interaction for larger $M$

\[Tabl3\]

We will derive the asymptotic behaviour of the process $\xi(t)$ from the asymptotic behaviour of the process of potentials. In turn the study of the process of potentials is greatly facilitated by analysis of the following auxiliary process \[eqv(t)\] v\_k(t)=u\_k(t)-m(t), k=1,…,M. Observe that $v(t)$ also forms a Markov chain on $\{0,1,2,\dots\}$ and for each $t$ there is a $k$ such that $v_k(t)=0$. Loosely speaking, the quantities $v_k(t)$, $k=1$, …, $M$, represent what happens ‘on the top of the growth profile’.

It turns out that in the asymmetric case there is a single class of recurrent states to which the chain eventually falls and then stays in forever. As we show later, this class is a certain subset of the set $\{0,1,2\}^M$ containing the origin $(0,\ldots,0)$. Thus a certain ‘stability’ of the process of potentials is observed as time goes to infinity.

In particular, it yields, as we show, that there will not be long queues in the system if $M$ is odd; however, this does not prevent occurrence of relatively long queues if $M$ is even. For instance, if $M$ is even, then one can easily imagine the profile with peaks at even sites, and 