I am modelling a torrent-like p2p network for a video-on-demand application. My goal is to know, for any given time T, how many seeds and leeches does the network need to keep its steady-state.
Ideally, this translates as a chart where X axis means time T (say, in seconds, or days); Y axis means amount of peers; and there's one curve showing the amount of seeds along time, and other showing the amount of leechers along time. Then, playing around with parameters should modify this curves and allow for multiple simulations (i.e: if the average upload bandwidth upload of a peer doubles, what happens to the required number of seeds at time = 765?).
Studying the model described in items 3.0 and 3.1 here: http://conferences.sigcomm.org/sigcomm/2004/papers/p444-qiu1.pdf brought me to 7 key parameters:
time (t); rate of new peers' arrival (λ); average upload bandwidth of a peer (µ); average download bandwidth of a peer (c); download abortion rate (mostly due to impatience) (θ); altruism rate (rate at which peers leave the system, indicating how much they keep seeding after having completed their downloads) (γ); effectiveness of file sharing, a discrete number between 0 and 1 (η).
I tried Wolfram Alpha to derive the formula given by the paper and arrive at an equation for x(t), but it also brought up the arbitrary constant k1 (for y(t) too), which I don't know what to do with it. All I want is to have the abovementioned chart with its 2 curves, like what is done in section V here: http://www.netlab.tkk.fi/tutkimus/shok-fi/publ/p2pvodglobe.pdf
I thought of using semi-arbitrary parameters inspired in those found on these papers (i λ = 0.20 // γ = 0.20 // θ = 0.20 // c = 1024 // µ = 512 // η = 0.85), but suppose I need their functions too in order to make the simulations work.
Can anyone help me clarify my next steps, shine me some light, and help me get closer to this curves and the simulations they'll allow?