Stochastic models of multi-channel particulate transport with blockage

Abstract

Particle conveying channels may be bundled together. The limited carrying capacity of the constituent channels may cause the bundle to be subject to blockages. If coupled, the blockage of one channel causes an increase in the flux entering the others, leading to a cascade of failures. Once all the channels are blocked, no additional particles may enter the system. If the blockages are of finite duration, the system reaches a steady state with an exiting flux that is reduced compared to the incoming one. We propose a stochastic model consisting of N channels, each with a blocking threshold of N particles. Particles enter the system's open channels according to a Poisson process, with an equally distributed input flux of intensity Λ. In an open channel the leading particle exits at a rate μ and a blocked channel unblocks at a rate [Formula: see text], where [Formula: see text]. We present and explain the methodology of an analytical description of the behavior of bundled channels. This leads to exact expressions for the steady-state output flux, for [Formula: see text], which promises to extend to arbitrary N and N. The results are applied to compare the efficiency of conveying a particulate stream of intensity Λ using a single, high capacity (HC) channel with multiple channels of a proportionately reduced low capacity (LC). The HC channel is more efficient at low input intensities, while the multiple LC channels have a higher throughput at high intensities. We also compare [Formula: see text] coupled channels, each of capacity N = 2 with the corresponding number of independent channels of the same capacity. For [Formula: see text], if [Formula: see text], the coupled channels are always more efficient. Otherwise the independent channels are more efficient for sufficiently large Λ.

Recurrence dynamics of particulate transport with reversible blockage: From a single channel to a bundle of coupled channels

“…The physical assumption of constant transit and deblocking times τ and τ b , respectively, is responsible for strong memory effects which prevent analytical solutions for general N from being obtained. We therefore recently introduced Markovian models [29,30], where the average transit and deblocking times are stochastic variables given by exponential distributions of intensity µ and µ * , respectively. The kinetic description of the Markovian model is given by a set of differential equations for the time evolution of the state probabilities P (i, t) with i ∈ [0 · · · N ] giving the number of particles in the channel.…”

“…The same procedure can be carried out for N = 3 using Eqs. (30) and (31), but the resulting expression for µ is considerably more complex. For general N we therefore propose the following ansatz, taking a similar form as the mapping for N = 2:…”

“…If all the elementary steps are Markov processes, the time evolution of the state probabilities can be described by systems of linear differential equations. In accordance with this approach, we recently introduced Markovian models of blockage [29,30] for which exact solutions can be obtained.…”

First-passage times of multiple diffusing particles with reversible target-binding kinetics

“…As binding and unbinding events of all particles are independent from each other and thus asynchronized, finding the probability density H K,N (t) of the T K,N is a challenging open problem. Note that the above problem of impatient particles resembles some stochastic models of multi-channel particulate transport with blockage [80][81][82].…”

Diffusion-controlled reactions triggered by multiple particles with reversible target-binding kinetics

“…As binding and unbinding events of all particles are independent from each other and thus asynchronized, finding the probability density H K,N (t) of the T K,N is a challenging open problem. Note that the above problem of impatient particles resembles some stochastic models of multi-channel particulate transport with blockage [78][79][80] .…”