Toward modeling of a single file broadcasting in a closed network

One fundamental question in modern communication is how to broadcast a file in an Internet overlay of peers with limited bandwidth. One prominent solution, already realised practically, is ´chunk casting´. The big file is divided into a large number of small chunks, each of which are broadcasted. Because of their small size, a peer can quickly upload such a chunk and their populations start to grow very rapidly in a large system. This makes also plausible that the pattern of the very first chunk replications can have big effect on the relative sizes of the chunk populations later on. On the other hand, the chunk replication process quickly takes a form of deterministic process, well described by some differential equations. The role of the short initial phase is to create initial conditions for the deterministic phase. We furnish an approax to proceed this passage. Due to randomness in the initial phase of evolution, the initial conditions for the differential equations are random. To find their distributions, the initial phase have to be solved. We give a simple example when this can be done. Assuming just two chunks and random encounters, the initial evolution can be coupled with Polya urn model with well known distribution. Then the corresponding differential equations can be numerically solved and results compared with simulations of the true discrete model. Results seem to match reasonably well already for systems of few hundred nodes. A similar scheme is discussed also for arbitrary number of chunks.