Age-Threshold Slotted ALOHA for Optimizing Information Freshness in Mobile Networks

In the early stages of AoI analysis for large-scale networks, many researchers adopted the traditional analytical framework of slotted ALOHA. For example, [4] characterized and optimized the average AoI by optimal tuning the channel access probability. [5] optimized the peak AoI using First-Come-First-Serve (FCFS) and Last-Come-First-Serve (LCFS) strategies, showing that with optimal packet arrival rate and channel access probability, the peak AoI scales linearly with the number of access nodes. [6] considered a multi-channel ALOHA network, enhancing AoI performance via joint radio access control and resource allocation. [7] explores the average AoI performance in a heterogeneous random access network where stochastic arrival nodes coexist with generate-at-will nodes. The function of irregular repetition slotted ALOHA (IRSA) on various AoI performance metrics has been demonstrated in [8]. The dynamic frame slotted ALOHA (FSA) has been studied in [9]. These efforts have alleviated the performance bottlenecks of the AoI metric in large-scale networks. However, the primary objective behind the conventional ALOHA-like protocol was to mitigate channel contention, thereby enhancing transmission success probability or throughput. It was not explicitly designed to cater to the distinct attributes of the AoI metric. Hence, it becomes essential to explore random access protocols that closely align with the attributes of the AoI metric.

To design an age-aware random access network, it is essential to (a) mitigate network interference, (b) minimize the waiting time for information packets at the transmitters, and (c) ensure equalization of the update packet reception intervals. A straightforward strategy to address these needs is to prioritize nodes with lower timeliness for updates, allowing nodes with higher information freshness to remain silent temporarily. Based on this idea, several works [10, 11, 12, 13, 14, 15] extended the Slotted ALOHA (SA) into an age-aware variant. Specifically, a threshold-based age-dependent random access protocol was proposed in [10, 11], where each node accesses the channel only when its instantaneous AoI exceeds a predetermined threshold. A distributed transmission strategy was proposed in [12] based on the age gain, defined as the reduction of instantaneous AoI when packets are successfully delivered. An Index-Prioritized Random Access scheme was proposed in [13], where nodes access the channel according to their indices that reflect the urgency of the update. Minislotted threshold ALOHA, designed to minimize the average AoI in the high payload scenario, is investigated in [14]. The dynamic FSA with age gain threshold is considered in [15].

The aforementioned research endeavors predominantly concentrate on the single-cell setting, where multiple nodes transmit data to a central access point. However, it is imperative to consider the ad-hoc network scenario for a more comprehensive understanding. Another noteworthy aspect is that the interference in the above references is modeled by the collision model[17], wherein concurrent transmissions always lead to communication failures. Such a model neglects essential physical elements in wireless channels, such as fading, path loss, and spatial network topology, oversimplifying the co-channel interference effects. In contrast, the SINR model is preferable to the update packet verdict on the receiver end, as it can adequately account for these critical features in the ad-hoc network scenario.

Consequently, a line of recent studies has been carried out, analyzing and optimizing AoI in wireless networks on the basis of the SINR model[18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30]. Specifically, the upper and lower bounds of the time-average AoI in Poisson bipolar networks are provided in [18] based on the favorable/dominant system argument, and the upper bound is improved by reconstruction of the dominant system in [19]. The authors in [20] derived the average peak AoI with different packet arrival patterns, i.e., the time-triggered and event-triggered traffic for large-scale wireless networks, while the authors in [21] and [22] analyze the AoI statistics under different buffer sizes and package management disciplines. The authors in [23] present a comprehensive study on the interplay between throughput and AoI in a cellular-based IoT network. Explicit optimal expression of peak AoI of slotted ALOHA, and its optimal parameters are derived in [24], the distinctions and similarities between the optimal parameters for the peak AoI and average AoI are extensively discussed in [25]. The energy-AoI tradeoff is further explored in [26]. The authors in [27] examine the AoI performance of random access networks under frame-slotted ALOHA protocol. Locally adaptive strategies for optimizing AoI have been studied in [28], [29], and [30], where the channel access probability, update rate, and transmit power at each node are adjusted according to the local communication environment to minimize AoI.

The existing results primarily pertain to the conventional slotted ALOHA protocol, whilst the effects of the age threshold remain unexplored. Although a recent work [16] has investigated the effect of TSA on AoI performance in static Poisson bipolar networks, the analysis involves complicated expressions of the SINR meta distribution, prohibiting it from optimizing TSA to reap the full potential of this protocol. In this paper, we consider the AoI optimization under TSA in a high mobility case, with which we can establish tractable analysis and obtain closed-form expression of the optimal network configuration, AoI performance limits, and scaling law that provide insights into network designs.

The main contributions of this paper are summarized below.

• •

  We develop a mathematical framework for analyzing the impact of the TSA protocol on the AoI performance of random access networks. We establish a fixed-point equation to characterize the transmission success probability, encompassing key features such as the update rate, age threshold, channel fading, deployment density, and co-channel interference.

• •

  We derive analytical expressions of the mean peak AoI and time-average AoI over the typical link. Our analysis culminates in an optimization strategy to minimize these AoI metrics, achieved by a joint optimization of the update rate and age threshold. Specifically, we investigate the performance limits of the mean peak AoI under TSA protocol and identify the corresponding parameter pairs that enable its minimization. We also propose an alternative iterative algorithm to achieve the optimal parameter configuration that minimizes the time-average AoI. Additionally, we derive closed-form approximations for these optimal parameters, which are asymptotically accurate.

• •

  We establish a scaling law of AoI in random access networks. Notably, it reveals that although the optimal mean peak AoI and time-average AoI increase linearly with the deployment density, the growth rate of time-average AoI under TSA is only half of that under SA.

• •

  Considering the inherent instability of ALOHA-like networks, especially under conditions of high interference levels and traffic load, we propose parameter tuning strategies for both mean peak AoI and time-average AoI in the presence of a bistable region, aiming to bolster the robustness of the TSA protocol network.

To better highlight the technical contributions of this paper, we summarize existing literature that dealt with age-threshold policy in Table I.

We consider a random access network comprised of source-destination pairs. The source nodes are scattered according to a homogeneous Poisson point process (HPPP) $\Phi_{\mathrm{s}}=\{X_{i}\}_{i=1}^{\infty}$ of density $\lambda$. Each source is paired with a destination in the distance $r$ and oriented in a random direction. According to the displacement theorem[31, 32], locations of the destination nodes constitute an independent HPPP $\Phi_{\mathrm{d}}$ of the same density. We add a receiver at the origin to the point process $\Phi_{\mathrm{d}}$. We also add its tagged transmitter, denoted by $X_{0}$, to the point process $\Phi_{\mathrm{s}}$. We refer to this link pair as the typical one. ¹¹1This setting is a large-scale analog to the classical model of Random Networks [33], where the fixed distance between each transmitter-receiver pair represents the average value. It is noteworthy that the analysis developed in this article can be extended to investigate the AoI performance in networks with centralized infrastructures where multiple sources can associate with a common destination and are located at random distances from the destinations [34, 35]. Based on Slivnyak’s theorem[31, 32], we can concentrate on analyzing the performance of the typical link. A sequence of status information is generated at each source, encapsulated into information packets, and transmitted over a shared spectrum. When a source node sends out information packets, it transmits at a fixed power. We assume the channel between any pair of nodes is affected by the Rayleigh fading with a unit mean, which varies independently across time slots, and path loss that follows the power law. We also assume the received signal is subjected to white Gaussian thermal noise. The main notations used throughout this article are summarized in Table II.

In this network, time is segmented into equal-length intervals, in which each slot is the minimum duration required to transmit an information packet (and receive feedback from the destination). We assume the network is synchronized[36, 37]. Each source adopts the generate-at-will policy [38] for the status updating. Particularly, if a node decides to transmit, it will generate a new sample at the beginning of the time slot and send the information packet to the destination immediately. By the end of the same time slot, the packet is successfully decoded if the received SINR exceeds a decoding threshold; otherwise, the transmission fails, and in that case, the packet is dropped. The delivery of packets, therefore, incurs a delay of one time slot. Additionally, the positions of the links vary according to the high mobility model [24, 39], i.e., spatial locations of the link pairs are (re)generated independently in each time slot according to the same Poisson bipolar network model. ²²2By leveraging the framework of [40], one can extend the results obtained in this paper to investigate AoI in a mobile network with spatial correlations over time.

We employ the TSA protocol for status updating. Specifically, every source node stays inactive until its AoI reaches a predefined threshold, denoted by $A$, upon which the source node turns on the status updating mode: it generates a fresh sample with probability $\eta$ at the beginning of the subsequent time slot, encapsulates that information into a packet, and immediately sends it to the destination. If the received SINR surpasses a decoding threshold $\theta$, the transmission succeeds; the receiver then feeds back an ACK to the source, and the age is reset to one.³³3In this paper, we make the commonly adopted assumption that the ACK feedback is error-free [39, 11, 12]. It is, nonetheless, noteworthy that the analysis can be extended to account for the feedback error of ACK messages. Otherwise, the source will generate a new sample in the next time slot with probability $\eta$. This probability is usually referred to as the update rate.

We put the main focus on the notion of AoI in this paper. As depicted in Fig. 1, AoI grows linearly with time in the absence of new updates at the destination, and it drops to the time elapsed since the generation of the latest packet received. Formally, the age evolution process over the typical link can be written as follows [2, 38]

where $G_{0}(t)$ denotes the generation time of the latest packet successfully delivered over this link.

In this paper, we use the mean peak AoI and time-average AoI over the typical link as our performance metrics, defined respectively as [21]

where $T_{0}(n)$ denotes the time slot at which the $n$-th packet is successfully delivered, and

Since the point processes formed by the TSA protocol still exhibit ergodicity and stationarity, the statistical characteristics of AoI performance for the typical link are identical to those of the other links.

Since all the source nodes use the same spectrum for packet delivery, each transmission would be affected by others owing to co-channel interference. As such, at the typical destination, the received SINR at time slot $t$ is[41]

where $h_{i0}(t)\sim\exp(1)$ represents the small-scale fading between transmitter $i$ and the typical receiver, $\alpha$ is the path-loss exponent, $\|\cdot\|$ stands for the Euclidean norm, $\rho$ denotes the signal-to-noise ratio (SNR) between the transmit power of each source node and the noise power, and $v_{i}(t)$ is a binary function, where $v_{i}(t)=1$ indicates that transmitter $i$ initiates a packet transmission at time slot $t$, and $v_{i}(t)=0$ otherwise.

According to the transmission policy, an information packet is successfully delivered if the received SINR exceeds a decoding threshold $\theta$. Therefore, the probability of successfully transmitting a packet from the typical source node can be written as [32]

Under the high mobility random walk model, the received $\text{SINR}_{i}(t)$ of each transmitter $i$, $i\in\mathbb{N}$, is independent and identically distributed (i.i.d.) across time $t$. By symmetry, the transmission success probability is also identical across the transmitters. As such, we drop the time index from $p_{\mathrm{s}}(t)$ in the sequel. The following theorem provides a formal characterization of this quantity.

The expression in (6) accounts for the impact of several key factors, including the update rate, age threshold, path loss, and deployment density, on the transmission success probability. As opposed to other variants of SA, e.g., the FSA [27], the transmission success probability under TSA is given in the form of a fixed-point equation, which steams from the spatial-temporal interactions amongst the source nodes across the network.

The parameter $c$ in (6) reflects the level of spatial contention [43], measuring the network’s capability of spatial reuse by quantifying how fast the transmission success probability deteriorates when the interferers’ density increases.

The solution(s) to fixed-point equation (6) can be obtained via fixed-point iterations. Specifically, we denote by $p_{\mathrm{s},1}\in(0,1]$ and $p_{\mathrm{s},n}$ the initial point and estimation of transmission success probability after the $n$th iteration, respectively. The iteration process can be expressed as follows

While the convergence of the fixed-point iteration is immune to the initial value, depending on the network configuration, different starting points may lead to different solutions. Therefore, we investigate the conditions of different distributions of roots of (6) in the following.

Notably, in the scenario that (6) has three distinct roots, if $p_{\mathrm{s},0}\in(0,p_{\mathrm{s}}^{M})$ or $p_{\mathrm{s},0}\in(p_{\mathrm{s}}^{M},1]$, the fixed-point iteration (8) results in $p_{\mathrm{s}}^{L}$ and $p_{\mathrm{s}}^{H}$, respectively. However, regardless of the initial value of $p_{\mathrm{s},0}$, there is no guarantee from (8) that the ultimate solution would be $p_{\mathrm{s}}^{M}$.

A pictorial example is provided in Fig. 2, where we set the parameters that satisfy $\lambda cr^{2}>\frac{4}{\eta}$ and showcase six distinct curves, arranged based on the value of the age threshold. Specifically, they are: $(i)$ $A=0$, $(ii)$ $A=10\in(0,A_{l})$, $(iii)$ $A=A_{l}\approx 31$, $(iv)$ $A=50\in(A_{l},A_{h})$, $(v)$ $A=A_{h}\approx 135$, and $(vi)$ $A=150\in(A_{h},+\infty)$. Let us take the case of $A=50\in(A_{l},A_{h})$ as an example. In this case, the region of $p_{\mathrm{s},n}$ has four possible scenarios: (1) $0<p_{\mathrm{s},n}<p_{\mathrm{s}}^{L}$; (2) $p_{\mathrm{s}}^{L}<p_{\mathrm{s},n}<p_{\mathrm{s}}^{M}$; (3) $p_{\mathrm{s}}^{M}<p_{\mathrm{s},n}<p_{\mathrm{s}}^{H}$; and (4) $p_{\mathrm{s}}^{H}<p_{\mathrm{s},n}<1$. In the first and second scenarios, $p_{\mathrm{s},n}$ will gradually approach $p_{\mathrm{s}}^{L}$. In contrast, $p_{\mathrm{s},n}$ tends toward $p_{\mathrm{s}}^{H}$ in the third and fourth cases. As such, in this example, there are two candidate points, namely, $p_{\mathrm{s}}^{L}$ and $p_{\mathrm{s}}^{H}$, that $p_{\mathrm{s},n}$ can converge to. In a similar vein, we can show that $p_{\mathrm{s},n}$ only converges to $p_{\mathrm{s}}^{H}$ (or $p_{\mathrm{s}}^{L}$) in other cases.

In the terminology of random network analysis [44], $p_{\mathrm{s}}^{L}$ and $p_{\mathrm{s}}^{H}$ are steady-state points of the fixed-point equation (6) since they are attainable by controlling the input to the fixed-point iteration in (8). In contrast, $p_{\mathrm{s}}^{M}$ is not a steady-state point in this context.

To this end, we define the following region of network parameter configurations under which the fixed-point equation of transmission success probability has steady-state solutions, and we term these ranges the stable regions.

• •

  Bistable region:⁴⁴4In the context of ALOHA-like random access networks, the bi-stable characteristic refers to a phenomenon where the system’s performance (measured by, e.g., transmission success probability) can exist in one of two statistically stable equilibrium points under certain conditions[45, 44, 46, 47]. The bi-stable region is the unstable region in practice, since the system will easily converge to the state where every transmitter has a small transmission success probability. In this case, data transmissions often fail, significantly deteriorating the performance of real-time applications.

  $$\mathcal{B}=\left\{\left(\lambda,\theta,r,\alpha,\rho,A,\eta\right)\Big{|}\lambda cr^{2}>\frac{4}{\eta},A_{l}<A<A_{h}\right\},$$

  in which the network has two steady-state points $p_{\mathrm{s}}^{L}$ and $p_{\mathrm{s}}^{H}$.

• •

  Monostable region: $\mathcal{M}=\mathcal{M}_{p_{\mathrm{s}}^{H}}\cup\mathcal{M}_{p_{\mathrm{s}}^{L}}=\bar{\mathcal{B}}$ ($\bar{\mathcal{B}}$ is the complement of set $\mathcal{B}$); in this case, the network has one steady-state point, which can be further partitioned into the following.

  (i) High-efficiency region:

  $$\mathcal{M}_{p_{\mathrm{s}}^{H}}=\left\{\left(\lambda,\theta,r,\alpha,\rho,A,\eta\right)\big{|}\lambda cr^{2}\leq\frac{4}{\eta}~{}\text{and}~{}\lambda cr^{2}>\frac{4}{\eta},A\geq A_{h}\right\},$$

  in which the iteration converges to $p_{\mathrm{s}}^{H}$.

  (ii) Low-efficiency region:

  $$\mathcal{M}_{p_{\mathrm{s}}^{L}}=\left\{\left(\lambda,\theta,r,\alpha,\rho,A,\eta\right)\big{|}\lambda cr^{2}>\frac{4}{\eta},A\leq A_{l}\right\},$$

  in which the iteration converges to $p_{\mathrm{s}}^{L}$.

The following corollary summarizes the properties of the steady-state points in terms of the configuration parameters.

It is intuitive that in densely deployed networks, the transmission success probability will benefit (more) from the TSA protocol due to its capability in mitigating channel contention. This can also be observed from Fig. 2 by comparing the value of transmission success probability at $A=0$ with those at $A>0$.

Using the transmission success probability, we can derive closed-form expressions for the mean peak AoI and time-average AoI as follows.

Leveraging the above theorem, we can investigate different regimes of the age threshold to demystify its influence on the AoI performance. Firstly, if we set the age threshold $A$ to be zero, the results in Theorem 2 reduce to the mean peak AoI and time-average AoI in large-scale networks under the SA protocol[10], i.e.,

Consequently, we can compare the AoI achieved under the SA and TSA protocols to identify the operating regimes where TSA outperforms SA.

We first compare the mean peak AoI attained under the SA and TSA protocols for $A>0$, which yields

where $(a)$ follows from the mean value theorem, in which $\xi_{\mathrm{p}}\in[\frac{1}{1+A\eta p_{\mathrm{s}}},1]$. As such, when $\lambda cr^{2}\geq\frac{1+A\eta p_{\mathrm{s}}^{\mathrm{TSA}}}{\eta}$, we have

Following similar steps, we can compare the time-average AoI attained under the SA and TSA protocols as follows:

where $(a)$ follows from the mean value theorem, in which $\xi_{\mathrm{a}}\in[\frac{1}{1+A\eta p_{\mathrm{s}}},1]$. When $\lambda cr^{2}\geq\frac{(1+A)(1+A\eta p_{\mathrm{s}}^{\mathrm{TSA}})}{2A\eta}$, we have

The inequalities (17) and (19) demonstrate that the TSA has the potential of outperforming SA in reducing AoI.

However, if we raise the age threshold to an excessively large value, i.e., $A\to\infty$, then we have

implying that a large age threshold will be detrimental to the AoI.

To this end, it is apparent that the age threshold, as well as the update rate, shall be adequately tuned to reveal the full potential of TSA. We will detail the approaches to obtain the optimal parameters in the following sections. For ease of exposition, we start our derivations without considering the influence of the bi-stable region. In Section VI, we will demonstrate how to adjust the parameters to ensure network stability while concurrently optimizing the AoI performance in the presence of a bi-stable region.

The following theorem presents the optimal update rate for a given age threshold.

Lemma 1 shows that the optimal update rate shall be tuned to the maximum, i.e., $\hat{\eta}^{*}_{|A}=1$, when the $\lambda cr^{2}\leq 1+Ap_{\mathrm{s},*}$. This is because when the nodes are sparsely deployed, each receiver has a high chance of successfully receiving updates. As such, the AoI performance can be effectively improved by increasing update rates. In contrast, when the nodes are deployed densely, each source shall decrease its update rate to circumvent channel contention and promote timely information delivery across the network.

The following theorem characterizes the optimal age threshold for a given update rate.

Lemma 2 indicates that when the interference level is high, i.e., $\lambda cr^{2}\eta>1$, imposing an age threshold to the ALOHA-like protocol is beneficial for reducing mean peak AoI. Conversely, when the interference level is low, the conventional SA protocol is preferable in decreasing the mean peak AoI.

By integrating results from the above, the theorem below presents a closed-form result for the optimal pair of the update rate and age threshold that minimizes the mean peak AoI.

From (31), we observe that the optimal age threshold increases with the optimal update rate to mitigate the induced interference. Moreover, when $\lambda cr^{2}\leq 1$, the optimal TSA configuration reduces to the conventional SA. In addition to these, a (perhaps) more important message conveyed by Theorem 3 is that by appropriately adjusting the update rate, SA can attain the same optimal mean peak AoI as TSA. In other words, TSA and SA are equivalent in minimizing the mean peak AoI.

First, we optimize the update rate under a fixed age threshold, where the structural result is summarized as follows.

The condition that $\lambda cr^{2}\leq 1+Ap_{\mathrm{s},*}$ can be expressed as

This inequality indicates that if the transmission success probability is lower than $\exp\left(-1-\theta r^{\alpha}\rho^{-1}\right)$ when $\eta=1$, the update rate should be reduced to ensure the transmission success probability stabilizes at this value to optimize the time-average AoI. Because the age threshold always contributes to the transmission success probability, the optimal update rate goes up with the increase of the age threshold.

A comparison between (25) and (33) reveals that for a given age threshold, the optimal update rates that minimize the mean peak AoI and time-average AoI admit the same structure. This finding seems surprising, as the two performance metrics have very different analytical expressions (cf. Theorem 1). And the reason boils down to that, in essence, the core of optimizing the update rate for mean peak AoI and time-average AoI lies at maximizing the spatial throughput $\eta p_{\mathrm{s}}$.

In a similar vein, we explore the optimal structure of the age threshold under a fixed update rate. The following result provides a systematic approach to achieving this.

According to Lemma 4, we note that the optimal age threshold is $\bar{A}^{*}_{|\eta}=0$ when

indicating that when network interference is mild, one shall not impose any waiting duration at the source nodes but employ an SA-like protocol for the status updating. In contrast, TSA protocol is preferable when interference is severe, as it reduces concurrent transmissions, thus mitigating potential contentions. As a result, the nodes with urgent updating demands will benefit from a better transmission environment, which facilitates age performance.

The condition in (39) can be equivalently written as

This implies that TSA outperforms SA in terms of AoI when the update frequency is relatively high. Furthermore, when $\lambda cr^{2}>\tfrac{1}{2}\exp(-\theta r^{\alpha}\rho^{-1})$, TSA protocol consistently achieves a better time-average AoI than SA. Notably, these observations are in line with those drawn in [12], which are concluded based on a collision channel model.

While it is possible to solve equation (38) numerically, obtaining a closed-form solution (which could provide design insights) is challenging. In that respect, we leverage (tight) upper and lower bounds of the time-average AoI to facilitate our optimization. Specifically, we can derive a lower bound to the average AoI as

and an upper bound as

Then, we take a derivative of $\bar{\Delta}^{\text{LB}}_{0}$ (and/or $\bar{\Delta}^{\text{UB}}_{0}$) with respect to $A$ and assign it to zero, which gives

By solving this equation, we obtain a closed-form expression for the suboptimal age threshold

where

Correspondingly, we have a closed-form expression, given in (46), for the time-average AoI under the suboptimal threshold. It is noteworthy that the difference between the bounds and the time-average AoI is at most $\frac{1}{2}-\frac{1}{2(1+A\eta p_{\mathrm{s}})}\in[0,\frac{1}{2})$, hence the suboptimal solution differs from the optimal one by (at most) a constant. Indeed, by comparing (14) and (42), we can see that the time-average AoI approaches the upper bound when the age threshold $A$ becomes large. Therefore, the solution given in (44) is asymptotically optimal.

The previous subsections have explored the optimization of the update rate (resp. age threshold) given the age threshold (resp. update rate). This part will investigate how to jointly adjust the update rate and age threshold to further reduce the time-average AoI.

Since the age threshold and update rate have a composite influence on the time-average AoI (cf. (6) and (14)), it is challenging to characterize the optimal values of these factors explicitly. As such, based on Lemmas 3 and 4, we propose an iterative approach, presented in Algorithm 1, to jointly optimize the age threshold and update rate. Specifically, let $i$ denote the iteration index, while $\bar{\eta}^{*}_{i}$ and $\bar{A}^{*}_{i}$ represent the update rate and age threshold in the $i$th iteration, respectively. As detailed in Algorithm 1, we initialize $\bar{\eta}^{*}_{1}$ by a randomly chosen value from $(0,1]$, and then obtain $\bar{A}^{*}_{1}$ using Lemma 4. Based on $\bar{\eta}^{*}_{1}$ and Lemma 3, we further calculate $\bar{A}^{*}_{2}$ and $\bar{\eta}^{*}_{2}$. Subsequently, we can compute each update of $\bar{\eta}_{i}^{*}$ and $\bar{A}^{*}_{i}$ for $i\geq 2$. The iterations repeat until reaching the termination condition $|\bar{\Delta}_{0,i}-\bar{\Delta}_{0,i-1}|<\epsilon$, where $\epsilon$ is an error accuracy.