Backhaul Rate Exceedance Probability (BREP)

• BREP is a threshold-based metric that measures the probability that instantaneous backhaul capacity exceeds the aggregate ingress traffic in multi-layer networks.
• It enables rigorous performance evaluation in SAGIN, UAV, IAB, and hybrid FSO/THz frameworks by mapping random SNR/SINR distributions to rate exceedance events.
• Analytical models employing stochastic geometry and outage formulations leverage BREP to optimize load sharing, resource partitioning, and system reliability.

Searching arXiv for papers on Backhaul Rate Exceedance Probability and closely related backhaul rate coverage formulations. Backhaul Rate Exceedance Probability (BREP) is a threshold-based backhaul sufficiency metric. In its explicit relay-perspective formulation for the uplink in a satellite-aerial-ground integrated network (SAGIN), BREP is the probability that the instantaneous HAP–satellite backhaul rate exceeds the instantaneous aggregate user–HAP access rate, namely

$$\mathcal{P}_{\mathrm{BREP}}=\mathbf{Pr}\bigg\{\sum_{i\in \Phi_u} B_{\mathrm{RF}}\log_2\!\left(1+\gamma_{H,i}\right) < B_{\mathrm{FSO}}\log_2\!\left(1+\gamma_s\right)\bigg\}.$$

In this form, the threshold is not fixed; it is the random aggregate ingress traffic generated on the access side. In adjacent backhaul literatures, the same underlying question appears through user rate coverage, backhaul-link rate exceedance derived from SNR or SINR coverage, or backhaul-constrained admission proxies, rather than under a uniform name (Tarhouni et al., 27 Jul 2025).

1. Definition and scope

BREP is not a single universally fixed object across the backhaul literature. In the SAGIN relay model, it is an end-to-end relay sufficiency probability: the HAP must forward, in real time, the aggregate traffic that it currently receives from users, so the comparison is between one random rate and another random rate, not between a random rate and a fixed target (Tarhouni et al., 27 Jul 2025).

In terrestrial and aerial backhaul work, the closest explicit analog is often a conventional rate exceedance probability built from a Shannon-rate model. For low-altitude UAV backhaul, the backhaul rate is

$\mathcal R=b\log_2(1+\mathrm{SINR}),$

so the rate exceedance event at target rate $\rho$ is

$$\mathbb{P}(\mathcal R>\rho)=\mathbb{P}\!\left(\mathrm{SINR}>2^{\rho/b}-1\right).$$

This is substantively a BREP, even though the paper uses the language of backhaul success probability and expected backhaul rate rather than the term BREP itself (Galkin et al., 2017).

In integrated access and backhaul (IAB), the closest object is frequently an end-to-end service threshold event. For UEs served through a wirelessly backhauled SBS, the rate is explicitly bottlenecked by backhaul through

$$R_u=\min\!\left(R_{\text{access},u},R_{\text{backhaul},k}\right),$$

and the principal reliability metric becomes

$\rho=\Pr(R_u\ge \beta),$

which is equivalent to a joint access-and-backhaul rate exceedance event for SBS-served users. This is not a pure backhaul-only BREP, but it embeds a backhaul threshold event exactly (Madapatha et al., 17 Feb 2025).

A recurrent source of ambiguity is therefore definitional. In some settings BREP refers to the aggregate backhaul link; in others it refers to the per-user backhaul-supported rate; in others it refers to an end-to-end user service probability in which backhaul enters through a minimum operator; and in relay-perspective SAGIN it refers to whether instantaneous backhaul capacity exceeds instantaneous aggregate ingress traffic. The literature is consistent on the bottleneck logic, but not on a single canonical metric name.

2. Mathematical structure

Two threshold structures dominate the literature. The first is the fixed-threshold form,

$\Pr\!\left(R_{\text{backhaul}}\ge \beta\right),$

which is typically converted to an SNR or SINR exceedance event by the Shannon mapping. The UAV backhaul analysis is the clearest example: all BREP content is recoverable from the derived SINR CCDF by substituting $\theta_\rho=2^{\rho/b}-1$ into the coverage expressions (Galkin et al., 2017).

The second is the random-threshold form, where the “target” is itself stochastic. The SAGIN relay model adopts exactly this structure: $$R_{\mathrm{acc}}=\sum_{i\in\Phi_u} B_{\mathrm{RF}}\log_2(1+\gamma_{H,i}), \qquad R_{\mathrm{bh}}=B_{\mathrm{FSO}}\log_2(1+\gamma_s),$$ and

$$\mathcal{P}_{\mathrm{BREP}}=\Pr\{R_{\mathrm{acc}}<R_{\mathrm{bh}}\}.$$

This is not a $\mathcal R=b\log_2(1+\mathrm{SINR}),$0 two-hop DF rate. It is a many-to-one relay sufficiency condition: $\mathcal R=b\log_2(1+\mathrm{SINR}),$1 The paper explicitly contrasts this with average-rate metrics such as the average access data rate and average backhaul data rate, and states that average backhaul superiority supports long-term operation because the HAP may buffer traffic, whereas BREP captures stricter real-time forwarding requirements (Tarhouni et al., 27 Jul 2025).

IAB analyses introduce a third structure: load-coupled thresholding. In RIS-assisted IAB, the total spectrum is partitioned as

$\mathcal R=b\log_2(1+\mathrm{SINR}),$2

with per-child backhaul bandwidth

$\mathcal R=b\log_2(1+\mathrm{SINR}),$3

and an intended backhaul rate model of the form

$\mathcal R=b\log_2(1+\mathrm{SINR}),$4

A backhaul exceedance event is therefore load dependent even before access coupling is imposed (Madapatha et al., 17 Feb 2025).

This suggests that BREP is best understood as a family of exceedance probabilities defined after specifying three ingredients: the backhaul rate object, the threshold object, and the load-sharing rule. Without those three elements, the term remains underspecified.

3. Embedding in IAB and backhaul-limited user service

IAB literature makes the backhaul bottleneck operationally visible. In one finite mmWave IAB model, the SBS aggregate backhaul rate is

$\mathcal R=b\log_2(1+\mathrm{SINR}),$5

while the SBS-served user rate is

$\mathcal R=b\log_2(1+\mathrm{SINR}),$6

The paper’s formal metric is downlink rate coverage probability, $\mathcal R=b\log_2(1+\mathrm{SINR}),$7, but the first argument in the SBS term is exactly a per-user backhaul-supported exceedance condition after threshold conversion. Equal partition and load-based partition alter the backhaul threshold through $\mathcal R=b\log_2(1+\mathrm{SINR}),$8, so BREP becomes either fixed-share or load-conditioned depending on the scheduler (Saha et al., 2017).

A later mmWave IAB framework generalizes this logic to integrated resource allocation (IRA) and orthogonal resource allocation (ORA). Under IRA, the backhaul fraction allocated to an SBS is

$\mathcal R=b\log_2(1+\mathrm{SINR}),$9

and the SBS-served user rate contains the bottleneck term

$\rho$0

Under ORA, a static fraction $\rho$1 is reserved for access, and the backhaul-limited term becomes

$\rho$2

These are not labeled BREP, but they are explicit backhaul-supported rate objects and their threshold exceedance probabilities are embedded in the main rate-coverage theorems (Saha et al., 2019).

A central design consequence follows directly from these models: densifying SBSs does not monotonically improve rate coverage in IAB, because the rate of the wireless backhaul links between MBS and SBS becomes the limiting factor. The literature states that there exists a critical volume of cell-load beyond which the gains provided by the IAB-enabled network disappear and performance converges to that of the traditional macro-only network (Saha et al., 2018).

In this sense, BREP is the missing explicit label for a phenomenon already central to IAB analysis: the threshold probability that the shared wireless backhaul can sustain the traffic induced by offloading and densification.

4. Stochastic-geometry and outage frameworks

BREP has been analyzed or reconstructed under several SG regimes. The SAGIN relay model uses spherical stochastic geometry (SSG) because dynamic topology and interference analysis are required on spherical shells rather than on a plane. Ground users follow a spherical homogeneous PPP, LEO satellites follow a spherical homogeneous BPP, and the typical HAP relays to the nearest satellite. The paper derives analytical expressions for the average access data rate, the average backhaul data rate, and BREP, and presents a closed-form expression for the end-to-end performance metric BREP in series-integral form (Tarhouni et al., 27 Jul 2025).

For low-altitude UAV backhaul, the ground-station network is modeled as a homogeneous PPP, the serving GS is the nearest GS, the UAV uses a directional antenna, and the backhaul success probability is derived by conditioning on LOS or NLOS service and integrating over the serving distance distribution

$\rho$3

Since the backhaul rate is

$\rho$4

the exact BREP is obtained directly by substituting $\rho$5 into the paper’s SINR coverage expression. This is a textbook example of BREP emerging as a CCDF transformation of a link-level stochastic-geometry result (Galkin et al., 2017).

Hybrid FSO/THz backhaul adopts an outage-theoretic rather than SG formulation. The FSO and THz branches are in parallel, the AP selects the branch under hard or soft switching, and the main reliability object is backhaul outage. Under hard switching,

$\rho$6

so a BREP interpretation is immediate: $\rho$7 The same complement-of-outage interpretation applies to the soft-switching hybrid-backhaul model and to the end-to-end DF relaying system (Singya et al., 2022).

These frameworks share a common analytical pattern. A random geometry or fading mechanism generates an SNR or SINR distribution; a rate model maps that distribution to a threshold event; and BREP is the resulting tail probability. What differs across the literature is whether the dominant randomness is spatial load, relay topology, channel impairment, or branch selection.

5. Environmental, admission, and uncertainty drivers

BREP is highly sensitive to upstream variables that may not themselves be “rate” metrics. Rain-rate statistics are the clearest example. In microwave and mmWave backhaul planning, the relevant upstream quantity is the 1-minute rain rate exceeded for $\rho$8 of time,

$\rho$9

with special emphasis on

$$\mathbb{P}(\mathcal R>\rho)=\mathbb{P}\!\left(\mathrm{SINR}>2^{\rho/b}-1\right).$$0

the 1-minute rain rate exceeded for $$\mathbb{P}(\mathcal R>\rho)=\mathbb{P}\!\left(\mathrm{SINR}>2^{\rho/b}-1\right).$$1 of an average year. The literature states that many attenuation recommendations use $$\mathbb{P}(\mathcal R>\rho)=\mathbb{P}\!\left(\mathrm{SINR}>2^{\rho/b}-1\right).$$2 directly, that terrestrial attenuation under ITU-R P.530 is calculated as a function of $$\mathbb{P}(\mathcal R>\rho)=\mathbb{P}\!\left(\mathrm{SINR}>2^{\rho/b}-1\right).$$3, and that the relevant pipeline is

$$\mathbb{P}(\mathcal R>\rho)=\mathbb{P}\!\left(\mathrm{SINR}>2^{\rho/b}-1\right).$$4

For RF links above roughly $$\mathbb{P}(\mathcal R>\rho)=\mathbb{P}\!\left(\mathrm{SINR}>2^{\rho/b}-1\right).$$5 GHz, a $$\mathbb{P}(\mathcal R>\rho)=\mathbb{P}\!\left(\mathrm{SINR}>2^{\rho/b}-1\right).$$6 storm causes $$\mathbb{P}(\mathcal R>\rho)=\mathbb{P}\!\left(\mathrm{SINR}>2^{\rho/b}-1\right).$$7–$$\mathbb{P}(\mathcal R>\rho)=\mathbb{P}\!\left(\mathrm{SINR}>2^{\rho/b}-1\right).$$8 rain attenuation, so even short links may suffer fades or outage. This makes high-percentile rain-tail estimation an upstream determinant of BREP rather than a merely descriptive climatological input (Aoki, 2016).

Wireless backhaul uncertainty can also be abstracted without an instantaneous backhaul-rate model. In a transmitter-selection framework with uncertain wireless backhaul, each transmitter has a Bernoulli activity state

$$\mathbb{P}(\mathcal R>\rho)=\mathbb{P}\!\left(\mathrm{SINR}>2^{\rho/b}-1\right).$$9

The paper does not define BREP, but it derives threshold-performance results under backhaul link activity knowledge unavailable (BKU) and available (BKA). The asymptotic secrecy-outage floors are $$R_u=\min\!\left(R_{\text{access},u},R_{\text{backhaul},k}\right),$$0 under BKU and $$R_u=\min\!\left(R_{\text{access},u},R_{\text{backhaul},k}\right),$$1 under BKA, so the corresponding exceedance ceilings are $$R_u=\min\!\left(R_{\text{access},u},R_{\text{backhaul},k}\right),$$2 and $$R_u=\min\!\left(R_{\text{access},u},R_{\text{backhaul},k}\right),$$3. This suggests a general reliability-gated exceedance principle: activity-state knowledge fundamentally changes the tail probability of any thresholded performance variable that is multiplied by backhaul availability (Wafai et al., 2022).

A further proxy appears in count-constrained backhaul models. In an uplink two-tier network, an FAP with backhaul capacity $$R_u=\min\!\left(R_{\text{access},u},R_{\text{backhaul},k}\right),$$4 admits only up to $$R_u=\min\!\left(R_{\text{access},u},R_{\text{backhaul},k}\right),$$5 macro users after serving femto users, and the blocked macro-user count is

$$R_u=\min\!\left(R_{\text{access},u},R_{\text{backhaul},k}\right),$$6

The natural exceedance event is

$$R_u=\min\!\left(R_{\text{access},u},R_{\text{backhaul},k}\right),$$7

equivalently $$R_u=\min\!\left(R_{\text{access},u},R_{\text{backhaul},k}\right),$$8. This is not a bit-rate BREP, but it is an exact count-based exceedance proxy for backhaul-constrained admission (Jalali et al., 2014).

These examples show that BREP often depends on hidden layers: heavy-rain tail estimation, binary backhaul activity states, or admission caps. The exceedance event is defined at the rate layer, but its dominant uncertainty may originate elsewhere.

6. Interpretation, design use, and limitations

BREP is most useful when it is separated from average-rate metrics. The SAGIN relay analysis states explicitly that average backhaul data rate greater than average access data rate supports long-term operation because the HAP may buffer traffic, whereas BREP captures stricter short-term or real-time requirements. The difference is operationally important: a system may satisfy an average-rate inequality and still exhibit poor BREP because instantaneous aggregate ingress can exceed instantaneous backhaul capacity (Tarhouni et al., 27 Jul 2025).

This distinction also explains why rms-type fit metrics or mean-rate summaries can be misleading. In rain-limited backhaul, the critique of ITU-R P.837-6 is that conventional global fit statistics can look acceptable while the model is poor at identifying regions with very high $$R_u=\min\!\left(R_{\text{access},u},R_{\text{backhaul},k}\right),$$9, which are precisely the regions most relevant to outage-capacity collapse. The paper argues that rms error is dominated by variance, that methods with similar rms can differ materially in spatial bias distribution, and that for planning the bias pattern in heavy-rain regions matters more than rms alone. A plausible implication is that BREP estimation requires tail-aware validation rather than average-fit validation (Aoki, 2016).

Design use follows directly from the chosen formulation. In SAGIN, BREP increases with higher HAP transmit power, more satellites, and lower satellite altitude, but with diminishing returns in power. The paper further uses BREP as a design constraint through minimum $\rho=\Pr(R_u\ge \beta),$0 heatmaps for target values such as $\rho=\Pr(R_u\ge \beta),$1 and $\rho=\Pr(R_u\ge \beta),$2; for satellites at altitude $\rho=\Pr(R_u\ge \beta),$3 km, achieving $\rho=\Pr(R_u\ge \beta),$4 can require less than $\rho=\Pr(R_u\ge \beta),$5 dBW, whereas $\rho=\Pr(R_u\ge \beta),$6 requires about $\rho=\Pr(R_u\ge \beta),$7 dBW (Tarhouni et al., 27 Jul 2025).

In IAB, the principal design levers are bandwidth partition and load sharing. Equal partition, instantaneous load-based partition, and average load-based partition induce different backhaul thresholds, and the literature states that load-aware partitioning yields better rate coverage while an optimal access/backhaul split exists. The corresponding BREP interpretation is immediate: the backhaul tail probability improves when the partition rule tracks the instantaneous or average load more closely (Saha et al., 2018).

The main limitation of the term is therefore not mathematical but semantic. “BREP” may denote a pure backhaul-link CCDF, a per-user backhaul-supported rate CCDF, a joint access-and-backhaul rate coverage event, or an aggregate relay sufficiency probability. Careful specification of the threshold, the shared resource model, and the end-to-end bottleneck structure is essential. Without that specification, the same acronym can refer to distinct, though closely related, reliability objects.