TITLE: Duration and critical groups order in sandpile models and chip firing games
QUESTION [5 upvotes]: The famous chip firing game (which is closely related to sandpile models) goes like this: 

Place chips at the vertices of a graph. REPEATEDLY: If a vertex $v$ of
  degree $d_{v}$ has at least $d_{v}$ chips, then it "fires" a chip
  along each of its incident edges.  If there is vertex which can fire,
  the game is over.

Remarkably, whether a game terminates only depends on the graph and the initial configuration of chips (and not on the vertices chose to fire during the game). Even more remarkably, the length of the game (the number of fired vertices) does not depend on the actual play but rather only on the graph and the initial configuration.
Tardos and Bjorner-Lovasz-Shor have given well-known bounds for the length of the game.
On the other hand, there is a vast literature which studies the "critical group" (aka "sandpile group") of a graph, which is strictly speaking for a slightly different but related game (the so-called dollar game). 

QUESTION: I am wondering if there is some connection between the duration of the
  chip firing game and the order of the critical group.

This is based on the vague intuition that the larger the group, the more opportunities the game has to terminate and so it can be expected to end rather sooner than later. But I may be completely off the mark here (for instance, I am mixing both games in my reasoning...).
Any thoughts on this highly appreciated.

REPLY [2 votes]: Here is an example that ought to convince you that no bound like this can exist. Let $\ell(G,N)$ (for $N$ sufficiently small) denote the maximum length of a game on the graph $G$ that terminates and involves $N$ chips. Let $P_n$ denote the path graph on $n$ vertices and $S_n$ the star graph on $n$ vertices. It's not hard to see that $\ell(S_n,N) = N$ while $\ell(P_n,N) = \Theta(N^2)$. However, because they are trees, both $P_n$ and $S_n$ have trivial critical groups.