two clients in the arriving pair, one joins the shortest queue, the other its left neighbouring queue, the two choices being equally likely. In case [**(A3)**]{}, of the three clients in an arriving triple, one joins the shortest queue, the others its left and right neighbouring queues, the choices being equally likely.

Our goal is to describe the long time behaviour of the growth process, or, equivalently, to describe the limiting queueing profile of the network. It should be noted that the method of proof in this paper is purely combinatorial. This is in contrast with  [@ShchVolk], where the results are proved by combining the [martingale]{} techniques from [@fmm] with some probabilistic techniques used in the theory of reinforced processes from [@Tarres] and [@V2001].

Observe that the model considered here can be viewed as a randomized *Tetris* game, and hence it can possibly be analyzed using the techniques of max-plus algebra as well; see [@BM] and Section 1.3 of [@HOW] for details.

For the sake of completeness, let us mention another limit case of the growth process specified by transition probabilities (\[growth\]): namely, the limit process arising as $\beta{\rightarrow}\infty$. It is easy to see that the limit process in this 