Sharp exponent in the occupation-time bound

Determine whether the occupation-time estimate for an open Jackson network with at least three balanced nodes can be improved from a bound proportional to $(n+1)^{k_0}$ to one proportional to $(n+1)^2$, where $k_0$ is the number of balanced nodes.

Background

Theorem 1 proves that when an open Jackson network has no overloaded nodes and has k0≥3k_0\geq 3 balanced nodes, the network is transient and the expected total time spent in the region where the total population of balanced nodes is at most nn is bounded by a constant times (n+1)k0(n+1)^{k_0}.

The paper compares this estimate with the order-n2n^2 occupation-time behavior of simple random walk and Brownian motion in dimensions at least three. The authors therefore conjecture that the exponent k0k_0 is not optimal and that a dimension-independent quadratic bound may hold.

References

We do not expect the exponent~$k_0$ in the last bound to be sharp: by analogy with the simple random walk and the Brownian motion in dimension~$k_0\geq 3$, for which the expected total time spent in a ball of radius~$n$ is of order~$n2$, our conjecture is that this bound should hold with $(n+1)2$ in place of $(n+1){k_0}$.

— Recurrence and transience of open Jackson networks with balanced nodes  (2609.31183 - Popov, 25 Sep 2026) in Introduction and results, immediately after Theorem 1