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We consider a nonlinear SISO system that is a cascade of a scalar "bottleneckentrance" and an arbitrary Hurwitz positive linear system. This system entrainsi.e. in response to a $T$-periodic inflow every solution converges to a unique$T$-periodic solution of the system. We study the problem of maximizing theaveraged throughput via controlled switching. The objective is to choose aperiodic inflow rate with a given mean value that maximizes the averagedoutflow rate of the system. We compare two strategies: 1) switching between ahigh and low value, and 2) using a constant inflow equal to the prescribed meanvalue. We show that no switching policy can outperform a constant inflow rate,though it can approach it asymptotically. We describe several potentialapplications of this problem in traffic systems, ribosome flow models, andscheduling at security checks.

No Switching Policy is Optimal for a Positive Linear System with a Bottleneck Entrance