  We study a stochastic particle system which models the time evolution of the
ranking of books by online bookstores (e.g., Amazon). In this system, particles
are lined in a queue. Each particle jumps at random jump times to the top of
the queue, and otherwise stays in the queue, being pushed toward the tail every
time another particle jumps to the top. In an infinite particle limit, the
random motion of each particle between its jumps converges to a deterministic
trajectory. (This trajectory is actually observed in the ranking data on web
sites.) We prove that the (random) empirical distribution of this particle
system converges to a deterministic space-time dependent distribution. A core
of the proof is the law of large numbers for {\it dependent} random variables.
