Throughput optimality of BalancedSplitting-π

Determine whether the BalancedSplitting-π scheduling policy in the multiserver-job queueing model is throughput optimal whenever the auxiliary helper-set policy π is throughput optimal; specifically, establish that the mean response time under BalancedSplitting-π remains finite for all loads ρ<1 as defined by the model with classes i=1,…,C, class-specific server needs n_i, service time distributions D_i, and the Balanced Splitting Framework partition {A_i} and H described in Section 3.

Background

BalancedSplitting-π is introduced within the Balanced Splitting Framework (BSF), which statically partitions the server set into class-specific subsets A_i that serve only class-i jobs in FCFS order, and a helper set H that uses an auxiliary policy π. Throughput optimality is defined as stability (finite mean response time) whenever the load ρ is less than one.

In the Stability subsection, the authors provide a sufficient condition (Proposition 1) for stability but note that static partitioning can cause loss of capacity due to the inability to reassign jobs across A_i subsets, raising uncertainty about general throughput optimality. They later show asymptotic (many-server) regimes where the helper routing probability vanishes, but the general question of throughput optimality remains unresolved.