An Age of Information Characterization of SPS

Abstract

We derive a closed-form approximation of the stationary distribution of the Age of Information (AoI) of the semi-persistent scheduling (SPS) protocol which is a core part of NR-V2X, an important standard for vehicular communications. While prior works have studied the average AoI under similar assumptions, in this work we provide a full statistical characterization of the AoI by deriving an approximation of its probability mass function. As result, besides the average AoI, we are able to evaluate the age-violation probability, which is of particular relevance for safety-critical applications in vehicular domains, where the priority is to ensure that the AoI does not exceed a predefined threshold during system operation. The study reveals complementary behavior of the age-violation probability compared to the average AoI and highlights the role of the duration of the reservation as a key parameter in the SPS protocol. We use this to demonstrate how this crucial parameter should be tuned according to the performance requirements of the application.

Figures (11)

• Figure 1: All-to-all communication. Each node monitors its own time process, which is of interest to all neighboring nodes. Consequently, every node must communicate its observations to all other nodes, leading to an all-to-all communication.
• Figure 2: Channel occupancy pattern of node $v$ by frame. A time-slotted communication channel is split into frames. Each frame consists of five slots, which are numbered and called positions. The position in which node $v$ transmits in each frame $x$ is marked blue and described by $D^{v}(x)$.
• Figure 3: Example of realization of the AoI $\Delta(t)$. The time is structured into frames of $m$ slots. At the beginning of each frame, a packet is generated (sampled from $\Phi(t)$) and afterwards transmitted. The green checks mark signify singleton transmissions while the red crosses mark collided transmissions. The solid line shows the AoI at the receiver. In addition, the dashed line shows the channel access delay. For this example with a singleton transmission in frame $k(t_1)$, respectively $k(t_2)$, and a collided transmission in the previous frame, the collision duration $C(k(t_2))$ equals zero according to \ref{['def:collidedFrames']}.
• Figure 4: Example of operation for SPS. (a) shows the transmission pattern and reservation counter for nodes A, B, C and D during a generic frame $x$ and (b) shows the subsequent frame $x+1$. In frame $x$, the reservation counters of node C and D are greater than zero, therefore they transmit in frame $x + 1$ in the same position as in frame $x$ and the reservation counter is decremented by one. In contrast, the reservation counters of node A and B are both zero in frame $x$, therefore both nodes perform a reselection and select new positions for their transmissions in frame $x+1$. The new positions are selected uniformly from the empty positions (orange) in frame $x$. The number of remaining reservations is reset to a new value drawn from a uniform distribution. In the example shown, both nodes A and B select position 3 for their new transmission. Their transmissions collide, and the collision can potentially last for several frames until one of the nodes performs a reselection.
• Figure 5: Segments contributing to the AoI. The transmission pattern of node $v$ is illustrated frame by frame. The transmissions are shown here as circles. Green squares signify singleton transmissions and red circles collided transmissions. In addition, we mark the start and end of a reservation using inward-pointing triangles. Based on \ref{['def:frameRecursive']} and \ref{['def:collidedReselections']}, we label the final transmission of the last reservations before $k(t)$ (a) and, $k(t)-1$ (b), respectively. Thus, the distance between two reservations, for example, the first and second reservation before frame $k(t)$, is $mB(\xi_{1}(k(t)))$. According to definition \ref{['def:AoI']}, the AoI is the time between the sampling instance of the last singleton and the time of observation $t$. In this example the last singleton is in frame $\xi_{W(k(t))}(k(t))$(a) respectively $\xi_{W(k(t)-1)}(k(t)-1)$(b) and the sampling occurred at the beginning of this frame. The last singleton transmission before $t$ is always the last transmission within a reservation. Thus, the AoI is characterized by the sum of $w$ reservation durations (in slots) plus the slots already passed in frame $k(t)$.

Theorems & Definitions (16)

• Theorem 1
• Lemma 1
• Lemma 2
• proof
• proof : Proof of Theorem \ref{['thm:segmentation']}
• proof
• Lemma 3
• proof
• Lemma 4
• proof