# Optimizing the Throughput of Particulate Streams Subject to Blocking

## TL;DR

This paper models and analyzes the maximum throughput of particulate streams subject to temporary blocking using a circular Markov model, revealing conditions under which throughput is optimized at finite particle entry rates.

## Contribution

It introduces an exact Markovian model for throughput in blocked channels and identifies conditions for maximum throughput at finite input intensities.

## Key findings

- Maximum throughput occurs at finite input rates when blockage clearance rate is sufficiently low.
- Steady state throughput can be mapped to a non-Markovian model with fixed transit and blockage times.
- First and second moments of exiting particles are maximized at finite input flux for certain blockage processes.

## Abstract

Filtration, flow in narrow channels and traffic flow are examples of processes subject to blocking when the channel conveying the particles becomes too crowded. If the blockage is temporary, which means that after a finite time the channel is flushed and reopened, one expects to observe a maximum throughput for a finite intensity of entering particles. We investigate this phenomenon by introducing a queueing theory inspired, circular Markov model. Particles enter a channel with intensity $\lambda$ and exit at a rate $\mu$. If $N$ particles are present at the same time in the channel, the system becomes blocked and no more particles can enter until the blockage is cleared after an exponentially distributed time with rate $\mu^*$. We obtain an exact expression for the steady state throughput (including the exiting blocked particles) for all values of $N$. For $N=2$ we show that the throughput assumes a maximum value for finite $\lambda$ if $\mu^*/\mu < 1/4$. The time-dependent throughput either monotonically approaches the steady state value, or reaches a maximum value at finite time. We demonstrate that, in the steady state, this model can be mapped to a previously introduced non-Markovian model with fixed transit and blockage times. We also examine an irreversible, non-Markovian blockage process with constant transit time exposed to an entering flux of fixed intensity for a finite time and we show that the first and second moments of the number of exiting particles are maximized for a finite intensity.