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Most existing stochastic models on age of information (AoI) focus on a single shared server serving status update packets from $N>1$ sources where each packet update stream is Poisson, i.e., single-hop scenario. In the current work, we study a two-hop edge computing system for which status updates from the information sources are still Poisson but they are not immediately available at the shared edge server, but instead they need to first receive service from a transmission server dedicated to each source. For exponentially distributed and heterogeneous service times for both the dedicated servers and the edge server, and bufferless preemptive resource management, we develop an analytical model using absorbing Markov chains (AMC) for obtaining the distribution of AoI for any source in the system. Moreover, for a given tagged source, the traffic arriving at the shared server from the $N-1$ un-tagged sources, namely the interference traffic, is not Poisson any more, but is instead a Markov modulated Poisson process (MMPP) whose state space grows exponentially with $N$. Therefore, we propose to employ a model reduction technique that approximates the behavior of the MMPP interference traffic with two states only, making it possible to approximately obtain the AoI statistics even for a very large number of sources. Numerical examples are presented to validate the proposed exact and approximate models., Comment: 9 pages, 4 figures

We study push-based sampling and transmission policies for a status update system consisting of a general finite-state continuous-time Markov chain (CTMC) information source with known dynamics, with the goal of minimizing the average age of incorrect information (AoII). The problem setting we investigate involves an exponentially distributed delay channel for transmissions and a constraint on the average sampling rate. We first show that the optimum sampling and transmission policy is a 'multi-threshold policy', where the thresholds depend on both the estimation value and the state of the original process, and sampling and transmission need to be initiated when the instantaneous AoII exceeds the corresponding threshold, called the estimation- and state-aware transmission (ESAT) policy. Subsequently, we formulate the problem of finding the thresholds as a constrained semi-Markov decision process (CSMDP) and the Lagrangian approach. Additionally, we propose two lower complexity sub-optimum policies, namely the estimation-aware transmission (EAT) policy, and the single-threshold (ST) policy, for which it is possible to obtain these thresholds for CTMCs with relatively larger number of states. The underlying CSMDP formulation relies on the 'multi-regime phase-type' (MRPH) distribution which is a generalization of the well-known phase-type distribution, which allows us to obtain the distribution of time until absorption in a CTMC whose transition rates change with respect to time in a piece-wise manner. The effectiveness of the proposed ESAT, EAT and ST sampling and transmission policies are shown through numerical examples, along with comparisons with a baseline scheme that transmits packets according to a Poisson process in out-of-sync periods.

In multi-source status update systems, sources need to be scheduled appropriately to maintain timely communication between each of the sources and the monitor. A cyclic schedule is an age-agnostic schedule in which the sources are served according to a fixed finite transmission pattern, which upon completion, repeats itself. Such a scheme has a low $O(1)$ runtime complexity, which is desirable in large networks. This paper's focus is on designing transmission patterns so as to be used in massive scale networking scenarios involving a very large number of sources, e.g., up to thousands of IoT sources, with service time requirements and weights being heterogeneous in nature. The goal is to minimize the weighted sum age of information (AoI), called weighted AoI, when transmitting users' packets over a channel susceptible to heterogeneous packet errors. The main tool we use is a stochastic modeling framework using either Markov chains (MC) or moment generating functions (MGF), by which we obtain the weighted AoI for a given transmission pattern, which is not straightforward in the presence of packet drops. Using this framework, we provide a lower bound on the weighted AoI for the particular case of two sources, and also an algorithm to attain this lower bound. Then, by using the same framework, we design a cyclic scheduler for general number of sources with reasonable complexity using convex optimization and well-established packet spreading algorithms, and comparatively evaluate the proposed algorithm and existing age-agnostic scheduling schemes for general number of sources (resp.~two sources) when the lower bound is not available (resp.~when it is available). We present extensive numerical results to validate the effectiveness of the proposed approach.

Safety guarantees and security-latency problem of Nakamoto consensus have been extensively studied in the last decade with a bounded delay model. Recent studies have shown that PoW protocol is secure under random delay models as well. In this paper, we analyze the security-latency problem, i.e., how secure a block is, after it becomes k-deep in the blockchain, under general random delay distributions. We provide tight and explicit bounds which only require determining the distribution of the number of Poisson arrivals during the random delay. We further consider potential effects of recent Bitcoin halving on the security-latency problem by extending our results.

We study age of information (AoI) in a single-source dual-server status update system for the generate at will (GAW) scenario, consisting of an information source, dual servers, and a monitor. For this system, the method of stochastic hybrid systems (SHS) was used to obtain the mean AoI for the work-conserving ZW (zero wait) policy with out-of-order packet discarding at the monitor. In this paper, we propose a non-work-conserving F/P (freeze/preempt) policy for which the sampling and transmission process is frozen for an Erlang distributed amount of time upon each transmission, and out-of-order packets are preempted immediately at the source, rather than being discarded at the monitor upon reception. We use the absorbing Markov chain (AMC) method to obtain the exact distributions of AoI and also the peak AoI (PAoI) processes, for both ZW and F/P policies. Numerical results are presented for the validation of the proposed analytical model and a comparative evaluation of ZW and F/P policies., Comment: 10 pages, 7 figures

We consider a sensor that samples an $N$-state continuous-time Markov chain (CTMC)-based information source process, and transmits the observed state of the source, to a remote monitor tasked with timely tracking of the source process. The mismatch between the source and monitor processes is quantified by age of incorrect information (AoII), which penalizes the mismatch as it stays longer, and our objective is to minimize the average AoII under an average sampling rate constraint. We assume a perfect reverse channel and hence the sensor has information of the estimate while initiating a transmission or preempting an ongoing transmission. First, by modeling the problem as an average cost constrained semi-Markov decision process (CSMDP), we show that the structure of the problem gives rise to an optimum threshold policy for which the sensor initiates a transmission once the AoII exceeds a threshold depending on the instantaneous values of both the source and monitor processes. However, due to the high complexity of obtaining the optimum policy in this general setting, we consider a relaxed problem where the thresholds are allowed to be dependent only on the estimate. We show that this relaxed problem can be solved with a novel CSMDP formulation based on the theory of absorbing MCs, with a computational complexity of $\mathcal{O}(N^4)$, allowing one to obtain optimum policies for general CTMCs with over a hundred states.

In a communication setting where multiple sources share a single channel to provide status updates to a remote monitor, source transmissions need to be scheduled appropriately to maintain timely communication between each of the sources and the monitor. We consider age-agnostic scheduling policies which are advantageous due to their simplicity of implementation. Further, we focus on a special class of age-agnostic policies, called cyclic schedulers, where each source is scheduled based on a fixed cyclic pattern. We use weighted average age of information (AoI) to quantify the timeliness of communication. We develop a Markov chain formulation to compute the exact mean AoI for the case of two-source cyclic schedulers. Based on the obtained age expression, we develop an algorithm that generates near-optimal cyclic schedulers to minimize the weighted average AoI for two heterogeneous sources, in the presence of channel errors.

We study cyclic scheduling for generate-at-will (GAW) multi-source status update systems with heterogeneous service times and packet drop probabilities, with the aim of minimizing the weighted sum age of information (AoI), called system AoI, or the weighted sum peak AoI (PAoI), called system PAoI. In particular, we obtain well-performing cyclic schedulers which can easily scale to thousands of information sources and which also have low online implementation complexity. The proposed schedulers are comparatively studied against existing scheduling algorithms in terms of computational load and system AoI/PAoI performance, to validate their effectiveness., Comment: 7 pages, 5 figures

Age of incorrect information (AoII) has recently been proposed as an alternative to existing information freshness metrics for real-time sampling and estimation problems involving information sources that are tracked by remote monitors. Different from existing metrics, AoII penalizes the incorrect information by increasing linearly with time as long as the source and the monitor are de-synchronized, and is reset when they are synchronized back. While AoII has generally been investigated for discrete time information sources, we develop a novel analytical model in this paper for push- and pull-based sampling and transmission of a continuous time Markov chain (CTMC) process. In the pull-based model, the sensor starts transmitting information on the observed CTMC only when a pull request from the monitor is received. On the other hand, in the push-based scenario, the sensor, being aware of the AoII process, samples and transmits when the AoII process exceeds a random threshold. The proposed analytical model for both scenarios is based on the construction of a discrete time MC (DTMC) making state transitions at the embedded epochs of synchronization points, using the theory of absorbing CTMCs, and in particular phase-type distributions. For a given sampling policy, analytical models to obtain the mean AoII and the average sampling rate are developed. Numerical results are presented to validate the analytical model as well as to provide insight on optimal sampling policies under sampling rate constraints.

We study the scheduling problem for a multi-source single-server generate-at-will (GAW) status update system with sources having heterogeneous service times and weights, with the goal of minimizing the weighted sum age of information (AoI). In particular, we study \emph{age-agnostic} schedulers which rely only on the first two moments of the source service times and they are relatively easier to implement than their age-aware counterparts which make use of the actual realizations of the service times. In particular, we focus on age-agnostic cyclic schedulers with $O(1)$ runtime complexity where status updates from multiple sources are scheduled according to a fixed finite transmission pattern. We first develop an analytical method to obtain the exact average AoI of each source when a transmission pattern is given. Then, we derive the optimum transmission pattern in closed form for the specific case of two sources. For general number of sources, we propose a novel algorithm, called IS (Insertion Search), for constructing transmission patterns, and we show that IS is capable of producing the optimum pattern for two-source systems, and it outperforms other existing age-agnostic schemes, for the case of more than two sources. Numerical examples are presented to showcase the effectiveness of the proposed approach., Comment: 12 pages, 8 figures

We study a remote monitoring system in which a mutually independent and heterogeneous collection of finite-state irreducible continuous time Markov chain (CTMC) based information sources is considered. In this system, a common remote monitor queries the instantaneous states of the individual CTMCs according to a Poisson process with possibly different intensities across the sources, in order to maintain accurate estimates of the original sources. \color{black}Three information freshness models are considered to quantify the accuracy of the remote estimates: fresh when equal (FWE), fresh when sampled (FWS) and fresh when close (FWC). For each of these freshness models, closed-form expressions are derived for mean information freshness for a given source. Using these expressions, optimum sampling rates for all sources are obtained so as to maximize the weighted sum freshness of the monitoring system, subject to an overall sampling rate constraint. This optimization problem leads to a water-filling solution with quadratic worst case computational complexity in the number of information sources. Numerical examples are provided to validate the effectiveness of the optimum sampling policy in comparison to several baseline sampling policies., Comment: 10 pages, 6 figures

We consider an information update system consisting of $N$ sources sending status packets at random instances according to a Poisson process to a remote monitor through a single server. We assume a heteregeneous server with exponentially distributed service times which is equipped with a waiting room holding the freshest packet from each source referred to as Single Buffer Per-Source Queueing (SBPSQ). The sources are assumed to be equally important, i.e., non-weighted average AoI is used as the information freshness metric, and subsequently two symmetric scheduling policies are studied in this paper, namely First Source First Serve (FSFS) and the Earliest Served First Serve (ESFS) policies, the latter policy being proposed the first time in the current paper to the best of our knowledge. By employing the theory of Markov Fluid Queues (MFQ), an analytical model is proposed to obtain the exact distribution of the Age of Information (AoI) for each source when the FSFS and ESFS policies are employed at the server. Subsequently, a benchmark scheduling-free scheme named as Single Buffer with Replacement (SBR) that uses a single one-packet buffer shared by all sources is also studied with a similar but less complex analytical model. We comparatively study the performance of the three schemes through numerical examples and show that the proposed ESFS policy outperforms the other two schemes in terms of the average AoI and the age violation probability averaged across all sources, in a scenario of sources possessing different traffic intensities but sharing a common service time., Comment: 16 pages, 7 figures

We study age-agnostic scheduling in a non-preemptive status update system with two sources sending time-stamped information packets at random instances to a common monitor through a single server. The server is equipped with a waiting room holding the freshest packet from each source called "single-buffer per-source queueing". The server is assumed to be work-conserving and when the waiting room has two waiting packets (one from each source), a probabilistic scheduling policy is applied so as to provide Age of Information (AoI) differentiation for the two sources of interest. Assuming Poisson packet arrivals and exponentially distributed service times, the exact distributions of AoI and also Peak AoI (PAoI) for each source are first obtained. Subsequently, this analytical tool is used to numerically obtain the optimum probabilistic scheduling policy so as to minimize the weighted average AoI/PAoI by means of which differentiation can be achieved between the two sources. In addition, a pair of heuristic age-agnostic schedulers are proposed on the basis of heavy-traffic analysis and comparatively evaluated in a wide variety of scenarios, and guidelines are provided for scheduling and AoI differentiation in status update systems with two sources., Comment: 14 pages, 7 figures, 2 tables

Age of Information (AoI) and Peak AoI (PAoI) and their analytical models have recently drawn substantial amount of attention in information theory and wireless communications disciplines, in the context of qualitative assessment of information freshness in status update systems. We take a queueing-theoretic approach and study a probabilistically preemptive bufferless $M/PH/1/1$ queueing system with arrivals stemming from $N$ separate information sources, with the aim of modeling a generic status update system. In this model, a new information packet arrival from source $m$ is allowed to preempt a packet from source $n$ in service, with a probability depending on $n$ and $m$. To make the model even more general than the existing ones, for each of the information sources, we assume a distinct PH-type service time distribution and a distinct packet error probability. Subsequently, we obtain the exact distributions of the AoI and PAoI for each of the information sources using matrix-analytical algorithms and in particular the theory of Markov fluid queues and sample path arguments. This is in contrast with existing methods that rely on Stochastic Hybrid Systems (SHS) which obtain only the average values and in less general settings. Numerical examples are provided to validate the proposed approach as well as to give engineering insight on the impact of preemption probabilities on certain AoI and PAoI performance figures., Comment: 16 pages, 6 figures, 3 tables

Information freshness in IoT-based status update systems has recently been studied through the Age of Information (AoI) and Peak AoI (PAoI) performance metrics. In this paper, we study a discrete-time server arising in multi-source IoT systems which accepts incoming information packets from multiple information sources so as to be forwarded to a remote monitor for status update purposes. Under the assumption of Bernoulli information packet arrivals and a common geometric service time distribution across all the sources, we numerically obtain the exact per-source distributions of AoI and PAoI in matrix-geometric form for three different queueing disciplines: i) Non-Preemptive Bufferless (NPB) ii) Preemptive Bufferless (PB) iii) Non-Preemptive Single Buffer with Replacement (NPSBR). The proposed numerical algorithm employs the theory of Discrete-Time Markov Chains (DTMC) of Quasi-Birth-Death (QBD) type and is matrix analytical, i.e, the algorithm is based on numerically stable and efficient vector-matrix operations.Numerical examples are provided to validate the accuracy and effectiveness of the proposed queueing model. We also present a numerical example on the optimum choice of the Bernoulli parameters in a practical IoT system with two sources with diverse AoI requirements., Comment: 15 pages, 3 figures

We study the MAP/M/s+G queuing model with MAP (Markovian Arrival Process) arrivals, exponentially distributed service times, infinite waiting room, and generally distributed patience times. Using sample-path arguments, we propose to obtain the steady-state distribution of the virtual waiting time and subsequently the other relevant performance metrics of interest for the MAP/M/s+G queue by means of finding the steady-state solution of a properly constructed Continuous Feedback Fluid Queue (CFFQ). The proposed method is exact when the patience time is a discrete random variable and is asymptotically exact when it is continuous/hybrid for which case discretization of the patience time distribution and subsequently the steady-state solution of a Multi-Regime Markov Fluid Queue (MRMFQ) is required. Besides the steady-state distribution, we also propose a new method to approximately obtain the first passage time distribution for the virtual and actual waiting times in the $MAP/M/s+G$ queue. Again, using sample-path arguments, finding the desired distribution is also shown to reduce to obtaining the steady-state solution of a larger dimensionality CFFQ where the deterministic time horizon is to be approximated by Erlang or Concentrated Matrix Exponential (CME) distributions. Numerical results are presented to validate the effectiveness of the proposed method., Comment: 21 pages, 14 tables, 3 figures

Bufferless and single-buffer queueing systems have recently been shown to be effective in coping with escalated Age of Information (AoI) figures arising in single-source status update systems with large buffers and FCFS scheduling. In this paper, for the single-source scenario, we propose a numerical algorithm for obtaining the exact distributions of both the AoI and the peak AoI (PAoI) in (i) the bufferless $PH/PH/1/1/P(p)$ queue with probabilistic preemption with preemption probability $p$, $0 \leq p \leq 1,$ and (ii) the single buffer $M/PH/1/2/R(r)$ queue with probabilistic replacement of the packet in the queue by the new arrival with replacement probability $r$, $0 \leq r \leq 1$. The proposed exact models are based on the well-established theory of Markov Fluid Queues (MFQ) and the numerical algorithms are matrix-analytical and they rely on numerically stable and efficient vector-matrix operations. Moreover, the obtained exact distributions are in matrix exponential form, making it amenable to calculate the tail probabilities and the associated moments straightforwardly. Firstly, we validate the accuracy of the proposed method with simulations, and for sume sub-cases, with existing closed-form results. We then comparatively study the AoI performance of the queueing systems of interest under varying traffic parameters., Comment: 18 pages, 12 figures

A semi-centralized joint cell muting and user scheduling scheme for interference coordination in a multicell network is proposed under two different temporal fairness criteria. The main principle behind the proposed scheme is that a central entity selects a cell muting pattern out of a pattern set at a decision instant, and subsequently the un-muted base stations opportunistically schedule the users in the associated cells, both decisions made on a temporal-fair basis. Although some pattern sets are easily obtainable from static frequency reuse systems, we propose a more general pattern set construction algorithm in this paper. As for the first fairness criterion, all cells are assigned to receive the same temporal share with the ratio between the temporal share of a cell center section and that of the cell edge section is set to a fixed desired value for all cells. The second fairness criterion is based on the max-min temporal fairness for which the temporal share of the network-wide worst-case user is maximized. Numerical results are provided to validate the effectiveness of the proposed scheme for both criteria. The impact of choice of the cell muting pattern set is also studied through numerical examples for various cellular topologies.