of similar CSA in a continuous setting were undertaken in [@Sch], [@PenSch] and [@PenSch1].

In [@ShchVolk] we consider the following variants of neighbourhood &[**(A1)**]{}& U\_i={i}\
&[**(A2)**]{}& U\_i={i, i+1}\
&[**(A3)**]{}& U\_i={i-1, i, i+1} where, due to the periodic boundary conditions, $U_{M}=\{M,1\}$ in case [**(A2)**]{} and $U_{1}=\{M, 1, 2\},\,U_{M}=\{M-1, M, 1\}$ in case [**(A3)**]{} respectively. It is easy to see that for the growth process studied in this paper the case [**(A1)**]{} is trivial. Therefore, we will consider cases [**(A2)**]{} and [**(A3)**]{} only.

A stochastic process $u(t)=(u_1(t),\dots,u_{M}(t))$ formed by the sites’ potentials plays an important role in our asymptotic study of the growth process. It is easy to see that $u(t)$ is also a Markov chain, with transition probabilities given by $$\label{eq_tran}
\P\left(u_{i}(t+1)=u_{i}(t)+1_{\{i\in U_k\}}\right)=
  \begin{cases}0,&\text{if $u_k(t)>m(t)$},\\
     1/N_{\min}(t),&\text{if $u_k(t)=m(t)$},
  \end{cases}$$ for $k\in\M$. This process has the following rather obvious queueing interpretation (S. Foss, personal communications) explaining the title of the paper (originally titled ‘Random sequential adsorption at extremes’). Namely, consider a system with $M$ servers, with the clients arriving in bunches of $2$ in case [**(A2)**]{} and of $3$ in case [**(A3)**]{}. The quantity $u_i(t)$ is interpreted as the number of clients at server $i$ at time $t$. In case [**(A2)**]{}, of the 