A Dynamic Jamming Game for Real-Time Status Updates

We study timely status updates of a real-time system in an adversarial setting. The system samples a physical process, and sends the samples from the source (e.g., a sensor) to the destination (e.g, a control center) through a channel. For real-time monitoring/control tasks, it is crucial for the system to update the status of the physical process "timely". We measure the timeliness of status updates by the time elapsed since the latest update at the destination was generated at the source, and define the time elapsed as age of information, or age in short. To sabotage the system, an attacker aims to maximize the age by jamming the channel and hence causing delay in status updates. The system aims to minimize the age by judiciously choosing when to sample and send the updates. We model the ongoing repeated interaction between the attacker and the system as a dynamic game. In each stage game, the attacker chooses the jamming time according to the jamming time distribution, and the system responds by delaying the sampling according to the sampling policy. We prove that there exists a unique stationary equilibrium in the game, and provide a complete analytical characterization of the equilibrium. Our results shed lights on how the attacker sabotages the system and how the system should defend against the attacker.Comment: 7 pages, 3 figures, INFOCOM 2018 Workshop on Age of Informatio

How to Price Fresh Data

We introduce the concept of a fresh data market, in which a destination user requests, and pays for, fresh data updates from a source provider. Data freshness is captured by the {\it age of information} (AoI) metric, defined as the time elapsed since the latest update has reached the destination. The source incurs an operational cost, modeled as an increasing convex function of the number of updates. The destination incurs an age-related cost, modeled as an increasing convex function of the AoI. The source charges the destination for each update and designs a pricing mechanism to maximize its profit; the destination on the other hand chooses a data update schedule to minimize the summation of its payments to the source and its age-related cost. The interaction among the source and destination is hence game-theoretic. Motivated by the existing pricing literature, we first study a time-dependent pricing scheme, in which the price for each update depends on when it is requested. We show in this case that the game equilibrium leads to only one data update, which does not yield the maximum profit to the source. This motivates us to consider a quantity-based pricing scheme, in which the price of each update depends on how many updates have been previously requested. We show that among all pricing schemes in which the price of an update may vary according to both time and quantity, the quantity-based pricing scheme performs best: it maximizes the source's profit and minimizes the social cost of the system, defined as the aggregate source's operational cost and the destination's age-related cost. Numerical results show that the optimal quantity-based pricing can be 27% more profitable for the source and incurs 54% less social cost, compared with the optimal time-dependent pricing.Comment: to appear in WiOpt 201

Secure Status Updates under Eavesdropping: Age of Information-based Physical Layer Security Metrics

This letter studies the problem of maintaining information freshness under passive eavesdropping attacks. The classical three-node wiretap channel model is considered, in which a source aims to send its latest status wirelessly to its intended destination, while protecting the message from being overheard by an eavesdropper. Considering that conventional channel capacity-based secrecy metrics are no longer adequate to measure the information timeliness in status update systems, we define two new age of information-based metrics to characterize the secrecy performance of the considered system. We further propose, analyze, and optimize a randomized stationary transmission policy implemented at the source for further enhancing the secrecy performance. Simulation results are provided to validate our analysis and optimization.Comment: Submitted for possible publication. The first two authors contributed equally to this wor

A Non-Cooperative Multiple Access Game for Timely Updates

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

Coexistence of Age and Throughput Optimizing Networks: A Game Theoretic Approach

Real-time monitoring applications have Internet-of-Things (IoT) devices sense and communicate information (status updates) to a monitoring facility. Such applications desire the status updates available at the monitor to be fresh and would like to minimize the age of delivered updates. Networks of such devices may share wireless spectrum with WiFi networks. Often, they use a CSMA/CA based medium access similar to WiFi. However, unlike them, a WiFi network would like to provide high throughputs for its users. We model the coexistence of such networks as a repeated game with two players, an age optimizing network (AON) and a throughput optimizing network (TON), where an AON aims to minimize the age of updates and a TON seeks to maximize throughput. We define the stage game, parameterized by the average age of the AON at the beginning of the stage, and derive its mixed strategy Nash equilibrium (MSNE). We study the evolution of the equilibrium strategies over time, when players play the MSNE in each stage, and the resulting average discounted payoffs of the networks. It turns out that it is more favorable for a TON to share spectrum with an AON in comparison to sharing with another TON. The key to this lies in the MSNE strategy of the AON that occasionally refrains all its nodes from transmitting during a stage. Such stages allow the TON competition free access to the medium

Sampling for Remote Estimation through Queues: Age of Information and Beyond

The age of information, as a metric for evaluating information freshness, has received a lot of attention. Recently, an interesting connection between the age of information and remote estimation error was found in a sampling problem of Wiener processes: If the sampler has no knowledge of the signal being sampled, the optimal sampling strategy is to minimize the age of information; however, by exploiting causal knowledge of the signal values, it is possible to achieve a smaller estimation error. In this paper, we extend a previous study by investigating a problem of sampling a stationary Gauss-Markov process, namely the Ornstein-Uhlenbeck (OU) process. The optimal sampling problem is formulated as a constrained continuous-time Markov decision process (MDP) with an uncountable state space. We provide an exact solution to this MDP: The optimal sampling policy is a threshold policy on instantaneous estimation error and the threshold is found. Further, if the sampler has no knowledge of the OU process, the optimal sampling problem reduces to an MDP for minimizing a nonlinear age of information metric. The age-optimal sampling policy is a threshold policy on expected estimation error and the threshold is found. These results hold for (i) general service time distributions of the queueing server and (ii) sampling problems both with and without a sampling rate constraint. Numerical results are provided to compare different sampling policies.Comment: 21 pages, 10 figures. This document has been submitted in IEEE/ACM Transactions on Networkin

Sampling for Data Freshness Optimization: Non-linear Age Functions

In this paper, we study how to take samples at a data source for improving the freshness of received data samples at a remote receiver. We use non-linear functions of the age of information to measure data freshness, and provide a survey of non-linear age functions and their applications. The sampler design problem is studied to optimize these data freshness metrics, even when there is a sampling rate constraint. This sampling problem is formulated as a constrained Markov decision process (MDP) with a possibly uncountable state space. We present a complete characterization of the optimal solution to this MDP: The optimal sampling policy is a deterministic or randomized threshold policy, where the threshold and the randomization probabilities are characterized based on the optimal objective value of the MDP and the sampling rate constraint. The optimal sampling policy can be computed by bisection search, and the curse of dimensionality is circumvented. These age optimality results hold for (i) general data freshness metrics represented by monotonic functions of the age of information, (ii) general service time distributions of the queueing server, (iii) both continuoustime and discrete-time sampling problems, and (iv) sampling problems both with and without the sampling rate constraint. Numerical results suggest that the optimal sampling policies can be much better than zero-wait sampling and the classic uniform sampling

The Age of Information in Networks: Moments, Distributions, and Sampling

A source provides status updates to monitors through a network with state defined by a continuous-time finite Markov chain. An age of information (AoI) metric is used to characterize timeliness by the vector of ages tracked by the monitors. Based on a stochastic hybrid systems (SHS) approach, first order linear differential equations are derived for the temporal evolution of both the moments and the moment generating function (MGF) of the age vector components. It is shown that the existence of a non-negative fixed point for the first moment is sufficient to guarantee convergence of all higher order moments as well as a region of convergence for the stationary MGF vector of the age. The stationary MGF vector is then found for the age on a line network of preemptive memoryless servers. From this MGF, it is found that the age at a node is identical in distribution to the sum of independent exponential service times. This observation is then generalized to linear status sampling networks in which each node receives samples of the update process at each preceding node according to a renewal point process. For each node in the line, the age is shown to be identical in distribution to a sum of independent renewal process age random variables.Comment: This work was presented in part at the 2018 IEEE Infocom Age of Information Workshop. This version will be (more or less) the same as what will appear in the IEEE Transactions on Information Theory. This work was supported by NSF award 171704

Minimizing Age of Information in Cognitive Radio-based IoT Systems: Underlay or Overlay?

We consider a cognitive radio-based Internet-of-Things (CR-IoT) network consisting of one primary IoT (PIoT) system and one secondary IoT (SIoT) system. The IoT devices of both the PIoT and the SIoT respectively monitor one physical process and send randomly generated status updates to their associated access points (APs). The timeliness of the status updates is important as the systems are interested in the latest condition (e.g., temperature, speed and position) of the IoT device. In this context, two natural questions arise: (1) How to characterize the timeliness of the status updates in CR-IoT systems? (2) Which scheme, overlay or underlay, is better in terms of the timeliness of the status updates. To answer these two questions, we adopt a new performance metric, named the age of information (AoI). We analyze the average peak AoI of the PIoT and the SIoT for overlay and underlay schemes, respectively. Simple asymptotic expressions of the average peak AoI are also derived when the PIoT operates at high signal-to-noise ratio (SNR). Based on the asymptotic expressions, we characterize a critical generation rate of the PIoT system, which can determine the superiority of overlay and underlay schemes in terms of the average peak AoI of the SIoT. Numerical results validate the theoretical analysis and uncover that the overlay and underlay schemes can outperform each other in terms of the average peak AoI of the SIoT for different system setups

Coexistence of Age and Throughput Optimizing Networks: A Spectrum Sharing Game

We investigate the coexistence of an age optimizing network (AON) and a throughput optimizing network (TON) that share a common spectrum band. We consider two modes of long run coexistence: (a) networks compete with each other for spectrum access, causing them to interfere and (b) networks cooperate to achieve non-interfering access. To model competition, we define a non-cooperative stage game parameterized by the average age of the AON at the beginning of the stage, derive its mixed strategy Nash equilibrium (MSNE), and analyze the evolution of age and throughput over an infinitely repeated game in which each network plays the MSNE at every stage. Cooperation uses a coordination device that performs a coin toss during each stage to select the network that must access the medium. Networks use the grim trigger punishment strategy, reverting to playing the MSNE every stage forever if the other disobeys the device. We determine if there exists a subgame perfect equilibrium, i.e., the networks obey the device forever as they find cooperation beneficial. We show that networks choose to cooperate only when they consist of a sufficiently small number of nodes, otherwise they prefer to disobey the device and compete