Notification Rate (NR) in Networks & Contact Tracing

• Notification Rate (NR) is a metric quantifying how frequently notifications occur in systems such as network congestion control and digital contact tracing.
• In computer networks, NR is computed using packet arrival rates, marking probabilities, and uniformization to achieve stable congestion feedback.
• In digital contact tracing, NR reflects the combined success of exposure detection, diagnosis, and notification delivery, highlighting operational challenges.

Notification Rate (NR) quantifies the frequency or fraction at which prominent events generate notifications in a system. Within computer networking, NR most often refers to the rate at which network devices mark/droppackets as congestion signals, while in public health digital contact tracing/notification, NR describes the fraction of exposed contacts successfully notified. Though these arise in distinct domains, each application grounds its definition, measurement, and optimization of NR in site-specific methodologies and performance targets.

1. Definitions and Domain Contexts

Networking (FN Algorithm)

In Fast Congestion Notification (FN), every marked (ECN-marked) or dropped packet constitutes a congestion notification. Formally, if $I$ denotes the inter-notification time (seconds), then the long-run Notification Rate is:

$NR = \frac{1}{E[I]}$

Alternatively, if $\lambda$ (packets/s) is the arrival rate at the router, and $p_{\text{avg}}$ is the steady-state average marking or dropping probability, then:

$NR \approx \lambda \cdot p_{\text{avg}}$

This allows NR to serve as a tunable metric for congestion feedback to traffic sources (0909.1372).

Digital Contact Tracing

In digital contact tracing or exposure notification, NR is the fraction of true exposure events that either (A) generate a successful notification (as measured in field trials) or (B) traverse all prerequisite steps in the notification-delivery pipeline (adoption, detection, diagnosis, etc.). Mathematically, for $N$ contact events, and outcome indicator $1\{\cdot\}$ per exposure score $ES_i$,

$NR = \frac{1}{N}\sum_{i=1}^N 1\{ES_i \geq \Delta\}$

with $ES_i$ defined according to a relevant rule or threshold (Leith et al., 2020). In broader epidemiological analysis, NR is treated as

$NR = p_1 \cdot p_2 \cdot p_3 \cdot p_4 \cdot p_5$

corresponding to the product of probabilities for adoption, exposure detection, diagnosis, code entry, and notification delivery (Masel et al., 2023).

2. Core Algorithms and Analytical Formulations

FN: Marking Probability and NR Computation

FN's notification logic computes the initial per-packet marking probability as:

$$p_{\text{ini}} = \frac{(R - \mu) T + (Q_{\text{cur}} - Q_{\text{opt}})}{R T}$$

where $R$ is the average arrival rate, $\mu$ the link capacity, $T$ a time constant, $Q_{\text{cur}}$ instantaneous, and $Q_{\text{opt}}$ the optimal queue size (0909.1372).

Simple independent marking yields geometrically distributed inter-marking intervals; FN introduces a "uniformization" transform:

$$p_{\text{fin}} = \begin{cases} 1 & \text{if } count \cdot p_{\text{ini}} \geq 2 \ \dfrac{(count+1)\,p_{\text{ini}}}{2 - count \cdot p_{\text{ini}}} & \text{otherwise} \end{cases}$$

This enforces upper and lower bounds on intervals between notifications.

Digital Contact Tracing: Exposure Score and NR Calculation

Under field-tested exposure-notification rules, each close-contact pair results in an exposure score, e.g., for the Swiss DP-3T rule:

• $B_1 =$ minutes with attenuation $< t_1$
• $B_2 =$ minutes with $t_1 \leq$ attenuation $< t_2$
• $ES_{Swiss} = B_1 + 0.5 B_2$

Exposure is flagged if $ES_{Swiss} \geq \Delta$ minutes. The NR is then calculated as the base rate of exposures crossing threshold (Leith et al., 2020).

In modeling NR for epidemic reduction, NR is treated as a product of successes: app adoption by both parties ($p_1$, $p_2$), exposure detection ($p_3$), test/diagnosis ($p_4$), notification delivery ($p_5$), with overall transmission effects modulated by subsequent behavior change ($p_6$) (Masel et al., 2023).

3. Theoretical Analysis: Uniformization, Effect of Parameters, and Interpretation

Uniformization in FN

Without uniformization, marking/dropping is geometric, with mean inter-notification interval $1/p_{\text{ini}}$. Variability is high, leading to bursty or excessively sparse feedback. FN's uniformization approximates the marking interval as uniform, leading to mean interval $(1/p_{\text{ini}}) + 1/2$ packets. Resulting NR is:

$$NR_{unif} = \frac{(λ p_{\text{ini}})}{1 + \frac{1}{2} p_{\text{ini}}}$$

Low $p_{\text{ini}}$ yields $NR_{unif} \approx \lambda p_{\text{ini}}$ (matching the geometric case); as $p_{\text{ini}}$ increases, NR saturates, preventing back-to-back notifications.

Parameter Sensitivity and Operational Effects

• In FN, setting $Q_{\text{opt}}$ (target queue occupancy) higher or lower modulates the sensitivity of NR: a queue well above $Q_{\text{opt}}$ sharply raises the marking rate, tightening congestion control.
• If $Q_{\text{cur}} < Q_{\text{opt}}$, $p_{\text{ini}}$ and thus NR can be zero, suspending notifications and reducing unnecessary packet loss (0909.1372).

Contact Tracing: Thresholds and Detection Sensitivity

• Increasing the attenuation threshold ($t_2$) or lowering the duration threshold ($\Delta$) raises NR, but often at the cost of increased false positives (Leith et al., 2020).
• In environments like buses, Bluetooth attenuation poorly correlates with true proximity, leading to low NRs regardless of threshold tuning.

4. Empirical Measurement and Practical Impact

FN Queue Control

Simulation results with FN uniformization indicate a reduction in queue oscillations' amplitude by approximately 30% and a decrease in aggregate TCP window size standard deviation by 25%, compared to geometric marking at the same mean NR (0909.1372). This highlights the operational significance of NR regularity for traffic stability and congestion avoidance.

Exposure Notification: Field Trials

On a commuter bus, using the official Swiss DP-3T rule (thresholds of 50/55 dB, $\Delta = 15$ min), the NR was observed to be 0%, i.e., no exposure notifications, even when all pairs were within 2m for 15 minutes. Raising $t_2$ to 70 dB and lowering $\Delta$ to 10 min increases NR to only 8%. These findings demonstrate severe limitations in practical NR attainable via RSSI-based Bluetooth proximity in real-world, reflective environments (Leith et al., 2020).

A summary of results:

Method/Threshold	t2/duration	Observed NR
Swiss DP-3T Rule	t2=55 dB, 15m	0%
Single-thresh, t2=70 dB	15m	≃5%
Single-thresh, t2=70 dB	10m	≃8%

Population-Level Impact

The probabilistic chain linking adoption, risk detection, diagnosis, and notification defines real-world NR:

$\Delta R = (p_1 p_2 p_3 p_4 p_5) p_6$

To achieve a 26% reduction in $R(t)$ (epidemic scaling parameter), success at each of the six steps must approach 80%. The observed rates in many settings fell far short (e.g., $p_1, p_2 < 50\%$, $p_5$ often much lower with manual workflows) (Masel et al., 2023).

5. Factors Limiting and Enabling Notification Rate

Key limiting factors of NR (particularly in digital notification) include:

• Low adoption rates: Especially among high-risk subpopulations.
• Poor risk signal calibration: Bluetooth attenuation is an inadequate proxy for epidemiologically relevant contact.
• Manual-to-automated bottlenecks: Automation of test-result code entry and notification can improve $p_5$ from <2% to ≈10% in some deployments.
• Platform restrictions: Constraints imposed by privacy architectures and OS (e.g., GAEN/Apple iOS) impede background detection and prevent richer metadata exchange.
• Behavioral noncompliance: Downstream adherence to quarantine or self-isolation is typically far below required levels for epidemic suppression.

Proposed approaches to improve NR include:

• Opt-out, OS-level proximity/presence recording.
• Integration of exposure notification with high-value services (test result delivery, venue check-in).
• Continuous threshold recalibration against infection outcomes.
• Explicit measurement and iteration of failure rates at each notification pipeline step, aiming for ≥80% performance per stage (Masel et al., 2023).

6. Pseudocode Examples

FN Uniformization for Congestion Notification (0909.1372)

count = 0 for each arriving packet: measure Qcur, update R p_ini = ((R - mu) * T + (Qcur - Qopt)) / (R * T) p_ini = clamp(p_ini, 0, 1) if (count * p_ini >= 2): p_fin = 1 else: p_fin = (count + 1) * p_ini / (2 - count * p_ini) u = uniform_random_0_1() if u <= p_fin: mark_or_drop(packet) count = 0 else: enqueue(packet) count += 1

NR Calculation for Exposure Notifications (abstracted)

hits = 0 for i in 1...N: # Obtain (B1i, B2i) for thresholds (t1, t2) ES = compute_exposure_score(B1i, B2i, t1, t2) if ES >= Delta: hits += 1 NR = hits / N

Where compute_exposure_score is specified by the notification rule (Swiss DP-3T, threshold-based, etc.) (Leith et al., 2020).

7. Cross-Domain Insights and Limitations

NR encapsulates a critical system-level bottleneck in both engineered queuing and epidemic digital notification. In the FN context, deterministic and bounded notification intervals, as achieved by uniformization, enhance stability and fair bandwidth sharing while suppressing adverse synchronization. In digital contact tracing, NR functions as the composite efficacy metric for all layers of the notification pipeline; achieving sufficient NR is overwhelmingly multiplicative across points of failure. Practical deployments reveal that even small inefficiencies at any step can dramatically suppress aggregate NR, undermining system goals even before accounting for downstream behavior change.

A plausible implication is that for both networks and public health, optimizing NR requires tight integration of system design, robust feedback or detection mechanisms, and targeted efforts to address all major sources of failure separately rather than merely tuning thresholds or protocols in isolation.

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References:

(0909.1372, Leith et al., 2020, Masel et al., 2023)