---
abstract: 'In this paper we study asymptotic behaviour of a growth process generated by a semi-deterministic variant of the cooperative sequential adsorption model (CSA). This model can also be viewed as a particular example from queueing theory, to which John Kingman has contributed so much. We show that the quite limited randomness of our model still generates a rich collection of possible limiting behaviours.'
author:
- Vadim Shcherbakov and Stanislav Volkov
---

Queueing with neighbours
========================

###### Keywords

cooperative sequential adsorption, interacting particle systems, max-plus algebra, queueing, Tetris

###### AMS subject classification (MSC2010)

Primary 60G17, 62M30; Secondary60J20

Introduction
------------

Let $\M=\{1,2,\ldots,M\}$ be a lattice segment with periodic boundary conditions (that is, $M+1$ will be understood as $1$ and $1-1$ will be understood as $M$), where $M\geq 1$. The growth process studied in this paper is defined as a discrete-time Markov chain $(\xi_i(t),\, i\in\M,\, t\in {\mathbb{Z}}_{+}),$ with values in ${\mathbb{Z}}^{M}_{+}$ and specified by the following transition probabilities: \[tran\] &¶(\_[i]{}(t+1)=\_[i]{}(t)+1, \_j(t+1) = \_j(t) ji(t)) &\
&={

[rcl]{} 0, & & u\_i(t)&gt;m(t),\
1/N\_(t), & & u\_i(t)=m(t),

. & for $i\in\M$, where $$u_i(t)=\sum\limits_{j\in U_i} \xi_j(t),\,\, i\in\M,$$ $U_i$ is a certain neighbourhood of site $i$, $$\label{mt}
m(t)=\min_{k\in\M}u_k(t)$$ and $N_{\min}(t)\in\{1,2,\dots,M\}$ is the number of $u_i(t)$ 