Timely Lossless Source Coding for Randomly Arriving Symbols

We consider a real-time streaming source coding system in which an encoder observes a sequence of randomly arriving symbols from an i.i.d. source, and feeds binary codewords to a FIFO buffer that outputs one bit per time unit to a decoder. Each source symbol represents a status update by the source, and the timeliness of the system is quantified by the age of information (AoI), defined as the time difference between the present time and the generation time of the most up-to-date symbol at the output of the decoder. When the FIFO buffer is allowed to be empty, we propose an optimal prefix-free lossless coding scheme that minimizes the average peak age based on the analysis of discrete-time Geo/G/1 queue. For more practical scenarios in which a special codeword is reserved for indicating an empty buffer, we propose an encoding scheme that assigns a codeword to the empty buffer state based on an estimate of the buffer idle time.Comment: IEEE Information Theory Workshop (ITW) 201

Timeliness in Lossless Block Coding

We examine lossless data compression from an average delay perspective. An encoder receives input symbols one per unit time from an i.i.d. source and submits binary codewords to a FIFO buffer that transmits bits at a fixed rate to a receiver/decoder. Each input symbol at the encoder is viewed as a status update by the source and the system performance is characterized by the status update age, defined as the number of time units (symbols) the decoder output lags behind the encoder input. An upper bound on the average status age is derived from the exponential bound on the probability of error in streaming source coding with delay. Apart from the influence of the error exponent that describes the convergence of the error, this upper bound also scales with the constant multiplier term in the error probability. However, the error exponent does not lead to an accurate description of the status age for small delay and small blocklength. An age optimal block coding scheme is proposed based on an approximation of the average age by converting the streaming source coding system into a D/G/1 queue. We compare this scheme to the error exponent optimal coding scheme which uses the method of types. We show that maximizing the error exponent is not equivalent to minimizing the average status age.Comment: Data Compression Conference (DCC), 201

Optimal Source Codes for Timely Updates

A transmitter observing a sequence of independent and identically distributed random variables seeks to keep a receiver updated about its latest observations. The receiver need not be apprised about each symbol seen by the transmitter, but needs to output a symbol at each time instant $t$. If at time $t$ the receiver outputs the symbol seen by the transmitter at time $U(t)\leq t$, the age of information at the receiver at time $t$ is $t-U(t)$. We study the design of lossless source codes that enable transmission with minimum average age at the receiver. We show that the asymptotic minimum average age can be attained up to a constant gap by the Shannon codes for a tilted version of the original pmf generating the symbols, which can be computed easily by solving an optimization problem. Furthermore, we exhibit an example with alphabet \X where Shannon codes for the original pmf incur an asymptotic average age of a factor O(\sqrt{\log |\X|}) more than that achieved by our codes. Underlying our prescription for optimal codes is a new variational formula for integer moments of random variables, which may be of independent interest. Also, we discuss possible extensions of our formulation to randomized schemes and to the erasure channel, and include a treatment of the related problem of source coding for minimum average queuing delay.Comment: Added a missing reference, in IEEE Transactions on Information Theory, 202

Using Erasure Feedback for Online Timely Updating with an Energy Harvesting Sensor

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.Comment: To appear in the 2019 IEEE International Symposium on Information Theor

Partial Updates: Losing Information for Freshness

We consider an information updating system where a source produces updates as requested by a transmitter. The transmitter further processes these updates in order to generate $partial$ $updates$, which have smaller information compared to the original updates, to be sent to a receiver. We study the problem of generating partial updates, and finding their corresponding real-valued codeword lengths, in order to minimize the average age experienced by the receiver, while maintaining a desired level of mutual information between the original and partial updates. This problem is NP hard. We relax the problem and develop an alternating minimization based iterative algorithm that generates a pmf for the partial updates, and the corresponding age-optimal real-valued codeword length for each update. We observe that there is a tradeoff between the attained average age and the mutual information between the original and partial updates

Benefits of Coding on Age of Information in Broadcast Networks

Age of Information (AoI) is studied in two-user broadcast networks with feedback, and lower and upper bounds are derived on the expected weighted sum AoI of the users. In particular, a class of simple coding actions is considered and within this class, randomized and deterministic policies are devised. Explicit conditions are found for symmetric dependent channels under which coded randomized policies strictly outperform the corresponding uncoded policies. Similar behavior is numerically shown for deterministic policies

Who Should Google Scholar Update More Often?

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 $square$ $roots$ 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

Selective Encoding Policies for Maximizing Information Freshness

An information source generates independent and identically distributed status update messages from an observed random phenomenon which takes $n$ distinct values based on a given pmf. These update packets are encoded at the transmitter node to be sent to a receiver node which wants to track the observed random variable with as little age as possible. The transmitter node implements a selective $k$ encoding policy such that rather than encoding all possible $n$ realizations, the transmitter node encodes the most probable $k$ realizations. We consider three different policies regarding the remaining $n-k$ less probable realizations: $highest$ $k$ $selective$ $encoding$ which disregards whenever a realization from the remaining $n-k$ values occurs; $randomized$ $selective$ $encoding$ which encodes and sends the remaining $n-k$ realizations with a certain probability to further inform the receiver node at the expense of longer codewords for the selected $k$ realizations; and $highest$ $k$ $selective$ $encoding$ $with$ $an$ $empty$ $symbol$ which sends a designated empty symbol when one of the remaining $n-k$ realizations occurs. For all of these three encoding schemes, we find the average age and determine the age-optimal real codeword lengths, including the codeword length for the empty symbol in the case of the latter scheme, such that the average age at the receiver node is minimized. Through numerical evaluations for arbitrary pmfs, we show that these selective encoding policies result in a lower average age than encoding every realization, and find the corresponding age-optimal $k$ values.Comment: Submitted for publication, April 2020. Some text overlap with its conference version arXiv:2001.0997

Information Freshness in Cache Updating Systems

We consider a cache updating system with a source, a cache and a user. There are $n$ files. The source keeps the freshest version of the files which are updated with known rates $\lambda_i$. The cache downloads and keeps the freshest version of the files from the source with rates $c_i$. The user gets updates from the cache with rates $u_i$. When the user gets an update, it either gets a fresh update from the cache or the file at the cache becomes outdated by a file update at the source in which case the user gets an outdated update. We find an analytical expression for the average freshness of the files at the user. Next, we generalize our setting to the case where there are multiple caches in between the source and the user, and find the average freshness at the user. We provide an alternating maximization based method to find the update rates for the cache(s), $c_i$, and for the user, $u_i$, to maximize the freshness of the files at the user. We observe that for a given set of update rates for the user (resp. for the cache), the optimal rate allocation policy for the cache (resp. for the user) is a $threshold$ $policy$, where the optimal update rates for rapidly changing files at the source may be equal to zero. Finally, we consider a system where multiple users are connected to a single cache and find update rates for the cache and the users to maximize the total freshness over all users.Comment: Submitted for publicatio

Information Freshness for Timely Detection of Status Changes

In this paper, we aim to establish the connection between Age of Information (AoI) in network theory, information uncertainty in information theory, and detection delay in time series analysis. We consider a dynamic system whose state changes at discrete time points, and a state change won't be detected until an update generated after the change point is delivered to the destination for the first time. We introduce an information theoretic metric to measure the information freshness at the destination, and name it as generalized Age of Information (GAoI). We show that under any state-independent online updating policy, if the underlying state of the system evolves according to a stationary Markov chain, the GAoI is proportional to the AoI. Besides, the accumulative GAoI and AoI are proportional to the expected accumulative detection delay of all changes points over a period of time. Thus, any (G)AoI-optimal state-independent updating policy equivalently minimizes the corresponding expected change point detection delay, which validates the fundamental role of (G)AoI in real-time status monitoring. Besides, we also investigate a Bayesian change point detection scenario where the underlying state evolution is not stationary. Although AoI is no longer related to detection delay explicitly, we show that the accumulative GAoI is still an affine function of the expected detection delay, which indicates the versatility of GAoI in capturing information freshness in dynamic systems.Comment: 6 pages, 7 figure