Distribution of the Number of Retransmissions of Bounded Documents

Abstract

Retransmission-based failure recovery represents a primary approach in existing communication networks that guarantees data delivery in the presence of channel failures. Recent work has shown that, when data sizes have infinite support, retransmissions can cause long (-tailed) delays even if all traffic and network characteristics are light-tailed. In this paper we investigate the practically important case of bounded data units 0 ≤ L b ≤ b under the condition that the hazard functions of the distributions of data sizes and channel statistics are proportional. To this end, we provide an explicit and uniform characterization of the entire body of the retransmission distribution P[N b > n] in both n and b. Our main discovery is that this distribution can be represented as the product of a power law and Gamma distribution. This rigorous approximation clearly demonstrates the coupling of a power law distribution, dominating the main body, and the Gamma distribution, determining the exponential tail. Our results are validated via simulation experiments and can be useful for designing retransmission-based systems with the required performance characteristics. From a broader perspective, this study applies to any other system, e.g., computing, where restart mechanisms are employed after a job processing failure.

Distribution of the Number of Retransmissions of Bounded Documents

“…Note that this result appeared in Theorem 1 of [20]. The proof can be found in Section 4 of [9]; see also [8].…”

On Channel Failures, File Fragmentation Policies, and Heavy-Tailed Completion Times

“…Split routing, where disjoint fragments of the file are sent along multiple paths and the completion time is the time when the last fragment arrives, also retains a power-law completion time though the tail can be lightened with a larger index. Having a bounded file size distribution of course eliminates the heavy-tailed completion time; however, it is shown in [23], [24] that when the upper bound on the file size distribution is large, the completion time distribution retains a power-law body. To the best of our knowledge, this work is the first to show that heavytailed completion times are actually quite fragile, and can be removed by a large class of simple fragmentation schemes.…”

Time inhomogeneity in longest gap and longest run problems

“…Section 6 deals with what provided our initial motivation, the study of the tail of the total execution time X of a task like program or file transmission in a fault-tolerant computing environment working under the restart protocol, where the task needs to be completely restarted after failure. Here takes the role of the ideal task time, failures occur at the epochs of N and so X = + D. Earlier studies of similar problems are in Asmussen et al [3,4,5] and Jelenković et al [13,12]. The novelty here is the time-inhomogeneity.…”