Waiting Time Dependent Thresholds

• Waiting Time Dependent Thresholds are functional boundaries defined as explicit functions of elapsed time used to trigger system transitions in stochastic and queueing models.
• They are applied in various contexts, including queueing with eligibility delays, age-of-information scheduling, first-passage problems, and quantum communication to manage service and error thresholds.
• These thresholds provide actionable guidelines to optimize resource use, maintain system stability, and improve performance across networks, servers, and communication channels.

A waiting time dependent threshold is a functional or parametric boundary—fixed or state-adaptive—defined in terms of an explicit function of the elapsed waiting time since arrival or last system event, and used to trigger, gate, or modulate transitions in stochastic systems, queueing processes, scheduling policies, first-passage problems, and information-theoretic network models. This concept appears in diverse forms, including deterministic eligibility thresholds for queue overflow, time-dependent erasure thresholds in quantum communication, adaptive cutoff mechanisms for preemption or update in information age, and varying decision boundaries in stochastic diffusion and jump-diffusion models.

1. Definitions and Model Instantiations

The notion of a waiting time dependent threshold is instantiated in several canonical stochastic models:

• Queueing with Eligibility Delay: In the $N$-model queue with two servers, a deterministic threshold $T>0$ is imposed such that a type-1 job in queue 1 becomes eligible for service at server 2 only after waiting time $\ge T$, while prior to that it can be only served by server 1 (if idle). This renders the system’s evolution non-Markovian due to a discontinuity in the drift at the threshold (Kempen et al., 14 Nov 2025).
• Age-of-Information Optimal Control: In status update systems, the policy for scheduling new updates is described by a threshold function $w^*(\Delta) = \max\{0,\,\Delta^*-\Delta\}$—an “age-threshold”—depending on the instantaneous waiting time (age of the information), with possible additional cutoff for preemption if the service time exceeds a fixed threshold $\theta$ (Arafa et al., 2019).
• First-Passage Problems with Moving Boundaries: In one-dimensional diffusion or jump-diffusion models, the process $X_t$ is stopped the first time it exceeds a time-varying threshold $S(t)$, i.e., the first-passage time $\tau_S = \inf\{t \geq 0: X_t \geq S(t)\}$. Here, the threshold $S(t)$ is a deterministic function of time, possibly reflecting synaptic or biophysical adaptation (Desmettre et al., 31 Oct 2025, Khurana et al., 17 Dec 2024).
• Error Thresholds in Queue-Channels: In quantum Jackson networks, the probability that a quantum state decoheres (erased) is modeled as $p_e(w)$, an increasing function of the waiting/sojourn time $w$. A “waiting time dependent threshold” $W_{\rm th}$ may be defined so that $p_e(w) = 0$ for $w \le W_{\rm th}$ and $p_e(w) = 1$ for $w > W_{\rm th}$, or more generally, the system's communication capacity collapses beyond this delay, making $W_{\rm th}$ a hard system boundary (Mandalapu et al., 2022).

2. Stability and Performance in Queueing Models

The introduction of a deterministic waiting time dependent threshold in queueing models fundamentally modifies job eligibility for service, but does not alter the classical stability region. In the $N$-model queue, where type-1 jobs become eligible for service at server 2 only after waiting $T$ units, the joint process of queue lengths is positive recurrent (stable) if and only if: $$\lambda_1 + \lambda_2 < \mu_1 + \mu_2,\quad \lambda_2 < \mu_2$$ independent of $T > 0$. The threshold delays the overflow of jobs to the flexible server, but under heavy load, jobs routinely exceed $T$, causing the system to transition to the effective $N$-model regime without the threshold. This invariance of the stability region under deterministic waiting time thresholds is established via coupling arguments to $M/M/1$ lower bounds and to an upper-bound system where all type-1 jobs are delayed exactly $T$, showing stochastic dominance of the queue-length process (Kempen et al., 14 Nov 2025).

3. Threshold Policies in Age-Optimal Scheduling

In age-of-information (AoI) minimization for cloud update systems, waiting time dependent thresholds emerge as the optimal structure for deterministic scheduling policies. The scheduler, after completing an upload at time with age $\Delta$, waits for $w(\Delta)$ before initiating a new upload. The optimal policy has the form

$$w^*(t) = [\lambda^* - \mathbb{E}[T] - t]^+ = \begin{cases} \lambda^* - \mathbb{E}[T] - t, & t < \Delta^* \ 0, & t \geq \Delta^* \end{cases}$$

where $\Delta^* = \lambda^* - \mathbb{E}[T]$ is an explicit threshold derived from the auxiliary root-finding problem $g(\lambda^*) = 0$, functionally dependent on the service-time distribution and preemption cutoff $\theta$. If the instantaneous age is below the threshold, the scheduler waits; if above, it acts immediately. The threshold increases with increasing tail weight in the service-time distribution, allowing for more conservative scheduling when high waiting times are likely. The joint use of waiting and preemption via a cutoff yields strictly lower average AoI than either tool alone (Arafa et al., 2019).

4. Simulation and Analysis of First-Passage to Time-Dependent Thresholds

Time-dependent thresholds feature centrally in the analysis and simulation of first-passage times (FPT) for stochastic processes, particularly in neural models and finance. Exact simulation schemes for FPT to moving boundaries combine the Lamperti transform (reducing to unit diffusion) and Girsanov’s theorem (for likelihood reweighting under measure change). The acceptance-rejection method involves simulating a candidate FPT for standard Brownian motion to the time-varying boundary, then accepting the sample with a probability determined by an explicit functional of the drift and boundary derivatives: $$\eta(t) = \exp\left(-\int_0^t [\gamma_1(s) + \gamma_2(W_s)]\,ds\right)$$ where $\gamma_1(s)$ and $\gamma_2(x)$ encode the SDE coefficients and boundary derivatives. For jump-diffusions, the hybrid exact (HEx) method employs this machinery on inter-jump intervals and additional rejection steps at jump instants, consistently handling arbitrary deterministic boundaries $S(t)$. The time-dependent threshold is thus central in both generation and acceptance criteria for pathwise simulation (Khurana et al., 17 Dec 2024, Desmettre et al., 31 Oct 2025).

5. Information-Theoretic Queue-Channel Thresholds

Waiting time dependent erasure thresholds play a critical quantitative role in quantum and classical communication over queue-channels with decoherence effects. The erasure probability $p_e(w)$, which increases with sojourn time $w$, implies a threshold $W_{\rm th}$ such that all qubits delayed beyond this point are unusable. The channel’s classical capacity is then governed by: $C = \lambda\, \mathbb{E}[1 - p_e(W)]$ For step-type $p_e$, $C = \lambda\, \mathbb{P}(W < W_{\rm th})$. More generally, for tandem queues or Jackson networks, the system capacity collapses when mean sojourn time at any node exceeds the associated delay-tolerance threshold, or equivalently, when the traffic load at any link approaches the service rate. The practical impact is the need to optimize pumping rates and routing so that the average or effective waiting time at each node remains below its threshold, thus ensuring positive information transfer capacity (Mandalapu et al., 2022).

Model Context	Nature of Waiting Time Threshold	Primary System Effect
$N$-model queue (Kempen et al., 14 Nov 2025)	Eligibility threshold $T$	Delays overflow, does not shift stability
AoI scheduling (Arafa et al., 2019)	Policy threshold $\Delta^*$	Gates update decision, minimizes AoI
FPT simulation (Khurana et al., 17 Dec 2024)	Boundary $b(t)$ or $S(t)$	Triggers stopping, exact simulation
Quantum queue-channel (Mandalapu et al., 2022)	Erasure threshold $W_{\rm th}$	Determines capacity breakdown

6. Mathematical Methods and Computational Considerations

Waiting time dependent thresholds often induce discontinuities or path dependencies in stochastic systems:

• The queue-length process in the $N$-model with eligibility threshold $T$ becomes non-Markovian, necessitating bounding systems and coupling (stochastic dominance) to establish properties such as stability (Kempen et al., 14 Nov 2025).
• For optimal update scheduling, the threshold structure is derived analytically via calculus of variations and solved by fractional programming (Dinkelbach’s method) (Arafa et al., 2019).
• In exact simulation of SDE FPTs, explicit characterization of threshold-induced acceptance probabilities, and tractable upper bounds (thinning), are crucial for algorithmic efficiency and unbiasedness. Tuning of splitting/segment size is needed for curved or non-monotone thresholds (Khurana et al., 17 Dec 2024, Desmettre et al., 31 Oct 2025).
• In network information theory, thresholds directly condition the expectation in capacity formulas, and require solution of optimization problems for maximal rate under delay constraints (Mandalapu et al., 2022).

The computational cost of processing waiting time dependent thresholds is often manageable with carefully selected algorithm parameters (e.g., choice of Poisson thinning intensity $\kappa$ and segmentation $\epsilon$), but may increase sharply for heavy-tailed or rapidly changing threshold functions.

7. Applications and Operational Guidelines

Waiting time dependent thresholds arise naturally in:

• Flexible server systems: Used as policy controls for overflow or redirection, with the threshold acting as a tunable parameter to control resource sharing without destabilizing the network (Kempen et al., 14 Nov 2025).
• Status update and sensing: Scheduler thresholds tune responsiveness versus resource utilization, and joint threshold-cutoff policies strictly reduce information latency (Arafa et al., 2019).
• Biophysical modeling: Thresholds model firing in neurons with adaptive or time-varying excitability, requiring high-precision simulation methods for spike prediction (Desmettre et al., 31 Oct 2025, Khurana et al., 17 Dec 2024).
• Quantum networks: Design and operation must ensure that mean sojourn times at all network nodes are kept below hardware-determined thresholds to avoid total loss of informational fidelity (Mandalapu et al., 2022).

A key operational principle is to ensure that all system parameters—arrival rates, service rates, and routing probabilities—are chosen so as to maintain, at each relevant node or decision point, an average waiting or sojourn time below the critical threshold, thereby guaranteeing nonzero system capacity or positive recurrence.

In summary, waiting time dependent thresholds serve as critical control points and analytical constructs in modern stochastic modeling and information systems. They enable both sophisticated control strategies in scheduling and provide foundational limits in the stability and throughput of complex queueing and network systems.