Diminishing blocking interval in particle-conveying channel bundles

We consider a channel bundle consisting of N c parallel channels conveying a particulate flux. Particles enter these channels according to a homogeneous Poisson process and exit after a fixed transit time, τ. An individual channel blocks if N particles are simultaneously present. When a channel is blocked the flux previously entering it is redistributed evenly over the remaining open channels. We perform event driven simulations to examine the behaviour of an initially empty channel bundle with a total entering flux of intensity Λ. The mean blockage time of the kth channel is denoted by ⟨ t k ⟩ ,k = 1, ... , N c . For N = 1, as shown previously, the interval between successive blockages is constant, while for N > 1 an accelerating cascade, i.e. one in which the interval between successive blockages decreases, is observed. After an initial transient regime we observe a well-defined universal regime that is characterized by \hbox{$\Delta_k^{(N)} = (-1)^{N-1}\frac{[(N-1)!]^2}{(\Lambda\tau)^N}$} Δ k ( N ) = ( − 1 ) N − 1 [ ( N − 1 ) ! ] 2 ( Λ τ ) N where \hbox{$\Delta_k^{(1)}=\langle t_k \rangle-\langle t_{k-1}\rangle$}Δk(1)=⟨tk⟩−⟨tk−1⟩ and \hbox{$\Delta_k^{(j)}=\Delta_k^{(j-1)}-\Delta_{k-1}^{(j-1)}$}Δk(j)=Δk(j−1)−Δk−1(j−1) denotes the jth order difference.