P2P systems provide a scalable solution for distributing large files in a
network. The file is split into many chunks, and peers contact other peers to
collect missing chunks to eventually complete the entire file. The so-called
`rare chunk' phenomenon, where a single chunk becomes rare and prevents peers
from completing the file, is a threat to the stability of such systems.
Practical systems such as BitTorrent overcome this issue by requiring a global
search for the rare chunk, which necessitates a centralized mechanism.
We consider five different peer-to-peer file sharing systems with two chunks,
with the aim of finding chunk selection algorithms that have provably stable
performance with any input rate and assuming non-altruistic peers who leave the
system immediately after downloading the second chunk. We show that many
algorithms that first looked promising lead to unstable or oscillating
behavior. However, we end up with a system with desirable properties.
With $M(t):=\sup_{s\in[0,t]}A(s)-s$ denoting the running maximum of a
fractional Brownian motion $A(\cdot)$ with negative drift, this paper studies
the rate of convergence of $\mathbb {P}(M(t)>x)$ to $\mathbb{P}(M>x)$. We
define two metrics that measure the distance between the (complementary)
distribution functions $\mathbb{P}(M(t)>\cdot)$ and $\mathbb{P}(M>\cdot)$.
With $M(t):=\sup_{s\in[0,t]}A(s)-s$ denoting the running maximum of a
fractional Brownian motion $A(\cdot)$ with negative drift, this paper studies
the rate of convergence of $\mathbb {P}(M(t)>x)$ to $\mathbb{P}(M>x)$. We
define two metrics that measure the distance between the (complementary)
distribution functions $\mathbb{P}(M(t)>\cdot)$ and $\mathbb{P}(M>\cdot)$.