Abstract

Many service organizations pair a low-cost outsourcing pool with a more capable in-house team, where customers whose issues are not resolved by the outsourcing pool reenter the system for in-house service. A manager must decide how aggressively to use the outsourcing pool and how the in-house team should prioritize incoming versus reentrant customers to minimize long-run average delay and abandonment costs.
Rather than separately investigating reentrant flows (e.g., the Erlang-R model) and parallel routing (e.g., the N model), this paper develops an integrated framework for analyzing a hybrid network featuring both reentrant and parallel structures. We study this problem using a fluid approximation of the many-server queueing network with class- and pool-dependent service rates.
Our analysis reveals that the optimal solution is characterized by five qualitatively distinct operating regimes. A key finding is that achieving system-wide efficiency necessitates intentional idling of the outsourcing pool to regulate reentrant flows and mitigate cumulative delays in the downstream pool. For strictly convex costs, we prove that a simple admission rule for the outsourcing pool (which achieves the same effect as a threshold policy but requires no threshold computation), combined with a dynamic priority generalized $c\mu/\theta$ rule for the in-house pool, is asymptotically optimal for all five operating regimes.
We extend these results to linear costs, where the policy reduces to a threshold plus fixed-priority rule, and to general (possibly non-convex) costs via a target-allocation policy. Our results provide managers with simple, implementable rules for operating hybrid outsourcing systems and reveal that the counterintuitive practice of holding cheap capacity idle can substantially reduce total system costs.