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Sequential Hypothesis Criterion Based Optimal Caching Schemes Over Mobile Wireless Networks

Xi Zhang1, Qixuan Zhu1, and H. Vincent Poor2

1Department of Electrical & Computer Engineering

Texas A&M University, College Station, Texas 77843, USA

• 2Dept. of Electrical Engineering, Princeton University, NJ 08544, USA

June 2020

This work of Xi Zhang and Qixuan Zhu was supported in part by the U.S. National Science Foundation under Grants ECCS-1408601 and CNS-1205726, and the U.S. Air Force under Grant FA9453-15-C-0423. This work of H. Vincent Poor was supported in part by the U.S. National Science Foundation under Grant CIF-1513915.

Xi Zhang, Qixuan Zhu, and H. Vincent Poor

Outline

Sequential Hypothesis Criterion Based Optimal Caching Schemes in Wireless Networks

• Cache design: accurately estimate the future data popularity
• Proposed scheme: sequential hypothesis testing to estimate

the future Zipf exponent

• Exponentially bounded stopping time of sequential

hypothesis testing over Zipf distribution

• Bounds of stopping time
• Optimal solution of sequential hypothesis testing

Xi Zhang, Qixuan Zhu, and H. Vincent Poor

Motivations

Cache Design in Edge Computing Networks: Accurately estimate the future data popularity and proactively cache the popular data contents. Cache Design Problem Popularity Estimation Scheme Sequential Hypothesis Testing: (1) accept (2) reject (3) continue observations

Xi Zhang, Qixuan Zhu, and H. Vincent Poor

Sequential Hypothesis Testing

Multiple hypothesis testing: where Define a sequential hypothesis test

• 1. stopping rule
• 2. decision rule

Xi Zhang, Qixuan Zhu, and H. Vincent Poor

• I. Exponentially bounded stopping time:
• R. A. Wijsman, “Exponentially bounded stopping time of

sequential probability ratio tests for composite hypotheses,” The Annals of Mathematical Statistics, vol. 42, no. 6, pp. 1859–1869, 1971. Assumption B



Xi Zhang, Qixuan Zhu, and H. Vincent Poor

• I. Exponentially bounded stopping time:

First, we prove Assumption A (ii): where We define:

Xi Zhang, Qixuan Zhu, and H. Vincent Poor

• I. Exponentially bounded stopping time:

Thus, Then, we prove Assumption A (i):

Xi Zhang, Qixuan Zhu, and H. Vincent Poor

• I. Exponentially bounded stopping time:

Assumption A (iii): As long as

• holds. However, if

does not hold, we calculate the probability of is less than 1.

Xi Zhang, Qixuan Zhu, and H. Vincent Poor

• II. Bounds of stopping time:

Define as the probability of selecting Hypothesis j but Hypothesis k is true, for exponentially bounded sequential hypothesis testing as which is equivalent to for every

Xi Zhang, Qixuan Zhu, and H. Vincent Poor

• II. Bounds of stopping time:

Extend the results to converge r-quickly: For , a sequence of of random variables is converge r-quickly, to a constant if , where

Xi Zhang, Qixuan Zhu, and H. Vincent Poor

• II. Bounds of stopping time:

Define We have

Xi Zhang, Qixuan Zhu, and H. Vincent Poor

• III. Optimal Decision of Hypothesis Testing

The optimal solution is the optimal hypothesis that minimizes the problem

Xi Zhang, Qixuan Zhu, and H. Vincent Poor