Survival-Time Risk Beyond Low-Order Channel Models: Identifiability and Scheduling

Survival-Time Risk Beyond Low-Order Channel Models: Identifiability and Scheduling Ruifeng Zheng∗ , Zhihan Xu∗ , Pit Hofmann∗† , and Frank H. P. Fitzek∗† ∗ Deutsche Telekom Chair of Communication Networks, Dresden University of Technology, Germany † Centre for Tactile Internet with Human-in-the-Loop (CeTI), Dresden, Germany

arXiv:2611.03871v1 [cs.NI] 2 Nov 2026

{ruifeng.zheng, zhihan.xu, pit.hofmann, frank.fitzek}@tu-dresden.de

Abstract—This paper investigates whether low-order packetloss statistics suffice to assess survival-time risk under bursty packet losses, i.e., the probability that no update succeeds for a prescribed number of consecutive slots. Conventional models, such as the Gilbert model, characterize packet losses using the loss rate and mean burst length, but channels sharing these statistics may exhibit substantially different failure risks. We characterize the Markov channels one order above a given loworder fit that share its statistics and analyze their risks under acknowledgment-based scheduling with transmission budgets. We show that, with dense transmission attempts, risk is primarily governed by loss persistence beyond the modeled memory, and the worst-case channel need not be the most persistent one. Under limited budgets, retry spacing makes loss relapse probabilities important and can reverse the risk ordering of compatible channels. Numerical and measurement results show that low-order models can substantially underestimate the risk, highlighting the need for scheduling-dependent channel statistics. Index Terms—Survival time, bursty packet loss, Markov channel models, risk identifiability, transmission scheduling, model uncertainty.

I. I NTRODUCTION

I

NDUSTRIAL control and human-in-the-loop applications of the Tactile Internet, such as teleoperation, require timely and reliable information updates [1]. A prolonged interruption in successful packet delivery may violate the application’s survival-time requirement [2], even when the average packetloss rate is low. Assessing reliability therefore requires accounting for temporal correlations in packet losses rather than their average rate alone. In practice, bursty communication channels are commonly represented by low-order models, such as the Gilbert model [3] and first-order Markov models of fading [4], which provide tractable descriptions of packet losses using a small number of statistics, including the loss rate and mean burst length. However, different channels may share these statistics while exhibiting substantially different probabilities of survival-time violations. This work was funded by the German Research Foundation (DFG, Deutsche Forschungsgemeinschaft) as part of Germany’s Excellence Strategy – EXC 2050/2 – Project ID 390696704 – Cluster of Excellence “Centre for Tactile Internet with Human-in-the-Loop” (CeTI) of Dresden University of Technology. The authors also acknowledge the financial support by the Federal Ministry of Research, Technology and Space (BMFTR) of Germany in the program “Souverän. Digital. Vernetzt.” Joint project 6G-life, project identification number 16KIS2413K, the program “Verbundprojekt: Disruptive Kommunikationsparadigmen für technologische Souveränität, Resilienz und Shared Prosperity - Translation in Industrie und Aufbau innovativer Technologiedemonstratoren - CommUnity,” project identification number 16KISS012K.

The temporal structure of packet losses and its effect on reliability have been widely investigated [5]–[7]. Previous studies have examined Markov models of different orders using measured loss traces [8] and queueing analysis [9], as well as scheduling and remote estimation over Markovian channels with acknowledgment feedback [10]–[12]. Related work has also characterized information freshness, age violations, and estimation stability under correlated losses [13], [14]. Robust verification and control methods [15], [16] and reliability bounds under unknown dependence [17] can account for channel-model uncertainty. However, these approaches assume a given model or uncertainty set, not the set of channels a loworder fit cannot resolve. A fundamental question remains: to what extent can survival-time risk be determined from loworder channel statistics, and which additional statistics are needed for reliable risk assessment? This paper addresses this question by considering channels that share the same low-order packet-loss statistics but differ in their higher-order dependencies. We investigate their survival-time risks under acknowledgment-based scheduling with a finite transmission budget. The key observation is that the relevant channel statistics depend on the transmission schedule. When transmissions are attempted in every slot, consecutive packet losses primarily determine the failure risk. Under a limited transmission budget, however, successful channel opportunities during idle slots cannot reset the age of information. Consequently, the spacing between transmission attempts introduces additional dependence on loss relapses, potentially changing the relative reliability of channels with identical low-order statistics. The main contributions of this paper are as follows. (i) We characterize the Markov channels one order above a low-order fit as an explicit box whose maximum-entropy point is the fitted model. (ii) We bound the risk of every channel in this box and show that, with dense attempts, loss persistence beyond the modeled memory governs the risk to first order, while the worst case need not be the most persistent channel. (iii) We show that under a budget, retry spacing makes loss relapses matter, and construct two compatible channels whose risk order the budget reverses. (iv) We show by exact computation, fading simulations, and measured IEEE 802.15.4 links that low-order models can substantially underestimate the risk, even with a second-order fit. Full proofs, code, and data are available in [18].

II. S YSTEM M ODEL AND P ROBLEM S TATEMENT A. Channel and Compatible Class We consider a slotted communication link and denote by γt ∈ {0, 1} the potential packet-delivery outcome in slot t, where γt = 1 indicates success and γt = 0 indicates loss if a transmission is attempted. The latent channel evolves in every slot, including idle ones. We model {γt } as a stationary Markov process of order at most M , with context St = (γt−M , . . . , γt−1 ) and conditional success probability p(w) = Pr(γt = 1 | St = w),

η ≤ p(w) ≤ 1 − η,

(1)

where w ∈ {0, 1}M ranges over the contexts and η ∈ (0, 21 ) is a known bound that excludes deterministic transitions. Equivalently, the channel is characterized by its stationary (M + 1)-block distribution ν, with p(w) = ν(w1)/µ(w), where wa denotes the word w followed by a ∈ {0, 1} and µ(w) = ν(w0) + ν(w1) is the probability of context w. More generally, Pr(x) denotes the probability that consecutive outcomes form the word x; e.g., Pr(0j ) is the probability of j consecutive losses. Suppose a Markov model of order L < M is fitted using the stationary (L + 1)-block distribution c, i.e., c(x) = Pr(x) for words x of length at most L + 1. We assume c is known so as to separate model ambiguity from estimation error. For L = 1, these statistics determine the packet-loss rate and mean burst length, as in the Gilbert model. We define the compatible class CL (c; η) as the set of all stationary, η-positive Markov channels of order at most M sharing the same (L+1)-block distribution c. These channels are indistinguishable under the low-order statistics but may differ in their higher-order dependencies. We focus on M = L + 1, the smallest order at which such ambiguity arises, enabling an explicit characterization of the compatible class. B. Survival Time, Budget, and Risk We consider an application that tolerates at most D − 1 consecutive slots without a successful update, where D denotes the survival-time threshold [2]. In remote estimation, for example, the estimation error covariance is P̄ immediately after a successful update and evolves to Ψd (P̄ ) after d slots without one, where Ψ is the open-loop Lyapunov map [19]. If the application requires tr P ≤ Pmax and tr Ψd (P̄ ) is nondecreasing in d, the survival-time threshold is D = min{d : tr Ψd (P̄ ) > Pmax },

relapse 0 1 0 latent γt

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dense: dt

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every 2nd slot: dt

1

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slot t

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= D: failure 6

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Fig. 1. Latent channel outcomes γt and the resulting age dt under dense attempts and under attempts every second slot (D = 5; shaded: loss; circled: attempted slot).

For each attempt, the scheduler receives immediate and errorfree acknowledgment (ACK/NACK) feedback, so it observes γt only in slots with ut = 1. A success in an idle slot therefore goes unobserved and does not reset the age. In Fig. 1, for example, the success in a loss–success–loss pattern resets the age under dense attempts, whereas under attempts every second slot it falls into an idle slot and the age reaches D. The channel starts in its stationary regime, i.e., S1 is drawn from µ. A survival-time violation occurs when the age reaches D, which first happens at τ = min{t : dt ≥ D}.

(4)

The mission fails if this happens within its T slots, so the mission failure risk of a scheduling policy π under channel M is (5) RM (π) = PrπM (τ ≤ T ), and 1 − RM (π) is the corresponding mission reliability. The scheduler does not know the true channel, only the fitted model. We denote by πL the policy that minimizes the mission failure risk under the fitted L-th order model, subject to the transmission budget. For L = 1, the fitted model is characterized by the success probabilities after a loss and after a success and by their one-slot propagation, qo =

c(o1) , c(o)

f (x) = q0 + (q1 − q0 )x.

(6)

Since the scheduler observes the channel only when it transmits, it predicts a success probability of f δ−1 (qo ) from its most recent observation o, made δ slots earlier, where f j denotes the j-fold composition of f ; before the first observation, it predicts c(1). The policy πL then follows by finite-horizon dynamic programming over the state (t, d, r, o, δ), where r is the remaining budget, with ties broken in favor of idling.

(2)

where tr(·) denotes the trace and P the estimation error covariance. In each slot t, the scheduler at the transmitter selects ut ∈ {0, 1}, where ut = 1 indicates a transmission attempt. The scheduler has a budget of at most B attempts (e.g., to save energy [10]) over a mission of T slots, with the constraint enforced on every sample path. Let dt denote the number of slots since the last successful update, initialized as d0 = 0. Its evolution is ( 0, ut γt = 1, dt = (3) dt−1 + 1, otherwise.

C. Risk Identifiability The scheduler designs πL from the low-order statistics c, but the true channel may be any member of CL (c; η). The low-order statistics therefore determine the risk of πL only up to the interval [R− (c), R+ (c)], with R± (c) = max/min RM (πL )

(7)

M∈CL (c;η)

the largest and smallest risks over compatible channels, whereas the fitted model predicts the nominal risk Rnom = Rfit (πL ).

(8)

This ambiguity becomes critical when the interval contains the risk threshold. For a prescribed threshold ε, if R− (c) ≤ ε < R+ (c),

(9)

the same low-order statistics are consistent both with channels that meet the requirement RM (πL ) ≤ ε and with channels that violate it, so whether the requirement is met is not identifiable from c. Our objective is to characterize this ambiguity and to identify the additional channel statistics that resolve it under different transmission schedules. D. Structure of the Compatible Class The risk ambiguity arises because a survival-time violation typically depends on the channel over a window of D+1 slots, more than the L+1 slots that c describes. Over such windows, compatible channels can differ in higher-order dependencies that c does not determine. For M = L +1, these dependencies are captured by the probabilities xm = ν(0m0), m ∈ {0, 1}L , that two losses are separated by the pattern m. Lemma 1 (Box structure). For M = L Q + 1, the compatible hi class CL (c; η) is affinely equivalent to m [xlo m , xm ] in the coordinates xm , where xlo m = max{ηc(0m), c(m0) − (1 − η)c(1m)}, xhi m = min{(1 − η)c(0m), c(m0) − ηc(1m)}. The remaining probabilities are ν(0m1) = c(0m) − xm , ν(1m0) = c(m0) − xm , and ν(1m1) = c(1m) − c(m0) + xm . The fitted model corresponds to xnom = c(0m)c(m0)/c(m) m and, if η-positive, uniquely maximizes the entropy rate over the class. Proof: Stationarity and compatibility with c fix the row and column sums of each 2 × 2 table ν(· m ·), leaving one free hi parameter xm . The η-positivity constraints yield [xlo m , xm ]. The entropy rate H(γL+2 | γ1:L+1 ) is at most H(γL+2 | γ2:L+1 ), fixed by c, with equality iff the first and last symbols are conditionally independent given m, i.e., at xnom m . The coordinates xm describe two kinds of higher-order dependencies. The coordinate x0L = Pr(0L+2 ) determines the persistence beyond the modeled memory x0L ρ = Pr(γt = 0 | St = 0M ) = , (10) c(0L+1 ) the probability that a run of more than L losses continues. By Lemma 1, ρ ranges over [ρlo , ρhi ], where ρlo = max{η, 1 − (1 − η)β}, ρhi = min{1 − η, 1 − ηβ}, and β = c(10L )/c(0L+1 ). The fitted model gives the nominal value ρnom = c(0L+1 )/c(0L ). The other coordinates xm , m ̸= 0L , are probabilities of loss relapses, i.e., losses that recur after at least one intervening success. For L = 1, the only relapse coordinate is x1 = c(01)(1 − b), where b = p(01) is the success probability after a loss followed by a success and 1 − b is the relapse probability. As a running example throughout the paper, we consider L = 1, M = 2, η = 0.05, and (c(00), c(01), c(10), c(11)) = (0.12, 0.08, 0.08, 0.72), i.e., a packet-loss rate of 0.2 and

a mean burst length of 2.5, and, unless stated otherwise, 57 11 , 500 ] missions of T = 60 slots with D = 12. Here, x0 ∈ [ 250 1 11 and x1 ∈ [ 250 , 250 ], so ρ and b range independently over 11 19 , 20 ] and [0.45, 0.95], respectively, and the fitted model has [ 30 ρnom = 35 and b = 0.9. III. D ENSE ATTEMPTS : O NE S CALAR D ECIDES We first consider B ≥ T , for which dense attempts, i.e., attempts in every slot, minimize the age pathwise and are thus optimal under every compatible channel, so the risk interval [R− (c), R+ (c)] is the range of the optimal risk over the class. A. Risk Bounds for Open-Loop Schedules To relate the risk to the coordinates of Lemma 1, we bound it by loss-pattern probabilities for any open-loop schedule, which attempts in a fixed set S ⊆ {1, . . . , T } of slots and has risk R(S). The age reaches D in slot s + D ≤ T when a success at s ∈ S0 = {0} ∪ S, or the mission start for s = 0, is followed by losses at all attempted slots in (s, s + D]. We denote this eventPby As and the expected number of such events by ΛS = s Pr(As ), which approximates R(S) when violations are rare. Theorem 1 (Risk bounds for open-loop schedules). For any open-loop schedule S, ΛS (1 − JS κS ) ≤ R(S) ≤ ΛS ,

(11)

where κS = maxs∈S, w Pr(As | Ss = w) and JS = maxs∈S0 |{s′ ∈ S : s + D < s′ ≤ T − D}|, with the maxima taken over s such that s + D ≤ T . Proof: Continue the age process beyond the first violation and let It = 1{dt = D}. Then It = 1Piff t = s + D for some T s ∈ S0 and As occurs, so ΛS =PE t=1 It ; as the age grows by one per slot, R(S) = Pr( t It ≥ 1). For two hits at t < t′ , the reset at s′ = t′ − D must occur after t. Since It is determined by the history through s′ − 1, the Markov ′ −1 ) ≤ κS . Since there are at property gives Pr(As′ | FsP ′ most JS eligible later hits, t<t′ E[It It ] ≤ JS κS ΛS . The union bound and the second Bonferroni inequality establish the result. B. Worst-Case Persistence With dense attempts, S = {1, . . . , T }, and we omit the subscript S. For M = L + 1 ≤ D, a run of more than L losses continues with probability ρ, so Pr(A0 ) = Pr(0D ) = c(0M )ρk and Pr(As ) = Pr(10D ) = c(0M )(1 − ρ)ρk for s = 1, . . . , n, where k = D − M and n = T − D. Hence Λ = c(0M )ρk [1 + n(1 − ρ)].

(12)

Moreover, Jκ ≤ θM Λ with θM = maxv: vM =1 Pr(0M | v)/(c(0M )(1 − ρ)), since κ ≤ ρk maxvM =1 Pr(0M | v) and J = (T − 2D)+ ≤ n, so 0 ≤ Λ − R ≤ θM Λ2 by Theorem 1. As all compatible channels share c(0M ), ρ thus determines the dense-attempt risk, and the extremes of Λ over [ρlo , ρhi ] locate the best and worst compatible channels, to leading order in the

risk R (dense attempts)

10 10

−1

10

−2

10

−3

10

Proposition 1 (Periodic attempts and decimated loss runs). For a stationary binary process {γt } and D ≥ g,  (g)  ΛSg = Pr(g) (0Dg ) 1 + ng (1 − ρDg ) , (13)

0

D = 10 T = 10 T = 30

nominal (L = 1 model)

−4

0.4

0.5

0.6

0.7

where Dg = ⌊D/g⌋ and ng = ⌊(T − D)/g⌋. For M = L + 1, P (1) (M ) ρj = ρ for j ≥ M , whereas ρ1 = m∈{0,1}L xm /c(0).

T = 60 T = 200 0.8

0.9

persistence ρ Fig. 2. Exact dense-attempt risk versus ρ, the loss probability after two losses, for several mission lengths T in the running example (D = 10, b = 0.7; markers: exact maxima; dotted lines: ρ⋆ from Theorem 2 for n > k).

rare-failure regime θM Λ ≪ 1, whereas the relapse coordinates affect only higher-order terms. Theorem 2 (Worst-case persistence). For dense attempts with k ≥ 1, define φ(ρ) = ρk [1+n(1−ρ)], so that Λ = c(0M )φ(ρ). (i) If n ≤ k, φ is increasing, and Λ is maximized at ρhi . (ii) If n > k, φ is unimodal with maximum at ρ⋆ = k(n + 1)/[(k + 1)n]. Over the compatible class, Λ is maximized at ρw = min{max{ρ⋆ , ρlo }, ρhi } and minimized at an endpoint. (iii) The most persistent channel maximizes Λ for every T ≥ D if and only if k ≥ ρhi /(1−ρhi ). (iv) For any compatible ρ′ < ρhi , φ(ρhi )/φ(ρ′ ) ≥ (ρhi /ρ′ )k (1 − ρhi )/(1 − ρ′ ). Consequently, in the rare-failure regime, R+ /Rnom and R+ /R− grow at least geometrically with D, with respective per-slot factors ρhi /ρnom and ρhi /ρlo . Proof: Claims (i)–(iii) follow from the sign of φ′ (ρ) = ρk−1 [k(n + 1) − (k + 1)nρ]. Claim (iv) follows from (1 + nx)/(1 + ny) ≥ x/y for 0 < x ≤ y and Theorem 1; see [18] for details. Theorem 2 reveals a persistence–frequency tradeoff. Increasing ρ prolongs loss bursts but, for fixed c(0M ), decreases their entry rate c(0M )(1−ρ). This tradeoff can place the worst case at an interior ρ rather than at ρhi , as Fig. 2 shows for D = 10. The most persistent channel is the worst there only for T = 10, where n ≤ k, whereas for T = 60 the exact risk peaks at ρ = 0.910, close to ρ⋆ = 0.907, with R+ = 0.289 against Rnom = 0.042 and R− = 1.3×10−3 . The fitted model, although the maximum-entropy member of the class, is thus not generally the worst case. IV. B UDGETED ATTEMPTS : T HE S TATISTIC C HANGES When B < T , the scheduler must leave some slots idle, and πL is in general optimal only under the fitted model. As in Fig. 1, the risk depends on attempted losses, which idle slots may separate, so ρ alone need not decide it. A. Periodic Attempts and Decimated Loss Runs Periodic attempts every g slots, i.e., Sg = {g, 2g, . . .} ∩ [1, T ], are the simplest schedules with idle slots. They observe the channel only through the g-decimated process (g) (g) γi = γig , with law Pr(g) and run persistences ρj = Pr(g) (0j+1 )/ Pr(g) (0j ), i.e., the probabilities that the next attempt fails after j consecutive attempted losses.

Proof: With As as in Theorem 1, A0 requires losses at the Dg attempts in (0, D]. Each eligible As , s ∈ Sg , requires a success followed by Dg attempted losses. Stationarity therefore gives ΛSg = Pr(g) (0Dg ) + ng Pr(g) (10Dg ), yielding Eq. (13). For M = L + 1, Pr(0j ) = c(0M )ρj−M for j ≥ M . Moreover, P (M ) Pr(γt = 0, γt+M = 0) = m ν(0m0), which yields ρ1 . Thus, periodic attempts probe the loss-run tail of the decimated channel, as dense attempts probe that of γt through ρ in Eq. (12). The low-order fit fixes the first L run persistences (g) (M ) of γt but, in general, not ρ1 for g > L. In particular, ρ1 combines persistence and relapse coordinates, so the relapses now enter the leading term. In the running example, varying b changes the dense-attempt risk by at most about 2% at any 11 lowering b from 0.95 to 0.45 fixed ρ, whereas at ρ = 30 multiplies the risk of two-slot periodic attempts by 34.6 and that of πL with B = 15 by 9.05. The effect of spacing can also be characterized asymptotically through the matrix Qg (w, w′ ) = Pr(St+g+1 = w′ | St+1 = w) on contexts ending in a loss, with intermediate outcomes marginalized, and its spectral radius σg . Here σ1 = ρ, σg ≥ ρg , and Qg is entrywise positive for g ≥ M . For fixed g, as D → ∞, if log(1 + T − D) = o(D) and either g = 1 or g ≥ M, 1 1 lim log R(Sg ) = log σg (14) D→∞ D g by Perron–Frobenius theory and Theorem 1; see [18]. Per attempt, σg thus takes the role of ρ but also depends on the relapse coordinates, and a channel with larger ρ but smaller σg than another is riskier under dense attempts but safer under g-spaced ones for large D. B. Closed-Loop Scheduling The same mechanism extends to policies that react to ACK/NACK feedback, such as πL . For a deterministic policy π, the attempts in (s, s + D] after a success at slot s with observation history h form a deterministic retry pattern Rπ (s, h) as long as they all fail. We denote by Es,h,w the event of a success at s before any violation, with history h and context Ss+1 = w, where s = 0 again represents the mission start. A partition of {τ ≤ T } by the last success before τ and the Markov property give R(π) = Λπ , where X Λπ = Pr(Es,h,w ) s,h,w s+D≤T

× Pr γRπ (s,h) = 0 | Ss+1 = w .

(15)

In the running example with B = 15, the retry pattern of πL at the mission start is {5, 7, 9, 10, 11, 12}, which becomes denser as the deadline approaches, so both relapses and persistence enter its loss probability.

πL , B = 15

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C2 C1

C2 0.4

B = 60 (both b) B = 15, b = 0.95

C1

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0.9

persistence ρ Fig. 3. Exact risk of πL versus ρ for B = 60 and B = 15 at b = 0.45 and 0.95 (T = 60, D = 12; circles: B = 60; squares: B = 15).

C. Reversal of the Risk Order Even at moderate D, a budget can reverse the risk order of compatible channels. In the running example, we consider C1 and C2 with (ρ, b) = (0.44, 0.95) and ( 11 30 , 0.45), so C1 is more persistent, whereas C2 relapses more often. Proposition 2 (Budget reverses the risk order). For T = 60 and D = 12, the exact risks, rounded for display, satisfy 1) R(C1 ) = 9.10 × 10−4 > R(C2 ) = 1.66 × 10−4 for dense attempts (πL , B = 60); 2) R(C1 ) = 3.46 × 10−3 < R(C2 ) = 4.45 × 10−2 for periodic attempts (g = 2, B = 30); 3) R(C1 ) = 1.30 × 10−2 < R(C2 ) = 6.04 × 10−2 for πL (B = 15). Proof: The risks are probabilities of absorption by slot T in finite Markov chains over channel context, age, and scheduler state, computed exactly in rational arithmetic; see [18]. The reversal reflects the statistics each schedule observes. The run persistences of the 2-decimated channel are lower (2) for C1 at every depth, from ρ1 = 0.284 versus 0.44 (Proposition 1) to their limit σ2 = 0.238 versus 0.433, so by Eq. (14) dense and two-slot periodic attempts rank C1 and C2 oppositely for large D, and for n = T − D = 48 already for every even D from 4 to 40. Under the fitted model, 70% of πL ’s retries after a NACK are two slots apart, as in item 2). In Fig. 3, b barely affects the risk with dense attempts (B = 60) but matters with B = 15, where C1 and C2 swap their order. On a 47 × 21 grid in (ρ, b), reducing B from 60 to 15 moves the grid worst case from (0.925, 0.95), near ρ⋆ = 0.928, to (0.899, 0.45), a less persistent channel that relapses more. Thus, under a budget, the loss-run tail at the attempted slots determines the risk, and no schedule-independent scalar, such as ρ or a Gilbert–Elliott parameter [20], ranks compatible channels by risk under every schedule. V. N UMERICAL AND M EASUREMENT-BASED E VALUATION This section quantifies how far the low-order fit misjudges the risk over the compatible class and outside it. A. Ambiguity over the Compatible Class With dense attempts in the running example, the fitted model reports Rnom = 1.46×10−2 (D = 12) and 2.07×10−4 (D = 20), against worst cases of 0.242 and 0.142 on the

loss rate

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Fig. 4. Worst compatible-to-nominal risk ratio R+ /Rnom across Gilbert fits (M = 2 class, T = 60, η = 10−4 ).

47×21 grid of Sec. IV-C, refined in ρ; with B = 15 at D = 12, it reports 5.97 × 10−2 against 0.286. It thus underestimates the worst case by 4.8× to 680×, and by Theorem 2(iv) the gap grows with D. Closing the dense-attempt gap requires an accurate estimate of ρ, since ∂ log Λ/∂ρ ≈ 14.3 at ρ = ρnom and D = 12, so an error of 0.05 in ρ changes Λ locally by a factor of about two. For dense attempts, the closed-form certificate Λmax = c(0M )φ(ρw ) bounds R+ by Theorems 1 and 2; at 0.253 and 0.143, it exceeds the grid worst case by less than 5% and 1%. A rectangular robust MDP [16] returns 0.80–0.98, because it lets each context’s success probability range over its projected interval and be chosen adversarially in every slot, as is the initial context, and thus discards the coupling of p(00) and p(10) enforced by Lemma 1. In Fig. 4, we vary the fit over loss rates from 0.5% to 20% and mean burst lengths from 1.25 to 5, each at the smallest D (6 to 42) for which it certifies Rnom ≤ ε = 10−4 with dense attempts; the same D is used for B = 15. We use η = 10−4 here, since η = 0.05 leaves the class empty for 27 of these 36 fits. With dense attempts, the worst compatible channel then has 46 to 5.3 × 103 times the nominal risk, so R− ≤ ε < R+ for every fit, and with B = 15 it has 4 to 640 times that of πL . Many worst cases lie on the boundary of the class and depend on η, including the dense ones with mean burst length 5 and loss of at least 1%, where n ≤ k, as in Theorem 2(i). B. Channels Outside the Class Real channels need not have order M = L + 1. Gilbert– Elliott channels [20], for instance, are in general not of finite order. For each eB ∈ [0.65, 1], the one with loss probabilities (eG , eB ) = (0, eB ) and matched transitions shares the Gilbert fit of the running example and hence πL ; eB = 1 gives the Gilbert channel. A loss reveals the bad state, so the persistence is 0.6 at every burst depth, and the dense risks agree within 0.3%. With B = 15, however, the risk grows from 5.97×10−2 at eB = 1 to 0.127 at eB = 0.65, as successes in the bad state create relapses, which lower b from 0.90 to 0.51. We also threshold Rayleigh fading with the Jakes spectrum [21] at normalized Doppler fD , and composite fading at fD = 0.03 with log-normal shadowing of 6 dB and a correlation time of 200 slots, to a given loss rate. For T = 60

TABLE I T RACE - DRIVEN DENSE - ATTEMPT RISK R AND FITTED - TO - TRACE RISK RATIOS FOR FADING CHANNELS AND MEASURED LINKS

Dense attempts Channel

Loss

Rayleigh, fD =0.01 5% Rayleigh, fD =0.03 5% Shadowed 5% Shadowed 1% Shadowed 1% FlockLab 12% FlockLab 19% †

πL , B = 15

D

R

L=1

L=2

L=1

20 10 18 12 16 15 18

0.018†

1.7 1.1 0.30 0.086 0.015 0.17 0.10

1.5 0.54 0.32 0.21 0.053 0.39 0.26

1.7 1.4 0.40 0.19 0.034 0.26 0.18

0.025 0.016 0.0025 0.0010 0.050 0.080

Below R− of the M = 2 class (η = 10−4 ); all other risks lie in [R− , R+ ].

and D chosen per channel, Table I reports the trace-driven risk and the fitted-to-trace risk ratios of order-L fits; πL is designed on the Gilbert fit. The error follows the run persistences of Proposition 1 and is two-sided. In Rayleigh fading, persistence first falls with burst depth, and the Gilbert fit is conservative. Under shadowing, it grows with depth, from 0.76 to 0.92 at 5% loss, and for D ≥ 12 the fit underestimates the risk by a factor that grows with D, up to 68× at 1% loss and D = 16. One more model order does not remove the error, as the secondorder fit is still 19× too low in that case and 1.8× too low in Rayleigh fading with fD = 0.03 and D = 10, where the Gilbert fit is nearly exact. Since 1 − b ≤ 0.011, relapses are rare, and the budgeted errors follow the dense ones. We further use IEEE 802.15.4 traces of the FlockLab testbed [22], namely 100-strobe tests on the 171 directed TelosB links with 1–25% loss and at least 500 tests, with one slot per strobe and missions sliding within tests. The records are too short for 10−4 , so D is the smallest survival time at which the Gilbert fit certifies Rnom ≤ 10−2 with dense attempts. The measured dense risk then exceeds the certified 10−2 on 55% of the links and the nominal risk on 64%, by up to 12×, and by more than two standard errors over tests on 42% and 47%, respectively. It is significantly lower on 20%, all of which have mean burst length below 1.9, and the second-order fit is too low on 85%, significantly on 49%. With B = 15, the nominal risk of πL is significantly too low on 33% of the links, by up to 5.4×, and significantly too high on 36%. The dense error grows with loss rate and burstiness, with rank correlations of 0.93 and 0.92, and the M = 2 class brackets the measured dense risk on the burstiest link of each of four loss bands, including the two links in Table I. Variation between the pooled tests may also contribute to these errors. VI. C ONCLUSION This paper examined how far low-order packet-loss statistics determine survival-time risk under acknowledgment-based scheduling with a transmission budget, and bounded this risk for every channel in the compatible class of higher-order Markov channels. With dense attempts, the persistence beyond the modeled memory decides the risk to first order; as higher persistence makes bursts longer but rarer, the most persistent channel need not be the worst. With a limited budget, spaced retries bring relapse probabilities into the risk and can reverse

the risk order of compatible channels. Compatible channels can be orders of magnitude riskier than the fit predicts, and on fading channels and measured links the fit can underestimate the risk even with one more model order. Risk should therefore be assessed from the loss-run tail that the schedule observes, not from the fit. The structural results assume channels one order above the fit; for channels of unknown order, estimating this tail from limited traces is left for future work. R EFERENCES [1] F. H. P. Fitzek et al., Eds., Tactile Internet with Human-in-the-Loop. Academic Press, 2021. [2] 3GPP, “Service requirements for cyber-physical control applications in vertical domains; Stage 1,” 3rd Generation Partnership Project, TS 22.104, v19.2.0, Jun. 2024. [3] E. N. Gilbert, “Capacity of a burst-noise channel,” Bell Syst. Tech. J., vol. 39, no. 5, pp. 1253–1265, 1960. [4] M. Zorzi, R. R. Rao, and L. B. Milstein, “Error statistics in data transmission over fading channels,” IEEE Trans. Commun., vol. 46, no. 11, pp. 1468–1477, 1998. [5] T. Hößler, M. Simsek, and G. P. Fettweis, “Mission reliability for URLLC in wireless networks,” IEEE Commun. Lett., vol. 22, no. 11, pp. 2350–2353, 2018. [6] L. Scheuvens et al., “State-aware resource allocation for wireless closedloop control systems,” IEEE Trans. Commun., vol. 69, no. 10, pp. 6604– 6619, 2021. [7] U. Singh, S. R. Khosravirad, and O. Tirkkonen, “Survival mode transmission strategy for URLLC services over time-correlated Rayleigh fading channel,” IEEE Trans. Commun., vol. 74, pp. 12 894–12 910, 2026. [8] M. Yajnik et al., “Measurement and modelling of the temporal dependence in packet loss,” in Proc. IEEE INFOCOM, 1999, pp. 345–352. [9] X. Yu, J. W. Modestino, and X. Tian, “The accuracy of Markov chain models in predicting packet-loss statistics for a single multiplexer,” IEEE Trans. Inf. Theory, vol. 54, no. 1, pp. 489–501, 2008. [10] G. Yao, A. M. Bedewy, and N. B. Shroff, “Age-optimal low-power status update over time-correlated fading channel,” IEEE Trans. Mobile Comput., vol. 22, no. 8, pp. 4500–4514, 2023. [11] M. Dutta and R. Singh, “Optimal scheduling policies for remote estimation of autoregressive Markov processes over time-correlated fading channel,” in Proc. IEEE Conf. Decis. Control (CDC), 2023, pp. 6455– 6462. [12] X. Wang et al., “Partially observable restless bandits for age-optimal scheduling over Markov channels,” IEEE Trans. Mobile Comput., vol. 25, no. 11, pp. 20 997–21 012, 2026. [13] J. Wu et al., “Kalman filtering over Gilbert–Elliott channels: Stability conditions and critical curve,” IEEE Trans. Autom. Control, vol. 63, no. 4, pp. 1003–1017, 2018. [14] Q. Guan and X. Xu, “Impact of channel memory on the data freshness,” IEEE Commun. Lett., vol. 27, no. 1, pp. 80–84, 2023. [15] T. Quatmann et al., “Parameter synthesis for Markov models: Faster than ever,” in Proc. Int. Symp. Autom. Technol. Verif. Anal. (ATVA), ser. LNCS, vol. 9938, 2016, pp. 50–67. [16] W. Wiesemann, D. Kuhn, and B. Rustem, “Robust Markov decision processes,” Math. Oper. Res., vol. 38, no. 1, pp. 153–183, 2013. [17] Z. Ge and E. A. Jorswieck, “Reliability in statistically dependent networks: Bounds, linear programming, and scalability,” IEEE Trans. Commun., vol. 74, pp. 2731–2746, 2026. [18] R. Zheng et al., “survival-time-risk: Code, data, and full proofs,” GitHub repository, 2026. [Online]. Available: https://github.com/ ruifeng105/survival-time-risk [19] B. Sinopoli et al., “Kalman filtering with intermittent observations,” IEEE Trans. Autom. Control, vol. 49, no. 9, pp. 1453–1464, 2004. [20] E. O. Elliott, “Estimates of error rates for codes on burst-noise channels,” Bell Syst. Tech. J., vol. 42, no. 5, pp. 1977–1997, 1963. [21] D. J. Young and N. C. Beaulieu, “The generation of correlated Rayleigh random variates by inverse discrete Fourier transform,” IEEE Trans. Commun., vol. 48, no. 7, pp. 1114–1127, 2000. [22] R. Jacob et al., “Dataset: Wireless link quality estimation on FlockLab – and beyond,” in Proc. 2nd Workshop Data Acquisition To Analysis (DATA), 2019, pp. 57–60.

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