Probabilistic Latency Analysis (PLA)

• Probabilistic Latency Analysis (PLA) is a modeling framework that represents latency as a probability distribution, enabling analysis of delay probabilities rather than worst-case bounds.
• It employs discrete-state Markov chains, hypoexponential models, and closed-form formulas to quantify metrics like message delivery ratios, retransmission delays, and jitter.
• PLA is practically applied to optimize ROS 2 DDS configurations, cloud robotics, 5G systems, and replicated storage by tuning system parameters based on probabilistic insights.

Probabilistic Latency Analysis (PLA) denotes a class of analytical and data-driven methods in which latency is treated as a random variable or probability distribution rather than only as a single deterministic or worst-case quantity. In the ROS 2 Data Distribution Service (DDS) setting, PLA is a discrete-state, probabilistic model of how ROS 2’s DDS reliability mechanisms—heartbeats, ACK/NACK, and retransmissions—interact with packet loss and fragmentation to produce latency and jitter, and it aims to predict, not just bound, the distribution of delays (Lee et al., 14 Aug 2025). Closely related formulations appear in cloud robotics, 5G user-plane analysis, replicated key-value stores, and probabilistic DAG response-time analysis, where the central outputs are deadline-miss probabilities, quantiles, steady-state latency distributions, or response-time distributions rather than only a worst-case upper bound (Chen et al., 2024, Skocaj et al., 2023, Ali, 2020, Gao et al., 2 Apr 2025). A deterministic baseline also exists in static placement-type calculi, where latency is tracked as a type-level upper bound and the authors explicitly propose moving to probability distributions for latency weights (Reinhard, 2020).

1. Conceptual scope and probabilistic quantities

Across the cited work, PLA replaces the question “what is the worst possible latency?” with questions such as “what is the probability that latency exceeds a threshold?”, “what is the steady-state distribution of retransmission delay?”, or “what is the response-time distribution induced by branching and interference?” In cloud robotics, the fundamental event is a deadline miss, with $\epsilon_{ij} \triangleq P(L_{ij} > D)$ for an interface-server pair, and the effective system latency under replicated requests is the minimum of several random latencies (Chen et al., 2024). In replicated key-value stores, the central quantity is the inconsistency or staleness probability

$$p_t = \Pr[\text{read at offset } t \text{ returns a stale value}],$$

which depends jointly on quorum sizes and latency distributions (Ali, 2020). In 5G, user-plane latency is modeled as a Hypoexponential random variable, enabling probabilistic statements such as $\Pr\{L \le L_{\max}\} \ge 1 - \epsilon$ (Skocaj et al., 2023). In probabilistic DAG analysis, the output is a full probabilistic response-time distribution with correctness guarantees rather than a single conservative bound (Gao et al., 2 Apr 2025).

The ROS 2 DDS formulation makes this scope especially explicit. Its outputs are the Message Delivery Ratio (MDR) in steady-state, the probability distribution and expectation of retransmission latency, and jitter as latency variance or standard deviation (Lee et al., 14 Aug 2025). The latency of interest is end-to-end retransmission latency, defined as the time from a publish event until the message is fully acknowledged, including all retransmission cycles driven by heartbeats. Physical propagation and queuing delays are explicitly excluded; the analysis isolates latency induced by loss and DDS retransmission logic (Lee et al., 14 Aug 2025).

A recurring implication is that PLA is not limited to one mathematical style. The ROS 2 DDS paper uses a discrete-time Markov chain over unacknowledged messages; the 5G paper uses hypoexponential distributions and Bayesian probabilistic regression; the key-value-store paper uses exponential order statistics and closed-form inconsistency probabilities; the p-DAG paper uses probabilities of longest-path realizations and conservative aggregation into a response-time distribution (Lee et al., 14 Aug 2025, Skocaj et al., 2023, Ali, 2020, Gao et al., 2 Apr 2025).

2. ROS 2 DDS as a canonical PLA formulation

In ROS 2, DDS runs over UDP with RTPS, and reliability is implemented through periodic heartbeats, AckNack messages, and selective retransmissions. With QoS Reliability = RELIABLE and History = KEEP_ALL, the publisher stores all sent messages in a history cache, periodically sends Heartbeat messages, receives AckNack responses listing missing sequence numbers and fragments, and retransmits the missing data. In lossy wireless networks, the interaction among heartbeat period $h$, publish period $r$, message-to-MTU ratio $m$, and packet delivery rate $p$ is nontrivial, especially because DDS is unaware of transport-layer fragmentation: losing one UDP fragment of a large RTPS message causes retransmission of the entire RTPS message (Lee et al., 14 Aug 2025).

The core state variable is

$X = \text{number of unacknowledged RTPS messages},$

with distribution $P_k = \Pr(X = k)$. System evolution is modeled as a discrete-time Markov chain driven by two event types: publish events at times $T_{\mathrm{pub}^{(i)}} = i \cdot r$ and heartbeat events at times $$p_t = \Pr[\text{read at offset } t \text{ returns a stale value}],$$0. Each event applies an operator to the state distribution: $$p_t = \Pr[\text{read at offset } t \text{ returns a stale value}],$$1 or $$p_t = \Pr[\text{read at offset } t \text{ returns a stale value}],$$2 (Lee et al., 14 Aug 2025).

For a publish, one ROS 2 message is split into

$$p_t = \Pr[\text{read at offset } t \text{ returns a stale value}],$$3

UDP packets. If exactly $$p_t = \Pr[\text{read at offset } t \text{ returns a stale value}],$$4 out of $$p_t = \Pr[\text{read at offset } t \text{ returns a stale value}],$$5 packets fail, the paper uses

$$p_t = \Pr[\text{read at offset } t \text{ returns a stale value}],$$6

and the publish operator becomes

$$p_t = \Pr[\text{read at offset } t \text{ returns a stale value}],$$7

Intuitively, a publish adds a Binomial($$p_t = \Pr[\text{read at offset } t \text{ returns a stale value}],$$8) number of new unacknowledged units to the existing backlog (Lee et al., 14 Aug 2025).

For a heartbeat, retransmission can only occur if both the Heartbeat and the AckNack are delivered, with success probability $$p_t = \Pr[\text{read at offset } t \text{ returns a stale value}],$$9. If control-plane delivery fails, the number of unacknowledged messages remains unchanged. If it succeeds, retransmissions are batched into UDP packets with

$\Pr\{L \le L_{\max}\} \ge 1 - \epsilon$0

messages per packet when possible, and the number of retransmission packets for $\Pr\{L \le L_{\max}\} \ge 1 - \epsilon$1 unacknowledged messages is

$\Pr\{L \le L_{\max}\} \ge 1 - \epsilon$2

The transition kernel $\Pr\{L \le L_{\max}\} \ge 1 - \epsilon$3 encodes the probability that $\Pr\{L \le L_{\max}\} \ge 1 - \epsilon$4 unacknowledged messages become $\Pr\{L \le L_{\max}\} \ge 1 - \epsilon$5 after retransmission, and the heartbeat operator is

$\Pr\{L \le L_{\max}\} \ge 1 - \epsilon$6

This construction explicitly bridges middleware-level events—publish scheduling, heartbeat scheduling, ACK/NACK processing, retransmission batching—with transport-level events such as packet loss, fragmentation into $\Pr\{L \le L_{\max}\} \ge 1 - \epsilon$7 publish packets and $\Pr\{L \le L_{\max}\} \ge 1 - \epsilon$8 retransmission packets, and loss of heartbeat and AckNack packets (Lee et al., 14 Aug 2025).

3. Steady-state latency distributions, metrics, and empirical validation

Because publish and heartbeat events are periodic, the ROS 2 DDS model computes a periodic steady state rather than a single stationary vector. The steady-state distributions after publish events are defined as

$\Pr\{L \le L_{\max}\} \ge 1 - \epsilon$9

where each $h$0 is the distribution of unacknowledged messages immediately after a publish in one phase of the periodic cycle. The paper computes this periodic fixed point numerically by iterating through the deterministic event sequence, initializing from $h$1, and stopping when the distance between successive cycles is below a threshold $h$2 (Lee et al., 14 Aug 2025).

From $h$3, PLA defines the steady-state Message Delivery Ratio as

$h$4

Here $h$5 is the probability that immediately after a publish there are zero unacknowledged messages. Latency is then obtained from the incremental probability that all backlog is cleared at the $h$6-th heartbeat: $h$7 If $h$8 is the average time offset from the current publish to the next heartbeat, completion at the $h$9-th heartbeat occurs at

$r$0

and the phase-wise average latency is

$r$1

The overall average latency is $r$2, and jitter is the steady-state standard deviation

$r$3

The offset $r$4 is analyzed separately for the cases $r$5, $r$6, and $r$7, because heartbeat stopping and restarting behavior changes with the relation between periods (Lee et al., 14 Aug 2025).

Validation used two Ubuntu 22.04 machines with ROS 2 Humble and Fast DDS, connected by a direct Ethernet link with packet delivery rate controlled via Linux tc. The study varied message-to-MTU ratio $r$8, publish interval $r$9 ms, heartbeat interval $m$0 ms, and packet delivery rate $m$1, yielding $m$2 scenarios. For each scenario, 5000 ROS 2 messages were sent. Across all 270 scenarios, the average errors were 0.91 percentage points mean absolute error for MDR, 1.82% relative error for average latency, and 4.57% relative error for jitter (Lee et al., 14 Aug 2025).

The reported parameter effects are central to the practical value of PLA. Higher $m$3 yields fewer retransmissions, higher MDR, and lower latency and jitter. MDR is much more sensitive to message size $m$4 than to heartbeat interval $m$5, with a sharp drop as $m$6 rises, especially beyond about $m$7 MTU. Latency and jitter are highly sensitive to $m$8, with roughly linear growth for fixed $m$9, and a notable convex pattern when $p$0. Under the paper’s strict reliability assumptions, setting $p$1 emerges as an effective choice to reduce latency and jitter. The paper recommends keeping $p$2 and ideally $p$3 to avoid fragmentation and burst retransmissions (Lee et al., 14 Aug 2025).

4. PLA beyond ROS 2 DDS

In replicated key-value stores, PLA is used to quantify the latency-consistency trade-off in Dynamo-style partial quorums. Write latencies $p$4 and read latencies $p$5 are modeled as i.i.d. exponential random variables, and the object of analysis is not only latency but the probability that a read started $p$6 time units after write completion returns a stale value: $p$7 The paper derives a closed-form expression for the inconsistency probability by combining exponential order statistics, the dynamic quorum size $p$8, and conditional probabilities over the first $p$9 responding servers. For three-way replication, explicit formulas are given for cases such as $X = \text{number of unacknowledged RTPS messages},$0, $X = \text{number of unacknowledged RTPS messages},$1, and $X = \text{number of unacknowledged RTPS messages},$2, showing that worst-case combinatorial quorum arguments can be substantially looser than the probabilistic analysis (Ali, 2020).

In cloud robotics, latency reliability is defined around a deadline $X = \text{number of unacknowledged RTPS messages},$3, and the key quantity for an interface-server pair is $X = \text{number of unacknowledged RTPS messages},$4. If the system sends replicated requests over multiple independent interfaces and servers and uses the first response, then

$X = \text{number of unacknowledged RTPS messages},$5

and under independence

$X = \text{number of unacknowledged RTPS messages},$6

This is the mathematical basis of the paper’s claim that replicated servers with uncorrelated failures can exponentially reduce the probability of missing a deadline. The same framework motivates an interface-server assignment problem and is validated by experiments showing reduced mean latency and P99 latency in both a cloud-connected driving experiment and an indoor human-tracking task (Chen et al., 2024).

In 5G, the downlink user-plane latency is analytically decomposed into scheduling time, packet transmission time, HARQ delay, and retransmissions. The resulting latency is modeled as a Hypoexponential distribution, a sum of exponential stages with different rates, and this distribution is then connected to probabilistic regression, anomaly detection, and forecasting using Bayesian Learning and Machine Learning on Graphs. The paper validates the analytical hypoexponential formulation against empirical measurements and reports probabilistic regression with $X = \text{number of unacknowledged RTPS messages},$7, time-series forecasting with $X = \text{number of unacknowledged RTPS messages},$8, and spatial forecasting in which GraphSAGE outperforms a DNN baseline ($X = \text{number of unacknowledged RTPS messages},$9 versus $P_k = \Pr(X = k)$0) (Skocaj et al., 2023).

In parallel real-time systems, probabilistic DAG tasks yield response-time variability because the graph structure itself changes with probabilistic branches. The analysis of probabilistic DAG tasks avoids scenario enumeration by identifying the set $P_k = \Pr(X = k)$1 of all longest paths that can become critical in some scenario, computing the probability $P_k = \Pr(X = k)$2 that each path is realized and is longest, and then constructing a full probabilistic response-time distribution with correctness guarantees. The resulting conditional response-time bound is

$P_k = \Pr(X = k)$3

where $P_k = \Pr(X = k)$4 is the worst-case interfering workload when $P_k = \Pr(X = k)$5 is longest. Experiments report computation cost reduced by six orders of magnitude relative to enumeration-based analysis, with 1.04% average deviation and below 5% for most p-DAGs (Gao et al., 2 Apr 2025).

5. Deterministic baselines, assumptions, and recurring misconceptions

A common misconception is that probabilistic latency analysis is simply a softer form of worst-case response-time analysis. The placement-type calculus for geo-distributed computation shows the distinction sharply. There, latency is a deterministic type-level quantity: each remote message from peer type $P_k = \Pr(X = k)$6 to $P_k = \Pr(X = k)$7 contributes a natural-number weight $P_k = \Pr(X = k)$8, and the soundness theorem states that if a term has type-level latency bound $P_k = \Pr(X = k)$9 and reduces to a value with runtime latency $T_{\mathrm{pub}^{(i)}} = i \cdot r$0, then $T_{\mathrm{pub}^{(i)}} = i \cdot r$1. The Outlook then explicitly proposes refining the approach by using probability distributions for the latency weights $T_{\mathrm{pub}^{(i)}} = i \cdot r$2 instead of natural numbers, which is precisely the transition from deterministic static latency tracking to PLA (Reinhard, 2020).

Another misconception is that PLA necessarily models all sources of delay. The ROS 2 DDS formulation explicitly excludes physical propagation and queuing delays and isolates latency induced by loss and DDS retransmission logic (Lee et al., 14 Aug 2025). The 5G hypoexponential model focuses on downlink user-plane latency and intentionally excludes core network and transport delays (Skocaj et al., 2023). FogROS2-PLR relies on independence between paths and servers and treats latency distributions as sufficiently stable for optimization, while noting that sudden shifts can temporarily invalidate estimates (Chen et al., 2024). The probabilistic key-value-store analysis assumes i.i.d. exponential read and write latencies and has no explicit crash or Byzantine failure modeling (Ali, 2020). The p-DAG analysis uses safe upper bounds on joint events and Graham’s bound, so probabilities and response times are conservative rather than exact in the strictest sense (Gao et al., 2 Apr 2025).

These assumptions are not defects unique to PLA; they delimit what each formulation is intended to capture. A plausible implication is that PLA should be read as model-based probabilistic reasoning under explicit structural assumptions, not as an all-purpose substitute for measurement or as a universal hard-real-time guarantee.

6. Applications, optimization roles, and research outlook

The practical role of PLA is optimization under uncertainty. In ROS 2 DDS, practitioners can estimate packet delivery rate $T_{\mathrm{pub}^{(i)}} = i \cdot r$3, choose message size $T_{\mathrm{pub}^{(i)}} = i \cdot r$4, evaluate candidate $T_{\mathrm{pub}^{(i)}} = i \cdot r$5 pairs, and use the dynamic-programming steady-state computation to obtain MDR, average latency, and jitter before final empirical validation. The paper’s specific guidance is to keep messages at or below MTU when possible and to consider $T_{\mathrm{pub}^{(i)}} = i \cdot r$6 under strict reliability (Lee et al., 14 Aug 2025).

In cloud robotics, PLA is operationalized through multi-interface and multi-server replication, explicit routing through non-default interfaces, and interface-server selection that minimizes deadline-miss risk. The reported results include P99 latency improved by up to 3.7x compared to using one service provider in a driving experiment, and mean latency reduced by 36% with P99 reduced by 33% on a Stretch 3 human-tracking task (Chen et al., 2024). In 5G, the probabilistic formulation supports predictive QoS, anomaly detection, and forecasting from operator-accessible KPIs, including an anomaly-detection result with 36 true positives, 2 false negatives, 0 false positives, and 717 true negatives on a stadium-event dataset (Skocaj et al., 2023). In replicated storage, the analytical formulas make it possible to tune quorum parameters against both latency and consistency SLOs (Ali, 2020). In probabilistic DAG analysis, the distributional response-time view allows substantial reduction in the number of cores required relative to deterministic Graham-style analysis (Gao et al., 2 Apr 2025).

The research outlook is correspondingly broad. The placement-type calculus explicitly points toward probability distributions for latency weights and toward combining static analysis with actual latency measurements collected via monitoring and retrofitted in the type system using methods from continuous integration (Reinhard, 2020). Taken together, these works suggest a common trajectory: latency analysis is moving from single-point bounds toward compositional models that combine protocol structure, stochastic latency mechanisms, measured distributions, and optimization objectives. PLA, in that sense, is not one algorithm but a family of methods for deriving distributions, tail probabilities, and probabilistic guarantees from the interaction between communication structure and uncertainty.