  In this work we study multi-server queues on a Euclidean space. Consider $N$
servers that are distributed uniformly in $[0,1]^d$. Customers (users) arrive
at the servers according to independent Poisson processes of intensity
$\lambda$. However, they probabilistically decide whether to join the queue
they arrived at, or move to one of the nearest neighbours. The strategy
followed by the customers affects the load on the servers in the long run. In
this paper, we are interested in characterizing the fraction of servers that
bear a larger load as compared to when the users do not follow any strategy,
i.e., they join the queue they arrive at. These are called overloaded servers.
In the one-dimensional case ($d=1$), we evaluate the expected fraction of
overloaded servers for any finite $N$ when the users follow probabilistic
nearest neighbour shift strategies. Additionally, for servers distributed in a
$d$-dimensional space we provide expressions for the fraction of overloaded
servers in the system as the total number of servers $N\rightarrow \infty$.
Numerical experiments are provided to support our claims. Typical applications
of our results include electric vehicles queueing at charging stations, and
queues in airports or supermarkets.
