Asymptotic Approximation of the Move-To-Front Search Cost Distribution and Least-Recently-Used Caching Fault Probabilities

Limits and rate of convergence for the distribution of search cost under the move-to-front rule. I 2 (t) = Z log(t) 1 e ?x x (log(t=x)) d dx + Z t log t e ?x x (log(t=x)) d dx def = I 21 (t) + I 22 (t) (7.66) Here, the asymptotic behavior of I 21 (t) is determined by I 21 (t) (log(t)) d Z log(t) 1 e ?x x dx (log(t)) d Z 1 1 e ?x x dx; (7.67) and I 21 (t) (log(t= log(t))) d Z log(t) 1 e ?x x dx (log(t)) d Z 1 1 e ?x x dx: (7.68) The estimate for I 22 (t) is given by I 22 (t) (log(t= log(t))) d 1 t log t = o(log(t) d) as t ! 1: (7.69) Finally, by combining equations (7.61-7.69) it follows I 1 (t) ? I 2 (t) (log(t) x dx as t ! 1 = (log(t)) d ; where the last equality follows from 16], pp. 946, equation 8.367 (12). This nishes the proof for the case d 0. For the case ?1 < d < 0 the inequalities in (7.63) and (7.64) are going to be reversed, but the asymptotic behavior of I 11 is still going to be the same. Since (7.65) still holds I 1 (t) will behave asymptotically the same as for the d 0 case. Similarly, the inequalities in (7.67) and (7.68) are going to be reversed, but the asymptotic behavior of I 21 (t) is unchanged. The asymptotic upper bound for I 22 (t) is given by I 22 (t) 1 t Z t log(t) 1 x (log(t=x)) d dx = 1 t Z log (t= log t) 0 u d du = 1 t (log t= log t) d+1 d + 1 = o((log t) d) as t ! 1: This implies that I 2 (t) will have the same asymptotics as in the d 0 case, and therefore (7.70) holds. This completes the proof of the lemma. 3 29 Proof of Lemma 5: It is easy to compute that Z t 0 1 ? e ?x x (log(t=x)) d dx ? (log(t)) d+1 d + 1 = Z 1 0 1 ? e ?x x (log(t=x)) d dx ? Z t 1 e ?x x (log(t=x)) d dx def = I 1 (t) ? I 2 (t): (7.61) By changing the variable …