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 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.

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crucial for latency-sensitive IoT applications, such as industrial automation and intelligent transportation, because outdated information can lead to delayed and inaccurate response.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.

In this paper, for discrete time status updating system with infinite size, we derive the explicit expression of average age of information (AoI) and its stationary distribution. Notice that there is no packet loss in an infinite size system, we successfully characterize the random dynamics of discrete AoI using a two-dimensional state vector, which simultaneously tracks the value of AoI and the age of packet currently under service. A two-dimensional stochastic process is constituted and we completely solve all the stationary probabilities. Then, as one of the marginal distributions the stationary distribution of AoI is obtained, with which the average AoI is also determined.

In this paper, we consider the age of information (AoI) of a discrete time status updating system, focusing on finding the stationary AoI distribution assuming that the Ber/G/1/1 queue is used. Following the standard queueing theory, we show that by invoking a two-dimensional state vector which tracks the AoI and packet age in system simultaneously, the stationary AoI distribution can be derived by analyzing the steady state of the constituted two-dimensional stochastic process. We give the general formula of the AoI distribution and calculate the explicit expression when the service time is also geometrically distributed. The discrete and continuous AoI are compared, we depict the mean of discrete AoI and that of continuous time AoI for system with M/M/1/1 queue. Although the stationary AoI distribution of some continuous time single-server system has been determined before, in this paper, we shall prove that the standard queueing theory is still appliable to analyze the discrete AoI, which is even stronger than the proposed methods handling the continuous AoI.