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 trace-based 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 was focused on deriving explicit expressions for the cache content distribution by using a Markov chain analysis [1].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 [6].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 [6] 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 (1 − e −p k T ). ( 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 [3], [7], [13], [14].All of these TTL approximations have been shown to be (very) accurate by means of numerical examples, but except for LRU in [8], 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 [1], while the second, called h-LRU, was recently introduced in [13].In fact, a TTL approximation for h-LRU was also introduced in [13], 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 [13] for h > 2. Numerical results for the TTL approximation for LRU(m) also suggest that it is asymptotically exact.
• 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.This paper is complemented by a technical report [10] that contains most of the proofs.

II. REPLACEMENT ALGORITHMS
We consider two families of cache replacement algorithms: h-LRU, introduced and called k-LRU in [13], and LRU(m), introduced in [1], [9].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 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 for = 1, . . ., h.Further for ∈ {1, . . ., h}, the average time spent in is and (2), this implies that when observing the system at a random point in time, that item k is in list ≥ 1 with probability 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 : 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 [5], [15].A MAP is characterized by two d×d matrices (D 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 1 ) = 0 and φ (k) e = 1.Note, the request rate λ k of item k can be expressed as 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 Due to the block structure of P M AP k , its steady state vector (π k,0 , πk,1 , . . ., πk,h ) obeys for = 1, . . ., h , where the matrices R k,s can be computed recursively as 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 where e is a column vector of ones.The hit probability h in list can subsequently be computed as for = 0, . . ., h.

B. TTL-approximation for h-LRU
1) IRM setting: As in [13], 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 ) be the stationary vector of Ph,k , then the balance equations imply: 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 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 , due to the PASTA property.These fixed point equations can be generalized without much effort to renewal arrivals as explained in [10,Appendix B].
The following property is proven in [10, Appendix D], where we also show that T 1 < T 2 < . . .< T h must hold to have a fixed point.
) which coincides with the hit probability of the so-called refined model for 2-LRU presented in [13,Eqn (9)].For h > 2 only an approximation that relied on an additional approximation of independence between the h lists was presented in [13, see Eqn (10)].In Figure 1 we plotted the ratio between our approximation and the one based on (10) of [13].The results indicate that the difference grows with h.We show in [10, Appendix C] that it typically decreases as the popular items gain in popularity.
As (11) 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 [13] 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.
Eq. ( 10) of [13] (err) Eq. ( 11 of the form e −p k T by e 1 .The fixed point equation for determining T h is found as where λ k is the request rate of item k and

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 an iterative procedure that update the values T in a roundrobin fashion.This iterative procedure is described in [10, 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 . 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 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 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 result 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 [10,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 Cheapproximation could rely on a more direct argument that uses the closed-form expression for the steady state distribution of LRU, as in [8].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: 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.We 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 having a request time greater or equal to Θ(t) is in the cache.Let H(t) be the sum of the popularities of items in the cache: Our approximation of the probability for item k to have a request time after b, is given by the following ODE (for b < t): 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 : probabilities of having items requested after b is equal to m.The approximation of the hit ratio for LRU is then given by 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 [9], which is to consider P α,b (t): The proof of the theorem, detailed in [10, Appendix E1], relies on classical results of stochastic approximation.It uses the fact that • 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: 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 ).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, We then define x k, ,b (t) that is 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 (14).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, where θ (t) = sup{b : 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 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 t) is always true, but this does not hold for > 2. This leads to: A similar reasoning can be applied to obtain an ODE for The hit probability of list used in Theorem 1 is then where the variables x k, ,b satisfy the above ODE.The proof of Theorem 1 in the case of h-LRU is very similar as the one for LRU and uses the same stochastic 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 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: where θ (t) and ζ b (t) are defined by: 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 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 a virtual request time b belongs to depends non-continuously on ρ (the list ζ b is a discrete quantity).In [10, 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 and 3 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 [10, 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 times.
This theorem complements the results of Jelenkovic and Radovanovic who showed in [12], [11] 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 [4, 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.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 [9].In fact, a generalization to a network of caches would be fairly straightforward for the class of RAND(m) policies.
where the entry (j, j ) of D (k) 1

1 ) 8 Fig. 1 .
Fig.1.Ratio of the approximation of the hit rate for h-LRU under the IRM model based on(11) and (10) of[13] 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.

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

Fig. 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.

Figure 6
Figure6compares 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.

ACCURACY
OF THE TWO APPROXIMATIONS FOR THE HIT PROBABILITY OF h-LRU UNDER THE IRM MODEL WITH A ZIPF-LIKE POPULARITY DISTRIBUTION WITH α = 0.8.SIMULATION IS BASED ON 10 RUNS OF 10 3 n REQUESTS WITH A WARM-UP PERIOD OF 33%.
. 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).