No Switching Policy Is Optimal for a Positive Linear System With a Bottleneck Entrance

We consider a nonlinear SISO system that is a cascade of a scalar “bottleneck entrance” and an arbitrary Hurwitz positive linear system. This system entrains, i.e., in response to a <inline-formula> <tex-math notation="LaTeX">$T$ </tex-math></inline-formula>-periodic inflow every solution converges to a unique <inline-formula> <tex-math notation="LaTeX">$T$ </tex-math></inline-formula>-periodic solution of the system. We study the problem of maximizing the averaged throughput via controlled switching. The objective is to choose a periodic inflow rate with a given mean value that maximizes the averaged outflow rate of the system. We compare two strategies: 1) switching between a high and low value and 2) using a constant inflow equal to the prescribed mean value. We show that no switching policy can outperform a constant inflow rate, though it can approach it asymptotically. We describe several potential applications of this problem in traffic systems, ribosome flow models, and scheduling at security checks.

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