Analysis Methodology for Age of Information under Sequence Based Scheduling

TL;DR

The AoI performance in a system where each user generates packets periodically to send to a common access point (AP) for status updating is analyzed, and an algorithm to optimize the construction parameters of the sequence scheme is proposed.

Abstract

We focus on the Age of Information (AoI) performance in a system where each user generates packets periodically to send to a common access point (AP) for status updating. To avoid heavy overhead, we assume that channel sensing, feedback information from the AP, and time synchronization are not available in the system. We adopt a multi-access scheme called the sequence scheme, where each user is assigned a periodic binary sequence to schedule their transmissions. In our previous work [18], we have thoroughly studied the AoI performance under sequence scheme when the period of schedule sequences, $L$, is equal to the status generating period, $T$. The results can be extended to the case where $T>L$. However, the case of $T<L$ is not covered by [18]. Therefore, in this paper, we concentrate on analyzing the AoI performance in the case of $T<L$, which is more challenging and requires different approaches. We conduct in-depth analysis on this case and develop a mathematical tool based on integer partitions to facilitate the analysis. We derive low-complexity closed-form expressions for two scenarios under $T<L$. Based on the obtained analytical results, we propose an algorithm to optimize the construction parameters of the sequence scheme. Finally, we compare our proposed sequence scheme with two commonly used baselines, and show that our proposed scheme outperforms the baselines in terms of AoI performance while consuming less energy.

Figures (14)

• Figure 1: Illustration for the sequence scheme. The two users are assigned with sequences $\bm{s}_1=[1\ 1\ 0\ 0]$ and $\bm{s}_2=[1\ 0\ 1\ 0]$, respectively. The sequence period equals $L=4$, and the frame length equals $T=3$. It follows that the length of a superframe equals $\beta=12$.
• Figure 2: Illustration for the time evolution of $A_{\bm{\tau}}(t)$. As observed, $A_{\bm{\tau}}(t)$ grows linearly over time in the absence of successful update delivery. Conversely, when a packet sent by user $i$ is delivered successfully, then $A_{\bm{\tau}}(t)$ immediately drops to the service time of this packet.
• Figure 3: Illustration of 1-positions. In the given scenario, $L=15, T=3$, $\beta=15$, and the sequence used is $\bm{v}_2$ in \ref{['example_s2']}, which is of Hamming weight $w=3$. Within a superframe, we have $w'=w=3$, and $\mathcal{I}'_i=\mathcal{I}_i=\{0,7,11 \}$. The $w'$ "1"s correspond to $\sigma_0=0, \sigma_1=1, \sigma_2=2$, respectively. Thus, $\mathcal{D}=\{0,1,2 \}$.
• Figure 4: Illustration of sequence $\bm{s}_i$, $\mathsf{sf0}$-word, $\mathsf{sf}$-word and an $r$-partition of $w$, and two classes of distances between two "1"s, $d^e_0,d^e_1,\ldots,d^e_{r-1}$ and $\ell_0, \ell_1, \ldots, \ell_{w-1}$. The sequence $\bm{s}_i$ used is $\bm{v}_2$ in \ref{['example_s2']}, which is of Hamming weight $w=3$ and length $L=15$. Under the given event $e$, the number of successful "1"s within $\bm{s}_i$ is $r=2$, and their positions are marked in yellow. This event can be represented by the $\mathsf{sf}$-word $\mathsf{sfs}$, which can be mapped to the partition $\bm{c}=(1,1,0)$ under the mapping $\theta$.
• Figure 5: Illustration of $Y_j^e$ and $X_j^e$. In the given scenario, $T=3$, $L=4$, $\beta=12$. Within a superframe, there are three AoI drops in total. The time slots that experience AoI drops are marked in green. The slot-level inter-departure times between two consecutive drops within a superframe are 5,4,3, respectively. The frame-level inter-departure times between two consecutive drops within a superframe are 3,6,3, respectively.

Theorems & Definitions (22)

• Definition 1
• Definition 2
• Example 1
• Definition 3
• Lemma 1
• Theorem 2
• Lemma 3
• Remark 1
• Lemma 4
• Lemma 5