Peak Blocking Turn Number (PTN)

• PTN is defined as the entrance intensity at which throughput peaks before blockage events degrade performance in finite-capacity systems.
• The phenomenon is modeled using circular Markov and non-Markovian frameworks where arrival, service, and clearance rates critically determine channel behavior.
• PTN insights guide optimal operational protocols in filtration, transport, and queuing systems by balancing inflow and blockage trade-offs.

The Peak Blocking Turn Number (PTN) refers to the phenomenon, observed in finite-capacity transport and queuing systems subject to temporary blocking, where system throughput exhibits a non-monotonic dependence on the particle entrance intensity. Specifically, PTN denotes the value of entrance intensity $\lambda$ at which the exit rate (throughput) of particles through a channel achieves its maximum before degradation due to excessive blocking events. This behavior is significant in processes such as filtration, single-file transport in narrow channels, and queueing networks with capacity constraints, where channel obstruction leads to throughput saturation or decline under high input intensity.

1. Stochastic Blocking Channel Model

The mathematical foundation for the PTN phenomenon is a circular Markov model representing a finite-capacity channel. Particles enter the channel at rate $\lambda$ and exit independently at rate %%%%2%%%%. The channel has capacity $N$: when $N$ particles occupy the channel, no additional particles can enter, and the system enters a blocked state. The blocking period is exponentially distributed with rate $\mu^*$, after which all $N$ particles are flushed from the channel simultaneously, resetting it to the empty state.

States are indexed by the particle count $k$ $(0 \leq k \leq N)$, with transition structure:

• For $k < N$, arrivals at rate $\lambda$ increase $k$, and each particle exits at rate $\mu$ (aggregated exit rate $k\mu$ decreases $k$).
• At $k = N$, no arrivals are possible; exit out of the blocked state occurs at rate $\mu^*$ (all $N$ particles exit together).

The time evolution is governed by the forward Kolmogorov equation $dP_N/dt = P_N Q_N$, with generator matrix $Q_N$ reflecting the above transitions. The steady-state probability of the blocked state ($\pi_N$) is

$\pi_N = C_N \lambda^N,$

where

$$C_N = \left[\lambda^N + \mu^* \sum_{j=0}^{N-1} \frac{N!}{(j+1)(N-j-1)!} \mu^j \lambda^{N-j-1}\right]^{-1}.$$

2. Throughput as a Function of Entrance Intensity

The channel throughput, $j_N(\lambda, t)$, is the expected exit rate at time $t$: $$j_N(\lambda, t) = \sum_{k=1}^{N-1} k \mu\, \pi_k(t) + N \mu^*\, \pi_N(t).$$ In steady state, conservation of probability flux gives the equivalent formula

$j_N(\lambda) = \lambda (1 - \pi_N).$

For $N = 2$, explicit calculation yields: $$j(\lambda) = \frac{\lambda (2\lambda + \mu)}{\lambda^2/\mu^* + 2\lambda + \mu}.$$

As $\lambda \to 0$, throughput increases linearly ($j(\lambda) \approx \lambda$), since blockages are rare. For large $\lambda$, blocking dominates and throughput saturates at $j(\lambda \rightarrow \infty) = N\mu^*$, determined by the clearing of channel blockages.

3. Existence and Location of the PTN

Maximization of steady-state throughput as a function of $\lambda$ reveals the existence criteria and value of PTN. For $N = 2$, the condition $dj/d\lambda = 0$ yields: $$\left(\frac{\lambda}{\mu}\right) = \frac{\sqrt{r} + 2r}{1-4r},\quad \text{with}\ r = \mu^*/\mu.$$ A real, positive root exists if and only if $r < 1/4$. Thus, a non-trivial, finite-$\lambda$ maximum (PTN) exists only when the blockage clearing rate is sufficiently small relative to the exit rate:

• If $\mu^*/\mu < 1/4$, increasing intensity $\lambda$ beyond a certain point incurs too frequent blockages, degrading overall throughput; the system exhibits a peak (PTN) at intermediate $\lambda$.
• If $\mu^*/\mu \geq 1/4$, throughput increases monotonically with $\lambda$ to its large-$\lambda$ asymptote, and no PTN exists.

This identifies a universal throughput–blocking tradeoff in such systems.

4. Time-Dependent Throughput and Transient Peaks

Analysis of the time-dependent throughput $j(\lambda, t)$ uncovers two regimes:

• For certain parameters, $j(\lambda, t)$ monotonically approaches its steady-state value.
• For other parameters (notably low $\mu^*/\mu$), $j(\lambda, t)$ overshoots, attaining a transient maximum at finite time $t_{\mathrm{max}}$ before declining to steady state.

In the $N = 2$ case, the threshold for such dynamic maxima is given by

$$\mu^*_b(\lambda) = \frac{\mu}{2}\left[1 - \sqrt{\frac{\mu}{2\lambda + \mu}}\right].$$

If $\mu^*/\mu < 1/2$, the process may exhibit this time-local PTN signature. This demonstrates that PTN can be a transient as well as a steady-state property, with its existence and magnitude controlled by the relative rates $\lambda$, $\mu$, and $\mu^*$.

5. Correspondence with Non-Markovian and Irreversible Models

The PTN phenomenon persists in related non-Markovian frameworks where transit and/or blockage times are deterministic (i.e., fixed rather than exponentially distributed). In a model with fixed transit time $\tau$ and blockage duration $\tau_b$, for $N = 2$, the steady-state throughput is

$$j_\infty = \frac{\lambda (2 - e^{-\lambda \tau})}{\lambda \tau_b (1 - e^{-\lambda\tau}) + 2 - e^{-\lambda \tau}}.$$

Although direct substitution $\tau = 1/\mu$, $\tau_b = 1/\mu^*$ does not map Markovian to deterministic models exactly, effective renormalization of parameters aligns their qualitative behaviors: both exhibit a PTN at intermediate $\lambda$ provided clearance is slow relative to service.

In an irreversible blocking scenario—where the channel, once blocked, does not reopen, and transit time is fixed to $\tau$—the total number of exiting particles over a finite operating window $(0, t_s)$ is

$m(t_s) = \int_{t=\tau}^{t_s} j(t) dt.$

For $1 < t_s < 2$ (units $\tau=1$), the solution is

$m(t_s) = e^{-\lambda} - e^{-\lambda t_s},$

with its maximum at

$\lambda = \frac{\ln t_s}{t_s - 1}.$

For long $t_s$, Laplace analysis confirms both the mean and variance of exiting particle counts vanish as $\lambda \to 0$ or $\lambda \to \infty$, attaining a maximum for finite $\lambda$. Notably, the maximizing $\lambda$ for variance can exceed that for the mean, implying optimal operation points vary by chosen metric.

6. Operational Implications and Generalizations

The existence of PTN has significant implications for the design and control of queueing, transport, and filtration systems subject to crowding-induced blocking. It provides a rigorous criterion for optimal loading: excessive entrance intensity can sharply degrade throughput, while too low intensity underutilizes capacity. This is particularly relevant for processes in which flushing blockages is slow or costly relative to normal transit dynamics. The correspondence across Markovian, semi-deterministic, and irreversible models suggests PTN-type maxima are robust features of a broad class of intermittent transport systems.

A practical implication is that throughput or related performance metrics (such as mean or variance of exiting particles) can be optimized by tuning the entering flux to the empirically or theoretically derived PTN. This guides operational protocols in processes where channel clogging or jamming is recurrent, such as membrane filtration, microfluidic particle sorting, and certain communication or traffic networks.

7. Summary Table of Peak Throughput Existence Conditions

Model Variant	Condition for PTN (Max at Finite $\lambda$)	PTN Characteristic
Markovian, reversible, $N=2$	$\mu^*/\mu < 1/4$	Throughput maximized at finite $\lambda$
Markovian, irreversible	Always (finite $t_s$)	Exit count and variance maximized at finite $\lambda$
Non-Markovian, deterministic	Slow clearance or short time horizon	Similar qualitative behavior

These results establish the generality of the PTN phenomenon across a range of stochastic and deterministic blocking models, characterizing a fundamental tradeoff in the management of blocked, finite-capacity channels.