation results, it is shown that both age and energy efficiency can be improved by the proposed optimal policy compared with two benchmark schemes. Also, age can be effectively reduced at the expense of higher energy cost, and more emphasis on energy consumption leads to higher average age at the same energy efficiency. Overall, the tradeoff between average age and energy efficiency is identified.

This paper focuses on the average age of information~(AoI) and average energy consumption for the packetized transmission in status update systems, where the energy consumptions for both update sensing and transmission are considered. We equally segment each sensed \(k\)-bits update packet into \(N\) subpackets, and each \(k_p\)-bits subpacket is encoded as an \(n_p\)-bits block by using the LDPC codes and is transmitted to the monitor over an additive white Gaussian noise channel. The expressions for the average AoI and average energy consumption are derived and expressed as functions of block error rate and the number of the subpackets \(N\). Numerical simulations show that when the channel condition is good enough, the packetized transmission is energy efficient and does not effect the average AoI. However, when the channel condition is bad, the packetization scheme is age efficient but energy inefficient.

This paper explores the age-energy region in a wireless powered communication network (WPCN), where an Internet of things (IoT) device works in a ``harvest-then-transmit'' protocol, where the IoT device first harvests energy from radio-frequency (RF) signals transmitted by a hybrid access point (HAP) via wireless power transfer (WPT), and then transmits real-time update using the harvested energy to the HAP. Achievable age-energy region and age-energy function are defined as metrics to measure the trade-off between the age of information (AoI) and the storable energy at the IoT device. Moreover, the corresponding optimal transmission policy is also designed by solving an AoI minimization problem formulated in terms of the storable energy at the IoT device, which also characterizes the boundaries of the system's age-energy region. As the problem is non-convex, it is first transformed into equivalent ones, and then solved with the closed-form solution. Numerical results show that the transmit power at the HAP and the distance between the HAP and the IoT device have larger effects on the storable energy rather than the system AoI. The data size of the update packet has a larger effect on the system AoI rather the storable energy. The minimal SNR threshold has a bilateral influence on the A-E region, and the impact of the maximal achievable transmit power of the IoT device on the A-E region is not obvious.

Vehicular networking offers the promise of greatly improving transportation safety but has stringent requirements on information age as well as information reachability, where the later refers to the range over which information is propagated. We consider an idealized model of a one-dimensional vehicular networks and show that there is a basic trade-off between these two metrics: a smaller age can be obtained by reducing the reachability of information. We apply this to two current technologies: Cellular V2X (C-V2X) and Dedicated Short Range Communication (DSRC) and derive an equation which characterizes the trade-off between these two metrics for both technologies. In the case of exponential path loss and negligible noise, this relationship becomes a fixed invariant ratio. Given this relationship, under high congestion, these two protocols trade-off these metrics differently. C-V2X tends to achieve a smaller age while DSRC tends to maintain a larger reachability. The idealized model is also applied to analyze the steady state of rate control and power control mechanisms such as those in the SAE standard J2945/1. We show that the ratio of age and reachability is still governed by the same trade-off curve: rate control tries to maintain a large reachability while power control helps improve the age.

We consider a resource-constrained updater, such as Google Scholar, which wishes to update the citation records of a group of researchers, who have different mean citation rates (and optionally, different importance coefficients), in such a way to keep the overall citation index as up to date as possible. The updater is resource-constrained and cannot update citations of all researchers all the time. In particular, it is subject to a total update rate constraint that it needs to distribute among individual researchers. We use a metric similar to the age of information: the long-term average difference between the actual citation numbers and the citation numbers according to the latest updates. We show that, in order to minimize this difference metric, the updater should allocate its total update capacity to researchers proportional to the \emph{square roots} of their mean citation rates. That is, more prolific researchers should be updated more often, but there are diminishing returns due to the concavity of the square root function. More generally, our paper addresses the problem of optimal operation of a resource-constrained sampler that wishes to track multiple independent counting processes in a way that is as up to date as possible.

The rapid growth of mobile devices has spurred the development of crowd-learning applications, which rely on users to collect, report and share real-time information.
A critical factor of crowd-learning is information freshness, which can be measured by a metric called age-of-information (AoI). Moreover, recent advances in machine learning and abundance of historical data have enabled crowd-learning service providers to make precise predictions on user arrivals, data trends and other predictable information. These developments lead to a fundamental question: Can we improve information freshness with predictions in mobile crowd-learning? In this paper, we show that the answer is affirmative. Specifically, motivated by the age-optimal Round-Robin policy, we propose the so-called ``periodic equal spreading'' (PES) policy. Under the PES policy, we first reveal a counter-intuitive insight that the frequency of prediction should not be too often in terms of AoI improvement. Further, we analyze the AoI performances of the proposed PES policy and derive upper bounds for the average age under i.i.d. and Markovian arrivals, respectively. In order to evaluate the AoI performance gain of the PES policy, we also derive two closed-form expressions for the average age under uncontrolled i.i.d. and Markovian arrivals, which could be of independent interest. Our results in this paper serve as a first building block towards understanding the role of predictions in mobile crowd-learning.

Caching has been regarded as a promising technique to alleviate energy consumption of sensors in Internet of Things (IoT) networks by responding to users' requests with the data packets stored in the edge caching node (ECN). For real-time applications in caching enabled IoT networks, it is essential to develop dynamic status update strategies to strike a balance between the information freshness experienced by users and energy consumed by the sensor, which, however, is not well addressed. In this paper, we first depict the evolution of information freshness, in terms of age of information (AoI), at each user. Then, we formulate a dynamic status update optimization problem to minimize the expectation of a long term accumulative cost, which jointly considers the users' AoI and sensor's energy consumption. To solve this problem, a Markov Decision Process (MDP) is formulated to cast the status updating procedure, and a model free reinforcement learning algorithm is proposed, with which the challenge brought by the unknown of the formulated MDP's dynamics can be addressed. Finally, simulations are conducted to validate the convergence of our proposed algorithm and its effectiveness compared with the zero-wait baseline policy.

Due to the flexibility and low operational cost, dispatching unmanned aerial vehicles (UAVs) to collect information from distributed sensors is expected to a promising solution in Internet of Things (IoT), especially for time-critical applications. How to maintain the information freshness is a challenging issue. In this paper, we investigate the fresh data collection in UAV-assisted IoT networks. Particularly, the UAV flies towards the sensors to collect status update packets within a given duration while maintaining a non-negative residual energy. We formulate a Markov Decision Process (MDP) to find the optimal flight trajectory of the UAV and transmission scheduling of the sensors that minimizes the weighted sum of the age of information (AoI). A UAV-assisted data collection algorithm based on deep reinforcement learning is further proposed to overcome the curse of dimensionality. Extensive simulation results demonstrate that the proposed DRL-based algorithm can significantly reduce the weighted sum of the AoI compared to other baseline algorithms.

We study Age of Information (AoI) in a random access channel where a number of devices try to send status updates over a common medium. Assuming a time-slotted scenario where multiple transmissions result in collision, we propose a threshold-based lazy version of Slotted ALOHA and derive the time average AoI achieved by this policy. We demonstrate that the average AoI performance of the lazy policy is significantly better than Slotted ALOHA, and close to the ideal round robin benchmark.

We consider the problem of minimizing the time
average cost of sampling and transmitting status updates by users
over a wireless channel subject to average Age of Information
constraints (AoI). Errors in the transmission may occur and
the scheduling algorithm has to decide if the users sample a
new packet or attempt for retransmission of the packet sampled
previously. The cost consists of both sampling and transmission
costs. The sampling of a new packet after a failure imposes an
additional cost in the system. We formulate a stochastic optimization
problem with time average cost in the objective under
time average AoI constraints. To solve this problem, we apply
tools from Lyapunov optimization theory and develop a dynamic
algorithm that takes decisions slot-by-slot. The algorithm decides
if a user: a) samples a new packet, b) transmits the old one, c)
remains silent. We provide optimality guarantees of the algorithm
and study its performance in terms of time average cost and AoI
through simulation results.

We consider a network of selfish nodes that would like to minimize the age of their updates at the other nodes. The nodes send their updates over a shared spectrum using a CSMA/CA based access mechanism. We model the resulting competition as a non-cooperative one-shot multiple access game and investigate equilibrium strategies for two distinct medium access settings (a) collisions are shorter than successful transmissions and (b) collisions are longer. We investigate competition in a CSMA/CA slot, where a node may choose to transmit or stay idle. We find that medium access settings exert strong incentive effects on the nodes. We show that when collisions are shorter, transmit is a weakly dominant strategy. This leads to all nodes transmitting in the CSMA/CA slot, therefore guaranteeing a collision. In contrast, when collisions are longer, no weakly dominant strategy exists and under certain conditions on the ages at the beginning of the slot, we derive the mixed strategy Nash equilibrium.

As the most well-known application of the Internet of Things (IoT), remote monitoring is now pervasive. In these monitoring applications, information usually has a higher value when it is fresher. A new metric, termed the age of information (AoI), has recently been proposed to quantify the information freshness in various IoT applications. This paper concentrates on the design and analysis of age-oriented random access for massive IoT networks. Specifically, we devise a new stationary threshold-based age-dependent random access (ADRA) protocol, in which each IoT device accesses the channel with a certain probability only when its instantaneous AoI exceeds a predetermined threshold. We manage to evaluate the average AoI of the proposed ADRA protocol mathematically by decoupling the tangled AoI evolution of multiple IoT devices and modelling the decoupled AoI evolution of each device as a Discrete-Time Markov Chain. Simulation results validate our theoretical analysis and affirm the superior age performance of the proposed ADRA protocol over the conventional age-independent scheme.

We study a network with a central controller collecting random updates from power-limited sensors. The time-varying channels between sensors and the central controller are modeled as ergodic Markov chains while packet-loss may happen due to decoding error. We measure the data freshness from the central controller by the metric Age of Synchronization (AoS), i.e., the average time elapsed since information about a sensor becomes desynchronized. To minimize the average AoS under all aforementioned bandwidth and power constraints, we first relax the hard bandwidth limit and decouple the multi-sensor problem into a single-sensor constrained Markov decision process (CMDP), which is then solved through linear programming (LP). We then propose an asymptotic optimal scheduling policy to solve the original hard bandwidth constrained problem. It is revealed that sensors are more likely to send updates under better channel states and higher AoS to save energy and avoid packet-loss.

We consider a distributed IoT network, where each node wants to minimize its age of information and there is a cost to make any transmission. A collision model is considered, where any transmission is successful from a node to a common monitor if no other node transmits in the same slot. There is no explicit communication/coordination between any two nodes. Under this distributed competition model, the objective of this paper is to find a distributed transmission strategy for each node that converges to an equilibrium that depends on the past observations seen by each node. A simple update strategy is shown to converge to an equilibrium, that is in fact a Nash equilibrium for a suitable utility function, that captures all the right tradeoffs for each node. In addition, the price of anarchy for the utility function is shown to approach unity as the number of nodes grows large.

Freshness of information is an important requirement in many real-time applications. The age of information
(AoI), a metric for measuring the freshness of information,
is the time elapsed since the generation of the last successful
update received by the destination. We consider M sources
(users) updating their statuses over a block-fading multiple access
channel to a base station (BS). At the start of each fading block,
the BS acquires perfect information about channel power gain
realization of all the users in the block. Using this information,
a centralized scheduling policy at the BS in a block decides
whether to idle or select a user to receive a packet from, in
order to minimize a long-term weighted average AoI across all
users subject to long-term average power constraint at each user.
In this work, we propose a simple age-independent stationary
randomized policy (AI-SRP), with which the minimum achievable
weighted average AoI across the users is at most two times the
AoI of the optimal policy.