Push-Pull Scheduler (PPS) Overview

• Push-Pull Scheduler (PPS) is a dual-access mechanism that integrates pull-initiated state refresh with push-initiated urgent reporting.
• It employs belief tracking, information gain, and collision-aware resource allocation to minimize the Age of Incorrect Information.
• PPS frameworks balance scheduled and contention-based accesses to reduce drift and anomaly detection delays in digital twin and switching applications.

Searching arXiv for the core PPS papers and closely related work to ground the article and disambiguate the acronym across domains. Searching for "A Combined Push-Pull Access Framework for Digital Twin Alignment and Anomaly Reporting" and related PPS terminology on arXiv. Push-Pull Scheduler (PPS) denotes a class of scheduling and medium-access mechanisms that jointly manage pull-initiated and push-initiated service. In its most explicit arXiv usage, PPS is a medium access framework for digital twin alignment and anomaly reporting: pull-updates are requested by the digital twin to reduce drift from the physical system, while push-updates are sensor-initiated reports of urgent events such as anomalies (Chiariotti et al., 29 Aug 2025). A broader conceptual lineage comes from pull-based packet scheduling in input-queued switches, where scheduling is driven by target outflow profiles rather than by backlog stabilization alone; in that setting, the distinction between push and pull is the distinction between maximizing carried load and tracking desired service traces, and it motivates hybrid designs that combine both objectives (Dua et al., 2010).

1. Conceptual basis

PPS is organized around a duality of initiation and objective. In the digital-twin setting, pull-updates follow a request from the digital twin to the sensors, and push-updates are sent directly by the sensors because they represent urgent information, such as anomalies (Chiariotti et al., 29 Aug 2025). In the switching literature, the classical paradigm is to “push” as much traffic load through the switch as possible while controlling delay and keeping congestion from exploding, whereas pull scheduling “pulls” traffic streams through the switch so that actual departures adhere to desirable target outflow profiles (Dua et al., 2010).

These two lineages share a common systems interpretation. Pull mechanisms are model-driven or deadline-driven: they are triggered by state uncertainty, target service traces, or the need to refresh stale information. Push mechanisms are event-driven: they are triggered by backlog pressure, local alarms, or urgent state transitions. PPS exists precisely where neither mechanism is sufficient in isolation. In such settings, a scheduler must decide not only which entity to serve, but also why it is being served: to restore alignment with a planned process, or to report an exceptional event immediately.

A recurrent misconception is that PPS necessarily denotes a single algorithmic template. The available literature does not support that interpretation. Rather, PPS refers to a design principle in which the resource controller exposes both pull and push paths, then arbitrates between them using urgency measures tailored to the application domain.

2. Digital-twin PPS: system model and access structure

In the digital-twin formulation, the system consists of a set of sensors grouped into disjoint clusters mapped to virtual models. Let the full sensor set be $N$, with a subset $N_d \subseteq N$ used for drift monitoring and a subset $N_a \subseteq N$ used for anomaly reporting; $N_d$ and $N_a$ can overlap, and $N_a \cup N_d = N$ (Chiariotti et al., 29 Aug 2025). Each cluster evolves as a hidden Markov model with hidden state $y^{(i)}(k)$, and drift is defined by membership in a designated subset of drift states $Y_d^{(i)}$, through

$$z^{(i)}(k) = \mathbf{1}\{y^{(i)}(k) \in Y_d^{(i)}\}.$$

The anomaly process is per-sensor and binary. For each $n \in N_a$, the anomaly state $N_d \subseteq N$0 follows an independent Markov chain whose evolution depends on the anomaly appearance rate $N_d \subseteq N$1, the spontaneous resolution rate $N_d \subseteq N$2, and an indicator $N_d \subseteq N$3 of successful reporting in frame $N_d \subseteq N$4 (Chiariotti et al., 29 Aug 2025). This separation is central: drift is a cluster-level latent-state estimation problem, whereas anomaly handling is a per-node reporting problem.

Time is partitioned into frames of duration $N_d \subseteq N$5, each with $N_d \subseteq N$6 uplink resource elements (REs). Every frame is split into a pull subframe with $N_d \subseteq N$7 REs and a push subframe with $N_d \subseteq N$8 REs. Pull transmissions are orthogonal and scheduled by the base station; push transmissions are contention-based and use framed slotted ALOHA (FSA). A minimum allocation $N_d \subseteq N$9 per subframe avoids starvation (Chiariotti et al., 29 Aug 2025).

Aspect	Pull side	Push side
Initiation	DT-initiated via the base station	Sensor-initiated on anomaly
Access mode	Orthogonal scheduled transmissions	Grant-free FSA
Primary purpose	Reduce drift and refresh DT state	Report urgent anomalies

The pull subframe is assumed error-free, while the push subframe follows a pure collision channel model: each active device picks one of $N_a \subseteq N$0 REs uniformly at random, and collisions cause erasures (Chiariotti et al., 29 Aug 2025). This asymmetry is deliberate. Pull service is treated as reliable state acquisition; push service is treated as urgent but contention-limited access.

3. State variables, AoII metrics, and frame-level control

PPS is formulated around Age of Incorrect Information (AoII). For digital-twin drift, the AoII of cluster $N_a \subseteq N$1 after frame $N_a \subseteq N$2 is

$N_a \subseteq N$3

For anomaly detection, the AoII of sensor $N_a \subseteq N$4 is

$N_a \subseteq N$5

These metrics quantify not merely staleness, but the persistence of incorrect state estimates (Chiariotti et al., 29 Aug 2025).

The base station tracks two belief processes. First, it maintains a posterior $N_a \subseteq N$6 over each cluster HMM state, and from it the drift risk

$N_a \subseteq N$7

Second, it maintains a per-sensor belief PMF $N_a \subseteq N$8 over anomaly AoII, from which it defines the anomaly AoII violation probability

$N_a \subseteq N$9

The average drift urgency is

$N_d$0

PPS uses these quantities at three decision points per frame. First, it allocates resources between push and pull. In the Reactive Subframe Manager (RSM),

$N_d$1

with clipping to $N_d$2. In the Stable Subframe Manager (SSM), the push budget is incremented or decremented by one RE according to whether $N_d$3 exceeds a hysteresis threshold $N_d$4 (Chiariotti et al., 29 Aug 2025).

Second, it schedules the pull subframe by information gain. For a node $N_d$5 in cluster $N_d$6, the gain from adding $N_d$7 to the scheduled set $N_d$8 is

$N_d$9

where $N_a$0 is the posterior entropy of drift risk under observation $N_a$1 (Chiariotti et al., 29 Aug 2025). The scheduler iteratively allocates pull REs to the most informative sensors.

Third, it sets the push threshold $N_a$2. Given the approximate collision probability $N_a$3, PPS chooses the largest threshold in the admissible set

$N_a$4

or falls back to $N_a$5 if $N_a$6 is empty (Chiariotti et al., 29 Aug 2025). Thus, push access is not open to all active anomalies; it is selectively triggered by estimated urgency.

4. Optimization, empirical behavior, and implementation

The digital-twin PPS is designed around a constrained stochastic objective: minimize average drift AoII while meeting anomaly-detection and collision constraints, with per-frame resource conservation $N_a$7, minimum per-subframe allocations, and a collision cap $N_a$8 (Chiariotti et al., 29 Aug 2025). The policy is heuristic and approximate rather than an exact MDP solution, but it is built from explicit belief tracking, entropy-based pull scheduling, and collision-aware push thresholding.

In the reported simulations, frames contain $N_a$9 REs per $N_a \cup N_d = N$0 ms, the system has $N_a \cup N_d = N$1 drift-monitoring sensors and $N_a \cup N_d = N$2 anomaly sensors, clusters are $N_a \cup N_d = N$3 with $N_a \cup N_d = N$4 sensors each, the push collision cap is $N_a \cup N_d = N$5, the anomaly AoII risk threshold is $N_a \cup N_d = N$6 ms, and the minimum per-subframe allocation is $N_a \cup N_d = N$7 REs (Chiariotti et al., 29 Aug 2025). Against baselines including Maximum Age First (MAF), Cluster Risk Aware (CRA), FSA, and adaptive FSA (AFSA), the main reported findings are twofold. First, PPS reduces average drift AoII by over $N_a \cup N_d = N$8 with respect to state-of-the-art solutions while maintaining the same anomaly detection guarantees. Second, under a $N_a \cup N_d = N$9 ms average drift AoII constraint, PPS reduces the worst-case anomaly detection AoII from $y^{(i)}(k)$0 ms to $y^{(i)}(k)$1 ms (Chiariotti et al., 29 Aug 2025).

The implementation burden resides primarily at the base station. It must maintain HMM posteriors for cluster drift, per-node AoII PMFs for anomalies, evaluate information gains for pull scheduling, and compute collision-aware push thresholds. The evaluation notes that for binary measurements and small cluster sizes such as $y^{(i)}(k)$2, the pull-side information-gain computations are tractable (Chiariotti et al., 29 Aug 2025). The framework is also explicitly compatible with 3GPP-style OFDM PRB structures, since REs are mapped to pull and push subframes with preceding downlink control.

A plausible implication is that PPS should be understood less as a single scheduler than as a cross-layer control architecture. Its defining property is not one update rule, but the joint use of belief-state tracking, urgency estimation, and medium-access partitioning.

5. Pull-based switch scheduling and hybrid PPS in packet switching

An earlier and mathematically distinct precursor appears in packet scheduling for $y^{(i)}(k)$3 input-queued crossbar switches with virtual output queues (VOQs) (Dua et al., 2010). Each VOQ $y^{(i)}(k)$4 is associated with a target service trace specifying desirable departure times or inter-departure times. The cumulative target stream profile is

$y^{(i)}(k)$5

the cumulative received service trace is

$y^{(i)}(k)$6

and the deviation is

$y^{(i)}(k)$7

Positive deviation means the stream is leading; negative deviation means it is lagging. With convex per-stream deviation penalties $y^{(i)}(k)$8, the aggregate slot cost is

$y^{(i)}(k)$9

and the finite-horizon objective is

$Y_d^{(i)}$0

The service-configuration dynamics are

$Y_d^{(i)}$1

where $Y_d^{(i)}$2 is the chosen switch configuration and $Y_d^{(i)}$3 is the vector of target departures due in slot $Y_d^{(i)}$4. The paper develops dynamic-programming and myopic policies, including Maximum Sum of Lags (MSL) and Largest Lag First (LLF), as well as complexity reductions based on orthogonal configuration subsets and meta-queues. Full MSL requires an $Y_d^{(i)}$5 maximum-weight matching, whereas subset-restricted MSL-SS and LLF-SS achieve $Y_d^{(i)}$6 per-slot complexity (Dua et al., 2010). Some of these schedules are provably shown to achieve $Y_d^{(i)}$7 pull-throughput, meaning bounded lag in expectation for admissible i.i.d. target loads.

Within this framework, a hybrid PPS is introduced as a natural combination of push and pull objectives. Classical push scheduling uses queue-length weights $Y_d^{(i)}$8 to stabilize backlogs and achieve $Y_d^{(i)}$9 push-throughput. Pull scheduling uses urgency derived from lag relative to target outflow profiles. One natural hybrid is to select a matching $$z^{(i)}(k) = \mathbf{1}\{y^{(i)}(k) \in Y_d^{(i)}\}.$$0 that maximizes

$$z^{(i)}(k) = \mathbf{1}\{y^{(i)}(k) \in Y_d^{(i)}\}.$$1

where $$z^{(i)}(k) = \mathbf{1}\{y^{(i)}(k) \in Y_d^{(i)}\}.$$2 is a pull urgency. Two choices stated for $$z^{(i)}(k) = \mathbf{1}\{y^{(i)}(k) \in Y_d^{(i)}\}.$$3 are

$$z^{(i)}(k) = \mathbf{1}\{y^{(i)}(k) \in Y_d^{(i)}\}.$$4

and a time-to-next-target form based on the next desired departure time. The resulting variants include PPS-SS, PPS-RS, and PPS-pSEL($$z^{(i)}(k) = \mathbf{1}\{y^{(i)}(k) \in Y_d^{(i)}\}.$$5), which use fixed, randomized, or periodic subset selection while retaining $$z^{(i)}(k) = \mathbf{1}\{y^{(i)}(k) \in Y_d^{(i)}\}.$$6 per-slot complexity (Dua et al., 2010).

This switch-based lineage clarifies the conceptual meaning of “push-pull” in scheduling theory. Push is backlog-centric service under matching constraints; pull is target-trace-centric service under the same constraints. Hybrid PPS is therefore a weighted superposition of congestion control and distortion control.

6. Terminological scope, adjacent usages, and limitations

The acronym PPS is not unique to scheduling. In weakly supervised temporal video grounding, PPS denotes a “Pull-Push Scheme” for learning Gaussian mixture proposals; it is a loss composition built from pulling and pushing losses and is not a scheduler (Kim et al., 2023). In constrained multi-objective optimization, PPS denotes “Push and Pull Search,” a two-stage framework in which an unconstrained push stage is followed by a constraint-aware pull stage (Fan et al., 2017). Related but distinct push-pull designs also appear in dual-mode wireless communication with wake-up radios (Cavallero et al., 31 Jul 2025), age-of-information analysis for gossip protocols (Srivastava et al., 2024), and push/pull update propagation for GPU graph analytics (Salvador et al., 2020). The term therefore requires domain qualification.

The main technical limitations of the digital-twin PPS are explicit. Its analysis assumes independent anomaly processes, equal activation probability among nodes that may transmit under the threshold, collisions involving at most three nodes per RE, independent per-node belief updates, a pure collision channel in the push subframe, perfect downlink control decoding, and error-free pull transmissions; no global optimality proof is claimed (Chiariotti et al., 29 Aug 2025). The switch-based hybrid designs inherit a different limitation profile: full matching-based pull control is $$z^{(i)}(k) = \mathbf{1}\{y^{(i)}(k) \in Y_d^{(i)}\}.$$7, and reduced-complexity subset methods depend on subset choice under non-uniform target intensities, although randomized and periodic subset selection restore full admissible-region coverage at $$z^{(i)}(k) = \mathbf{1}\{y^{(i)}(k) \in Y_d^{(i)}\}.$$8 cost (Dua et al., 2010).

A second common misconception is that push-pull systems always favor immediate push service for urgent traffic. The literature shows a more structured trade-off. In the digital-twin setting, push access is thresholded to satisfy a collision budget, while pull access is allocated by information gain. In the switch setting, pull service can dominate when the central requirement is adherence to target service traces rather than backlog minimization. PPS is therefore best understood not as a bias toward one mode or the other, but as a formal mechanism for arbitrating between two distinct notions of urgency.