Fluid Token Generation Rate

• Fluid token generation rate is the continuous process of issuing tokens in systems through periodic, probabilistic, and queue-aware mechanisms.
• It is applied in networking, distributed ledgers, and LLM serving to manage buffering, transmission eligibility, and system responsiveness.
• Analytical models using fluid limits, Markov chains, and scheduling constructs ensure optimized performance, stability, and resource allocation.

Fluid token generation rate refers to the rate at which tokens—representing discrete permissions, data units, or computational artifacts—are produced, released, or streamed in a quasi-continuous fashion across diverse systems such as networking mechanisms, distributed ledgers, and real-time LLM serving architectures. In rigorous models, this rate is typically governed by periodic, probabilistic, or queue-aware mechanisms that translate microscopic stochastic behaviors into macroscopic deterministic evolution, often via fluid limit approximations, Markov chains, or scheduling-theoretic constructs. The properties and management of fluid token generation rates are central to performance, responsiveness, throughput, and stability in these domains.

1. Periodic and Fluid Token Generation in Queuing and Networking

A canonical example of fluid token generation arises in the token bucket mechanism, in which tokens are generated or replenished periodically every $\tau$ time units. For fixed-length packets, each period yields exactly one token, defining a token generation rate of %%%%1%%%%. The token bucket dynamics distinguish between cases where the buffer is non-empty—where arriving tokens are instantly consumed to dequeue waiting packets—and periods when the buffer is empty, allowing tokens to accumulate up to a maximum bucket capacity. This duality ensures both backlog draining and readiness to serve future bursts.

For variable packet sizes, the model extends by requiring that each packet consumes tokens proportional to its size. Replenishment events are then coupled with checks to determine if enough tokens have accumulated to serve the head-of-line packet, linking the fluid token generation rate directly with the transmission eligibility of diverse packet classes. A higher token generation rate reduces queue backlog and packet delay, while inadequately frequent token arrivals increase the risk of buffer overflow and packet loss. These behaviors are captured in discrete-time Markov chain (DTMC) formulations featuring explicit state updates that embed the periodic (fluid) generation mechanism as a “–1” decrement per period (Schioler et al., 2020).

2. Markovian and Fluid Limit Models for Stochastic Systems

Markovian models provide a mathematically rigorous framework for analyzing the interplay between token generation, arrivals, and buffer evolution. With system state defined by backlog %%%%2%%%% and token count $T(t)$, the evolution incorporates both periodic replenishments and stochastic arrivals (often modeled as Poisson or compound Poisson processes). The Markovian invariant $Q(t) \cdot T(t) = 0$—i.e., tokens and nonzero backlog do not coexist—enables a reduction to a single effective state variable.

Crucially, the transition equations in variables such as $S_n$ (with coordinate shifts) highlight the explicit embedding of periodic token generation:

$$S^+_n = \max \left\{ 0, \min\{ L+M, S_{n-1}^+ + a_n \} - 1 \right\}$$

where $a_n$ denotes arrivals during the $n$th token period. For heterogeneous flows, the “full state” model increases cardinality by tracking ordered sequences of buffered packet sizes, thereby maintaining fidelity to the impact of packet class on token consumption and overall performance.

In distributed ledgers modeled as directed acyclic graphs (DAGs) with batch arrivals and random proof-of-work (POW) delays, fluid token generation rates are governed by scaling the raw arrival process: $\lambda = N/\epsilon$ for batch size $N$ and step interval $\epsilon$. Taking the fluid limit ($\lambda \to \infty$, $\epsilon \to 0$), the stochastic process converges to delayed differential equations, capturing the deterministic evolution of free and pending tokens. For instance,

$\frac{df}{dt} = 1 - 2 \frac{f(t)}{l(t)}$

where $f(t)$ is the fluid-scaled free tips, and $l(t)$ is the total tips (Feng et al., 2023).

3. Impact of Heterogeneous Token Requirements and Delays

Variable packet sizes or random delays introduce significant non-uniformity into the effective service rate. In the token bucket filter, partitioning flows into size classes with Poisson intensities and distinct token consumption per packet leads to nontrivial queueing dynamics. Larger packets incur longer waiting times, as they require the fluid token generation process to accumulate a corresponding surplus. Analytical updates at token replenishment explicitly involve the head-of-line packet size, embedding a nonlinear relationship between token arrival and service eligibility.

In distributed ledgers, each token’s random POW delay $h_i$ (with probability $p_i$) ensures non-Markovian system evolution, necessitating the use of delayed partial differential equations. The evolution equations for tips and pending tips contain delayed dependency terms (e.g., states evaluated at $t-h_i$), directly reflecting how fluid token generation is affected by heterogeneous confirmation times.

4. Analytical, Simulation, and Scaling Results

Theoretical analysis in these domains yields stationary distributions (e.g., $\pi = \pi G$ for the DTMC, with $G = \exp(Q\tau)\cdot H$) and per-class performance metrics (backlog, delay, loss). Continuous-time measures are obtained by integrating state probabilities over the token period; for instance,

$$\mathbb{E}_{(A)} = \frac{1}{\tau} \int_0^{T} P\{ x(\eta) \in A \} d\eta$$

enables direct comparison between Markovian predictions and discrete-event simulations. Simulation frameworks such as TrueTime validate the analytical models, demonstrating that fluid (periodic) token generation rates produce system behavior in close agreement with theoretical predictions, with negligible discrepancies for sufficiently large systems (Schioler et al., 2020).

In distributed ledger models, simulations for parameters like $\lambda=400$, $N=20$ confirm that scaled trajectories (e.g., $F(t)/\lambda$) concentrate tightly around fluid limit solutions, reinforcing their relevance for protocol evaluation and tuning (Feng et al., 2023).

5. Scheduling and Resource Management in LLM Serving

In LLM serving systems under bursty request loads, fluid token generation rate describes the controlled, continuous delivery of tokens to users—balancing short time-to-first-token (TTFT) and consistent inter-token delay. Systems such as TokenFlow implement a buffer-aware, preemptive scheduling strategy: tokens are generated rapidly and buffered per request, and requests are dynamically prioritized based on output buffer occupancy and user consumption rates (Chen et al., 3 Oct 2025). Requests with large output buffers may be preempted to allocate resources to waiting requests, ensuring low TTFT and steady (fluid) per-user streaming.

Resource management is tightly integrated, with proactive key-value (KV) cache offloading (e.g., write-through policy, synchronous chunked writing, load-evict overlap) to GPU and CPU memory. These mechanisms minimize preemption overhead, prevent scheduling stalls, and ensure that actual generation matches real-time consumption.

Metric-centered evaluation incorporates both throughput and timeliness. Effective throughput quantifies user-centric token generation, weighing each token’s utility against buffer occupancy (e.g., tokens delivered when buffer $<\tau$ fully counted, otherwise decayed):

Metric	Definition	Reported Results
Effective Throughput	Weighted by token consumption immediacy	Up to 82.5% higher
TTFT (P99)	99th percentile time-to-first-token	Reduced by up to 80.2%

Thus, fluid token generation in LLM serving systems is achieved not by maximizing absolute generation speed, but by aligning generation with precise scheduling and buffer management so as to optimize responsiveness and resource utilization.

6. Computational Complexity and System Scalability

For full-state models—especially with variable token requirements or packet sizes—the state space expands rapidly, as quantified by derived upper bounds. For a minimum packet (token) size $m$ and set $|Z|$ of sizes, the state cardinality satisfies:

$\#(Z^*_L) \leq (|Z|^{1/m})^L$

where $L$ is the buffer limit. This demonstrates that scenarios with many small tokens incur combinatorial growth in the system’s effective configuration space, affecting both analytical tractability and simulation runtime.

In distributed ledger systems at the fluid limit, the complexity shifts from combinatorial state enumeration to numerical solutions of delay differential equations, whose dimension is determined by the number of POW duration types ($M$) and the batch size parameter ($N$). Empirical results indicate that, for high volume systems, the macroscopic deterministic limit accurately predicts protocol behavior and stability, obviating the need to simulate full microscopic dynamics (Feng et al., 2023).

7. System Design Implications and Broader Impact

The mechanisms governing fluid token generation rate are central to the design and analysis of modern streaming, networking, and distributed computing systems. Precise scheduling, buffer occupancy management, and explicit resource orchestration (e.g., preemptive scheduling, proactive memory transfer) enable systems to provide both high throughput and user-responsive performance, a requirement increasingly critical in interactive and distributed environments.

In distributed ledgers, tuning the arrival rate and POW difficulty using fluid models facilitates maintaining system stability and robust security—minimizing free/unconfirmed tips and mitigating the risk of attacks based on delay or backlog manipulation. In network Congestion Control and LLM serving, similar fluidic principles ensure fairness and QoS, supporting interactive demands and heterogeneous workloads.

The use of Markovian frameworks, delayed differential equations, and preemptive resource management represents the current methodological frontier for achieving optimized, fluid token generation across these disparate but fundamentally related domains.