## Abstract

The transmission optimization of VR video streaming can improve the quality of user experience, which includes content prediction optimization and caching strategy optimization. Existing work either focuses on content prediction or on caching strategy. However, in the end-edge-cloud system, prediction and caching should be considered together. In this paper, we jointly optimize the four stages of prediction, caching, computing and transmission in mobile edge caching system, aimed to maximize the user’s quality of experience. In terms of caching strategy, we design a caching algorithm VIE with unknown future request content, which can efficiently improve the content hit rate, as well as the durations for prediction, computing and transmission. The VIE caching algorithm is proved to be ahead of other algorithms in terms of delay. We optimize the four stages under arbitrary resource allocation and obtain the optimal results. Finally, under the real scenario, the proposed caching algorithm is verified by comparing with several other caching algorithms, simulation results show that the user’s QoE is improved under the proposed caching algorithm.

## Data Availability

Data will be made available on reasonable request.

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## Funding

This work was supported in part by Natural Science Foundation of Jiangxi Province (Grant No. 20202BAB212003), Special Fund for Postgraduate Innovation of Jiangxi Province(No. YC2020-S481), Science and technology project of Jiangxi Provincial Department of Education(GJJ210854, GJJ210817) Jiangxi University of Science and Technology Graduate Innovation Special Fund Project(XY2023-S240).

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## Appendices

### Appendix A: Proof of Lemma 1

The interval [\(\alpha \), \(\beta \)] is within a group, so there is \(\alpha <t \le \beta \) that makes OPT(*t*) = OPT(\(\beta \)).

Since \(te_{i}\) is deleted at both \(\alpha \)-th and \(\beta \)-th requests arrival, \(te_{i}\) must be cached in the middle of [\(\alpha \), \(\beta \)]. After the tile is removed from the buffer, the system clears its previous request data. In summary, \(te_{i}\) was requested at least \(\eta \) times in the interval [\(\alpha \), \(\beta \)].

The \(te_{i}\) is deleted at \(\beta \), which must have a \(\zeta _{i}\) in Definition 2. \(te_{i}\) is requested at least \(\eta \) times in the interval [\(\alpha \), \(\beta \)], so \((t_{0} - \zeta _{i})\) in Definition 2 is greater than \(\alpha \). In summary, In the interval [\(\alpha \), \(\beta \)], \(te_{i}\) that has been requested at least \(\eta \) times is deleted, then the remaining tiles in VIE(\(\beta \)) are requested at least \(\eta \) times in the interval [\(\alpha \), \(\beta \)].

We prove Lemma 1 in two cases.

Case 1: VIE(\(\beta - 1\)) \(\ne \) OPT(\(\beta - 1\)). VIE(\(\beta - 1\)) must have a tile \(te_{j}\) that does not belong to OPT(\(\beta - 1\)). In the interval [\(\alpha \), \(\beta \)], OPT needs to generate delay for all requests of \(te_{j}\). \(te_{j}\) is requested at least \(\eta \) times in [\(\alpha \), \(\beta \)], so OPT produces at least \(\eta \cdot (T_{e}d_{0})\) delay in [\(\alpha \), \(\beta \)].

Case 2: VIE(\(\beta - 1\)) \(=\) OPT(\(\beta - 1\)). VIE must download a tile \(te_{j} \notin \) OPT(\(\beta - 1\)) when deleting \(te_{i}\) at the arrival of the \(\beta \)-th request. According to Definition 1, there is a positive integer \(\lambda \) that causes \(\sum _{t=\beta - \lambda }^{\beta } te_{j}(t) \ge \sum _{t=\beta - \lambda }^{\beta } te^{*}(t) + \eta \). If \(\beta - \lambda \ge \alpha \), \(te_{j}\) is requested at least \(\eta \) times in the interval [\(\alpha \), \(\beta \)], so OPT produces at least \(\eta \cdot (T_{e}d_{0})\) delay in [\(\alpha \), \(\beta \)]. If \(\beta - \lambda <\alpha \), we have \(\sum _{t=\beta - \lambda }^{\alpha -1} te_{j}(t) <\sum _{t=\beta - \lambda }^{\alpha -1} te^{*}(t) + \eta \), so \(\sum _{t=\alpha }^{\beta } te_{j}(t) \ge \sum _{t=\alpha }^{\beta } te^{*}(t) \ge \eta \). OPT produces at least \(\eta \cdot (T_{e}d_{0})\) delay in [\(\alpha \), \(\beta \)].

In summary, Lemma 1 is proved.

### Appendix B: Proof of Lemma 2

If OPT(*t*) \(\ne \) VIE(*t*), there must exist a tile \(te_{j}\) belongs to VIE(*t*) but not to OPT(*t*) in the system. In a group, we use *r* to denote the *r*-th tile cached at the edge, and use \(dl_{r}\) to represent the time the *r*-th tile cached at the edge was deleted. We set \(dl_{0} = \alpha - 1\), and if *r* is the last stored tile, then \(dl_{r} = \beta \). *r* is stored on the edge in the interval [\(dl_{[r-1]} + 1, dl_{r}\)]. Assuming \(r = te_{j}\), then \(\sum \nolimits _{t=dl_{[r-1]} + 1}^{dl_{r}} te_{j}(t) \>\sum \nolimits _{t=dl_{[r-1]} + 1}^{dl_{r}} te_{i}(t) - \eta \). If \(\sum \nolimits _{t=dl_{[r-1]} + 1}^{dl_{r}} te_{j}(t) \le \sum \nolimits _{t=dl_{[r-1]} + 1}^{dl_{r}} te_{i}(t) - \eta \), then \(te_{i}\) will be stored, which contradicts the conditions of Definition 1.

Case 1: \(r = 1\). We set \(dl_{[r-1]} + 1 = \alpha \), then \(\sum \nolimits _{t=\alpha }^{dl_{r}} te_{j}(t) \>\sum \nolimits _{t=\alpha }^{dl_{r}} te_{i}(t) - \eta \).

We next consider \(r > 1\). VIE downloads \(te_{j}\) after the arrival of \(dl_{[r-1]}\)-th request, so there are \(te_{j}^{*}\) and \(\lambda \) that satisfy the following formula:

Case 2: \(r > 1\), and \(dl_{[r-1]} - \lambda \ge \alpha \).

Case 3: \(r > 1\), and \(dl_{[r-1]} - \lambda <\alpha \). We have:

According to the above formula, we can get \(\sum _{t=\alpha }^{dl_{[r-1]}} te_{j}(t) \ge \sum _{t=\alpha }^{dl_{[r-1]}} te_{j}^{*}(t)\). So we can get :

In summary, we select the latest deleted *r* in case 1 and case 3 as \(r_{1}\). \(te_{j}\) is requested at least \(\sum \nolimits _{t=\alpha }^{dl_{r_{1}}} te_{i}(t) - \eta \) times in the interval [\(\alpha \), \(dl_{r_{1}}\)], and is requested at least \(\sum \nolimits _{t=dl_{r_{1}} + 1}^{\beta } te_{i}(t)\) times in the interval [\(dl_{r_{1}} + 1\), \(\beta \)]. Therefore, OPT forwards at least \(\sum \nolimits _{t=\alpha }^{\beta } te_{i}(t) - \eta \) tile requests to the cloud between [\(\alpha \), \(\beta \)].

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Liu, Q., Chen, H., Li, Z. *et al.* Online Caching Algorithm for VR Video Streaming in Mobile Edge Caching System.
*Mobile Netw Appl* (2024). https://doi.org/10.1007/s11036-024-02291-2

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DOI: https://doi.org/10.1007/s11036-024-02291-2