Temporal Coverage@k in Wireless Networks

• Temporal Coverage@k is defined as the slot-indexed probability that the SIR exceeds a target T, capturing the effect of dynamic interference and transient queuing.
• The analysis integrates spatial Poisson point processes with discrete-time Geo/Geo/1 queue models to reveal both transient and steady-state network performance.
• This metric guides practical applications such as traffic-aware network planning and resource allocation to ensure reliable coverage under bursty traffic conditions.

Temporal Coverage@k denotes the probability that a user in a wireless cellular network experiences a signal-to-interference ratio (SIR) exceeding a target threshold $T$ at a discrete time slot %%%%1%%%%, under dynamic traffic conditions. It quantifies the time-varying reliability of wireless coverage, accounting for both spatial placement of base stations (BSs) and temporal traffic fluctuations arising from stochastic packet arrivals and queuing at each BS. Temporal Coverage@k serves as a key metric for performance analysis and design in networks with temporally varying interference, as demonstrated in stochastic geometry frameworks coupling spatial Poisson point processes (PPP) and temporal queue evolution (Yang et al., 2018).

1. Formal Definition and System Model

Temporal Coverage@k is formally defined as the time-slot-indexed SIR coverage probability:

$$P_k(T) = P\{\mathrm{SIR} \geq T \text{ at slot } k\}$$

where SIR for a typical user is expressed as the ratio of received signal power to cumulative interference from active transmitters in slot $k$. The system model incorporates:

• Spatial PPP for Base Stations: BSs are positioned according to a homogeneous PPP $\Phi_b$ of intensity $\lambda_b$ on $\mathbb{R}^2$, capturing network irregularity.
• Typical UE and Association: A representative (“typical”) user at the origin associates to its nearest BS at distance $R_0$ with PDF $$f_{R_0}(r) = 2\pi\lambda_b\, r \exp(-\pi\lambda_b r^2)$$.
• Propagation and Fading: Path-loss exponent $\alpha > 2$ and independent Rayleigh fading $h_x\sim \exp(1)$.
• Temporal Traffic Model: Time is slotted. Each BS independently receives a packet at each slot according to a Bernoulli process with probability $\xi$. Each BS implements an infinite-buffer FIFO queue.
• Transmission Attempt and Departure: Active BSs with nonempty queues attempt to transmit. A transmission succeeds if SIR $\geq T$.

The queue at each BS is modeled as a discrete-time Geo/Geo/1 system, with packet arrivals at rate $\xi$ and service probability $\mu$ per slot, both geometrically distributed.

2. Derivation of Temporal Coverage@k

The coverage probability at slot $k$, $P_k(T)$, is derived via stochastic geometry and queueing analysis (Yang et al., 2018). For a typical user,

1. Queue Activity: The probability that a BS is active (i.e., its queue is non-empty) at slot $k$ is

$$p_a(k) := P\{Q(k) > 0\} = \frac{\xi}{\mu}\left(1 - \delta^k\right)$$

where $\mu$ is the service probability, and $\delta := 1 - \xi - \mu$.

2. Conditional SIR Probability: Conditioning on a realization of the PPP and activity indicators, the SIR coverage probability at distance $r$ is

$$P\{\mathrm{SIR} \geq T \;|\; R_0 = r\} = \exp\left(-\pi\lambda_b r^2 p_a(k)\, \delta Z(T, \delta)\right)$$

with $\delta = 2/\alpha$, and

$$Z(T, \delta) = \int_{1}^{\infty} \frac{T v^{-1}}{1 + T v^{-1}} v^{\delta - 1}\, dv$$

3. Marginalization Over User Distribution: Integrating over the distribution of $R_0$ yields the closed-form for Temporal Coverage@k:

$$P_k(T) = \frac{1}{1 + p_a(k)\, \delta Z(T, \delta)}$$

All terms are explicit functions of system parameters $(\lambda_b, \alpha, \xi, T)$ and the time slot index $k$ via $p_a(k)$, which reflects queue transience and system loading.

3. Dynamical Evolution and Asymptotic Behavior

The evolution of Temporal Coverage@k over time reflects the initial transient and eventual steady-state of the BS queues:

• At $k=0$, all BSs are inactive ($p_a(0)=0$), thus $P_0(T)=1$ for all $T$, as no interference occurs.
• As $k$ increases, $p_a(k)$ monotonically increases from $0$ to the steady-state $\xi/\mu$. Correspondingly, $P_k(T)$ decreases from $1$ to

$$P_\infty(T) = \frac{1}{1 + (\xi/\mu)\, \delta Z(T, \delta)}$$

indicating the long-run coverage probability under stationary traffic.

The transient regime is particularly relevant for traffic step changes or bursty events. Traffic loads $\xi$ near or above the service probability $\mu$ (sub-medium or heavy load regimes) precipitate rapid growth in interference and thus coverage degradation.

4. Regime-Specific Characteristics

Behavior of Temporal Coverage@k varies by operational regime (Yang et al., 2018):

• Sub-medium Load: For $\xi < \mu$ but not $\xi \ll 1$, $p_a(k)$ exhibits two-phase growth—rapid early increase, then slower convergence as $\delta^k$ decays.
• Very Light Load: If $\xi \ll \mu$, $p_a(k) \approx \frac{\xi}{\mu}(1 - \delta^k) \ll 1$ and

$P_k(T) \approx 1 - p_a(k)\, \delta Z(T, \delta)$

Outage $$1 - P_k(T) \approx \frac{\xi}{\mu}\, \delta Z(T, \delta)$$ is approximately linear in the packet arrival rate $\xi$ and in early-time $(1 - \delta^k)$.

The “temporal” nature of coverage is especially pronounced at low loads, as interference contributions from active queues are infrequent and queue dynamics drive coverage.

5. Numerical Computation and Implementation

Evaluation of Temporal Coverage@k in practical analysis proceeds as follows:

Step	Operation	Details
1	Solve for steady-state $\mu$	$\mu = P_\infty(T)$ via fixed-point
2	Compute $\rho = \xi / \mu$, $\delta$	$\delta = 1 - \xi - \mu$
3	For each $k$, $p_a(k) = \rho(1 - \delta^k)$
4	Compute $Z(T, \delta)$ (closed-form/hypergeometric)
5	Calculate $P_k(T) = 1 / [1 + p_a(k)\, \delta Z(T, \delta)]$

This pipeline enables computation of the temporal coverage trajectory for arbitrary system and traffic parameters, bridging queueing-theoretic and spatial stochastic characterizations.

6. Relevance and Applications

Temporal Coverage@k is integral for analyzing and optimizing wireless networks with bursty, time-variant demand. It provides insight into:

• Traffic-Aware Network Planning: Evaluating how queue transients due to traffic surges or light loads impact user-perceived SIR and performance.
• Resource Allocation: Guiding BS deployment, scheduling, and traffic engineering based on transient and steady-state coverage probabilities.
• Performance Benchmarks: Interpreting SIR requirements under realistic, rather than static, interference patterns. For example, SIR requirements may differ by more than 10 dB between light and heavy traffic to maintain equivalent coverage levels (Yang et al., 2018).

Temporal Coverage@k thus forms a critical metric for the design and dimensioning of cellular systems in the presence of stochastic traffic, informing protocols and policies responsive to highly variable network loads.