  The Persistent-Idle (PI) load distribution policy was recently introduced as
an appealing alternative to current low-communication load balancing
techniques. In PI, servers only update the dispatcher when they become idle,
and the dispatcher always sends jobs to the last server that reported being
idle. PI is unique in that it does not seek to push the server queue lengths
towards equalization greedily. Rather, it aggressively pulls the servers away
from starvation. As a result, PI's analysis requires different tools than other
load balancing approaches. So far, PI was proven to achieve the stability
region for Bernoulli arrivals and deterministic and constant service
capacities.
  Our main contribution is proving that PI achieves the stability region in a
model with batch arrivals and random service capacities. Proving this result
requires developing tighter bounds on quantities of interest and proving the
fulfillment of a generalized version of the state-dependent drift criteria
previously considered. We also present PI-Split, which allows batches to be
split among idle servers and prove its stability. Finally, we conduct
simulations comparing PI and PI-Split to other load balancing policies.
Simulation results indicate that both achieve excellent performance for the
model under consideration for a wide range of parameters.
