Retransmission Delays With Bounded Packets: Power-Law Body and Exponential Tail

摘要

Retransmissions serve as the basic building block that communication protocols use to achieve reliable data transfer. Until recently, the number of retransmissions was thought to follow a geometric (light-tailed) distribution. However, recent work shows that when the distribution of the packet sizes have infinite support, retransmission-based protocols may result in heavy-tailed delays and possibly zero throughput even when the aforementioned distribution is light-tailed. In reality, however, packet sizes are often bounded by the maximum transmission unit (MTU), and thus the aforementioned result merits a deeper investigation. To that end, in this paper, we allow the distribution of the packet size $L$ to have finite support. Under mild conditions, we show that the transmission duration distribution exhibits a transition from a power-law main body to an exponential tail. The timescale to observe the power-law main body is roughly equal to the average transmission duration of the longest packet. The power-law main body, if significant, may cause the channel throughput to be very close to zero. These theoretical findings provide an understanding on why some empirical measurements suggest heavy tails. We use these results to further highlight the engineering implications of distributions with power-law main bodies and light tails by analyzing two cases: 1) the throughput of on–off channels with retransmissions, where we show that even when packet sizes have small means and bounded support the variability in their sizes can greatly impact system performance; 2) the distribution of the number of jobs in an $M/M/infty$ queue with server failures. Here, we show that retransmissions can cause long-range dependence and quantify the impact of the maximum job sizes on the long-range dependence.