In the “$b$-ball random juggling” of [Warrington ’05] he put a maximum height $n$ on the throws (otherwise chosen with uniform probability) and worked out the long-term probability that the juggler finds herself in one of the ($n$ choose $b$) possible juggling states. I'll reverse this digraph and introduce a parameter $q$ that governs whether we move a ball vs. skip it and consider the next. Then I'll solve this model exactly, by counting matrices over a finite field with $q$ elements.