  In this paper, we study a distribution of labeled particles on a continuous
ring. It arises in three different ways, all related to the multi-type TASEP on
a ring. We prove formulas for the probability density function for some
permutations and give conjectures for a larger class. We give a complete
conjecture for the probability of two particles i, j being next to each other
on the cycle, for which we prove some cases. We also find that two natural
events associated to the process have exactly the same probability expressed as
a Vandermonde determinant. It is unclear whether this is just a coincidence or
a consequence of a deeper connection.
