Resetting-induced instability in queues fed by a search process in an interval

This paper investigates a queuing system with limited servers fed by a search process in a bounded domain subject to stochastic resetting, identifying a critical threshold resetting rate that determines whether resetting expands or shrinks the parameter regions for steady-state convergence and demonstrating that this threshold grows exponentially with the number of servers.

The Gist

Imagine a busy warehouse (the target) where workers are constantly trying to deliver packages (the resources). These packages are brought by a delivery driver (the searcher) who drives randomly around a city block (the interval) looking for the warehouse door.

Once the driver finds the door, they drop off a package, drive back to their starting point to reload, and go out searching again. Meanwhile, inside the warehouse, a team of workers (the servers) is busy unpacking and processing these packages.

The big question this paper asks is: Will the warehouse eventually fill up with an endless pile of packages, or will the workers keep up with the deliveries and reach a steady, manageable level?

The answer depends on two main things:

1. How fast the driver finds the door.
2. How many workers are inside.

The "Reset" Twist

In this story, the driver has a special trick: Stochastic Resetting. This means that every now and then, randomly, the driver gets a sudden urge to give up on their current path and instantly teleport back to their starting point to try again.

Usually, in physics, we think "resetting" is a good thing. If you are looking for something in a huge, empty field, stopping and restarting from the beginning can actually help you find it faster. It's like realizing you're walking in circles and deciding to just go back to the start.

However, this paper discovers a surprising twist: In a busy warehouse system, resetting can sometimes make things worse.

The Two Scenarios

1. The "Too Long" City Block (Long Interval)

Imagine the city block is very long.

• Without Resetting: If the driver starts far away, it takes them a long time to find the warehouse. They deliver packages slowly. The workers inside have plenty of time to process them, so the pile of packages stays manageable.
• With Resetting: If we add the "teleport back" rule, the driver might find the warehouse faster on average. They deliver packages more frequently.
• The Problem: If the driver delivers packages too fast, the workers inside can't keep up. The pile of packages starts to grow uncontrollably, eventually overflowing the warehouse.
• The Finding: For long city blocks, adding resetting can actually shrink the "safe zone." It turns a situation where the warehouse was stable into one where it overflows.

2. The "Short" City Block (Short Interval)

Now, imagine the city block is very short.

• Without Resetting: The driver is already close to the warehouse. They find it quickly. If they start too close, they might deliver packages so fast that the workers can't keep up, causing an overflow.
• With Resetting: If the driver starts very close, resetting forces them to go back to the start, which actually slows down their delivery rate.
• The Benefit: This "slow down" can be a good thing! It gives the workers inside a chance to catch up. In this specific case, resetting expands the "safe zone," allowing the system to stay stable even if the driver starts in a spot that would have caused a disaster before.

The "Tipping Point"

The authors found a specific "tipping point" (a threshold) that determines which of these two effects happens:

• If the city block is shorter than this point, resetting helps stabilize the warehouse.
• If the city block is longer than this point, resetting destabilizes it and causes overflow.

The "More Workers" Rule

The paper also looked at what happens if you hire more workers (increase the number of servers).

• You might think hiring more workers makes the system more robust.
• However, the paper found that as you add more workers, the "resetting rate" needed to actually help the system grows exponentially.
• The Analogy: Imagine you have a small team of 5 workers. A little bit of "resetting" (slowing the driver down) might help them. But if you have a massive team of 1,000 workers, you would need an enormous amount of resetting to make a difference. In fact, for large teams, it becomes incredibly difficult for resetting to help; it's much more likely to just mess things up and cause an overflow.

Summary

This paper is a warning to system managers: Just because a strategy (like resetting) makes a search process faster, doesn't mean it makes the whole system more stable.

• If you have a small team and a short search area, resetting might help you stay organized.
• If you have a large team or a long search area, forcing the searcher to reset often can actually cause the system to collapse under the weight of too many arrivals.

The authors provide mathematical formulas to tell you exactly where that line is drawn, so you know when to use resetting and when to avoid it.

Technical Summary

Technical Summary: Resetting-induced Instability in Queues Fed by a Search Process in an Interval

Problem Statement
The paper addresses the long-term stability of resource accumulation in systems where resources are delivered by a search process subject to stochastic resetting. Specifically, it investigates a "target-centric" model where a diffusing particle in a one-dimensional (1D) bounded interval $[0, L]$ searches for an absorbing target at the origin. Upon finding the target, the particle delivers a resource, returns to its starting position $x_0$ to reload (a refractory period), and restarts the search. This cycle generates a sequence of "burst" events (arrivals) that feed a queueing system with a finite number of servers ($c$) and exponential service times (consumption).

The core problem is determining how the introduction of stochastic resetting—where the particle is instantaneously returned to $x_0$ at a rate $r$—affects the convergence of the number of resources in the queue to a steady state. While stochastic resetting is known to optimize first-passage times (FPT) in unbounded domains or specific subregions of bounded domains, the paper explores whether this acceleration of the search process inadvertently destabilizes the queueing system by increasing the arrival rate beyond the service capacity, leading to resource blow-up.

Methodology
The authors combine classical queuing theory with the statistics of first-passage processes under stochastic resetting.

1. Mapping to Queuing Theory: The search-and-capture process is mapped to a $G/M/c$ queueing system. The inter-arrival times of the queue are determined by the sum of the FPT (with resetting) and the refractory period. The service times are Markovian (exponentially distributed) with rate $\mu$.
2. Convergence Condition: The system converges to a steady state if and only if the traffic intensity $\rho = \lambda / (c\mu) < 1$, where $\lambda$ is the mean arrival rate.
3. Analytical Derivation:
   • The authors utilize the Laplace transform of the FPT density for a diffusing particle in a bounded interval with stochastic resetting to derive the mean FPT, $T_r(x_0)$.
   • They derive the critical starting position $x^*_r$ as a function of the resetting rate $r$, interval length $L$, number of servers $c$, and other parameters. Convergence occurs if the initial position $x_0 > x^*_r$.
   • They analyze the behavior of $x^*_r$ and the minimal interval length $L^*_r$ required for convergence under varying $r$.
4. Parameter Space Analysis: The study examines two primary scenarios:
   • Fixed resetting rate $r$ with varying interval length $L$.
   • Fixed interval length $L$ with varying resetting rate $r$.

Key Contributions and Results

1. Critical Starting Position with Resetting:
   The authors derive an explicit expression for the critical starting position $x^*_r$ (Eq. 20) that ensures convergence. If the search starts at $x_0 > x^*_r$, the queue converges; otherwise, the number of resources blows up. Unlike the non-resetting case where $x^*_0 \to 0$ as $L \to \infty$, the presence of resetting results in a non-zero limit $x^{hl}_r$ for the half-line, implying that convergence is not guaranteed for all spatial configurations even in large domains unless the number of servers increases.

2. Threshold Interval Length ($L_{eq}$):
   For a fixed resetting rate, the authors identify a threshold interval length $L_{eq}$ (defined implicitly in Eq. 24).

   • For $L < L_{eq}$, stochastic resetting expands the convergence zone (widening the range of $x_0$ that leads to stability).
   • For $L > L_{eq}$, stochastic resetting shrinks the convergence zone, making the system more prone to instability.
     This indicates a transition where the benefit of resetting (expediting search) becomes detrimental to queue stability in longer intervals.

3. Threshold Resetting Rate ($r_{eq}$ and $r_{max}$):
   For a fixed interval length, the authors identify a critical resetting rate $r_{eq}$ where the effects of resetting switch from shrinking to expanding the convergence zone.

   • $r_{max}$: There exists a specific rate $r_{max}$ that maximizes the critical starting position $x^*_r$, thereby minimizing the convergence zone (maximizing instability).
   • $r_{eq}$: This is the rate at which the convergence zone with resetting equals the zone without resetting.
   • Scaling with Servers: The paper finds that $r_{max}$ scales linearly with the number of servers ($c$), while $r_{eq}$ scales as a power law ($c^\beta$ with $\beta > 2$). Consequently, as the number of servers increases, the resetting rate required to improve (expand) the convergence region grows exponentially, whereas the rate that worsens convergence grows only linearly.

4. Instability Mechanism:
   The study demonstrates that in scenarios where the search process is intrinsically positive recurrent (finite MFPT without resetting), introducing stochastic resetting can reduce the MFPT. In the context of the queue, this reduction in inter-arrival time increases the traffic intensity $\rho$. If $\rho$ exceeds 1, the system shifts from a convergent steady state to unbounded growth (blow-up).

Significance and Claims
The paper claims to provide a theoretical framework linking search dynamics with stochastic resetting to the stability of resource accumulation in queuing systems. Its primary significance lies in identifying that stochastic resetting, while generally beneficial for search efficiency, can induce resetting-induced instability in resource-constrained systems.

The authors conclude that for systems with a large number of servers, it becomes increasingly difficult for stochastic resetting to expand the regions of convergence. Instead, the inclusion of resetting is more likely to be detrimental, shrinking the parameter space where the system remains stable. This challenges the assumption that optimizing search times (via resetting) is universally beneficial for target-centric accumulation models, highlighting a trade-off between search speed and system stability.

The work does not propose new experimental protocols or applications but rather offers analytical results and implicit equations for determining stability boundaries in $G/M/c$ systems driven by resetting search processes. Future directions suggested include extending the model to $G/G/c$ systems, multiple targets, and interacting searchers.