Optimal timer-based caching policies for general arrival processes

Abstract

In this paper, we analyze the hit performance of cache systems that receive file requests with general arrival distributions and different popularities. We consider timer-based policies, with differentiated timers over which we optimize. The optimal policy is shown to be related to the monotonicity of the hazard rate function of the interarrival distribution. In particular, for decreasing hazard rates, timer policies outperform the static policy of caching the most popular contents. We provide explicit solutions for the optimal policy in the case of Pareto-distributed inter-request times and a Zipf distribution of file popularities, including a compact fluid characterization in the limit of a large number of files. We compare it through simulation with classical policies, such as Least Recently Used and discuss its performance.

Proofs

We provide here the analysis of the asymptotic behavior of the optimal policy in the heavy-tailed case.

1.1 Asymptotic caching fraction

We begin by establishing bounds on the number of objects M cached by the optimal policy, as defined in (20). From this definition, we first note that

$$\begin{aligned} p \frac{(\alpha - 1)}{\alpha } S_N(\beta )&< M^{-\beta }, \end{aligned}$$

(37)

$$\begin{aligned} p \frac{(\alpha - 1)}{\alpha } S_N(\beta )&\geqslant (M+1)^{-\beta } \quad \text { if } M<N. \end{aligned}$$

(38)

From here we can establish the following:

Lemma 3

$$\begin{aligned} 1 - \frac{C}{M}&< \frac{S_M(-\beta (\alpha -1))}{M^{1 + \beta (\alpha -1)}}, \end{aligned}$$

(39)

$$\begin{aligned} 1 - \frac{C}{M}&\geqslant \frac{S_M(-\beta (\alpha -1))}{M(M+1)^{\beta (\alpha -1)}} \quad \text { if } M<N. \end{aligned}$$

(40)

Proof

Substituting the optimal $u^*$ as a function of price in the constraint (18),

$$\begin{aligned} M - C&= \sum _{n=1}^M (1-u_m^*)\\&= \sum _{n=1}^M\left( \frac{p(\alpha - 1)}{\alpha \lambda _n} \right) ^{\alpha - 1} \\&= \left( \frac{p(\alpha - 1)}{\alpha }S_N(\beta ) \right) ^{\alpha - 1} \sum _{n=1}^M n^{\beta (\alpha -1)}.\\&= \left( \frac{p(\alpha - 1)}{\alpha }S_N(\beta ) \right) ^{\alpha - 1} S_M(-\beta (\alpha -1)).\\ \end{aligned}$$

Applying the bound (37), we arrive at

$$\begin{aligned} M - C < M^{-\beta (\alpha -1)}S_M(-\beta (\alpha -1)), \end{aligned}$$

which dividing by M yields (39). Establishing (40) for the case $M<N$ is analogous. $\square$

Proof of Proposition 1

Recall that, by definition (22), $x_0: = \limsup _{N\rightarrow \infty } \frac{M_N}{N} \in [c,1]$.

Suppose first that $x_0<1$. In this case, for large enough N we will have $M_N < N$, so both bounds in Lemma 3 apply, yielding

$$\begin{aligned} \frac{S_{M_N}(-\beta (\alpha -1))}{M_N(M_N+1)^{\beta (\alpha -1)}} \leqslant 1 - \frac{cN}{M_N} < \frac{S_{M_N}(-\beta (\alpha -1))}{M_N^{1 + \beta (\alpha -1)}}. \end{aligned}$$

Now, since $M_N\rightarrow \infty$, applying the equivalence (16) for $\gamma = -\beta (\alpha -1)$ we see that both bounds converge to the same limit $\frac{1}{1+\beta (\alpha -1)}$. This implies that $\frac{M_N}{N}$ has a limit $x_0$ [sharpening the condition (22)], satisfying in particular

$$\begin{aligned} 1 - \frac{c}{x_0} = \frac{1}{1+\beta (\alpha -1)} \quad \Rightarrow \quad x_0 = c \frac{1+\beta (\alpha -1)}{\beta (\alpha -1)}. \end{aligned}$$

A necessary condition for this result to be within our assumed case is that

$$\begin{aligned} c < \frac{\beta (\alpha -1)}{1+\beta (\alpha -1)}=c_{\alpha ,\beta }. \end{aligned}$$

(41)

Suppose instead that $x_0=1$. We first derive a necessary condition for this case to occur. Choosing a subsequence $N'$ that satisfies $\frac{M_{N'}}{N'} \rightarrow 1$, and applying the bound (39), we have

$$\begin{aligned} 1 - c \leqslant \lim _N \frac{S_{M_N}(-\beta (\alpha -1))}{{M_N}^{1 + \beta (\alpha -1)}} = \frac{1}{1+\beta (\alpha -1)}, \end{aligned}$$

from which we conclude that

$$\begin{aligned} c \geqslant \frac{\beta (\alpha -1)}{1+\beta (\alpha -1)}= c_{\alpha ,\beta }, \end{aligned}$$

(42)

the opposite situation to the one considered in (41). We claim, furthermore, that here as well the $\limsup$ is an actual limit. Indeed, if $x_1$ is the limit of any other subsequence $\frac{M_{N''}}{N''}<1$, we can repeat the argument in the previous case leading to

$$\begin{aligned} x_1 = c \frac{\beta (\alpha -1)+1}{\beta (\alpha -1)} \geqslant 1, \end{aligned}$$

using (42); therefore $x_1=1$, the only limit point. Note furthermore that this situation of convergence to unity strictly from below can only occur if $c= c_{\alpha ,\beta }$, the border case.

If, instead, we have strict inequality in (42), then it must be that for $N\geqslant N_0$ we have $M_N=N$, i.e., all files are stored. $\square$

1.2 Asymptotic optimal policy

We provide here the proof of Theorem 3, by separating the discussion in the two cases considered above.

1.2.1 Case $c<c_{\alpha ,\beta }$

This is the case where $x_0 = \frac{c}{c_{\alpha ,\beta }} = \lim \frac{M_N}{N} < 1$.

Suppose $x<x_0$; then $\lfloor x N\rfloor < M_N$ for large N. Invoking (24) and the bounds (37) and (38), we write, for $n\leqslant M_N$,

$$\begin{aligned} 1- \left( \frac{n}{M_N}\right) ^{\beta (\alpha -1)} < u_n^{*,N} \leqslant 1- \left( \frac{n}{M_N+1}\right) ^{\beta (\alpha -1)}. \end{aligned}$$

(43)

Noting that

$$\begin{aligned} \frac{\lfloor x N\rfloor }{M_N} = \frac{\lfloor x N\rfloor }{N}\frac{N}{M_N} \mathop {\longrightarrow }_{N} \frac{x}{x_0}, \end{aligned}$$

we obtain

$$\begin{aligned} u^{*,N}(x) \mathop {\longrightarrow }_{N} 1 - \left( \frac{x}{x_0}\right) ^{\beta (\alpha -1)}, \quad x < x_0. \end{aligned}$$

If, instead, $x > x_0$, then $\lfloor x N\rfloor > M_N$ for large N and $u^{*,N}(x)=0$. Both cases can therefore be summarized in the expression (27) which is a special case of (25). It is readily verified by integration that the constraint (26) is verified.

1.2.2 Case $c> c_{\alpha ,\beta }$

In this case, we have that for large N, $M_N=N$, i.e., we cache all files with positive probability given by

$$\begin{aligned} u_n^{*,N} = 1- \left( \frac{p_N S_N(\beta ) (\alpha - 1)}{\alpha } n^\beta \right) ^{\alpha - 1}. \end{aligned}$$

Substitution into the constraint $\sum _n (1-u_n) = N -cN$ leads to

$$\begin{aligned} \left( \frac{p_N S_N(\beta ) (\alpha - 1)}{\alpha } \right) ^{\alpha - 1} \sum _{n=1}^N n^{\beta (\alpha -1)} = N (1-c), \end{aligned}$$

from which we can solve for

$$\begin{aligned} \left( \frac{p_N S_N(\beta ) (\alpha - 1)}{\alpha } \right) ^{\alpha - 1} =\frac{N(1-c)}{S_N(-\beta (\alpha -1))}, \end{aligned}$$

(44)

leading to the expression

$$\begin{aligned} u_n^{*,N} = 1- \frac{N(1-c)}{S_N(-\beta (\alpha -1))}n^{\beta (\alpha -1)}. \end{aligned}$$

(45)

Taking now $n=\lfloor x N \rfloor$ we have

$$\begin{aligned}&u^N(x) \sim 1- \frac{(1-c)N^{1+\beta (\alpha -1)}}{S_N(-\beta (\alpha -1))} x^{\beta (\alpha -1)} \\&\quad \mathop {\longrightarrow }_{N} 1- {(1-c)(1+\beta (\alpha -1))}x^{\beta (\alpha -1)} . \end{aligned}$$

This is now the expression (28), again of the form (25) and once more satisfying (26).

1.3 Asymptotic optimal cost

We provide here a proof of Theorem 4, again distinguishing the two relevant cases. We will establish directly the relationships in (30). A standard calculation (omitted) verifies that they coincide with the integral in (29) for the two cases (27) or (28) of the optimal policy.

1.3.1 Case $c<c_{\alpha ,\beta }$, $x_0 < 1$.

Going back to (43), we derive in this case the inequalities

$$\begin{aligned} \left( \frac{n}{M_N+1}\right) ^{\beta \alpha } \leqslant \left( 1-u_n^{*,N}\right) ^\frac{\alpha }{\alpha -1} < \left( \frac{n}{M_N}\right) ^{\beta \alpha }. \end{aligned}$$

Substituting this into the optimal objective (17), denoted $H^*_N$, leads to upper and lower bounds. Focusing for brevity on the lower bound, we have

$$\begin{aligned} H^*_N&= \frac{1}{S_N(\beta )} \sum _{n=1}^{M_N} \frac{1}{n^\beta }\left[ 1-\left( 1-u_n^{*,N}\right) ^\frac{\alpha }{\alpha -1}\right] \\&> \frac{1}{S_N(\beta )} \sum _{n=1}^{M_N} \frac{1}{n^\beta }\left[ 1-\left( \frac{n}{M_N}\right) ^{\alpha \beta }\right] \\&=\frac{N^{-\beta }}{S_N(\beta )} \sum _{n=1}^{M_N} \left( \frac{N}{n}\right) ^\beta \left[ 1-\left( \frac{n}{N}\frac{N}{M_N}\right) ^{\alpha \beta }\right] \\&=\frac{N^{1-\beta }}{S_N(\beta )}\left\{ \sum _{n=1}^{M_N} \left[ \frac{N}{n}\right] ^\beta \frac{1}{N} - \left[ \frac{N}{M_N}\right] ^{\alpha \beta } \sum _{n=1}^{M_N} \left[ \frac{n}{N}\right] ^{(\alpha -1)\beta } \frac{1}{N}\right\} . \end{aligned}$$

Consider now the (convergent) integrals

$$\begin{aligned} \int _0^{x_0} x^{-\beta }\hbox {d}x&= \frac{x_0^{1-\beta }}{1-\beta }, \text { and } \\ \int _0^{x_0} x^{(\alpha -1)\beta }\hbox {d}x&= \frac{x_0^{(\alpha -1)\beta + 1 }}{(\alpha -1)\beta + 1 }, \end{aligned}$$

and their respective Riemann sums corresponding to the uniform partition n/N, with $n=1,\ldots , \lfloor x_0 N \rfloor$; they correspond exactly to the sums in the expression above except for the upper limit, but since $M_N\sim x_0 N$ the limits still coincide with the respective integrals. Thus, one can invoke (16) to find the limit of the above lower bound. A similar procedure with the upper bound shows it is tight, yielding the limiting objective

$$\begin{aligned} H^*&= (1-\beta )\left\{ \frac{x_0^{1-\beta }}{1-\beta } - \frac{1}{x_0^{\alpha \beta }} \frac{x_0^{(\alpha -1)\beta + 1 }}{(\alpha -1)\beta + 1 }\right\} \nonumber \\&= x_0^{1-\beta } \left[ 1 - \frac{1-\beta }{(\alpha -1)\beta + 1}\right] \nonumber \\&= x_0^{1-\beta } \frac{\alpha \beta }{\alpha \beta + 1 -\beta }, \end{aligned}$$

(46)

where we recall that $x_0 = \frac{c}{c_{\alpha ,\beta }}$, the latter given by (41).

1.3.2 Case $c>c_{\alpha ,\beta }$

Quoting from (45),

$$\begin{aligned} 1 - u_n^{*,N} = \frac{N(1-c)}{S_N(-\beta (\alpha -1))}n^{\beta (\alpha -1)}, \end{aligned}$$

from which we have

$$\begin{aligned} 1 - (1- u_n^{*,N})^\frac{\alpha }{\alpha -1} = 1 - \left[ \frac{N(1-c)}{S_N(-\beta (\alpha -1))}\right] ^\frac{\alpha }{\alpha -1} n^{\alpha \beta }. \end{aligned}$$

Multiplying by $S_N(\beta )^{-1} m^{-\beta }$ and adding in $n=1,\ldots , N$ yields the optimal objective

$$\begin{aligned} H^*_N&= 1 - \frac{1}{S_N(\beta )} \left[ \frac{N(1-c)}{S_N(-\beta (\alpha -1))}\right] ^\frac{\alpha }{\alpha -1} S_N(-(\alpha -1)\beta ) \\&= 1 - \frac{1}{S_N(\beta )} \left[ N(1-c)\right] ^\frac{\alpha }{\alpha -1}[S_N(-(\alpha -1)\beta )]^{-\frac{1}{\alpha -1}}. \end{aligned}$$

We can now invoke (16) for each of the terms to write

$$\begin{aligned} H^*_N&\sim 1 - \frac{1-\beta }{N^{1-\beta }} \left[ N(1-c)\right] ^\frac{\alpha }{\alpha -1} \left[ \frac{1+\beta (\alpha -1)}{N^{1+\beta (\alpha -1)}}\right] ^\frac{1}{\alpha -1}. \end{aligned}$$

Terms in N cancel out, and we reach the limit

$$\begin{aligned} H^* = 1 - (1-\beta )(1-c)^\frac{\alpha }{\alpha -1}(1+\beta (\alpha -1))^\frac{1}{\alpha -1}. \end{aligned}$$

(47)

In both cases, the optimal cost is consistent with the fluid version of the objective,

$$\begin{aligned} V(u)= (1-\beta ) \int _0^1 x^{-\beta } \left[ 1 - (1 - u(x))^\frac{\alpha }{\alpha - 1}\right] \hbox {d}x, \end{aligned}$$

(48)

evaluated at the limiting policies u(x) given in (27) or (28).

1.4 Asymptotic prices

It is also interesting to find the behavior of the Lagrange multipliers (prices) for large N. We find that prices are of the order of $N^{-1}$, with constant depending on each case.

In the case of $c<c_{\alpha ,\beta }$, referring to (37)–(38), and using $M_N \sim x_0 N$, we get

$$\begin{aligned} p_N&\sim M_N^{-\beta } \frac{\alpha }{\alpha -1}\frac{1}{S_N(\beta )} \sim x_0^{-\beta } \frac{\alpha }{\alpha -1}N^{-\beta } \frac{1-\beta }{N^{1-\beta }} = \frac{1}{N}x_0^{-\beta } \frac{\alpha (1-\beta )}{\alpha -1}. \end{aligned}$$

For $c>c_{\alpha ,\beta }$, referring to (44), we have

$$\begin{aligned} \left( p_N\frac{(\alpha - 1)}{\alpha } \frac{N^{1-\beta }}{1-\beta }\right) ^{\alpha - 1} \sim \frac{N(1-c)(1+\beta (\alpha -1))}{N^{1+\beta (\alpha -1)}}; \end{aligned}$$

leading to

$$\begin{aligned} p_N \sim \frac{1}{N} \frac{\alpha (1-\beta )}{\alpha -1} \left[ (1-c)(1+\beta (\alpha -1))\right] ^\frac{1}{\alpha -1}. \end{aligned}$$