Nicolas Gast

Benny Van Houdt

Asymptotically Exact TTL-Approximations of the Cache Replacement Algorithms LRU(m) and h-LRU

Computer system and network performance can be significantly improved by caching frequently used information. When the cache size is limited, the cache replacement algorithm has an important impact on the effectiveness of caching. In this paper we introduce time-to-live (TTL) approximations to determine the cache hit probability of two classes of cache replacement algorithms: the recently introduced h-LRU and LRU(m). These approximations only require the requests to be generated according to a general Markovian arrival process (MAP). This includes phase-type renewal processes and the IRM model as special cases.

We provide both numerical and theoretical support for the claim that the proposed TTL approximations are asymptotically exact. In particular, we show that the transient hit probability converges to the solution of a set of ODEs (under the IRM model), where the fixed point of the set of ODEs corresponds to the TTL approximation. We further show, by using synthetic and tracebased workloads, that h-LRU and LRU(m) perform alike, while the latter requires less work when a hit/miss occurs. We also show that, as opposed to LRU, h-LRU and LRU(m) are sensitive to the correlation between consecutive inter-request times.

I. INTRODUCTION

Caches form a key component of many computer networks and systems. A large variety of cache replacement algorithms has been introduced and analyzed over the last few decades. A lot of the initial work existed in deriving explicit expressions for the cache content distribution using a Markov chain analysis [START_REF] Aven | Stochastic Analysis of Computer Storage[END_REF]. This approach, however, is not always feasible: Even if explicit expressions can be obtained, they are often only applicable to analyze small caches, because of the time it takes to evaluate them. This gave rise to various approximation algorithms to compute cache hit probabilities and most notably to time-to-live (TTL) approximations.

The first TTL approximation was introduced for the least recently used (LRU) policy under the Independent reference model (IRM) by Che et al. in [START_REF] Che | Hierarchical web caching systems: modeling, design and experimental results[END_REF]. The main idea behind this approximation is that an LRU cache behaves similar to a TTL cache. In a TTL cache, when an item enters the cache, it sets a deterministic timer with initial value T . When this timer expires the item is removed from the cache. If an item is requested before its timer expires, its timer is reset to T . When T is fixed, an item with popularity p k is present in the cache at a random point in time with probability 1 -e -p k T and N k=1 1 -e -p k T is the average number of items in the cache. The Che approximation [START_REF] Che | Hierarchical web caching systems: modeling, design and experimental results[END_REF] consists in approximating an LRU cache of size m by a TTL cache with characteristic time T (m), where T (m) is the unique solution of the fixed point equation

m = N k=1
(1 -e -p k T ).

(

) 1
The above TTL approximation for LRU can easily be generalized to renewal requests as well as to other simple variations of LRU and RANDOM under both IRM and renewal requests, as well as to certain network setups [START_REF] Berger | Exact analysis of TTL cache networks[END_REF], [START_REF] Fofack | Analysis of TTLbased cache networks[END_REF], [START_REF] Martina | A unified approach to the performance analysis of caching systems[END_REF], [START_REF] Rosensweig | Approximate models for general cache networks[END_REF]. All of these TTL approximations have been shown to be (very) accurate by means of numerical examples, but except for LRU in [START_REF] Fricker | A versatile and accurate approximation for LRU cache performance[END_REF], no theoretical support was provided thus far.

In this paper we introduce TTL approximations for two classes of cache replacement algorithms that are variants of LRU. The first class, called LRU(m), dates back to the 1980s [START_REF] Aven | Stochastic Analysis of Computer Storage[END_REF], while the second, called h-LRU, was recently introduced in [START_REF] Martina | A unified approach to the performance analysis of caching systems[END_REF]. In fact, a TTL approximation for h-LRU was also introduced in [START_REF] Martina | A unified approach to the performance analysis of caching systems[END_REF], but this approximation relies on an additional approximation of independence between the different lists when h > 2. As we will show in the paper, this implies that the approximation error does not reduce to zero as the cache becomes large.

In this paper we make the following contributions:

• We present a TTL approximation for LRU(m) and h-LRU that is valid when the request process of an item is a Markovian arrival process (MAP). This includes any phase-type renewal process and the IRM model. In the special case of the IRM model, we derive simple closedform expressions for the fixed point equations.

• Our TTL approximation for h-LRU can be computed in linear time in h and appears to be asymptotically exact as the cache size grows, in contrast to the TTL approximation in [START_REF] Martina | A unified approach to the performance analysis of caching systems[END_REF] for h > 2. Numerical results for the TTL approximation for LRU(m) also suggest that it is asymptotically exact.

• We prove that, under the IRM model, the transient behavior of both h-LRU and LRU(m) converges to the unique solution of a system of ODEs as the cache size goes to infinity. Our TTL approximations correspond to the unique fixed point of the associated systems of ODEs. This provides additional support for the claim that our TTL approximations are asymptotically exact and is the main technical contribution of the paper.

• We validate the accuracy of the TTL approximation. We show that h-LRU and LRU(m) perform alike in terms of the hit probability under both synthetic and trace-based workloads, while less work is required for LRU(m) when a hit/miss occurs.

• We indicate that both h-LRU and LRU(m) can exploit correlation in consecutive inter-request times of an item, while the hit probability of LRU is insensitive to this type of correlation.

The paper is structured as follows. We recall the definitions of LRU(m) and h-LRU in Section II. We show how to build and solve the TTL-approximation for LRU(m) in Section III-A, and for h-LRU in Section III-B. We demonstrate the accuracy of the TTL-approximation for any finite time period in Section IV. We compare LRU(m) and h-LRU in Section V, by using synthetic data and real traces. We conclude in Section VI.

II. REPLACEMENT ALGORITHMS

We consider two families of cache replacement algorithms: h-LRU, introduced and called k-LRU in [START_REF] Martina | A unified approach to the performance analysis of caching systems[END_REF], and LRU(m), introduced in [START_REF] Aven | Stochastic Analysis of Computer Storage[END_REF], [START_REF] Gast | Transient and steady-state regime of a family of list-based cache replacement algorithms[END_REF]. Both operate on a cache that can store up to m items and both are variants of LRU, which replaces the least-recently-used item in the cache. One way to regard LRU is to think of the cache as an ordered list of m items, where the i-th position is occupied by the i-th most-recentlyused item. When a miss occurs, the item in the last position of the list is removed and the requested item is inserted at the front of the list. If a hit occurs on the item in position i, item i moves to the front of the list, meaning the items in position 1 to i -1 move back one position.

The h-LRU replacement algorithm: h-LRU manages a cache of size m by making use of h -1 additional virtual lists of size m (called list 1 to list h -1) in which only meta-data is stored and one list of size m that correspond to the actual cache (called list h). Each list is ordered, and the item in the ith position of list is the ith most-recently-used item among the items in list . When item k is requested, two operations are performed:

• For each list in which item k appears (say in a position i), the item k moves to the first position of list and the items in positions 1 to i -1 move back one position.

• For each list in which item k does not appear but appears in list -1, item k is inserted in the first position of list , all other items of list are moved back one position and the item that was in position m of list is discarded from list .

List 1 of h-LRU behaves exactly as LRU, except that only the meta-data of the items is stored. Also, an item can appear in any subset of the h lists at the same time. This implies that a request can lead to as many as h list updates. Note that there is no need for all of the h lists to have the same size m.

The LRU(m) replacement algorithm: LRU(m) makes use of h lists of sizes m 1 , . . . , m h , where the first few lists may be virtual, i.e., contain meta-data only. If the first v lists are virtual we have m v+1 + • • • + m h = m (that is, only the items in lists v + 1 to h are stored in the cache). With LRU(m) each item appears in at most one of the h lists at any given time. Upon each request of an item:

• If this item is not in the cache, it moves to the first position of list 1 and all other items of list 1 move back one position. The item that was in position m 1 of list 1 is discarded.

• If this item is in position i of a list < h, it is removed from list and inserted in the first position of list + 1.

All other items of list + 1 move back one position and the item in the last position of list + 1 is removed from list + 1 and inserted in the first position of list . All previous items from position 1 to i -1 of list move back one position.

• If this item is in position i of list h, then this item moves to the first position of list h. All items that are in position 1 to i -1 of list h move back one position. When using only one list, LRU(m) coincides with LRU, and therefore with 1-LRU.

III. TTL-APPROXIMATIONS

A. TTL-approximation for LRU(m) 1) IRM setting: Under the IRM model the string of requested items is a set of i.i.d. random variables, where item k is requested with probability p k . As far as the hit probability is concerned this corresponds to assuming that item k is requested according to a Poisson process with rate p k .

The TTL-approximation for LRU(m) exists in assuming that, when an item is not requested, the time it spends in list is deterministic and independent of the item. We denote this characteristic time by T . Let t n be the n-th time that item k is either requested or moves from one list to another list (where we state that an item is part of list 0 when not in the cache). Using the above assumption, we define an h + 1 states discrete-time Markov chain (X n ) n≥0 , where X n is equal to the list id of the list containing item k at time t n .

With probability e -p k T the time between two requests for item k exceeds T . Hence, the transition matrix of (X n ) n is

P k =        0 1 e -p k T1 0 1 -e -p k T1 . . . . . . . . . e -p k T h-1 0 1 -e -p k T h-1 e -p k T h 1 -e -p k T h        .
The Markov chain X n is a discrete-time birth-death process. Hence, its steady state vector (π k,0 , π k,1 , . . . , π k,h ) obeys

π k, = π k,0 -1 s=1 (1 -e -p k Ts ) s=1 e -p k Ts = π k,0 e p k T -1 s=1 (e p k Ts -1), (2)
for = 1, . . . , h. Further for ∈ {1, . . . , h}, the average time spend in is

E[t n+1 -t n |X n = ] = T t=0 e -p k t dt = 1 -e -p k T p k ,
and [START_REF] Baccelli | Elements of queueing theory: Palm Martingale calculus and stochastic recurrences[END_REF], this implies that when observing the system at a random point in time, that item k is in list ≥ 1 with probability

E[t n+1 -t n |X n = 0] = 1/p k . Combined with
π k, E[t n+1 -t n |X n = ] h j=0 π k,j E[t n+1 -t n |X n = j] = (e p k T1 -1) . . . (e p k T -1) 1+ h j=1 (e p k T1 -1) . . . (e p k Tj -1)
The expected number of items part of list is the sum of the previous expression over all items k. As for the Che approximation, setting this sum equal to m leads to the following set of fixed point equations for T 1 to T h :

m = n k=1 (e p k T1 -1) . . . (e p k T -1) 1 + h j=1 (e p k T1 -1) . . . (e p k Tj -1) . (3)
An iterative algorithm used to determine a solution of this set of fixed point equations is presented in Appendix A. In the next section we generalize this approximation to MAP arrivals.

2) MAP arrivals: We now assume that the times that item k is requested are captured by a Markovian Arrival Process (MAP). MAPs have been developed with the aim of fitting a compact Markov model to workloads with statistical correlations and non-exponential distributions [START_REF] Casale | Building accurate workload models using markovian arrival processes[END_REF], [START_REF] Telek | A minimal representation of markov arrival processes and a moments matching method[END_REF]. A MAP is characterized by two d×d matrices (D

(k) 0 , D (k) 1 ), where the entry (j, j ) of D (k) 1
is the transition rate from state j to j that is accompanied by an arrival and the entry (j, j ) of D (k) 0 is the transition rate from state j to j (with j = j ) without arrival. Let φ (k) such that φ (k) (D 0 ev (k) implies that item k is requested according to a phase-type renewal process characterized by (v (k) , D (k) 0 ). Extending the previous section, we define a discrete-time Markov chain (X n , S n ), where X n is the list in which item k appears and S n is the state of the MAP process at time t n . This Markov chain has d(h + 1) states and its transition probability matrix P M AP k is given by

        0 (-D (k) 0 ) -1 D (k) 1 e D (k) 0 T1 0 A k,1 . . . . . . . . . e D (k) 0 T h-1 0 A k,h-1 e D (k) 0 T h A k,h         where A k, = T t=0 e D (k) 0 T dtD (k) 1 = (I -e D (k) 0 T )(-D (k) 0 ) -1 D (k) 1 .
Due to the block structure of P M AP k , its steady state vector (π k,0 , πk,1 , . . . , πk,h ) obeys

πk, = πk,0 s=1 R k,s , (4)
for = 1, . . . , h , where the matrices R k,s can be computed recursively as

R k,h = A k,h-1 (I -A k,h ) -1 , (5)
R k, = A k, -1 I -R k, +1 e D (k) 0 T +1 -1 , (6)
for = 1, . . . , h -1 and h > 1.

We also define the average time (N k, ) j,j that item k spends in state j in (t n , t n+1 ) given that X n = ( , j), for j, j ∈ {1, . . . , d}. Let N k, be the matrix with entry (j, j ) equal to (N k, ) j,j , then

N k, = T t=0 e D (k) 0 t dt = (I -e D (k) 0 T )(-D (k) 0 ) -1 , for ≥ 1 and N k,0 = (-D (k) 0 ) -1 . The fixed point equations for T 1 to T h given in (3) generalize to m = n k=1 πk, N k, e h j=0 πk,j N k,j e , ( 7
)
where e is a column vector of ones. The hit probability h in list can subsequently be computed as

h = 1 n k=1 λ k n k=1 πk, N k, D (k) 1 e h j=0 πk,j N k,j e , (8)
for = 0, . . . , h.

B. TTL-approximation for h-LRU

1) IRM setting: As in [START_REF] Martina | A unified approach to the performance analysis of caching systems[END_REF], our approximation for h-LRU is obtained by assuming that an item that is not requested spends a deterministic time T in list , independently of the identity of this item. For now we assume that T 1 < T 2 < . . . < T h . We will show that the fixed point solutions for T 1 to T h always obey these inequalities.

We start by defining a discrete-time Markov chain (Y n ) n≥0 by observing the system just prior to the time epochs that item k is requested. The state space of the Markov chain is given by {0, . . . , h}. We say that Y n = 0 if item k is not in any of the lists (just prior to the nth request). Otherwise, Y n = if item k is in list , but is not in any of the lists + 1 to h. In short, the state of the Markov chain is the largest id of the lists that contain item k.

If Y n = , then with probability 1 -e -p k T , item k is requested before time T in which case we have Y n+1 = +1. Otherwise, due to our assumption that T ≥ T -1 ≥ . . . ≥ T 1 we have Y n+1 = 0 as in this case the item was discarded from all lists. Therefore the transition probability matrix Ph,k of the

h + 1 state Markov chain (Y n ) n≥0 is given by        e -p k T1 1-e -p k T1 e -p k T2
1-e -p k T2 . . . . . .

e -p k T h 1-e -p k T h e -p k T h 1-e -p k T h        . (9) Let π(h,k) = (π (h,k) 0 , . . . , π(h,k) h
) be the stationary vector of Ph,k , then the balance equations imply:

π(h,k) = ξ π(h,k) 0 s=1
(1 -e -p k Ts ),

for = 1, . . . , h, where ξ = 1 for < h and ξ h = e p k T h .

The probability π(h,k)

h that item k is in the cache just before a request (which by the PASTA property is also the steady-state probability for the item to be in the cache) can therefore be expressed as

h s=1 (1 -e -p k Ts ) h s=1 (1 -e -p k Ts ) + e -p k T h 1 + h-1 =1 s=1 (1 -e -p k Ts ) . ( 11
)
Due to the nature of h-LRU, T 1 can be found from analyzing LRU, T 2 from 2-LRU, etc. Thus, it suffices to define a fixed point equation for T h . Under the IRM model this is simply m = n k=1 π(h,k) h , due to the PASTA property. These fixed point equations can be generalized without much effort to renewal arrivals as explained in Appendix B.

The following property is proven in Appendix D, where we also show that T 1 < T 2 < . . . < T h must hold to have a fixed point.

Proposition 1. The fixed point equation

m = n k=1 π (h,k) h has a unique solution T h which is such that T h > T h-1 . When h = 2 Equation (11) simplifies to (1 -e -p k T1 )(1 - e -p k T2 )/(1 -e -p k T1 + e -p k T2
) which coincides with the hit probability of the so-called refined model for 2-LRU presented in [START_REF] Martina | A unified approach to the performance analysis of caching systems[END_REF]Eqn (9)]. For h > 2 only an approximation that relied on an additional approximation of independence between the h lists was presented in [12, see Eqn [START_REF] Jelenkovic | Asymptotic insensitivity of leastrecently-used caching to statistical dependency[END_REF]]. In Figure 1 we plotted the ratio between our approximation and the one based on [START_REF] Jelenkovic | Asymptotic insensitivity of leastrecently-used caching to statistical dependency[END_REF] of [START_REF] Martina | A unified approach to the performance analysis of caching systems[END_REF]. The results indicate that the difference grows with h. We show in Appendix C that it typically decreases as the popular items gain in popularity.

As [START_REF] Jelenković | Least-recently-used caching with dependent requests[END_REF] does not rely on the additional independence approximation, we expect that its approximation error is smaller and even tends to zero as m tends to infinity. This is confirmed by simulation and we list a small set of randomly chosen examples in Table I to illustrate.

2) MAP arrivals: For order d MAP arrivals, characterized by (D 11) and ( 10) of [START_REF] Martina | A unified approach to the performance analysis of caching systems[END_REF] as a function of the cache size for various values of h with n = 1000 items with a Zipf-like popularity distribution with α = 0.8.

h Simul.

Eq. ( 10) of [START_REF] Martina | A unified approach to the performance analysis of caching systems[END_REF] (err) Eq. ( 11 of the form e -p k T by e D (k) 0 T (-D

(k) 0 ) -1 D (k) 1 and 1 -e -p k T by (I -e D (k) 0 T )(-D (k) 0 ) -1 D (k)
1 . The fixed point equation for determining T h is found as

m = n k=1 (π (h,k) h-1 + π(h,k) h )(I -e D (k) 0 T h )(-D (k) 0 ) -1 e 1/λ k , (12)
where λ k is the request rate of item k and

π(h,k) = π (h,k) 0 s=1 (I -e D (k) 0 Ts )(-D (k) 0 ) -1 D (k) 1 Ξ , for = 1, . . . , h, where Ξ = I for < h and Ξ h = (I -(I -e D (k) 0 T h )(-D (k) 0 ) -1 D (k) 1 ) -1 . Finally, let ν (k) be the stochastic invariant vector of (-D (k) 0 ) -1 D (k)
1 , that is, its d entries contain the probabilities to be in state 1 to d immediately after an arrival. Hence,

π(h,k) 0 can be computed by noting that h =0 π(h,k) = ν (k) .

IV. ASYMPTOTIC EXACTNESS OF THE APPROXIMATIONS

In this section, we give evidences that the approximations presented in the previous section are asymptotically exact as the number of items tends to infinity. We first provide numerical evidence. We then show that the transient behavior of LRU(m) and h-LRU converges to a system of ODEs. By using a change of variable, these ODE can be transformed into PDEs whose fixed points are our TTL-approximations.

A. Numerical procedure and validation

For LRU(m), the fixed point of Equation ( 7) can computed by a iterative procedure that update the values T in a round-robin fashion. This iterative procedure is described in Appendix A and works well for up to h ≈ 5 lists but can be slow for a large number of lists. The computation for h-LRU is much faster and scales linearly with the number of lists: by construction, the first h -1 lists of a h-LRU cache behave like an (h-1)-LRU cache. Once T h-1 has been computed, the right-hand side of the fixed point equation ( 12) is increasing in T h and can therefore be easily computed with a complexity that does not depend on h.

1) Accuracy for LRU(m): We show that our TTLapproximation for LRU(m) is accurate by comparing the approximation with a simulation. We assume that the interrequest times of item k follow a hyperexponential distribution with rate zp k in state one and p k /z in state two, while the popularity distribution is a Zipf-like distribution with parameter α, i.e., p k = (1/k α )/ n i=1 1/i α . Correlation between consecutive inter-request times is introduced using the parameter q ∈ (0, 1]. More precisely, let (D

(k) 0 , D (k) 1 ) equal p k -z 0 0 -1/z , q z 1/z [z 1]/(z + 1) -(1 -q)D (k) 0
.

The squared coefficient of variation (SCV) of the inter-request times of item k is given by 2(z 2 -z + 1)/z -1 and the lag-1 autocorrelation of inter-request times of item k is

ρ 1 = (1 -q) (1 -z) 2 2(1 -z) 2 + z .
In other words the lag-1 autocorrelation decreases linearly in q and setting q = 1 implies that the arrival process is a renewal process with hyperexponential inter-request times.

Table II compares the accuracy of the model with time consuming simulations (based on 5 runs of 2 • 10 6 requests). We observe a good agreement between the TTL approximation and simulation that tends to improve with the size of the system (i.e., when n increases from 100 to 1000).

2) Accuracy for h-LRU: For the IRM model the TTL approximation was already validated by simulation in Table I

B. Asymptotic behavior and TTL-approximation

In this subsection, we construct two systems of ODEs that approximate the transient behavior of LRU(m) and h-LRU. These approximations become exact as the popularity of the most popular item decreases to zero: Theorem 1. Let H (t) be the sum of the popularity of the items of list and h (t) be the corresponding ODE approximation (Equation (18) for h-LRU and Equation (22) for LRU(m)). Then: for any time T , there exists a constant C such that

E sup t≤T / √ max k p k |H (t) -h (t)| ≤ C max k p k ,
where C does not depend on the probabilities p 1 . . . p n , the cache size m or the number of items n.

Our proof of this results is to use an alternative representation of the state space that allows us to use techniques from stochastic approximation. We present the main ideas in this paper while the technical details are provided in Appendix E.

We associate to each item k a variable τ k (t) that is called the request time of item k at time t and an additional variable that tracks if an item appears in a list. Our approximation is given by an ordinary differential equation (ODE) on x k,b (t) that is an approximation of the probability that τ k (t) is greater than b while appearing in a list . A more natural representation would be to consider the time since the last request. Our ODE approximation would then be replaced a partial differential equation (PDE) by replacing the ODE in x k, ,b by a PDE in y k, ,s , where y k, ,s (t) = x k, ,t-s (t). However, when working directly with the PDE, the proofs are much more complex. In each case, we show that the fixed point of the PDE corresponds to the TTL-approximation of LRU(m) and h-LRU presented in Sections III-A and III-B.

To ease the presentation, we present the convergence result when the arrivals follow an IRM model, where each item k has a probability p k of being requested at each time step. This proof can be adapted to the case of MAP arrivals but at the price of more complex notations. Indeed, for IRM, our system of ODEs is given by the variables x k, ,b (t) which are essentially an approximation of the probability for item k to be in a list while having been requested between b and t. If the arrival process of an item is modeled by a MAP with d states, then our approximation would need to consider x k, ,b,j (t) which would approximate the probabilities for item k to be in state j, in list and having being requested between b and t. A detailed proof for the case of MAP arrivals is beyond the scope of this paper, both because of space constraints and for the sake of clarity of the exposition.

1) LRU: We first construct the ODE approximation for LRU. In this simpler case the proof of the validity of the Che-approximation could rely on a more direct argument that uses the close-form expression of the steady state distribution of LRU, as in [START_REF] Fricker | A versatile and accurate approximation for LRU cache performance[END_REF]. Yet, the ideas presented in this section serve to illustrate the more complex cases of h-LRU and LRU(m).

The request time of an item k evolves as follows:

τ k (t + 1) = τ k (t) if k is not requested t + 1 if k is requested. ( 13
)
At time 0, τ k (0) = -i if the item is in the ith position in the cache and τ k (0) = -(m + 1) if the item is not in the cache.

The cache contains m items. Let us denote 1 by Θ(t) = sup{b : n k=1 1 {τ k (t)≥b} ≥ m} the request time of the mth most recently requested item. When using LRU, an item that has a request time greater or equal to Θ(t) is in the cache. We denote by H(t) the sum of the popularities of items in the cache:

H(t) = n k=1 p k 1 {τ k (t)≥Θ(t)} .
Our approximation of the probability for item k to have a request time after b, is given by the following ODE (for b < t):

ẋk,b (t) = p k (1 -x k,b (t)). ( 14
)
with the initial conditions that for t > 0, x k,t (t) = 0 and for t = 0, x k,b (0) = 1 {τ k (t)≥b} . Similarly to the stochastic system, we define θ(t) = sup{b :

n k=1 x k,b (t)
≥ m}, which is the time for which the sum of the approximated 1 Throughout the paper 1 {A} is the indicator function of an event A. It is equal to 1 if A is true and 0 otherwise. probabilities of having items requested after b is equal to m. The approximation of the hit ratio for LRU is then given by

h(t) = n k=1 p k x k,θ(t) (t).
Once these variables have been defined, the key ingredient of the proof is to use the same changes of variables as in the proof of Theorem 6 of [START_REF] Gast | Transient and steady-state regime of a family of list-based cache replacement algorithms[END_REF], which is to consider P α,b (t):

P α,b (t) = a 1-α n k=1 (p k ) α 1 {τ k (t)≥b} ,
f α,b (ρ) = a 1-α ( k (p k ) α+1 ) -aρ α+1,a (t).
The proof of the theorem, detailed in Appendix E1, relies on classical results of stochastic approximation. It uses the fact that

• the function f is Lipchitz-continuous • f is the E [P α,b (t + 1) -P α,b (t) | P (t)] = f α,b (P (t))
• The second moment of the variation of P (t) is bounded:

E P (t+1)-P (t) 2
∞ |P ≤ a. Note that Equation ( 14) can be transformed into a PDE by considering the change of variable y k,s (t) = x k,t-s (t). The quantity y k,s (t) is an approximation of the probability for an item k to have been requested between t -s and t. The set of ordinary differential Equations ( 14) can then be naturally transformed in the following PDE:

∂ ∂t y k,s (t) = p k (1 -y k,s (t)) - ∂ ∂s y k,s (t). (15)
The fixed point y of the PDE can be obtained by solving the equation ∂ ∂t y = 0. This fixed point satisfied y k,s = 1 -e -p k s . For this fixed point, the quantity T = t -θ satisfies m = n k=1 (1 -e -p k T ). This equation is the same as the Cheapproximation, given by Equation (1).

2) h-LRU: The construction for LRU can be extended to the case of h-LRU by adding to each item h variables L k, (t) ∈ {true, false}. For item k and a list , L k, (t) equals true if item k was present in list just after the last request2 of item k and false otherwise. Similarly to the case of LRU, we define the quantity Θ (t) to be the request time of the least recently requested item that belongs to list at time t, that is,

Θ (t) = sup{b : n k=1 1 {τ k (t)≥b∧L k, (t)} ≥ m }.
We then define a quantity x k, ,b (t) that is meant to be an approximation of the probability for item k to have τ k (t) ≥ b and L (t) = true.

As L 1 (t) is always equal to true, the ODE approximation for x k,1,b (t) is the same as [START_REF] Telek | A minimal representation of markov arrival processes and a moments matching method[END_REF]. Moreover, this implies that Θ 1 (t) ≥ Θ (t) for ≥ 2. For the list = 2, the approximation is obtained by considering the evolution of L 2 (t). After a request,

L 2 (t+1) is true if τ k (t) ≥ Θ 1 (t) or if (τ k (t) ≥ Θ 2 (t)
and L 2 (t) = true). Both these events occur if (τ k (t) ≥ Θ 1 (t) and L 2 (t) = true) as Θ 1 (t) ≥ Θ 2 (t). This suggests that, if the item k is requested, then, in average L k,2 (t + 1) is approximately x k,1,θ1(t) + x k,2,θ2(t) -x k,2,θ1(t) , which leads to the following ODE approximation for x k,2,b :

ẋk,2,b = p k (x k,2,θ2(t) + x k,1,θ1(t) -x k,2,θ1(t) -x k,2,b ), (16)
where θ (t) = sup{b :

n k=1 x k, ,b (t) ≥ m } for ∈ {1, 2}.
The formulation for the third list and above is more complex. In Section III-B, we showed that the computation of the fixed point is simple because the quantities T of the fixed point satisfy T 1 ≤ T 2 • • • ≤ T h . However, for the stochastic system, we do not necessarily have 3 Θ (t) ≥ Θ +1 (t) when ≥ 2, which implies that the ODE approximation for h-LRU has 2 h-1 terms.

Applying the reasoning of L k,2 to compute L k, ( ≥ 3) involves computing the probability of (τ

k (t) ≥ Θ -1 (t) and L k, -1 (t) = true) or (τ k (t) ≥ Θ (t) and L k, (t) = true). When Θ (t) ≤ Θ -1 (t), both these events occur if (τ k (t) ≥ Θ -1 (t) and L k, (t) = L k, -1 (t) = true).
This suggest that the ODE for x k, ,b (t) has to involve a term x k,{ -1, },θ -1 (t) (t), that is an approximation for the item k to have a request time after θ -1 (t) and such that L k, -1 (t) = L k, (t) = true. Note, for = 2 we have x k,{ -1, },b (t) = x k, ,b (t) as L k,1 (t) is always true, but this does not hold for > 2. This leads to:

ẋk, ,b = p k (x k, ,θ (t) + x k, -1,θ -1 (t) -x k,{ -1, },max{θ -1 (t),θ (t)} -x k, ,b ), (17)
A similar reasoning can be applied to obtain an ODE for x k,{ -1, },b (t) as a function of x k,{ -1, },b (t), x k,{ -2, -1, },b (t) and x k,{ -2, },b (t). For example, for = 3 this approximation becomes

ẋk,{2,3},b (t) = x k,2,θ2(t) +x k,3,θ1(t) -x k,{2,3},θ1(t) -x k,{2,3},b as L k,1 (t) is always true.
The hit probability of list used in Theorem 1 is then

h (t) = n k=1 x k, ,θ (t) (t), (18)
where the variables x k, ,b satisfy the above ODE.

3 When h = 3 lists, the variables Θ (t) are not always ordered. For example, consider the case of four items {1, 2, 3, 4} and m 1 = m 2 = m 3 = 3. If initially the three caches contain the three items 1, 2, 3. Then, after a stream of requests: 4, 4, 3, 2, 1, the cache 1 and 3 will contain the items {1, 2, 3} while the cache 2 will contain {1, 2, 4}. This implies that t

-3 = Θ 2 (t) < Θ 3 (t) = Θ 1 (t) = t -2.
The proof of Theorem 1 in the case of h-LRU is very similar as the one for LRU and uses the same stochastic approximation argument. Moreover, as for LRU, the ODE (16) can be transformed into a PDE by using the change of variables y k, ,s (t) = x k, ,t-s (t) and T (t) = t -θ (t). This PDE has a unique fixed point that corresponds to Section III-B.

3) LRU(m): The construction of the approximation and the proof for the case of LRU(m) is more involved because of discontinuities in the dynamics. We replace the request time by a quantity that we call a virtual request time that is such that the m h items that have the largest virtual request times are in list h. The next m h-1 are in list h -1, etc. The virtual request time of an item changes when this item is requested. If the item was in list h or h -1 prior to the request, its virtual request time becomes t + 1. If the item was in a list ∈ {0 . . . h -2}, its virtual request time becomes the largest virtual request time of the items in list + 2.

The approximation of the distribution of virtual request times is then given by an ODE on the quantities x k,b (t) that are meant to be an approximation of the probability that the item k has a virtual request time after b:

ẋk,b (t) = p k (x k,θ ζ b (t) (t)-1 (t) -x k,b (t)), (19)
where θ (t) and ζ b (t) are defined by:

θ (t) = sup{b : n k=1 x k,b (t) ≥ m h + • • • + m } (20) ζ b (t) = max{ : θ (t) ≤ b} (21)
The intuition behind Equation ( 19) is that the changes in x k,b are due to the items that had a virtual request time prior to b and that now have a virtual request time b or after. This only occurs if the item had a virtual request time between θ ζ b (t)-1 and b and was requested, in which case its new virtual request time is θ ζ b (t)+1 ≥ b. Otherwise, if an item had a virtual request time prior to θ ζ b (t)-1 , then upon request it jumps to a list < ζ b (t) -1 and therefore its new virtual request time stays prior to b.

The hit ratio for LRU(m) used in Theorem 1 is given by

h (t) = n k=1 p k (x k,θ +1 (t) -x k,θ (t) )(t) (22)
The main difference between the proof for LRU(m) compared to the one of h-LRU is that the right-side of the differential equation ( 19) is not Lipschitz-continuous in ρ because the list in which an item that has an virtual request time b belongs to depends non-continuously on ρ (the list ζ b is a discrete quantity). In Appendix E3, we explain how to prove the convergence of the stochastic approximation algorithm by using one-sided Lipschitz-continuous functions.

V. COMPARISON OF LRU(M) AND H-LRU

In this section we compare the performance of LRU(m) and h-LRU in terms of the achieved hit probability when subject to IRM, renewal, MAP requests and trace-based simulation. A good replacement algorithm should keep popular items in the cache, but needs to be sufficiently responsive to changes in the popularity. As LRU(m) and h-LRU are clearly better suited to keep popular items in the cache than LRU, they perform better under static workloads (IRM). We demonstrate that they often also outperform LRU when the workload is dynamic.

A. Synthetic (static) workloads

For the synthetic workloads we restrict ourselves to LRU, 2-LRU and LRU(m, m). The latter two algorithms both use a cache of size m and additionally keep track of meta-data only for the m items in list 1.

Figure 2 depicts the hit probability as a function of the cache size when n = 1000, items follow a Zipf-like popularity distribution with parameter 0.8 under IRM and renewal requests (with z = 10, see Section IV-A1). Figure 3 shows the impact of having correlation between consecutive inter-request times (that is, q = 1/20 instead of q = 1 for z = 2, 10).

One of the main observations is that LRU(m, m) performs very similar to 2-LRU under IRM, renewal and MAP requests. In fact, 2-LRU performs slightly better, unless the workload is very dynamic (z = 10 and q = 1 case). Another important observation that can be drawn from comparing Figures 2 and3 is that the hit rate of both 2-LRU and LRU(m, m) significantly improves in the presence of correlation between consecutive inter-request times (that is, when q < 1), while LRU does not. Recall that LRU(m) needs to update at most one list per hit, as opposed to h-LRU. Thus, whenever both algorithms perform alike, LRU(m) may be more attractive to use.

Figure 4 shows that the hit rate of 2-LRU and LRU(m, m) both increase with increasing lag-1 autocorrelation and more importantly that the hit probability of LRU is completely insensitive to any correlation between consecutive inter-request times. Figure 4 further indicates that the hit probability also increases with ρ 1 when splitting the cache in two lists of equal size (although the gain is less pronounced). As indicated by the following theorem, whose proof is given in Appendix F, the insensitivity of LRU is a general result.

Theorem 2. Assume that the items' request processes are stationary, independent of each other and that the expected number of requests per unit time is positive and finite. Then, the hit probability of LRU only depends on the inter-arrival time distribution. In particular, it does not depend on the correlation between inter-arrival time.

This theorem complements the results of Jelenkovic and Radovanovic who showed in [START_REF] Jelenković | Least-recently-used caching with dependent requests[END_REF], [START_REF] Jelenkovic | Asymptotic insensitivity of leastrecently-used caching to statistical dependency[END_REF] that for dependent request processes, the hit probability is asymptotically, for large cache sizes, the same as in the corresponding LRU system with i.i.d. requests. Our insensitivity result is valid not just asymptotically but requires the request processes of the various items to be independent.

B. Trace-based simulation

To perform the trace-based simulations we rely on the same 4 IR cache traces as in [START_REF] Bianchi | Check before storing: What is the performance price of content integrity verification in LRU caching?[END_REF]Section 4]. In this section, we only report the result for the trace collected on Monday 18th Feb 2013. We also simulated the other traces and obtained very similar results.

The hit probability of LRU(m) with a split cache and/or virtual lists normalized by the LRU hit probability is depicted in Figure 5 as a function of the cache size m. It indicates that LRU(m) is more effective than LRU, especially when the cache is small. For small caches using a virtual list is better than splitting the cache and using both a virtual list and split cache offers only a slight additional gain. While not depicted here, we should note that using more virtual lists or splitting the cache in more parts sometimes result in a hit probability below the LRU hit probability for larger caches.

Figure 6 compares h-LRU with LRU(m) using virtual lists, where the hit probability is now normalized by the hit probability of LRU(m, m) to better highlight the differences. We observe that 2-LRU differs by less than 1% from LRU(m, m), while 5-LRU and LRU(m, m, m, m, m) differ by less than 2%. Given that h-LRU may require an update of up to h lists, while LRU(m) requires only one update in case of a hit, LRU(m) seems preferential in this particular case.

VI. CONCLUSION

In this paper, we developed algorithms to approximate the hit probability of the cache replacement policies LRU(m) and h-LRU. These algorithms rely on an equivalence between LRU-based and TTL-based cache replacement algorithms. We showed numerically that the TLL-approximations are very accurate for moderate cache sizes and appear asymptotically exact as the cache size grows. We also provide theoretical support for this claim, by establishing a bound between the transient dynamics of both policies and a set of ODEs whose fixed-point coincides with the proposed TTL-approximation.

A possible extension of our results would be to study networks of caches in which LRU, LRU(m) or h-LRU is used in each node. Further, our TTL-approximation with MAP arrivals can be readily adapted to other policies such as FIFO(m) and RAND(m) introduced in [START_REF] Gast | Transient and steady-state regime of a family of list-based cache replacement algorithms[END_REF]. In fact, a generalization to a network of caches would be fairly straightforward for the class of RAND(m) policies.

T = T + x; T +1 = T +1 -x; 9 end 10 end
Algorithm 1: Iterative algorithm used to solve fixed point equations in [START_REF] Fofack | Analysis of TTLbased cache networks[END_REF].

B. h-LRU with renewal arrivals

The same approach as for the IRM model can be used to obtain a TTL approximation when the requests for item k follow a renewal process, characterized by a distribution with cumulative distribution function

F k (x). Let Fk (x) = 1 -F k (x).
In this case we get ( Ph,k ) j,0 = Fk (T min(h,j+1) ) and ( Ph,k ) j,min(h,j+1) = F k (T min(h,j+1) ). The hit probability for item k can therefore be expressed as

π(h,k) h = h s=1 F k (T s ) h s=1 F k (T s ) + Fk (T h ) 1 + h-1 j=1 j s=1 F k (T s ) , while for j = 1, . . . , h -1 we have π(h,k) j = Fk (T h ) j s=1 F k (T s ) h s=1 F k (T s ) + Fk (T h ) 1 + h-1 j=1 j s=1 F k (T s )
. 11) and ( 10) of [START_REF] Martina | A unified approach to the performance analysis of caching systems[END_REF] as a function of the cache size for various values of α with n = 1000 items with a Zipf-like popularity distribution with α.

The fixed point equation for determining T h is found as

m = n k=1 (π (h,k) h-1 + π(h,k) h ) T h x=0 xdF k (x) ∞ x=0 Fk (x)dx = n k=1 (π (h,k) h-1 + π(h,k) h ) T h - T h x=0 F k (x)dx ∞ x=0 Fk (x)dx , as (π (h,k) h-1 + π(h,k) h )
T h x=0 xdF k (x)
is the mean time that item k spends in the cache between two requests for item k and ∞ x=0 Fk (x)dx is simply the mean time between two requests.

C. Comparison of TTL-approximations for h-LRU

In Figure 1 we depicted the difference between our TTLapproximation for h-LRU under the IRM model based on [START_REF] Jelenković | Least-recently-used caching with dependent requests[END_REF] and [START_REF] Jelenkovic | Asymptotic insensitivity of leastrecently-used caching to statistical dependency[END_REF] of [START_REF] Martina | A unified approach to the performance analysis of caching systems[END_REF], where the popularity of the items followed a Zipf-like distribution with α = 0.8. Figure 7 depicts the impact of the parameter α of the Zipf-like distribution when h = 5. The difference between both approximations appears to grow as α decreases. In other words, the difference decreases as the popular items gain in popularity.

D. Some proofs for h-LRU under IRM 1) Proof of Proposition 1:

The fixed point equation for h ≥ 2 can be written as m = f h (T h ), where

f h (x) = n k=1 (1 -e -p k x ) h-1 s=1 e k,s h-1 s=1 e k,s + e -p k x 1 + h-2 j=1 j s=1 e k,s
, with e k,s = (1 -e -p k Ts ). The function f h (x) is clearly an increasing function in x and therefore m = f (x) has a unique solution T h . Further,

f h (T h-1 ) = n k=1 (1 -e -p k T h-1 ) h-1 s=1 e k,s h-1 s=1 e k,s + e -p k T h-1 1 + h-2 j=1 j s=1 e k,s < n k=1 h-1 s=1 e k,s h-1 s=1 e k,s + e -p k T h-1 1 + h-2 j=1 j s=1 e k,s = n k=1 π(h-1,k) h-1 = m, meaning T h > T h-1 .
2) Structure of the fixed point: Using induction we prove that the fixed point solutions obey T 1 < . . . < T h . We assume that T 1 < . . . < T h-1 (which trivially holds for h = 2) and show that the fixed point equation for T h does not have a solution for T h ∈ (0, T h-1 ). The key thing to note is that when T h ≤ T h-1 item k is part of list h -1 whenever it is part of list h. As such we still obtain a Markov chain by observing the system just prior to the request times, but now the last two rows of the transition probability matrix are both equal to

(e -p k T h-1 , 0, . . . , 0, e -p k T h -e -p k T h-1 , 1 -e -p k T h ). Let (π (h,k) 0 , . . . , π(h,k) h
) be the invariant vector of this modified Markov chain, then it is easy to see that

π(h,k) h-1 + π(h,k) h = π (h-1,k) h-1
, as lumping the last two states into a single state results in the matrix Ph-1,k . Hence the fixed point equation

n k=1 π(h,k) h = m cannot have a solution as n k=1 π(h,k) h < n k=1 (π (h,k) h-1 + π(h,k) h ) = n k=1 π(h-1,k) h-1 = m.
f α,b (ρ) = a 1-α ( k (p k ) α+1 ) -aρ α+1,a (t).
To prove the result, we use the following two lemmas, whose proofs are given below. The first one is a classical result from stochastic approximation that uses the fact that X(t) is noisy Euler discretization of a Lipchitz-continuous differential equation. The second lemma states that the popularity in the cache is a continuous function of ρ.

Lemma 1. Let f : P → span(P) be a Lipchitz continuous function with constant aL such that sup x∈P f (x) ∞ ≤ a ≤ 1 and f (x) -x ∈ P. Let X be a P-valued stochastic process adapted to a filtration F such that E [X(t + 1) -X(t)

| F t ] = f (X(t)) and E X(t + 1) -X(t) 2 ∞ ≤ a.
Then, the ODE ẋ = f (x) has a unique solution x X(0) that starts in X(0) and for any T > 0, Let us show that f satisfies the assumption of Lemma 1. It should be clear that f is Lipschitz-continuous with constant a. Moreover, P α,b (t) changes if the requested item has a request time prior to b. If this item is k, then P α,b (t + 1) = P α,b (t) + a 1-α (p k ) α . This shows that

E sup t≤T /a X(t) -x X(0) (t) 2 ∞ ≤ T (2L + 1) exp(2T L)a.
E [P α,b (t + 1) -P α,b (t) | F t ] = n k=1 p k a 1-α (p k ) α 1 {τ k (t)<b} = f α,b (P (t))
Last, the second moment of the variation of P (t) is bounded:

E P (t+1)-P (t) 2 ∞ |P = E sup α,b |P α,b (t+1)-P α,b (t)| 2 |F t = E |P 0,t (t+1)-P 0,t (t)| 2 |F t = n k=1 ap k = a.
This implies that for each T > 0, there exists a constant C such that E sup t≤T /a P (t) -ρ(t) 2 ∞ ≤ Ca/4. Lemma 2 concludes the proof for LRU.

Proof of Lemma 1. The solution of the ODE ẋ = f (x) that starts in X(0) satisfies x(t) = X(0)+ t s=0 f (x(s))ds. Let E(t) be such that

X(t) = X(0) + t-1 s=0 f (X(s)) + E(t) = X(0) + t-1 0 f (X( s ))ds + E(t).
We have:

X(t) -x(t) ∞ ≤ t s=0 f (X( s )) -f (x(s)) ∞ + E(t) ∞ ≤ aL t s=0 X( s ) -x(s) ∞ + E(t) ∞ ,
where we used that f is Lipschitz-continuous of constant La.

Let X(t) be the the piecewise-linear interpolation of X such that X(t) = X(t) when t ∈ Z + . We have:

X( s ) -x(s) ∞ ≤ X( s ) -X(s) ∞ + X(s) -x(s) ∞ ≤ a + X(s) -x(s) ∞ ,
where we used that f (x) ∞ ≤ a.

This shows that for any t ≤ T /a (with t ∈ Z + ):

X(t) -x(t) ∞ ≤ aL t s=0 X(s) -x(s) ∞ + a 2 Lt + E(t) ∞ ≤ exp(aLt)(a 2 Lt + sup s≤t E(s) ∞ ) ≤ exp(LT )(aLT + sup t≤T /a E(t) ∞ ),
using Gronwall's inequality.

By assumption,

E E(t + 1) -E(t) 2 ∞ | F t = var [ X(t + 1) -X(t) | F t ∞ ] ≤ E X(t + 1) -X(t) 2 2 | F t ≤ a 2 . As E(t) is a martingale, this implies that E sup t≤T /a E(t) 2 ∞ ≤ E E(T ) 2 ∞ ≤ aT.
Proof of Lemma 2. Let ρ, ρ ∈ P. By definition of P, there exist x and x such that ρ

α,b = a 1-α n k=1 (p k ) α x k,b and ρ α,b = a 1-α n k=1 (p k ) α x k,b .
Let θ, θ be such that ρ 0,θ = ρ 0,θ = m and assume without loss of generality that θ ≤ θ. As x k,b is non-increasing in b, this implies that x k,θ ≥ x k,θ . Hence, we have:

|ρ 1,θ -ρ 1,θ | = n k=1 p k (x k,θ -x k,θ ) ≤ n k=1 a(x k,θ -x k,θ ) = |ρ 0,θ -ρ 0,θ | ≤ ρ 0,θ -ρ 0,θ + ρ 0,θ -ρ 0,θ = |ρ 0,θ -ρ 0,θ | ≤ ρ -ρ ∞ . (23)
Therefore:

|g m (ρ) -g m (ρ )| = ρ 1,θ -ρ 1,θ ≤ |ρ 1,θ -ρ 1,θ | + ρ 1,θ -ρ 1,θ ≤ 2 ρ -ρ ∞ ,
where the last inequality comes from (23).

2) Generalization to h-LRU: The proof for h-LRU is almost identical to the proof for LRU. For simplicity, we focus on the case of 2-LRU. The proof is similar for h ≥ 3.

We define the quantities ρ α, ,b (t) and P α, ,b (t) by

ρ α, ,b (t) = a 1-α n k=1 (p k ) α x k, ,b (t); P α, ,b (t) = a 1-α n k=1 (p k ) α 1 {τ k (t)≥b∧L k, (t)}
and Equation ( 16) implies that

ρα,2,b = a(ρ α+1,2,θ2(t) + ρ α+1,1,θ1(t) -ρ α+1,2,θ1(t) -ρ α+1,2,b ). (24)
Lemma 2 implies that the quantity g m, (ρ) = ρ 1, ,θ , where θ is such that ρ 0, ,θ = m , is a Lipschitz function of ρ with constant 2. It follows that the right-side of the ODE Equation ( 24) is Lipschitz-continuous with constant 4a. As for LRU, the right side of Equation ( 24) is the average variation of P α,2,b and that the second moment of the variation is bounded by a. Lemma 1 concludes the proof for 2-LRU.

As for LRU, we can transform (16) into a PDE by using the change of variables y k, ,s (t) = x k, ,t-s (t) and T (t) = t -θ (t). For example, for = 2, the fixed point y of this PDE satisfies

0 = p k (y k,2,T2 + y k,1,T1 -y k,2,T1 -y k,2,s ) - ∂ ∂s y k,2,s .
The solution of this ODE in s is given by

y k,2,s = (y k,2,T2 -y k,2,T1 + y k,1,T1 )(1 -e -p k s ) (25) = y k,1,T1 1 + e -p k T2 -e p k T1 (1 -e -p k s ), (26)
where we use (25) for s = T 1 and s = T 2 to obtain (26).

In Section IV-B1, we have shown that y k,1,T1 = 1 -e -p k T1 where T 1 is such that n k=1 y k,1,T1 = m. One can verify that replacing y k,1,T1 by 1 -e -p k T1 in Equation ( 26) with s = T 2 leads to Equation [START_REF] Jelenković | Least-recently-used caching with dependent requests[END_REF].

3) LRU(m): We now highlight the main differences with the case of h-LRU. They are mainly due to the non-continuity of the right-side of the differential equation (19).

As before, let As for the the cases of LRU and h-LRU, one can verify that f α,b is the average variation of P α,b (t) during one time step and that the second moment of the average variation is bounded by a 2 . Moreover, if x is a solution of the differential equation ( 19

P α,b (t) = a 1-α n k=1 (p k ) α 1 {σ k (t)
), then ρ α,b (t) = n k=1 x k,b (t) is a solution of the differential equation ρ = f (ρ).
The next lemma states some key properties of the function f . In particular, (i) quantifies what we mean by partially onesided Lipschitz. Its proof is given below. Lemma 3. For any ρ, ρ ∈ P and α ≥ 1, we have:

(i) (ρ 0,b -ρ 0,b )(f 0,b (ρ) -f 0,b (ρ )) ≤ 2a ρ -ρ 2 ∞ ; (ii) f (ρ) ∞ ≤ a; (iii) |f α,b (ρ) -f α,b (ρ )| ≤ |f 0,b (ρ) -f 0,b (ρ )|+3 ρ -ρ ∞ .
Let us denote by V (t) ∈ P the vector defined by

V α,b (t) = P α,b (t + 1) -P α,b (t) -f α,b (P (t)). We have E [V (t) | F t ] = 0 and E V (t) 2 ∞ | F t ≤ a 2 . Moreover, the definition of ρ(t + 1) = ρ(t) + 1 0 f (ρ(t + s))ds implies that (P 0,b (t + 1) - ρ 0,b (t + 1)) 2 equals (P 0,b (t) -ρ 0,b (t)+V 0,b (t)+f 0,b (P (t))+ 1 0 f 0,b (ρ(t+s))ds) 2 = (P 0,b (t) -ρ 0,b (t)) 2 + V 0,b (t) + f 0,b (P (t)) + 1 0 f 0,b (ρ(t + s))ds 2 +2 P 0,b (t) -ρ 0,b (t) V 0,b (t) +2 P 0,b (t)-ρ 0,b (t) f 0,b (P (t))+ 1 0 f 0,b (ρ(t+s))ds (27)
In expectation, the second term is smaller than 9a 2 , the third is 0. For α = 0, the last term equals

2 1 0 P 0,b (t) -ρ 0,b (t) f 0,b (P (t)) + f 0,b (ρ(t + s)) ds = 2 1 0 P 0,b (t) -ρ 0,b (t + s) f 0,b (P (t)) + f 0,b (ρ(t + s)) ds + 2 1 0 ρ 0,b (t + s) -ρ 0,b (t) f 0,b (P (t)) + f 0,b (ρ(t + s)) ds ≤ 4a 1 0 P (t) -ρ(t + s) 2 ∞ ds + 2a 2 , (28)
where we use Lemma 3(i) to bound the first term of the equality and Lemma 3(ii) for the second.

As P α,b (b) = ρ α,b (b), Lemma 3(iii) implies that |P α,b (t)-ρ α,b (t)| ≤ t s=b |f 0,b (P ( s )) -f 0,b (ρ(s))| ds + 3 t s=b P ( s ) -ρ(s) ∞ ds + t-1 s=b |V α,b (s)| . (29)
As ζ b (t) is a decreasing function of time that can take at most h values, it can be shown that

Combining Equation ( 27), ( 28), ( 29) and (30) shows that E P (t)-ρ(t) ).

We then distinguish three cases: )

≤ (ρ 0,b -ρ 0,b )(ρ α+1,θ ζ b -1 -ρ α+1,θ ζ b -1 ) ≤ ρ -ρ 2 ∞ ,
where the last inequality comes from Lemma 2.

• The case ζ b < ζ b is symmetric.

This concludes the proof of (i). Point (ii) follows directly from Equation (31).

For point (iii), we can mimic the proof of Equation (23). By definition of P, there exists non-decreasing functions x and For each k, the requests of k are generated according to a stationary point process R k . For t < s, R k [t, s) is the number of requested of item k during a time interval [t, s]. Let ϑ k (t) be the time elapsed since the last request of item k. Without loss of generality, in the rest of the proof, we assume that the request process is simple (i.e. that with probability 1, the time between two consecutive requests of an item is never 0). If it is not the case, one can suppress any of the two request and obtain the same behavior of the LRU cache. Hence, the process (R k , ϑ k ) is a stationary marked point process that satisfies the Hypothesis 1.1.1 of [START_REF] Baccelli | Elements of queueing theory: Palm Martingale calculus and stochastic recurrences[END_REF].

As R is stationary, the probability that the item k is requested during a time interval [t, t + x] does not depend on t. Let Fk (x) denote this quantity. We have:

Fk (x) = P [R k [t, t + x] ≥ 1] = P [R k [0, x] ≥ 1] .
We also define F k (x) that is the probability that the time between two arrivals is smaller than x. As (R k , ϑ k ) is a stationary marked point process, this quantity is well defined and can be expressed as

F k (x) = P [R k [t, t + x] ≥ 1 | a request occurred a time t] = P [R k [0, x] ≥ 1 | a request occurred a time 0]
Note that the definition of F k (x) only requires the process R k to be stationary. When the process is a renewal process, F k (x) is the cumulative distribution function of the inter-request time.

By the inversion formula [2, Section 1.2.4], Fk can be expressed as a function of F k :

Fk (x) = λ k x 0 (1 -F k (t))dt, (33)
where

λ k = 1/ ∞ 0 (1 -F k (t)
)dt is the request rate of item k. This quantity only depends on F k and not on the correlation between two arrivals.

To conclude the proof, we remark that the probability that an item k is in the cache when it is requested can be expressed in terms of the functions F k and F for = k. Indeed, Let S n,-k be the set of permutation of {1 . . . k -1} {k + 1 . . . n} (i.e. all integers between 1 and n except k). An item is in the cache at time t if it is among the m items that were last requested. Hence, the probability for item k to be in the cache at time t is σ∈S n,-k P ϑ k (t) ≤ ϑ σ(m) (t), ϑ σ(1) (t) ≤ • • • ≤ ϑ σ(n-1) (t) . This event conditioned on the fact that item k is requested at time t is the probability that item k is in the cache when it is requested. Hence, the hit rate is:

k λ k σ∈S n,-k P ϑ k (t) ≤ ϑ σ(m) (t), ϑ σ(1) (t) ≤ • • • ≤ ϑ σ(n-1) (t) item k is requested at t .
This quantity can clearly be expressed as a function of the F k and Fk which by Equation (33) can be expressed solely as a function of the F k .

  0 and φ (k) e = 1. Note, the request rate λ k of item k can be expressed as λ k = θ (k) D

1 ) 8 Fig. 1 .

 181 Fig.1. Ratio of the approximation of the hit rate for h-LRU under the IRM model based on[START_REF] Jelenković | Least-recently-used caching with dependent requests[END_REF] and (10) of[START_REF] Martina | A unified approach to the performance analysis of caching systems[END_REF] as a function of the cache size for various values of h with n = 1000 items with a Zipf-like popularity distribution with α = 0.8.

  where a := max n k=1 p k . These variables are defined for α ∈ {0, 1, . . . } and b ∈ Z. The collection of variables {P α,b } α,b describes completely the state of the system at time t and live in an set of infinite dimension. Similarly, we define a set of functions ρ α,b by ρ α,b (t) = a 1-α n k=1 (p k ) α x k,b (t). The functions ρ α,b are solutions of the system of ODEs d/dtρ α,b (t) = f α,b (ρ), where:

Fig. 2 .

 2 Fig. 2. Hit probability as a function of the cache size for LRU, LRU(m, m) and 2-LRU under the IRM model and with hyperexponential inter-request times (with z = 10).

Fig. 3 .

 3 Fig. 3. Hit probability as a function of the cache size for LRU, LRU(m, m) and 2-LRU when subject to MAP arrivals (with z = 2, 10 and q = 1/20).

Fig. 4 .

 4 Fig. 4. Hit probability as a function of the lag-1 autocorrelation ρ 1 for LRU, LRU(m, m), LRU(m/2, m/2) and 2-LRU when subject to MAP arrivals (with z = 10).

Fig. 5 .

 5 Fig. 5. Hit probability as a function of the cache size for LRU(m) compared to LRU using trace-based simulation.

Fig. 6 .

 6 Fig.6.Hit probability as a function of the cache size for LRU(m, m, m, m, m) and h-LRU compared to LRU(m, m) using tracebased simulation.

4 Fig. 7 .

 47 Fig. 7. Ratio of the approximation of the hit rate for 5-LRU under the IRM model based on (11) and (10) of[START_REF] Martina | A unified approach to the performance analysis of caching systems[END_REF] as a function of the cache size for various values of α with n = 1000 items with a Zipf-like popularity distribution with α.

E. Proof of Theorem 1 1)

 1 LRU: Let us denote a = max n k=1 p k . For α ∈ {0, 1, . . . } and b ∈ Z, we define P α,b (t):P α,b (t) = a 1-α n k=1 (p k ) α 1 {τ k (t)≥b} . P α,b (t)is the sum of the popularity to the power α of all items that have a request time greater or equal to b at time t.The collection of variables {P α,b } α,b describe completely the state of the system at time t. They live in a set P, that is the set of infinite vectors such that there exists a vector x k,b that is bounded by 1, non-increasing in b and such thatP α,b = a 1-α n k=1 (p k ) α x k,b . P = (P α,b ) α,b : ∃(x k,b ) non-increasing in b, bounded by 1 such that for all α, b: P α,b = a 1-α n k=1 (p k ) α x k,b . We equip P with the L ∞ norm and denote ρ ∞ = sup α,b |ρ α,b | the norm of a vector ρ ∈ P. Similarly, we define a set of functions ρ α,b by ρ α,b (t) = a 1-α n k=1 (p k ) α x k,b (t). The functions ρ α,b are solutions of the system of ODEs d/dtρ α,b (t) = f α,b (ρ), where:

Lemma 2 .

 2 Let g m : P → [0, 1] be the function defined by g m (ρ) = ρ 1,θ , where θ = sup{b : ρ 0,b ≥ m}. The function g m (ρ) is Lipschitz-continuous on P with the constant 2.

  ≥b} , where a = max n k=1 p k . We also define f : P → span(P) by f α,b (ρ) = a(ρ α+1,θ ζ b -1 -ρ α+1,b ), where θ and ζ b are two functions of ρ that are defined by ρ 0,θ = m + • • • + m h and θ ζ b ≤ b < θ ζ b +1 .

  b (P ( s ))-f 0,b (ρ(s))|≤h t s=b f 0,b (P ( s ))-f 0,b (ρ(s)) + a t s=b P ( s )-ρ(s) ∞

Proof of Lemma 3 . 2 ∞- 1 )≤ a ρ -ρ 2 ∞

 3212 The function f : P → span(P) is given byf α,b (ρ) = a(ρ α+1,θ ζ b -1 -ρ α+1,b ),(31)where θ and ζ b are two functions of ρ that are defined byρ 0,θ = m + • • • + m h and θ ζ b ≤ b < θ ζ b +1 . (32)We begin by the proof of (i) which states that(ρ 0,bρ 0,b )(f 0,b (ρ) -f 0,b (ρ )) ≤ 2a ρ -ρ . Let ρ,ρ ∈ P and let ζ b and ζ b be defined as in Equation (32). We have(ρ 0,b -ρ 0,b )(f 0,b (ρ) -f 0,b (ρ )) = a(ρ 0,b -ρ 0,b )(ρ α+1,b -ρ α+1,b + ρ α+1,θ ζ b -1 -ρ α+1,θ ζ b + a(ρ 0,b -ρ 0,b )(ρ α+1,θ ζ b -1 -ρ α+1,θ ζ b -1

•- 1 ≤•- 1 >

 11 If ζ b = ζ b , then we can use Lemma 2 to show that we have ρ α+1,θ ζ b -1-ρ α+1,θ ζ b ρ -ρ ∞ , which implies that (ρ 0,b -ρ 0,b )(ρ α+1,θ ζ b -1 -ρ α+1,θ If ζ b > ζ b , then Equation (32) implies that ρ 0,b ≥ m ζ b +1 + • • • + m h > ρ 0,b . ζ b > ζ b also implies that ρ α+1,θ ζ b ρ α+1,θ ζ b -1 . Hence, (ρ 0,b -ρ 0,b )(ρ α+1,θ ζ b -1 -ρ α+1,θ ζ b -1

x

  . Assume without loss of generality thatθ ζ b ≤ θ ζ b which implies that x k,θ ζ b ≥ x k,θ ζ bfor all k ∈ {1 . . . n}. Thus:ρ α,θ ζ b -1 -ρ α,θ ζ b -1 = a 1-α n k=1 (p k ) α (x k,θ ζ b -x k,θ ) α (x k,θ ζ b -x k,θ ζ b ) + ρ -ρ ∞ = a 1-α n k=1 (p k ) α (x k,θ ζ b -x k,θ ζ b ) + ρ -ρ ∞ ≤ n k=1 p k (x k,θ ζ b -x k,θ ζ b ) + ρ -ρ ∞ ≤ ρ 0,θ ζ b -1 -ρ 0,θ ζ b -1 + 2 ρ -ρ ∞ F.Proof of the Insensitivity of LRU

  . Using the same numerical examples as for LRU(m) we demonstrate the accuracy of the TTL-approximation under MAP arrivals in Table III. Simulation results are based on 5 runs containing 2•10 6 requests each and are in good agreement with the TTL-approximation. OF FINDING A REQUESTED ITEM IN LIST FOR LRU(m). IN THIS EXAMPLE α = 0.8, h = 2 AND m 1 = m 2 = n/5 (i.e., 20 OR 200).

	n	q	z	method	h 0	h 1	h 2
	100	1	2	model	0.26898 0.19304 0.53798
				simul.	0.27021 0.19340 0.53639
			10	model	0.03712 0.05889 0.90399
				simul.	0.03723 0.06106 0.90171
	1000	1	2	model	0.22580 0.16262 0.61158
				simul.	0.22599 0.16256 0.61145
			10	model	0.03112 0.04963 0.91925
				simul.	0.03108 0.04969 0.91923
	1000 0.1	2	model	0.21609 0.14510 0.63881
				simul.	0.21603 0.14526 0.63870
			10	model	0.03006 0.02044 0.94950
				simul.	0.02984 0.02032 0.94985
					TABLE II
	ACCURACY OF PROBABILITY h n q z	method	h = 2	h = 3
		100	1	2	model	0.53619 0.54292
					simul.	0.53449 0.54150
				10	model	0.88249 0.83718
					simul.	0.87936 0.83449
		1000	1	2	model	0.61028 0.61605
					simul.	0.61016 0.61587
				10	model	0.90103 0.86300
					simul.	0.90071 0.86262
		1000 0.1	2	model	0.64744 0.65841
					simul.	0.64807 0.65899
				10	model	0.94935 0.94646
					simul.	0.94924 0.94632
					TABLE III
	ACCURACY OF HIT PROBABILITY FOR h-LRU WITH MAP ARRIVALS. IN

THIS EXAMPLE α = 0.8 AND m = n/5.

  By Gronwall's inequality, this is less than exp(9aht)14ha 2 t, which, when t is less than T /a this is less than Ca for C = 14hT exp(9T h). Lemma 2 concludes the result.

	2 ∞ ≤ 9ah	0	t E P ( s )-ρ(s)	2 ∞ ds + 14ha 2 t.

Note that, after a request, an item is always inserted in list 1. This implies L k,1 (t) = true.

APPENDIX

A. Fixed point computations

This subsection contains some details on the computation of the fixed point for LRU(m) and h-LRU when subject to MAP arrivals. For LRU(m) the fixed point equations in case of MAP arrivals are given by [START_REF] Fofack | Analysis of TTLbased cache networks[END_REF] and an iterative algorithm to compute the fixed point is presented in Figure 1. For h-LRU we first determine T 1 via ( 12) with h = 1, then we determine T 2 by considering the TTL approximation for 2-LRU with T 1 fixed, etc. In other words:

• for h-LRU computing T 1 , . . . , T h corresponds to solving h one dimensional problems • for LRU(m), computing T 1 . . . T h corresponds to solving a single h-dimensional one.

In practice, the computation for h-LRU is much faster.