(Samta Shukla+trade-off for delay and cost) Optimal Device-Aware Caching

Reference:14,20
flash cache
膜拜一下大神的Abstract

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1、Contribution

1)、offline caching: Damage-Aware REtention
即caching when the content request string is given.
优化每个文件最优的滞留时间。
2)、online caching:
即：caching when the request string is not given ahead of time.
First. 假定没有capacity constraint--->obtain the optimal file retention

A large cache assumption implies that there are no evictions and the cache misses are only because of the files expiring

Second. 缓存空间有限

2、假设

For tractability, we obtain results for Poisson file arrivals modulated with a suitable popularity distribution—such as ZipF popularity law [15], [16]—and exponentially distributed retention times in our analysis in Sections 4 and 5

we emulate a ICN/CCN environment by considering a non-uniform, non-constant (random) file retrieval costs which are independent of both arrivals
and retention processes.

3、在线优化

We now consider the case of online caching where the files are requested according to a distribution, however, the exact request string is not known to the policy apriori

Our goal is to design a policy that jointly finds the optimal retention times for all times for all files and the optimal eviction sequence in the event of a cache miss.

方法：①假设没有缓存限制；原因：[due to the large cache assumption, the objective function quickly converges to a steady state with increasing cache size. In other words,这样文件只因为TTL失效而删除，为②做准备 ]②在给定了文件的最优滞留时间情况下，获得最优的文件删除序列。

(2)、假设没有缓存限制

定义：

①Retention cost:After the file is retrieved, it is subsequently written to the
cache by incurring a deterministic retention (writing) cost f(m). Retention cost can be perceived as the storage cost and/or the cost of writing the file to the cache memory as every write incurs a physical damage that reduces cache
lifetime.

一些定义

①. Problem Formulation

瞅瞅人家大牛：

Definition 2(Optimal online policy) A policy is online optimal if it finds the values of the retention parameters for each file that minimizes the expected cache damage due to successive file writes under the constraint that the expected delay does not exceed ￥> 0

indicator variable:两次文件请求的时间间隔>文件缓存时间，且只与上述两种变量有关/每次请求文件，文件缓存时间独立同分布

the probability of missing file

the expacted damage: D; For the nth file request, we write the file with retention (Rn+1) if there is a miss (and we do not write otherwise)

其中，

damage function

expected delay constraint where (lamda) denotes the mean parameter for file retrieval cost

问题：

文件请求概率*文件missing概率*missing文件在缓存的消耗

公式处理：

由于retention time 与damage cost之间的function 不明确，通过数学处理将原问题转换为关于文件被请求时间间隔服从指数分布的系数+retention time指数分布的系数。

处理：
证明该目标函数为凸函数(二阶导在定义域内>0)--->Matlab convex包；
其中实验中retention time与damage cost 的函数(即ak、n的取值)分别取f=1/120R;f=100R^2;f=30R3

(2)、假设缓存限制

①.马尔科夫链
目的： Formulate the problem of finding an optimal eviction sequence as a sequential decision problem using the theory of Markov Decision Process(MDP).且文件被剔除的原因有：retention time expired+ cache capacity

S(t+1)应该加粗，表示状态；the goal is to find an optimal eviction sequence has changed to find the optimal eviction sequence U(t)

定义U(t)作为文件剔除序列：U(t)=0的情况：①没有需要剔除的文件；②缓存中有文件剔除，但文件没有被请求

注：马尔科夫部分没有看懂，文章中的马尔科夫的定义参考文献[15]
( http://infocom2003.ieee-infocom.org/papers/11_02.PDF)。具体如下：

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1. 问题定义

文章优化的问题：when the cache is full and the requested document is not in the cache;其中，p(j) denoted the probability of reference of doc(j)

2.马尔科夫框架

the resulting cache state St+1: in this case, the document is not necessarily evicted from the cache if the requested document is not in cache and the cache is full

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对应地，

a randomized replacement policies which are now defined: A randomized replacement policy π is a collection(πt, t=0,1,...) of mapping πt

对应地，

a probability measure Pπ defined through the following requirements: (11)means Rt ∈ St; and (12) means Rt �∈ St

3. The cost functions

只定义cost function，不管function的具体表达

the average cost

the basic problem

还是看不懂

注：

1.hit rate:

the probability of a cache hit

2.泊松分布与指数分布

• 指数分布与泊松分布的关系：泊松过程中，第K次随机事件与第K+1次随机事件出现的时间间隔服从指数分布

  
  
  

  指数分布的k阶中心距