Received samples of a stochastic process are processed by a server for delivery as updates to a monitor. Each sample belongs to a class that specifies a distribution for its processing time and a function that describes how the value of the processed update decays with age at the monitor. The class of a sample is identified when the processed update is delivered. The server implements a form of M/G/1/1 blocking queue; samples arriving at a busy server are discarded and samples arriving at an idle server are subject to an admission policy that depends on the age and class of the prior delivered update. For the delivered updates, we characterize the average age of information (AoI) and average value of information (VoI). We derive the optimal stationary policy that minimizes the convex combination of the AoI and (negative) VoI. It is shown that the policy has a threshold structure, in which a new sample is allowed to arrive to the server only if the previous update's age and value difference surpasses a certain threshold that depends on the specifics of the value function and system statistics.

Age of information (AoI), defined for an information source as the time elapsed since the latest received update was generated, is a recently proposed metric that quantifies the timeliness of information delivery in a communication system. This paper studies a fundamental problem of how the achievable AoI scales in mobile networks. Specifically, we consider a network consisting of n/2 source-destination (S-D) pairs and employ the protocol model to characterize interference incurred by concurrent transmissions. We consider a general class of scheduling policies potentially with the multi-hop transmission and the duplication of packets to multiple nodes. The analysis of AoI faces significant challenges due to potential out-of-order packet delivery, the inherent tradeoffs between packet-centric metrics (throughput and delay), and their unexplored relation to AoI. We first show that the average per-node AoI in static settings scales as [EQUATION]. In the case of networks with i.i.d. mobility, where the node locations vary independently over time, we introduce an episodic technique that allows us to establish lower bounds and design and analyze scheduling policies as constructive upper bounds. Our analytical results reveal that the average per-node AoI scales as [EQUATION] under i.i.d. mobility, which highlights that mobility can enhance timeliness. Finally, we show that, in a more general class of wireless network settings, one can design the age-minimal scheduling policy by balancing throughput and delay.

A status updating system is considered in which a source updates a destination over an erasure channel. The utility of the updates is measured through a function of their age-of-information (AoI), which assesses their freshness. Correlated with the status updates is another process that needs to be kept private from the destination. Privacy is measured through a leakage function that depends on the amount and time of the status updates received: stale updates are more private than fresh ones. Different from most of the current AoI literature, a post-sampling waiting time is introduced in order to provide a privacy cover at the expense of AoI. More importantly, it is also shown that, depending on the leakage budget and the channel statistics, it can be useful to retransmit stale status updates following erasure events without resampling fresh ones.

In cyber-physical systems (CPSs) and internet-of-things applications, various sensor-actuator pairs are deployed for control purposes which require timely online communication. The sensors are measuring information about the CPS, e.g., process systems, whereas the actuators are using the information to take control actions. These sensor-actuator pairs usually communicate via the same wireless medium and thus their transmissions need to be scheduled in time. When transmitting the process data, ashort blocklength source-channelcoding approach is employed to reduce data errors. We investigate the influence of the decision policy consisting of communication parameters and scheduling design on data freshness and accuracy of process monitoring systems. An age-of-information (AoI) metric is used to assess data timeliness, while the mean square error (MSE) is used to assess the precision of the predicted process values. We characterize the AoI and MSE with closed-form expressions for the blocklengths and accuracy levels, for special types of scheduling strategies, namely, round-robin and maximum-age scheduling. We optimize the coding strategies by showing anachievability regionof AoI and MSE. Other priority-based scheduling policies are also investigated. It is shown that the maximum-age policy provides excellent results in terms of AoI, while priority-based scheduling performs better in terms of MSE.

We consider the problem of privately updating a message out of K messages from N replicated and non-colluding databases. In this problem, a user has an outdated version of the message Ŵθof length L bits that differ from the current version Wθin at most f bits. The user needs to retrieve Wθcorrectly using a private information retrieval (PIR) scheme with the least number of downloads without leaking any information about the message index θ to any individual database. To that end, we propose a novel achievable scheme based on syndrome decoding. Specifically, the user downloads the syndrome corresponding to Wθ, according to a linear block code with carefully designed parameters, using the optimal PIR scheme for messages with a length constraint. We derive lower and upper bounds for the optimal download cost that match if the term ${\log _2}\left( \sum\nolimits_{i = 0}^f \binom L i \right)$ is an integer. Our results imply that there is a significant reduction in the download cost if $fθdirectly using classical PIR approaches without taking the correlation between Wθand Ŵθinto consideration.

We introduce the problem of timely private information retrieval (PIR) from$N$non-colluding and replicated servers. In this problem, a user desires to retrieve a message out of$M$messages from the servers, whose contents are continuously updating. The retrieval process should be executed in a timely manner such that no information is leaked about the identity of the message. To assess the timeliness, we use the age of information (AoI) metric. Interestingly, the timely PIR problem reduces to an AoI minimization subject to PIR constraints under asymmetric traffic. We explicitly characterize the optimal tradeoff between the PIR rate and the AoI metric (peak AoI or average AoI) for the case of$N=2,\ M=3$. Further, we provide some structural insights on the general problem with arbitrary$N,\ M$.

In this paper, we study a fresh data acquisition problem to acquire fresh data and optimize the age-related performance when strategic data sources have private market information. We consider an information update system in which a destination acquires, and pays for, fresh data updates from a source. The destination incurs an age-related cost, modeled as a general increasing function of the age-of-information (AoI). The source is strategic and incurs a sampling cost, which is its private information and may not be truthfully reported to the destination. To this end, we design an optimal (economic) mechanism for timely information acquisition by generalizing Myerson's seminal work. The goal is to minimize the sum of the destination's age-related cost and its payment to the source, while ensuring that the source truthfully reports its private information and will voluntarily participate in the mechanism. Our results show that, under some distributions of the source's cost, our proposed optimal mechanism can lead to an unbounded benefit, compared against a benchmark that naively trusts the source's report and thus incentivizes its maximal over-reporting.

We introduce the concept of fresh data trading, in which a destination user requests, and pays for, fresh data updates from a source provider, and data freshness is captured by the age of information (AoI) metric. Keeping data fresh relies on costly frequent data updates by the source, which motivates the source to price fresh data. In this work, the destination incurs an age-related cost, modeled as a general increasing function of the AoI. The source designs a pricing mechanism to maximize its profit, while the destination chooses a data update schedule to trade off its payments to the source and its age-related cost. Depending on different real-time applications and scenarios, we study both a finite-horizon model and an infinite-horizon model with time discounting. The key challenge of designing the optimal pricing scheme lies in the destination's time-interdependent valuations, due to the nature of AoI, and the infinite-dimensional dynamic optimization. To this end, we exploit three different dimensions in designing pricing by studying three pricing schemes: a time-dependent pricing scheme, in which the price for each update depends on when it is requested; a quantity-based pricing scheme, in which the price of each update depends on how many updates have been previously requested; and a simple subscription-based pricing scheme, in which the price per update is constant but the source charges an additional subscription fee. Our analysis reveals that (1) the optimal subscription-based pricing maximizes the source's profit among all possible pricing schemes under both finite-horizon and infinite-horizon models; (2) the optimal quantity-based pricing scheme is only optimal with a finite horizon; and (3) the time-dependent pricing scheme, under the infinite-horizon model with significant time discounting, is asymptotically optimal. Numerical results show that the profit-maximizing pricing schemes can also lead to significant reductions in AoI and social costs, and that a moderate degree of time discounting is enough to achieve a close-to-optimal time-dependent pricing scheme.

The proliferation of real-time applications has spurred much interest in data freshness, captured by the age-of-information (AoI) metric. When strategic data sources have private market information, a fundamental economic challenge is how to incentivize them to acquire fresh data and optimize the age-related performance. In this work, we consider an information update system in which a destination acquires, and pays for, fresh data updates from multiple sources. The destination incurs an age-related cost, modeled as a general increasing function of the AoI. Each source is strategic and incurs a sampling cost, which is its private information and may not be truthfully reported to the destination. The destination decides on the price of updates, when to get them, and who should generate them, based on the sources' reported sampling costs. We show that a benchmark that naively trusts the sources' reports can lead to an arbitrarily bad outcome compared to the case where sources truthfully report. To tackle this issue, we design an optimal (economic) mechanism for timely information acquisition following Myerson's seminal work. To this end, our proposed optimal mechanism minimizes the sum of the destination's age-related cost and its payment to the sources, while ensuring that the sources truthfully report their private information and will voluntarily participate in the mechanism. However, finding the optimal mechanisms may suffer from prohibitively expensive computational overheads as it involves solving a nonlinear infinite-dimensional optimization problem. We further propose a quantized version of the optimal mechanism that achieves asymptotic optimality, maintains the other economic properties, and enables one to tradeoff between optimality and computational overheads. Our analytical and numerical studies show that (i) both the optimal and quantized mechanisms can lead to an unbounded benefit under some distributions of the source costs compared against a benchmark; (ii) the optimal and quantized mechanisms are most beneficial when there are few sources with heterogeneous sampling costs.

The effects ofquantizationandcodingon the estimation quality of Gauss-Markov processes are considered, with a special attention to the Ornstein-Uhlenbeck process. Samples are acquired from the process, quantized, and then encoded for transmission using eitherinfinite incremental redundancy(IIR) orfixed redundancy(FR) coding schemes. A fixedprocessingtime is consumed at the receiver for decoding and sending feedback to the transmitter. Decoded messages are used to construct a minimum mean square error (MMSE) estimate of the process as a function of time. This is shown to be an increasing functional of theage-of-information(AoI), defined as the time elapsed since the sampling time pertaining to the latest successfully decoded message. Such functional depends on the quantization bits, codewords lengths and receiver processing time. The goal, for each coding scheme, is to optimize sampling times such that the long-term average MMSE is minimized. This is then characterized in the setting ofgeneral increasing functionals of AoI,not necessarily corresponding to MMSE, which may be of independent interest in other contexts. We first show that the optimal sampling policy for IIR is such that a new sample is generated only if the AoI exceeds a certainthreshold,while for FR it is such that a new sample is deliveredjust-in-timeas the receiver finishes processing the previous one.Enhancedtransmissions schemes are then developed in order to exploit the processing times to make new data available at the receiver sooner. For both IIR and FR, it is shown that there exists an optimal number of quantization bits that balances AoI and quantization errors, and hence minimizes the MMSE. It is also shown that for longer receiver processing times, the relatively simpler FR scheme outperforms IIR.

Optimization of information freshness in wireless networks has usually been performed based on queueing analysis that captures only the temporal traffic dynamics associated with the transmitters and receivers. However, the effect of interference, which is mainly dominated by the interferers' geographic locations, is not well understood. In this paper, we leverage a spatiotemporal model, which allows one to characterize the age of information (AoI) from a joint queueing-geometry perspective, for the design of a decentralized scheduling policy that exploits local observation to make transmission decisions that minimize the AoI. To quantify the performance, we also derive accurate and tractable expressions for the peak AoI. Numerical results reveal that: i) the packet arrival rate directly affects the service process due to queueing interactions, ii) the proposed scheme can adapt to traffic variations and largely reduce the peak AoI, and iii) the proposed scheme scales well as the network grows in size. This is done by adaptively adjusting the radio access probability at each transmitter to the change of the ambient environment.

We investigate the age-of-information (AoI) in the context of random access networks, in which transmitters need to send a sequence of information packets to intended receivers over shared spectrum. We establish an analytical framework that accounts for the key features of a wireless system, including the fading, path loss, network topology, as well as the spatial interactions amongst the queues. A closed-form expression is derived to quantity the network average AoI and its accuracy is verified via simulations. Our analysis unveils several unconventional behaviors of AoI in such a setting. For instance, even when the packet transmissions are scheduled in a last-come first-serve (LCFS) order whereby the newly incoming packets can replace the undelivered ones, the network average AoI may not monotonically decline with respect to the packet arrival rates, if the infrastructure is densely deployed. Moreover, the ALOHA protocol is shown to be instrumental in reducing the AoI when the packet arrival rates are high, yet it cannot contribute to decreasing the AoI in the regime of infrequent packet arrivals.

Federated learning (FL) is a machine learning model that preserves data privacy in the training process. Specifically, FL brings the model directly to the user equipments (UEs) for local training, where an edge server periodically collects the trained parameters to produce an improved model and sends it back to the UEs. However, since communication usually occurs through a limited spectrum, only a portion of the UEs can update their parameters upon each global aggregation. As such, new scheduling algorithms have to be engineered to facilitate the full implementation of FL. In this paper, based on a metric termed the age of update (AoU), we propose a scheduling policy by jointly accounting for the staleness of the received parameters and the instantaneous channel qualities to improve the running efficiency of FL. The proposed algorithm has low complexity and its effectiveness is demonstrated by Monte Carlo simulations.

In cyber-physical systems, as in 5G and beyond, multiple physical processes require timely online monitoring at a remote device. There, the received information is used to estimate current and future process values. When transmitting the process data over a communication channel, source-channel coding is used in order to reduce data errors. During transmission, a high data resolution is helpful to capture the value of the process variables precisely. However, this typically comes with long transmission delays reducing the utilizability of the data, since the estimation quality gets reduced over time. In this paper, the trade-off between having recent data and precise measurements is captured for a Gauss-Markov process. An Age-of-Information (AoI) metric is used to assess data timeliness, while mean square error (MSE) is used to assess the precision of the predicted process values. AoI appears inherently within the MSE expressions, yet it can be relatively easier to optimize. Our goal is to minimize a time-averaged version of both metrics. We follow a short blocklength source-channel coding approach, and optimize the parameters of the codes being used in order to describe an achievability region between MSE and AoI.

The effects of quantization and coding on the estimation quality of a Gauss-Markov, namely Ornstein-Uhlenbeck, process are considered. Samples are acquired from the process, quantized, and then encoded for transmission using either infinite incremental redundancy or fixed redundancy coding schemes. A fixed processing time is consumed at the receiver for decoding and sending feedback to the transmitter. Decoded messages are used to construct a minimum mean square error (MMSE) estimate of the process as a function of time. This is shown to be an increasing functional of the age-of-information, defined as the time elapsed since the sampling time pertaining to the latest successfully decoded message. Such (age-penalty) functional depends on the quantization bits, codeword lengths and receiver processing time. The goal, for each coding scheme, is to optimize sampling times such that the long term average MMSE is minimized. This is then characterized in the setting of general increasing age-penalty functionals, not necessarily corresponding to MMSE, which may be of independent interest in other contexts.

A downlink single-input single-output non-orthogonal multiple access setting is considered, in which a base station (BS) is communicating with two legitimate users in two possible scenarios of unsecure environments: existence of an external eavesdropper and communicating through an untrusted relay. For the first scenario, a number of trusted cooperative half-duplex relays is employed to assist with the BS's transmission and secure its signals from the external eavesdropper. Various relaying schemes are proposed and analyzed for that matter: cooperative jamming, decode-and-forward, and amplify-and-forward. For each scheme, secure beamforming signals are devised at the relays to maximize the achievable secrecy rate regions. For the second scenario, with the untrusted relay, achievable secrecy rate regions are derived for two different relaying schemes, compress-and-forward and amplify-and-forward, under two different modes of operation. In the first mode, coined passive user mode, the users receive signals from both the BS and the untrusted relay and combine them to decode their messages. In the second mode, termed the active user mode, the users transmit a cooperative jamming signal simultaneously with the BS's transmission to further confuse the relay. Focusing on half-duplex nodes, the users cannot receive the BS's signal while jamming the relay, i.e., while being active, and rely only on the signals forwarded to them by the relay. It is shown that the best relaying scheme highly depends on the system parameters, in particular the distances between the nodes, and also on the part of the secrecy rate region at which the system is to operate.

An energy-harvesting sensor node that is sending status updates to a destination is considered. The sensor is equipped with a battery of finite size to save its incoming energy, and consumes one unit of energy per status update transmission, which is delivered to the destination instantly over an error-free channel. The setting is online in which the harvested energy is revealed to the sensor causally over time after it arrives, and the goal is to design status update transmission times (policy) such that the long term average age of information (AoI) is minimized. The AoI is defined as the time elapsed since the latest update has reached at the destination. Two energy arrival models are considered: a random battery recharge (RBR) model, and an incremental battery recharge (IBR) model. In both models, energy arrives according to a Poisson process with unit rate, with values that completely fill up the battery in the RBR model, and with values that fill up the battery incrementally in a unit-by-unit fashion in the IBR model. The key approach to characterizing the optimal status update policy for both models is showing the optimality of renewal policies, in which the inter-update times follow a renewal process in a certain manner that depends on the energy arrival model and the battery size. It is then shown that the optimal renewal policy has an energy-dependent threshold structure, in which the sensor sends a status update only if the AoI grows above a certain threshold that depends on the energy available in its battery. For both the random and the incremental battery recharge models, the optimal energy-dependent thresholds are characterized explicitly, i.e., in closed-form, in terms of the optimal long term average AoI. It is also shown that the optimal thresholds are monotonically decreasing in the energy available in the battery, and that the smallest threshold, which comes in effect when the battery is full, is equal to the optimal long term average AoI.

A status updating communication system is examined, in which a transmitter communicates with a receiver over a noisy channel. The goal is to realize timely delivery of fresh data over time, which is assessed by an age-of-information (AoI) metric. Channel coding is used to combat the channel errors, and feedback is sent to acknowledge updates' reception. In case decoding is unsuccessful, a hybrid ARQ protocol is employed, in which incremental redundancy (IR) bits are transmitted to enhance the decoding ability. This continues for some amount of time in case decoding remains unsuccessful, after which a new (fresh) status update is transmitted instead. In case decoding is successful, the transmitter has the option to idly wait for a certain amount of time before sending a new update. A general problem is formulated that optimizes the codeword and IR lengths for each update, and the waiting times, such that the long term average AoI is minimized. Stationary deterministic policies are investigated, in which the codeword and IR lengths are fixed for each update, and the waiting time is a deterministic function of the AoI. The optimal waiting policy is then derived, and is shown to have a threshold structure, in which the transmitter sends a new update only if the AoI grows above a certain threshold that is a function of the codeword and IR lengths. Choosing the codeword and IR lengths is discussed in the context of binary symmetric channels.

Optimization of information freshness in wireless networks has usually been performed based on queueing analysis that captures only the temporal traffic dynamics associated with the transmitters and receivers. However, the effect of interference, which is mainly dominated by the interferersa geographic locations, is not well understood. In this paper, we leverage a spatiotemporal model, which allows one to characterize the age of information (AoI) from a joint queueing-geometry perspective, and design a decentralized scheduling policy that exploits local observation to make transmission decisions that minimize the AoI. Simulation results reveal that the proposed scheme not only largely reduces the peak AoI but also scales well with the network size.

A real-time status updating system is considered, in which an energy harvesting sensor is acquiring measurements regarding some physical phenomenon and sending them to a destination through an erasure channel. The setting is online, in which energy arrives in units according to a Poisson process with unit rate, with arrival times being revealed causally over time. Energy is saved in a unit-sized battery. The sensor is notified by the destination of whether updates were erased via feedback. Updates need to reach the destination successfully in a timely fashion, namely, such that the long term average age of information, defined as the time elapsed since the latest successful update has reached the destination, is minimized. First, it is shown that the optimal status update policy has a renewal structure: successful update times should constitute a renewal process. Then, threshold-greedy policies are investigated: a new update is transmitted, following a successful one, only if the age of information grows above a certain threshold; and if it is erased, then all subsequent update attempts are greedily scheduled whenever energy is available. The optimal threshold-greedy policy is then analytically derived.

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 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.

A visible light communication broadcast channel is considered, in which a transmitter luminaire communicates with two legitimate receivers in the presence of an external eavesdropper. A number of trusted cooperative half-duplex relay luminaires are deployed to aid with securing the transmitted data. Transmitters are equipped with single light fixtures, containing multiple light emitting diodes, and receiving nodes are equipped with single photo-detectors, rendering the considered setting as a single-input single-output system. Transmission is amplitude-constrained to maintain operation within the light emitting diodes' dynamic range. Achievable secrecy rate regions are derived under such amplitude constraints for this multi-receiver wiretap channel, first for direct transmission without the relays, and then for multiple relaying schemes: cooperative jamming, decode-and-forward, and amplify-and-forward. Superposition coding with uniform signaling is used at the transmitter and the relays. Further, for each relaying scheme, secure beamforming vectors are carefully designed at the relay nodes in order to hurt the eavesdropper and/or benefit the legitimate receivers. Superiority of the proposed relaying schemes, with secure beamforming, is shown over direct transmission. It is also shown that the best relaying scheme depends on how far the eavesdropper is located from the transmitter and the relays, the number of relays, and their geometric layout.

A two-hop energy harvesting communication network is considered, in which measurement updates are transmitted by a source to a destination through an intermediate relay. Updates are to be sent in a timely fashion that minimizes the age of information, defined as the time elapsed since the most recent update at the destination was generated at the source. The source and the relay communicate using energy harvested from nature, which is stored in infinite-sized batteries. Both nodes use fixed transmission rates, and hence updates incur fixed delays (service times). Two problems are formulated: an offline problem, in which the energy arrival information is known a priori, and an online problem, in which such information is revealed casually over time. In both problems, it is shown that it is optimal to transmit updates from the source just in time as the relay is ready to forward them to the destination, making the source and the relay act as one combined node. A recurring theme in the optimal policy is that updates should be as uniformly spread out over time as possible, subject to energy causality and service time constraints. This is perfectly achieved in the offline setting, and is achieved almost surely in the online setting by a best effort policy.

A downlink single-input single-output nonorthogonal multiple access system is considered in which a base station (BS) is communicating with two legitimate users in the presence of an external eavesdropper. A group of trusted cooperative half-duplex relay nodes, powered by the BS, is employed to assist the BS's transmission. The goal is to design relaying schemes such that the legitimate users' secrecy rate region is maximized subject to a total power constraint on the BS and the relays' transmissions. Three relaying schemes are investigated: cooperative jamming, decode-and-forward, and amplify-and-forward. Depending on the scheme, secure beamforming signals are carefully designed for the relay nodes that either diminish the eavesdropper's rate without affecting that of the legitimate users, or increase the legitimate users' rates without increasing that of the eavesdropper. The results show that there is no relaying scheme that fits all conditions; the best relaying scheme depends on the system parameters, namely, the relays' and eavesdropper's distances from the BS, and the number of relays. They also show that the relatively simple cooperative jamming scheme outperforms other schemes when the relays are far from the BS and/or close to the eavesdropper.

We consider an energy harvesting sensor that is sending measurement updates regarding some physical phenomenon to a destination. The sensor relies on energy harvested from nature to measure and send its updates, and is equipped with a battery of finite size to collect its harvested energy. The energy harvesting process is Poisson with unit rate, and arrives in amounts that fully recharge the battery. Our setting is online in the sense that the times of energy arrivals are revealed causally to the sensor after the energy is harvested; only the statistics of the arrival process is known a priori. Updates need to be sent in a timely manner to the destination, namely, such that the long term average age of information is minimized over the course of communication. The age of information is defined as the time elapsed since the freshest update has reached the destination. We first show that the optimal scheduling update policy is a renewal policy, and then show that it has a multi threshold structure: the sensor sends an update only if the age of information grows above a certain threshold that depends on the available energy.

We characterize delay minimal power scheduling policies in energy harvesting communication systems. We consider a continuous-time system, where the delay experienced by each bit is given by the time spent by the bit in the queue waiting to be transmitted to its receiver. We first consider a single-user channel, where the transmitter has a finite-sized battery to save its harvested energy. Data arrives during the course of communication and are saved in a finite data buffer as well. We find the optimal power policy that minimizes the average delay experienced by the bits subject to energy and data causality constraints. We characterize the optimal solution in terms of Lagrange multipliers, and calculate their values in a recursive manner. We show that, different from the existing literature, the optimum transmission power is not constant between the energy and data arrival events; the transmission power starts high, decreases linearly, and potentially reaches zero between energy and data arrivals. Intuitively, untransmitted bits experience cumulative delay due to the bits to be transmitted ahead of them, and hence the reason for transmission power starting high and decreasing over time. Next, we study a multiuser version of this problem, namely, a two-user broadcast channel, and characterize the optimal transmission policies that minimize the sum delay. For this setting, we consider the case, where the transmitter has an infinite-sized battery, and that all data packets intended for the receivers are available at the beginning of the communication session. We characterize the optimal solution in terms of Lagrange multipliers, and present an iterative solution that calculates their values. Our results show that in the optimal policy, both users may not be served simultaneously all the time; there may be times, where only one of the two users is served alone. We also show that the optimal policy may have gaps in transmission in between energy arrivals, where none of the users is served, echoing the results of the single-user setting.

We consider power scheduling policies for single-user energy harvesting communication systems, where the goal is to characterize online policies that maximize the long term average utility, for general concave and monotonically increasing utility functions. The transmitter relies on energy harvested from nature to send its messages to the receiver, and is equipped with a finite-sized battery to store its harvested energy. Energy packets are independent and identically distributed (i.i.d.) over time slots, and are revealed causally to the transmitter. We first characterize the optimal solution for the case of Bernoulli arrivals. Then, for general i.i.d. arrivals, we first show that fixed fraction policies, in which a fixed fraction of the battery state is consumed in each time slot, are within a constant multiplicative gap from the optimal solution for all energy arrivals and battery sizes. We then derive a set of sufficient conditions on the utility function to guarantee that fixed fraction policies are within a constant additive gap as well from the optimal solution. We then apply these results to a specific scenario where a sensor node collects samples from a Gaussian source and sends them to a destination node over a Gaussian channel, and the goal is to minimize the long term average distortion of the source samples received at the destination. We study two problem settings for this case: the first is when sampling is cost-free, and the second is when there is a sampling cost incurred whenever samples are collected. For the problem with sampling costs, the transmission policy can be bursty; the sensor may collect samples and transmit for only a portion of the time. Finally, we present an alternative analysis approach that is more tailored to these distortion problems to show that fixed fraction policies achieve an additive gap that is independent of the sampling cost.

We consider an energy harvesting two-hop network where a source is communicating to a destination through a relay. During a given communication session time, the source collects measurement updates from a physical phenomenon and sends them to the relay, which then forwards them to the destination. The objective is to send these updates to the destination as timely as possible; namely, such that the total age of information is minimized by the end of the communication session, subject to energy causality constraints at the source and the relay, and data causality constraints at the relay. Both the source and the relay use fixed, yet possibly different, transmission rates. Hence, each update packet incurs fixed non-zero transmission delays. We first solve the single-hop version of this problem, and then show that the two-hop problem is solved by treating the source and relay nodes as one combined node, with some parameter transformations, and solving a single-hop problem between that combined node and the destination.

We consider a mobile energy harvesting transmitter where movement is motivated by finding better energy harvesting locations. Movement comes with an energy cost expenditure, and hence there exists a tradeoff between staying at the same location and moving to a new one. On one hand, the transmitter may opt not to move and use all its available energy for transmission; on the other hand, it can choose to move to a potentially better location, spending some of its available energy during the movement process, and yet harvest larger amounts of energy at the new location and achieve higher throughput. In this paper, we characterize this tradeoff by designing throughput-optimal power allocation policies subject to energy causality constraints and moving costs. In our setup, the transmitter moves along a straight line, where two energy sources are located at the opposite ends of the line. We first study the case of a single energy arrival at both sources, and then generalize it to the case of multiple energy arrivals.

We consider online scheduling policies for single-user energy harvesting communication systems, where the goal is to characterize online policies that maximize the long term average utility, for some general concave and monotonically increasing utility function. In our setting, the transmitter relies on energy harvested from nature to send its messages to the receiver, and is equipped with a finite-sized battery to store its energy. Energy packets are independent and identically distributed (i.i.d.) over time slots, and are revealed causally to the transmitter. Only the average arrival rate is known a priori. We first characterize the optimal solution for the case of Bernoulli arrivals. Then, for general i.i.d. arrivals, we first show that fixed fraction policies [1] are within a constant multiplicative gap from the optimal solution for all energy arrivals and battery sizes. We then derive a set of sufficient conditions on the utility function to guarantee that fixed fraction policies are within a constant additive gap as well from the optimal solution.

We consider online scheduling for an energy harvesting communication system where a sensor node collects samples from a Gaussian source and sends them to a destination node over a Gaussian channel. The sensor is equipped with a finite-sized battery that is recharged by an independent and identically distributed (i.i.d.) energy harvesting process over time. The goal is to minimize the long term average distortion of the source samples received at the destination. We study two problems: the first is when sampling is cost-free, and the second is when there is a sampling cost incurred whenever samples are collected. We show that fixed fraction policies [1], in which a fixed fraction of the battery state is consumed in each time slot, are near-optimal in the sense that they achieve a long term average distortion that lies within a constant additive gap from the optimal solution for all energy arrivals and battery sizes. For the problem with sampling costs, the transmission policy is bursty; the sensor can collect samples and transmit for only a portion of the time.

We consider an energy harvesting two-way channel with decoding costs. In this system, each node spends energy to transmit data to the other user, and also to decode data coming from the other user; that is, each user divides its harvested energy for transmission and reception. The power needed for decoding the incoming data is a function of the incoming data rate. We determine the optimal offline power scheduling policies for both users that maximize the sum throughput of the system by a given deadline. We first consider the case with a single energy arrival at each user. We show that the transmission is limited by the user with the smaller energy. In this case, the user with larger energy may not consume all of its energy. We next consider the case with multiple energy arrivals at both users. We show that the optimal power allocations are non-decreasing over time, and they increase synchronously at both users. We then develop an iterative algorithm based on two-slot updates to obtain the optimal power allocations for both users.

We consider a two-user energy harvesting broadcast channel, and characterize the delay minimal transmission policies that minimize the total delay experienced by the data packets in the system. We consider a continuous time system where the delay experienced by each bit is given by the time spent by the bit in the queue waiting to be transmitted to its receiver. We consider the case where all data packets are available at the transmitter at the beginning of the communication session. We characterize the optimal solution in terms of the Lagrange multipliers, and present an iterative algorithm that optimally calculates their values. Our results show that in the optimal policy, both users may not be served simultaneously all the time; there may be times where only the strong user or only the weak user is served alone. We also show that the optimal policy may have gaps in transmission where none of the users is served until the next energy arrival.

We consider a two-way communication channel in which both users rely solely on energy harvested from nature. Each user incurs a processing cost per unit time as long as it communicates; that is, each user's energy consumption includes energy spent for transmission and energy spent for processing. We maximize the sum throughput by a given deadline subject to energy causality constraints. We first show that the optimal power policy is bursty; the two users communicate only during a portion of the time that is uniquely determined by their available energies and processing costs. We show that it is optimal for the two users to be fully synchronized; they turn on and exchange data during the same portion of time, and then turn off together. We first solve the single energy arrival case, and then extend it to solve the multiple energy arrival throughput maximization problem. We show that it is optimal for the users to communicate in a deferred fashion; users postpone their energy consumption to utilize later time slots first. We present an algorithm that gives the optimal deferred policy by iteratively applying a modified version of the single energy arrival result in a backward manner.

We consider the effects of decoding costs in energy-harvesting communication systems. In our setting, receivers, in addition to transmitters, rely solely on energy harvested from nature, and need to spend some energy in order to decode their intended packets. We model the decoding energy as an increasing convex function of the rate of the incoming data. In this setting, in addition to the traditional energy causality constraints at the transmitters, we have the decoding causality constraints at the receivers, where energy spent by the receiver for decoding cannot exceed its harvested energy. We first consider the point-to-point single-user problem where the goal is to maximize the total throughput by a given deadline subject to both energy and decoding causality constraints. We show that decoding costs at the receiver can be represented as generalized data arrivals at the transmitter, and thereby moving all system constraints to the transmitter side. Then, we consider several multiuser settings. We start with a two-hop network where the relay and the destination have decoding costs, and show that separable policies, where the transmitter's throughput is maximized irrespective of the relay's transmission energy profile, are optimal. Next, we consider the multiple access channel (MAC) and the broadcast channel (BC) where the transmitters and the receivers harvest energy from nature, and characterize the maximum departure region. In all multiuser settings considered, we decompose our problems into inner and outer problems. We solve the inner problems by exploiting the structure of the particular model, and solve the outer problems by water-filling algorithms.

In this paper, we examine a cognitive spectrum access scheme in which a secondary user exploits the primary feedback information. We consider an overlay model in which the secondary user accesses the channel by certain access probabilities that are function of the spectrum sensing metric. In setting our problem, we assume that the secondary user can receive the primary link's feedback automatic repeat request (ARQ), but through an erasure channel. This means that the primary feedback may either be received correctly or is erased with a certain erasure probability. We study the cognitive radio network from a queuing theory point of view. Access probabilities are determined by solving a secondary throughput maximization problem subject to a constraint on the primary queues' stability. Fortunately, our problem is convex and can be solved using standard optimization techniques. Our scheme yields improved results in the secondary throughput than the non-feedback based access scheme attributed to the efficient utilization of the primary user's unerased feedback messages.

In this paper, we examine a cognitive spectrum access scheme in which secondary users exploit the primary feedback information. We consider an overlay secondary network employing a random access scheme in which secondary users access the channel by certain access probabilities that are functions of the spectrum sensing metric. In setting our problem, we assume that secondary users can eavesdrop on the primary link's feedback. We study the cognitive radio network from a queuing theory point of view. Access probabilities are determined by solving a secondary throughput maximization problem subject to a constraint on the primary queues' stability. First, we formulate our problem which is found to be non-convex. Yet, we solve it efficiently by exploiting the structure of the secondary throughput equation. Our scheme yields improved results in, both, the secondary user throughput and the primary user packet delay as compared to the scheme where no feedback information is exploited. In addition, it comes very close to the optimal genie-aided scheme in which secondary users act upon the presumed perfect knowledge of the primary users' activity.