In this work, we generalize the Stochastic Hybrid Systems (SHSs) analysis of Age of information (AoI) to the Age of Incorrect Information (AoII) metric. Hierarchical ageing processes are adopted using the continuous AoII for the first time. Two different hierarchy schemes are considered: 1). A hierarchy of zero and linear ageing processes with different slopes; 2). A hierarchy of zero, linear and exponential ageing processes. We first modify the main result in Yates (2020) to provide a systematic way to analyze the continuous hierarchical AoII over unslotted real-time systems. The closed-form expressions of average hierarchical AoII are obtained in two typical scenarios with different channel conditions, i.e., an M/M/1/1 queue over noisy channels and two M/M/1/1 queues over collision channels. Moreover, under each scenario, we analyze the stability issue regarding positive recurrence and provide the stability conditions that ensure a steady-state average AoII. Finally, we compare the closed-form results between average AoI and AoII in the M/M/1/1 queue. The effects of different channel parameters on the average hierarchical AoII are also evaluated.

This paper considers a real-time scheduling problem disseminating status update of sensors timely from a base station (BS) to users over wireless fading channels. The freshness of the updated status is quantified using the Age of Incorrect Information (AoII) metric. The objective is to minimize the long-term AoII under the constraint of limited transmission power, which necessitates that only a subset of users can successfully receive updates in each time slot. To find an optimal transmission policy, we first model the AoII minimization problem as a multi-action multi-armed bandit problem. After that, we decompose the derived MAB problem into multiple sub-problems. For each sub-problem, we obtain an optimal policy based on multiple threshold, thereby establishing the indexability of the optimal problem. Building upon this, a novel multi-action productivity index (MAPI) is introduced to get the optimal transmission policy. However, due to the computational complexity of the relative value iteration (RVI) algorithm, the exact value of the MAPI remains difficult to determine. To address this challenge, a computationally efficient algorithm is proposed to approximate the indices and derive the transmission policy for each user. Compared with Whittle’s Index and Max Weight policies, MAPI-based policy demonstrates significant performance improvement, particularly in large-scale networks and when the transmission power separation interval is small.

We introduce the hybrid ageing problem in gossip networks, where each node has different ageing processes. This generalization brings two new issues: i) the traditional subset recursion method in [1] fails; ii) the existence of stationary average age penalty needs to be re-examined. To resolve issue i) and ii), we leverage a node-by-node SHSs analysis by introducing splitting Poisson processes to evaluate the average age penalty. We first analyze the hybrid ageing problem in two types of line networks, i.e., one-way and two-way gossip lines. In one-way gossip lines, we derive the closed-form expression of average age penalty in two different cases, where the ageing process of each node can be ordered or disordered (Definition 1). The closedform expressions of average age penalty are also derived in the two-way gossip line. Moreover, we show that the growth rate of age penalty is bounded by the arrival rates between gossip nodes in the one-way line to ensure the existence of average age penalty.

This paper extends the analysis of discrete age of information (AoI) to Ber/G/1/1 systems using the method of Probability Generation Functions (PGFs). The evolutions of AoI are characterized by defining a two-dimensional stochastic process. We derive the relationship between the PGF of arbitrary service time S and the system AoI in two different cases, where the packet service can or cannot be preempted by newly arrived packets. The closed-form expressions of the average AoI in both cases are obtained. Moreover, we prove that when fixing the mean of S, deterministic service time minimizes the system average AoI if packet service cannot be preempted. On the contrary, if preemption in the server is allowable, then deterministic S maximizes the average AoI.

We study a class of gossip networks where a source delivers fresh status updates through networks consisting of a set of gossiping nodes. Contrary to previous works, the source delivers updates subject to an age-dependent point process where the rates are related to the age process at each node. Moreover, our work divides the age-dependent gossip networks into two types, i.e., type-A and type-B, where the former allows only one node sampling from source and the latter allows all. We first prove the necessary Markovity and ergodicity of both age-dependent gossip networks and proceed with deriving the general form of unique stationary distributions. With these premises established, closed-form expressions of the stationary age moments for type-A networks with arbitrary topologies are derived with the help of SHSs, with three specified results given for the line, ring and fully-connected networks. Meanwhile, we prove that no closed-form expressions of stationary age moments can be obtained in type-B networks. An approximated model for arbitrary-connected type-B networks is proposed, where we formulate the stationary moment equations for the approximated SHSs under the guarantee of Lagrange stability. Then, we provide a moment closure method to solve two symmetric cases, i.e., fully-connected and ring networks, approximately and verify the effectiveness of our algorithm by comparing to the simulations.

We study a class of gossip networks where a single source delivers fresh status updates through networks consisting of a set of gossiping nodes. Contrary to previous works, the source delivers updates to nodes subject to an age-dependent point process where the update rates are related to the age process at each node. We derive the closed-form expressions of stationary age moments for the disconnected networks, and demonstrate the general procedure of moment analysis in the age-dependent gossip networks based on stochastic hybrid systems (SHSs). Considering the analytical difficulties in solving the infinite-dimensional stationary moment equations in more complex topologies, we provide two numerical methods to solve for the numerical values of stationary age moments approximately, and verify the effectiveness of these methods on the first three age moments in ring networks.

The continuous development of modern communication and computation gives rise to a large number of applications based on Internet of Things (IoT) technology. Considering the diverse uses and various service requirements, the information freshness of IoT data, which is measured by age of information (AoI), is crucial for latency-sensitive IoT applications, such as industrial automation and intelligent transportation, because outdated information can lead to delayed and inaccurate response. Under the assumptions of bufferless and no service preemption, in this paper we consider the scenario where multiple sources transmit packets through a common server and analyze the discrete AoI corresponding to each source. To facilitate the description of random AoI evolutions, we assign priorities to the sources and assume that when multiple sources generate new packets in one time slot, the packet with the highest priority is selected and served. We obtain the explicit expression of average AoI and calculate the general formula in several cases, where the relationships between average AoI and other system parameters are investigated detailly. Also, under a constraint imposed on average service time over all the sources, we consider minimizing average AoI of one source by finding the optimal service rates for the packet from each source. In particular, for minimizing one source’s average AoI in two-source systems, the pair of optimal service rates are determined completely. The results show that if the probability that one source’s packet obtains the service is large enough, then to minimize average AoI of this source, maximizing the service rate for its own packets is not the optimal. Some discussions on source priorities are given and finally we provide the numerical simulations for the obtained results.