Bidirectional Age of Incorrect Information (BAoII)

• BAoII is a performance metric that quantifies bidirectional correctness by measuring the penalty for outdated self and remote state information in dynamic virtual environments.
• It leverages continuous-time Markov chains to derive closed-form expressions, analyzing reset conditions and error-cycle durations to guide optimal update policies.
• Cost-sensitive optimization reveals a transmission-cost threshold, indicating when an always-transmit policy minimizes the total cost in immersive digital twin and Metaverse applications.

Bidirectional Age of Incorrect Information (BAoII) is a performance metric for status updates in virtual dynamic environments (VDEs) such as the Metaverse and digital twins (DTs), where interacting entities must maintain mutual awareness of themselves and of one another. It extends the Age of Incorrect Information (AoII) from a unidirectional setting to bidirectional information exchange by quantifying the time-dependent penalty incurred when an entity has incorrect or outdated knowledge about itself, the other entity, or the other entity’s knowledge of it. In the formulation introduced for VDEs, BAoII is analyzed through a continuous-time Markov chain (CTMC), yielding a closed-form long-term expression and a transmission-cost threshold that characterizes optimal update strategies (Schiavo et al., 17 Jul 2025).

1. Conceptual foundations

BAoII is motivated by interactive settings in which correctness is not exhausted by the statement “the latest packet is recent.” In a VDE, each entity has a true state, a self-view derived from local sensing, and a remote view of the other entity derived from received packets. From the viewpoint of entity 1, the information configuration relevant to BAoII is incorrect when at least one of the following is wrong: entity 1’s knowledge of its own state, entity 1’s knowledge of entity 2’s state, or entity 2’s knowledge of entity 1’s state. The metric therefore couples self-awareness, awareness of the other, and reciprocated awareness in a single freshness-correctness construct (Schiavo et al., 17 Jul 2025).

This coupling distinguishes BAoII from classical Age of Information (AoI) and from standard AoII. AoI is a unidirectional age metric, typically written as $\Delta(t)=t-u(t)$, and does not distinguish whether the information is still correct. AoII penalizes only periods during which the estimate is wrong, with the penalty growing since the information became incorrect. BAoII generalizes this idea to a bidirectional interaction: reset is governed by collective correctness conditions rather than by a single source-monitor relation (Schiavo et al., 17 Jul 2025).

A common misunderstanding is to treat BAoII as merely “two AoII processes, one in each direction.” The original formulation rejects that interpretation. For a given entity, reset depends on a joint correctness condition across multiple knowledge components, so the metric is stricter than a simple sum of two independent unidirectional incorrectness ages. This suggests a metric tailored to co-manipulation, immersive collaboration, and DT-based control, where “I know where you are” and “you know where I am” are operationally coupled (Schiavo et al., 17 Jul 2025).

2. System model and CTMC representation

The basic model contains two entities, indexed by $i\in\{1,2\}$. Their true states are $X_1(t)$ and $X_2(t)$. Each entity has a self-estimate, $\widehat{X}_1(t)$ and $\widehat{X}_2(t)$, obtained from local measurements, and a remote estimate, $\widetilde{X}_1(t)$ and $\widetilde{X}_2(t)$, obtained from packets sent by the other side. Time is slotted with normalized slot duration $1$, and the analytical development subsequently uses a CTMC representation (Schiavo et al., 17 Jul 2025).

Correctness is abstracted into binary indicators: $$\widehat{Y}_i(t)=X_i(t)\oplus \widehat{X}_i(t),\qquad \widetilde{Y}_i(t)=X_{3-i}(t)\oplus \widetilde{X}_i(t),$$ where $i\in\{1,2\}$0 is bitwise XOR. A value of $i\in\{1,2\}$1 means correct knowledge and a value of $i\in\{1,2\}$2 means incorrect knowledge. The global information state is the four-bit vector

$i\in\{1,2\}$3

Each entity measures its own state with rate $i\in\{1,2\}$4 and transmits with rate $i\in\{1,2\}$5, where $i\in\{1,2\}$6. Transmissions are always tied to a measurement event: “measure only” occurs at rate $i\in\{1,2\}$7, while “measure + transmit” occurs at rate $i\in\{1,2\}$8. Drift in the true state occurs with rate $i\in\{1,2\}$9 and makes the corresponding self-view incorrect; because the other entity’s remote view then also becomes stale, drift imposes logical constraints on reachable four-bit configurations (Schiavo et al., 17 Jul 2025).

Two classes of four-bit states are impossible: configurations of the form $X_1(t)$0 and $X_1(t)$1. After excluding impossible combinations and grouping the remaining configurations, the CTMC has nine states. The state space is

$X_1(t)$2

State	Coding	Brief meaning
$X_1(t)$3	$X_1(t)$4	complete correct information for both
$X_1(t)$5	$X_1(t)$6	entity 1 fully correct; entity 2 wrong about entity 1
$X_1(t)$7	$X_1(t)$8	entity 1 wrong on self; entity 2 wrong about entity 1
$X_1(t)$9	$X_2(t)$0	each entity correct on self, wrong about the other
$X_2(t)$1	$X_2(t)$2	entity 1 wrong about entity 2; entity 2 in complete error
$X_2(t)$3	$X_2(t)$4	both correct on self, each wrong about the other
$X_2(t)$5	$X_2(t)$6	entity 1 wrong on self and on entity 2
$X_2(t)$7	$X_2(t)$8	entity 1 wrong about entity 2; entity 2 fully correct
$X_2(t)$9	$\widehat{X}_1(t)$0	complete error for both

The CTMC transitions are induced only by drift and by measurement with or without immediate transmission. In the symmetric case, the measurement processes are independent Poisson processes with common rate $\widehat{X}_1(t)$1, the drifts are independent Poisson processes with rate $\widehat{X}_1(t)$2, and the transmission probability $\widehat{X}_1(t)$3 is common to both entities (Schiavo et al., 17 Jul 2025).

3. Formal definition of BAoII

For entity 1, BAoII is defined as a time-dependent penalty

$\widehat{X}_1(t)$4

where $\widehat{X}_1(t)$5 is logical OR and $\widehat{X}_1(t)$6 is an increasing penalty function. The logical structure is the core novelty: BAoII is zero only when all three relevant correctness bits are zero, and it is positive whenever any one of them is incorrect (Schiavo et al., 17 Jul 2025).

The paper adopts a linear penalty function,

$\widehat{X}_1(t)$7

with

$\widehat{X}_1(t)$8

Thus BAoII for entity 1 grows linearly with the elapsed time since the last instant at which entity 1 knew itself correctly, entity 2 knew itself correctly, and entity 1 knew entity 2 correctly. This makes the penalty a sawtooth process: it ramps during error periods and resets when the required collective correctness condition is restored (Schiavo et al., 17 Jul 2025).

The reset structure is viewpoint-dependent. For entity 1, the reset states are $\widehat{X}_1(t)$9 and $\widehat{X}_2(t)$0, because both satisfy $\widehat{X}_2(t)$1, $\widehat{X}_2(t)$2, and $\widehat{X}_2(t)$3. In $\widehat{X}_2(t)$4, only entity 2’s knowledge of entity 1 is wrong, so $\widehat{X}_2(t)$5 is a reset state for entity 1’s BAoII but not a full-system mutual-awareness state. Full system-wide correctness corresponds to $\widehat{X}_2(t)$6 (Schiavo et al., 17 Jul 2025).

This distinction matters conceptually. BAoII is bidirectional, but it is not purely global. It can be instantiated from the viewpoint of a specific entity while still preserving the coupling between both directions of interaction. A plausible implication is that application-level optimization may need to distinguish between per-entity mutual-awareness objectives and stricter system-wide consistency objectives.

4. Closed-form behavior and analytical results

The CTMC uses the rate matrix $\widehat{X}_2(t)$7, with off-diagonal entries given by drift and measurement/transmission events and diagonal terms satisfying $\widehat{X}_2(t)$8. Under ergodicity, the stationary distribution $\widehat{X}_2(t)$9 satisfies

$\widetilde{X}_1(t)$0

For the reset states, the paper reports

$\widetilde{X}_1(t)$1

$\widetilde{X}_1(t)$2

Conditioned on being in one of the reset states, the renewal-cycle starting probabilities are

$\widetilde{X}_1(t)$3

From $\widetilde{X}_1(t)$4 or $\widetilde{X}_1(t)$5, the first transition into an error state occurs only via drift, leading to the first-error states $\widetilde{X}_1(t)$6, $\widetilde{X}_1(t)$7, or $\widetilde{X}_1(t)$8 (Schiavo et al., 17 Jul 2025).

The paper then introduces expected hitting times $\widetilde{X}_1(t)$9 to return to a reset state from each non-reset state, together with conditional jump probabilities

$\widetilde{X}_2(t)$0

These yield a linear system of standard Markov-chain hitting-time equations, with $\widetilde{X}_2(t)$1. One explicit relation reported in the appendix is

$\widetilde{X}_2(t)$2

After solving the system and averaging over the first-error mixture, the mean error-cycle duration is

$\widetilde{X}_2(t)$3

which becomes, in the symmetric case $\widetilde{X}_2(t)$4,

$\widetilde{X}_2(t)$5

The long-term BAoII is then

$\widetilde{X}_2(t)$6

This closed form decreases with increasing $\widetilde{X}_2(t)$7 and with increasing $\widetilde{X}_2(t)$8 (Schiavo et al., 17 Jul 2025).

One notable feature is the absence of the drift rate $\widetilde{X}_2(t)$9 from the final symmetric expression. This does not mean drift is irrelevant; rather, the paper’s interpretation is that drift determines how often error periods are triggered, whereas the average duration of an error episode depends on how quickly measurements and transmissions resolve the episode. A common misconception is therefore to expect $1$0 to appear explicitly in the closed-form BAoII expression; in this model, it does not (Schiavo et al., 17 Jul 2025).

5. Cost-sensitive optimization and design implications

To couple correctness-freshness performance with resource expenditure, the model assigns a measurement cost $1$1 and a transmission cost $1$2. Since measurements occur at rate $1$3 and transmissions at rate $1$4, the average per-unit-time resource cost is

$1$5

The total cost is defined as

$1$6

which, in the symmetric case, becomes

$1$7

This makes the central design trade-off explicit: increasing $1$8 and $1$9 reduces BAoII but increases resource expenditure (Schiavo et al., 17 Jul 2025).

The key optimization result is a transmission-cost threshold. There exists

$$\widehat{Y}_i(t)=X_i(t)\oplus \widehat{X}_i(t),\qquad \widetilde{Y}_i(t)=X_{3-i}(t)\oplus \widetilde{X}_i(t),$$0

such that, if $$\widehat{Y}_i(t)=X_i(t)\oplus \widehat{X}_i(t),\qquad \widetilde{Y}_i(t)=X_{3-i}(t)\oplus \widetilde{X}_i(t),$$1, the optimal transmission policy is

$$\widehat{Y}_i(t)=X_i(t)\oplus \widehat{X}_i(t),\qquad \widetilde{Y}_i(t)=X_{3-i}(t)\oplus \widetilde{X}_i(t),$$2

Hence, when transmission is sufficiently cheap, every measurement should be sent immediately. If measurement and transmission costs satisfy $$\widehat{Y}_i(t)=X_i(t)\oplus \widehat{X}_i(t),\qquad \widetilde{Y}_i(t)=X_{3-i}(t)\oplus \widetilde{X}_i(t),$$3, the threshold can be rewritten as

$$\widehat{Y}_i(t)=X_i(t)\oplus \widehat{X}_i(t),\qquad \widetilde{Y}_i(t)=X_{3-i}(t)\oplus \widetilde{X}_i(t),$$4

This is the paper’s analytic criterion for deciding when “measure + always transmit” is cost-optimal (Schiavo et al., 17 Jul 2025).

The numerical evaluation clarifies how this trade-off behaves in different application regimes. BAoII decreases with larger $$\widehat{Y}_i(t)=X_i(t)\oplus \widehat{X}_i(t),\qquad \widetilde{Y}_i(t)=X_{3-i}(t)\oplus \widetilde{X}_i(t),$$5 and larger $$\widehat{Y}_i(t)=X_i(t)\oplus \widehat{X}_i(t),\qquad \widetilde{Y}_i(t)=X_{3-i}(t)\oplus \widetilde{X}_i(t),$$6, and the lowest BAoII arises for $$\widehat{Y}_i(t)=X_i(t)\oplus \widehat{X}_i(t),\qquad \widetilde{Y}_i(t)=X_{3-i}(t)\oplus \widetilde{X}_i(t),$$7 with large $$\widehat{Y}_i(t)=X_i(t)\oplus \widehat{X}_i(t),\qquad \widetilde{Y}_i(t)=X_{3-i}(t)\oplus \widetilde{X}_i(t),$$8. Under example packet sizes of up to $$\widehat{Y}_i(t)=X_i(t)\oplus \widehat{X}_i(t),\qquad \widetilde{Y}_i(t)=X_{3-i}(t)\oplus \widetilde{X}_i(t),$$9 bytes for measurement packets and about $i\in\{1,2\}$00 kByte for transmission packets, the total cost $i\in\{1,2\}$01 shows that for small or moderate measurement rates, such as up to approximately $i\in\{1,2\}$02 Hz, $i\in\{1,2\}$03 can minimize total cost, whereas at higher $i\in\{1,2\}$04 the transmission-cost term dominates and lower values of $i\in\{1,2\}$05 become preferable. When $i\in\{1,2\}$06, the range of costs for which $i\in\{1,2\}$07 is optimal shrinks as both $i\in\{1,2\}$08 and $i\in\{1,2\}$09 increase, while for a low measurement rate such as $i\in\{1,2\}$10 Hz, $i\in\{1,2\}$11 can remain optimal even for high costs because fewer transmissions occur per unit time (Schiavo et al., 17 Jul 2025).

These results map directly onto VDE design classes. Highly immersive or teleoperation scenarios require updates every $i\in\{1,2\}$12–$i\in\{1,2\}$13 ms ($i\in\{1,2\}$14–$i\in\{1,2\}$15 Hz); in the numerical examples, the allowable costs in this regime are low, between $i\in\{1,2\}$16 and $i\in\{1,2\}$17. DT-driven logistics allows updates every $i\in\{1,2\}$18–$i\in\{1,2\}$19 s ($i\in\{1,2\}$20–$i\in\{1,2\}$21 Hz), tolerating higher costs between $i\in\{1,2\}$22 and $i\in\{1,2\}$23. The model therefore suggests different operating points for immersive XR and slower DT processes, even though both are evaluated with the same BAoII formalism (Schiavo et al., 17 Jul 2025).

6. Position within freshness-correctness research and recognized limitations

BAoII belongs to a line of work that replaces purely temporal freshness with correctness-aware freshness. AoII was introduced as a metric of the form

$i\in\{1,2\}$24

with $i\in\{1,2\}$25 capturing information mismatch and $i\in\{1,2\}$26 capturing time dissatisfaction, and later studied under constrained control problems where optimal policies take threshold or randomized-threshold forms (Maatouk et al., 2019, Maatouk et al., 2020). BAoII preserves the age-of-incorrectness logic but adapts it to mutual awareness in bidirectional VDE interactions, using a logical OR across multiple correctness bits and a reset condition defined by collective correctness rather than by a single received update (Schiavo et al., 17 Jul 2025).

The broader literature shows how correctness-aware freshness has diversified. AoII has been used as the objective in semantic-aware downlink optimization with rate-splitting multiple access, where RSMA is reported to achieve lower AoII than SDMA under multi-user interference (Dizdar et al., 2023). For two-state Markov monitoring, adjacent semantic metrics such as Version Innovation Age (VIA) and Age of Incorrect Version (AoIV) distinguish version lag from duration of incorrectness, while AoII remains the canonical time-based incorrectness age (Salimnejad et al., 2024). In random access channels without feedback and in remote tracking of Markov sources, AoII has also been analyzed via closed-form renewal expressions and via Whittle-index scheduling, respectively, indicating that the metric family extends beyond pairwise VDEs to shared-medium and multi-source settings (Munari, 2023, Kriouile et al., 2021).

The recognized limitations of the BAoII formulation are explicit. The model contains only two entities; correctness is binary and XOR-based; drifts and measurements are Poisson processes; transmission decisions are memoryless and tied to measurement events; transmissions are abstracted as instantaneous with no explicit queueing, packet loss, or variable delay; and the penalty function is linear, $i\in\{1,2\}$27. The authors identify natural extensions to multi-entity systems, heterogeneous policies with different $i\in\{1,2\}$28 and $i\in\{1,2\}$29, non-Markovian dynamics, alternative penalty functions, and integration with security and game theory (Schiavo et al., 17 Jul 2025).

Another conceptual nuance is often overlooked: entity-specific BAoII is not identical to system-wide mutual-awareness age. For entity 1, both $i\in\{1,2\}$30 and $i\in\{1,2\}$31 are reset states, while full global correctness requires $i\in\{1,2\}$32 alone. This distinction suggests that future multi-agent generalizations may need separate notions of local bidirectional awareness and full-system consistency, rather than a single scalar metric.