Probability Factor

Description

The Probability Factor (PF) is the core computational element within the widely deployed Proportional Fair (PF) scheduling algorithm used in the downlink and uplink of cellular networks like LTE and 5G New Radio (NR). The scheduler, residing in the base station's Medium Access Control (MAC) layer, must decide which user equipment (UE) to serve in every Transmission Time Interval (TTI) and on each Physical Resource Block (PRB). The PF algorithm aims to achieve a multi-objective optimization: maximizing the total cell spectral efficiency (throughput) while maintaining a degree of fairness so that users with poor channel conditions (e.g., at the cell edge) still receive service. The PF metric for a given user 'i' at scheduling time 't' is calculated as: PF_i(t) = DRCi(t) / T_i(t), where DRCi(t) is the instantaneous data rate the user can support (based on its reported Channel Quality Indicator - CQI), and T_i(t) is the user's exponentially weighted average throughput over a past time window.

The algorithm works by continuously updating two key values for each active user. First, the instantaneous achievable rate (DRC) is derived from channel state information, which is frequently reported by the UE. This rate is high when the user experiences good radio conditions (e.g., high Signal-to-Interference-plus-Noise Ratio - SINR) and low when conditions are poor. Second, the average throughput (T_i) is updated using a moving average filter: T_i(t+1) = (1 - 1/t_c) * T_i(t) + (1/t_c) * R_i(t), where t_c is the time constant of the filter and R_i(t) is the actual data rate served to the user in the current TTI (which is zero if the user was not scheduled). The time constant t_c controls the fairness window; a larger t_c emphasizes long-term fairness, while a smaller t_c makes the scheduler more opportunistic.

In each scheduling decision, the base station computes the PF metric (DRC/T) for all candidate users. The user with the highest PF metric is granted the resource. This elegantly balances the two objectives: a user with a momentarily high DRC (due to a good channel) gets a high priority, promoting throughput. However, if a user has not been scheduled for a while, its average throughput T_i decays, increasing its PF metric and thus its chance of being scheduled next, which enforces fairness. The PF scheduler is adaptive and channel-aware, making it highly efficient for packet data services with bursty traffic. Its performance and parameters (like the filter time constant) are detailed in 3GPP performance study specifications (e.g., TR 37.470, TR 38.470) and it forms the basis for more advanced QoS-aware schedulers that incorporate multiple QoS classes and latency requirements.

Purpose & Motivation

The Proportional Fair scheduler and its Probability Factor were developed to address fundamental challenges in shared-channel packet scheduling for cellular data networks, moving beyond the limitations of simpler algorithms. Early scheduling approaches like Maximum CQI (Max-C/I), which always serves the user with the best instantaneous channel quality, maximize total cell throughput but are grossly unfair—users at the cell edge or in deep fade may be starved of resources indefinitely. Conversely, a purely round-robin scheduler is perfectly fair in time but ignores channel conditions, leading to very low spectral efficiency as resources are wasted on users with poor channels who can only support low data rates.

The PF algorithm was conceived to find an optimal trade-off between these two extremes of throughput and fairness, which is essential for commercial networks serving a mix of users with varying channel conditions and service expectations. It enables "multi-user diversity" gain by exploiting the fact that different users experience independent channel fluctuations over time; the scheduler can pick the user currently at a channel peak. The Probability Factor provides the mathematical mechanism to quantify this opportunity relative to the user's historical service, ensuring that a user waiting for a channel peak is eventually served.

Its introduction and standardization in 3GPP (from LTE Release 6 onwards) were driven by the need for efficient support of best-effort Internet-access data services (like web browsing and file downloads) over shared radio resources. The PF scheduler became the default or highly recommended algorithm for non-GBR (Guaranteed Bit Rate) bearers. It provided a robust, predictable, and implementable scheduling strategy that network vendors could optimize and operators could tune (via the averaging window parameter) to match their specific fairness-throughput policy goals, forming the foundation for later, more complex schedulers supporting latency-critical and QoS-differentiated services in 4G and 5G.