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Cited by 7Network Traffic and Congestion ControlAdvanced Queuing Theory AnalysisProbability and Risk Models

The DOI matches its Crossref record; 27 of the 27 references it cites were found in a registry or an index; 4 of the 19 sources its claims cite were checked for support.

Distribution of the Number of Retransmissions of Bounded Documents

Jelenković, Predrag R., Skiani, Evangelia D.·2015·Advances in Applied Probability
DOI 10.1239/aap/1435236982
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Abstract

Retransmission-based failure recovery represents a primary approach in existing communication networks that guarantees data delivery in the presence of channel failures. Recent work has shown that, when data sizes have infinite support, retransmissions can cause long (-tailed) delays even if all traffic and network characteristics are light-tailed. In this paper we investigate the practically important case of bounded data units 0 ≤ L b ≤ b under the condition that the hazard functions of the distributions of data sizes and channel statistics are proportional. To this end, we provide an explicit and uniform characterization of the entire body of the retransmission distribution ℙ[ N b > n ] in both n and b . Our main discovery is that this distribution can be represented as the product of a power law and gamma distribution. This rigorous approximation clearly demonstrates the coupling of a power law distribution, dominating the main body, and the gamma distribution, determining the exponential tail. Our results are validated via simulation experiments and can be useful for designing retransmission-based systems with the required performance characteristics. From a broader perspective, this study applies to any other system, e.g. computing, where restart mechanisms are employed after a job processing failure.

Overview

AI snapshot

Written from the paper’s full text.

TL;DR

For bounded data units retransmitted until a channel availability period exceeds their size, the paper gives a uniform asymptotic characterization of the entire retransmission-count distribution when data-size and channel-availability tails are related by a power law with a slowly varying factor. The distribution is shown to be a product of a power law and a gamma/exponential tail, with simulations supporting the approximation.

Key findings

  • Under \(P[L>x]=\ell(P[A>x]^{-1})P[A>x]^\alpha\), \(\alpha>0\), with \(\ell\) slowly varying, \(P[N_b>n]\) is uniformly approximated for all large \(n,b\) by a product of a power law and an incomplete gamma function \(\Gamma_1(\cdot,\alpha)\).
  • The power law dominates when \(-n\log P[A\le b]\downarrow 0\) (equivalently \(nP[A>b]\le C\)), while the geometric/exponential tail dominates when \(nP[A>b]\to\infty\); the transition point \(n_b\) satisfies \(n_b\log P[A\le b]\approx \alpha\log n_b\).
  • The power-law region grows as the distributions of \(L\) and \(A\) become lighter-tailed. Simulations with Weibull tails at \(b=8\), \(\alpha=4\) show the Gaussian-like \(k=2\) case follows almost entirely the power law, whereas the heavier \(k=1/2\) case has a smaller power-law main body.
  • For integer \(\alpha\) and \(\ell(x)\equiv 1\), the paper gives an exact expression for \(P[N_b>n]\) (Proposition 2).

Methods

  • The model uses i.i.d. channel availability periods \(\{A_i\}\); a data unit \(L_b\in[0,b]\) is retransmitted from the beginning until \(A_i>L_b\), with \(P[L_b\le x]=P[L\le x]/P[L\le b]\).
  • The analysis derives a general upper bound (Proposition 1), a uniform logarithmic-scale characterization (Theorem 2), and exact asymptotics (Theorems 3 and 4) under proportional hazard/power-law tail assumptions, plus special-case propositions.
  • Simulation validation uses \(10^8\) or more independent samples of \(N_{b,i}\), ensuring at least 100 occurrences in the lightest tail shown, with examples using exponential, Weibull, and gamma distributions.

Limitations

  • The paper states that the theorems require \(n\) and \(b\) sufficiently large; Theorem 4 needs more restrictive assumptions on \(\ell\) (eventually differentiable with \(\ell'(x)x/\ell(x)\to 0\)). It also notes the approximation can be accurate for relatively small \(b\) if \(A\) is light-tailed, and simulations show good agreement even for small \(n,b\).
  • For the \(\alpha<1\) case in Theorem 4, the proof omits details “to avoid unnecessary repetitions.”
  • Evident from the text: validation is simulation-only for selected distributions and parameters; no real network traces or empirical deployment data are analyzed.

Model-written summary. Check it against the paper before relying on it.

Integrity signals

Registry

Crossref · 10.1239/aap/1435236982

Cites

27 of 27 references verified0 citation errors · 0 not checkable

Statistics

Nothing to recompute.

Verification

Claims the paper makes

4 of 19 cited sources checked · 3 supported · 0 flagged · accuracy not yet measured

  • Premisecitation
    § 1. Introduction · p. 2

    The generic channel with failures model results in power law distributions when the distributions of L and A have a matrix exponential representation.

    “It was first recognized in [6] that this model results in power law distributions when the distributions of L ≡ L ∞ and A have a matrix exponential representation, and this result was rigorously proved and further generalized in [3], [11], and [16].”
    Quote in the paperSubstantial claim1 of 3 sources checked
    • cites [6] [6] Fiorini, P. M., Sheahan, R. and Lipsky, L. (2005). On unreliable computing systems when heavy-tails appear as a result of the recovery procedure. ACM SIGMETRICS Performance Evaluation Rev. 33, 15-17. - source
    • cites [3] [3] Asmussen, S. et al. (2008). Asymptotic behavior of total times for jobs that must start over if a failure occurs. Math. Operat. Res. 33, 932-944. - source
    • cites [11] [11] Jelenkovi´ c, P. R. and Tan, J. (2007). Can retransmissions of superexponential documents cause subexponential delays? In Proc. 26th IEEE INFOCOM'07 , IEEE, pp. 892-900. -
  • Premisecitation
    § 1. Introduction · p. 2

    Several well-known retransmission based protocols in different layers of networking architecture can lead to power law delays.

    “It has been shown that several well-known retransmission based protocols in different layers of networking architecture can lead to power law delays, e.g. ALOHA type protocols in MAC layer [12], [14] and end-to-end acknowledgments in transport layer [10], [16] as well as in other layers [11].”
    Quote in the paperNot a substantial claim1 of 3 sources checked
    • cites [12] [12] Jelenkovi´ c, P. R. and Tan, J. (2007). Is ALOHA causing power law delays? In Managing Traffic Performance in Converged Networks (Lecture Notes Comput. Sci. 4516 ). Springer, Berlin, pp. 1149-1160. -
    • cites [14] [14] Jelenkovi´ c, P. R. and Tan, J. (2009). Stability of finite population ALOHA with variable packets. Preprint. Available at http://arxiv.org/abs/0902.4481v2. - source
    • cites [10] [10] Jelenkovi´ c, P. R. and Tan, J. (2007). Are end-to-end acknowlegements causing power law delays in large multi-hop networks? In 14th Informs Applied Probability Conference , APS, pp. 9-11. -
  • Findingnone given
    § 2.1. Exact asymptotics · p. 10

    The number of retransmissions is geometrically distributed given the data size Lb.

    “First, notice that the number of retransmissions is geometrically distributed given the data size Lb ,”
    Quote in the paperNot a substantial claim
    • The authors state this as an assumption: The channel availability process {A, Ai} is independent and identically distributed.
    • The authors state this as an assumption: The channel is continuously available during periods {Ai} and fails between these periods.
  • Findingown data
    § 3. Simulation experiments · p. 23

    In Example 2, the exponential asymptote appears to fit well starting from nb ≈ α e^b, i.e. nb ≈ 6, 15, 100 for b = 1, 2, 4, respectively.

    “Roughly speaking, we can see from Figure 3(a) that the exponential asymptote appears to fit well starting from nb ≈ α e b , i.e. nb ≈ 6 , 15 , 100 for b = 1 , 2 , 4, respectively.”
    Quote in the paperNot a substantial claim
  • Findingown data
    § 3. Simulation experiments · p. 24

    In Example 4, the analytic approximation works nicely even for small values of n and b although the conditions in the theorems require n and b to be large.

    “Interestingly, our analytic approximation works nicely even for small values of n and b although the conditions in our theorems require n and b to be large.”
    Quote in the paperNot a substantial claim
  • Probably not a substantial claim. 13
  • It may state two separate claims as one. 10
  • The quote only partly states this claim. 1

  • Probably not a substantial claim. It may state two separate claims as one.

    Premisecitation
    § Abstract · p. 1

    When data sizes have infinite support, retransmission-based protocols can cause heavy-tailed behavior and possibly zero throughput even if data sizes and channel failures are light-tailed.

    “Recent work has shown that, when data sizes have infinite support, retransmissions can cause long (-tailed) delays even if all traffic and network characteristics are light-tailed.”
    • Also stated in: § 1. Introduction · p. 2 · hedged
      “Recent work [10]-[12], and [16] has shown that, when the data size distribution has infinite support, all retransmission-based protocols could cause heavy-tailed behavior and possibly result in zero throughput, regardless of how light-tailed the distributions of data sizes and channel failures are.”
    Quote in the paperSubstantial claim1 of 4 sources checked
    • cites [10] [10] Jelenkovi´ c, P. R. and Tan, J. (2007). Are end-to-end acknowlegements causing power law delays in large multi-hop networks? In 14th Informs Applied Probability Conference , APS, pp. 9-11. -
    • cites [11] [11] Jelenkovi´ c, P. R. and Tan, J. (2007). Can retransmissions of superexponential documents cause subexponential delays? In Proc. 26th IEEE INFOCOM'07 , IEEE, pp. 892-900. -
    • cites [12] [12] Jelenkovi´ c, P. R. and Tan, J. (2007). Is ALOHA causing power law delays? In Managing Traffic Performance in Converged Networks (Lecture Notes Comput. Sci. 4516 ). Springer, Berlin, pp. 1149-1160. -
    • cites [16] [16] Jelenkovi´ c, P. R. and Tan, J. (2013). Characterizing heavy-tailed distributions induced by retransmissions. Adv. Appl. Prob. 45, 106-138. Extended version available at http://arxiv.org/pdf/0709.1138.pdf - source

  • Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.

    Interpretationreasoning
    § 2.1. Exact asymptotics · p. 15

    Corollary 1 is an immediate consequence of Theorem 3 and represents a small generalization of Theorem 2.1 of [16].

    “The following corollary is an immediate consequence of Theorem 3 and it represents a small generalization of Theorem 2.1 of [16].”
    Quote in the paperNot a substantial claim1 of 1 sources checked
    • cites [16] [16] Jelenkovi´ c, P. R. and Tan, J. (2013). Characterizing heavy-tailed distributions induced by retransmissions. Adv. Appl. Prob. 45, 106-138. Extended version available at http://arxiv.org/pdf/0709.1138.pdf - source

      The retrieved passage from §4.6, “Proof of Theorem 2.5” (page unavailable), discusses a lemma based on Nagaev’s results. Searches for Corollary 1, Theorem 3, and the claimed generalization of Theorem 2.1 of [16] found no passage in our copy addressing that relationship. (LLM, conf=0.74)

      Experimental — accuracy not yet measured

  • The quote only partly states this claim.

    Findingown data
    § 1. Introduction · p. 3

    The approximation admits an explicit form as a product of a power law and the gamma distribution.

    “Surprisingly, the approximation admits an explicit form (see Theorems 3 and 4)”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The hazard functions of the distributions of data sizes and channel statistics are proportional.
    • The authors state this as an assumption: ℓ(x) is positive and locally bounded.

  • It may state two separate claims as one.

    Findingown data
    § 3. Simulation experiments · p. 24

    In Example 4, both asymptotes are needed to approximate the entire distribution well: the 'Initial Asymptote' fits well the first part of the distribution, whereas the 'Tail Asymptote' is inaccurate in the beginning but works well for the tail.

    “Hereby, we see from Figure 4(b) that both asymptotes are needed to approximate the entire distribution well, i.e. the 'Initial Asymptote' fits well the first part of the distribution, whereas the 'Tail Asymptote' is inaccurate in the beginning but works well for the tail.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim.

    Findingnone given
    § 2.1. Exact asymptotics · p. 8

    The characterization is explicit in that it is a product of a power law and the gamma distribution.

    “As noted earlier in the introduction, this characterization is explicit in that it is a product of a power law and the gamma distribution”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The hazard functions of the distributions of data sizes and channel statistics are proportional.
    • The authors state this as an assumption: ℓ(x) is positive and locally bounded.

  • Probably not a substantial claim. The paper’s conclusions may not depend on it.

    Premisecitation
    § 1. Introduction · p. 1

    The same restart mechanism is at the core of modern communication networks where retransmissions are used on all protocol layers to guarantee data delivery in the presence of channel failures.

    “In [11], it was noted that the same mechanism is at the core of modern communication networks where retransmissions are used on all protocol layers to guarantee data delivery in the presence of channel failures.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [11] [11] Jelenkovi´ c, P. R. and Tan, J. (2007). Can retransmissions of superexponential documents cause subexponential delays? In Proc. 26th IEEE INFOCOM'07 , IEEE, pp. 892-900. -

  • It may state two separate claims as one.

    Interpretationreasoning
    § 1. Introduction · p. 1

    Retransmission phenomena can lead to zero throughput and system instabilities.

    “In particular, retransmission phenomena can lead to zero throughput and system instabilities, and therefore need to be considered carefully for the design of fault tolerant systems.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim. The paper’s conclusions may not depend on it.

    Premisecitation
    § 1. Introduction · p. 1

    Power law number of retransmissions and delay occur whenever the hazard functions of the data and failure distributions are proportional.

    “Furthermore, [11] shows that the power law number of retransmissions and delay occur whenever the hazard functions of the data and failure distributions are proportional.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [11] [11] Jelenkovi´ c, P. R. and Tan, J. (2007). Can retransmissions of superexponential documents cause subexponential delays? In Proc. 26th IEEE INFOCOM'07 , IEEE, pp. 892-900. -

  • Probably not a substantial claim.

    Interpretationcitation
    § 1. Introduction · p. 2

    Such truncated power law distributions are characterized by a power law main body and an exponentially bounded tail.

    “Such distributions are characterized by a power law main body and an exponentially bounded tail.”
    Quote in the paperSubstantial claim

  • It may state two separate claims as one.

    Interpretationreasoning
    § 2.1. Exact asymptotics · p. 8

    When -n log P[A ≤ b] ↓ 0, the power law dominates; when -n log P[A ≤ b] → ∞, the integral determines the tail with the geometric leading term.

    “Note that when -n log P [ A ≤ b ] ↓ 0, the power law dominates, whereas when -n log P [ A ≤ b ] → ∞ , the integral determines the tail with the geometric (exponential) leading term.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim.

    Study design and conductown data
    § 3. Simulation experiments · p. 22

    10^8 or more independent samples were needed to ensure at least 100 independent occurrences in the lightest end of the tail presented in the figures, thus providing a good confidence interval.

    “This number of samples was needed to ensure at least 100 independent occurrences in the lightest end of the tail that is presented in the figures ( Nb,i ≥ n max), thus providing a good confidence interval.”
    Quote in the paperSubstantial claim

  • It may state two separate claims as one.

    Interpretationreasoning
    § 1. Introduction · p. 3

    In the logarithmic-scale approximation, the first term corresponds to the power law part n^-α and the second part describes the exponential (geometric P[A ≤ b]^n) tail.

    “Note that the first term in the preceding approximation corresponds to the power law part n -α of the distribution, while the second part describes the exponential (geometric P [ A ≤ b ] n ) tail.”
    Quote in the paperSubstantial claim

  • It may state two separate claims as one.

    Interpretationreasoning
    § 3. Simulation experiments · p. 23

    The transition from the power law main body to the exponential tail occurs roughly at nb ≈ ̄G(b)^{-1}; hence, the lighter the tail of the distribution of A, the larger the size of the power law region.

    “This increase in the power law region can be inferred from our theorems, which show that the transition from the power law main body to the exponential tail occurs roughly at nb ≈ ¯ G(b) -1 . Hence, the lighter the tail of the distribution of A , the larger the size of the power law region.”
    Quote in the paperSubstantial claim

  • It may state two separate claims as one.

    Premisecitation
    § 1. Introduction · p. 2

    The exponential behavior appears only for very small probabilities, often meaning that the number of retransmissions of interest may fall inside the region where the distribution behaves as a power law.

    “However, the exponential behavior appears only for very small probabilities, often meaning that the number of retransmissions of interest may fall inside the region where the distribution behaves as a power law.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim. It may state two separate claims as one.

    Interpretationown data
    § 3. Simulation experiments · p. 23

    Proposition 3 gives the exact asymptotic equation for the region n ≫ 1/ ̄G(b) and lends merit to Theorems 2 and 4.

    “Note that this proposition gives the exact asymptotic equation for the region n ≫ 1 / ¯ G(b) and lends merit to our Theorems 2 and 4.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The slowly varying function is ℓ(x) ≡ 1.
    • The authors state this as an assumption: b is fixed.

  • It may state two separate claims as one.

    Interpretationown data
    § 3. Simulation experiments · p. 23

    For small b, the power law asymptotics cover a smaller region of the distribution of Nb, and as n increases the exponential tail becomes more evident and eventually dominates; as b becomes large, the exponential tail becomes less distinguishable.

    “For both cases, we deduce that for small b the power law asymptotics cover a smaller region of the distribution of Nb and, as n increases, the exponential tail becomes more evident and eventually dominates. As b becomes large-recall that b → ∞ corresponds to the untruncated case where the power law phenomenon arises-the exponential tail becomes less distinguishable.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim. The paper’s conclusions may not depend on it.

    Premisecitation
    § 1. Introduction · p. 2

    The power law region will grow faster than exponential if the distributions of A and Lb are lighter than exponential.

    “It was argued in Example 3 of [11] that the power law region will grow faster than exponential if the distributions of A and Lb are lighter than exponential.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [11] [11] Jelenkovi´ c, P. R. and Tan, J. (2007). Can retransmissions of superexponential documents cause subexponential delays? In Proc. 26th IEEE INFOCOM'07 , IEEE, pp. 892-900. -

  • Probably not a substantial claim. The paper’s conclusions may not depend on it.

    Interpretationown data
    § 3. Simulation experiments · p. 23

    A point nb such that -nb log(1 - ̄G(b)) ≈ nb ̄G(b) = α log nb represents the transition from the power law to the exponential tail.

    “Informally, we could say that a point nb such that -nb log ( 1 - ¯ G(b)) ≈ nb ¯ G(b) = α log nb represents the transition from the power law to the exponential tail.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim. The paper’s conclusions may not depend on it.

    Interpretationreasoning
    § 2.1. Exact asymptotics · p. 8

    This may be the reason why the paper obtains accurate results in its simulation examples in Section 3 for small values of b.

    “This may be the reason why we obtain accurate results in our simulation examples in Section 3 for small values of b .”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim. The paper’s conclusions may not depend on it.

    Premisecitation
    § 2.1. Exact asymptotics · p. 9

    The incomplete gamma function can be easily computed using the well known asymptotic approximation in Section 6.5.32 of [1].

    “Remark 5. Note here that the incomplete gamma function Gamma1 (α, x) = ∫ ∞ x z α -1 e -z d z can be easily computed using the well known asymptotic approximation (see Section 6.5.32 in [1]), as x →∞ ,”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [1] [1] Abramowitz, M. and Stegun, I. A. (1964). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables . U.S. Government Printing Office, Washington, D.C. -

  • Probably not a substantial claim. The paper’s conclusions may not depend on it.

    Study design and conductown data
    § 1. Introduction · p. 3

    All of the preceding results are validated via simulation experiments in Section 3.

    “All of the preceding results are validated via simulation experiments in Section 3.”
    Quote in the paperNot a substantial claim
21 of 21

  • The authors state this as an assumption.

    The channel is of unit capacity.

    “Without loss of generality, we assume that the channel is of unit capacity.”

  • The authors state this as an assumption.

    The hazard functions of the distributions of data sizes and channel statistics are proportional.

    “under the condition that the hazard functions of the distributions of data sizes and channel statistics are proportional.”

  • The authors state this as an assumption.

    ℓ(x) is positive and locally bounded.

    “If not directly implied by our assumptions, ℓ(x) is assumed positive and locally bounded.”

  • The authors state this as an assumption.

    The channel availability process {A, Ai} is independent and identically distributed.

    “The channel dynamics are described by the independent and identically distributed (i.i.d.) channel availability process { A,Ai } i ≥ 1”

  • The authors state this as an assumption.

    The channel is continuously available during periods {Ai} and fails between these periods.

    “where the channel is continuously available during periods { Ai } and fails between these periods.”

  • The authors state this as an assumption.

    ℓ(x) is slowly varying.

    “where ℓ(x) is slowly varying”

  • The authors state this as an assumption.

    ℓ(x) is eventually differentiable with ℓ'(x)x/ℓ(x) → 0 as x → ∞.

    “and eventually differentiable with ℓ ′ (x)x/ℓ(x) → 0 as x →∞”

  • The authors state this as an assumption.

    The slowly varying function is ℓ(x) ≡ 1.

    “we need ℓ(x) ≡ 1”

  • The authors state this as an assumption.

    α is a positive integer.

    “α is a positive integer”

  • The authors state this as an assumption.

    b is fixed.

    “Let b be fixed.”
10 of 10

Self-check

Percentages, sums and reported differences re-computed from the paper’s own numbers.

Each contradiction between two passages listed here was confirmed by an independent check.

  • Incomplete gamma function arguments are reversed

    Minor

    Theorem 3 defines Gamma1(x, α) using x as the lower integration limit and α as the exponent parameter. Remark 5 gives the same integral as Gamma1(α, x), reversing those arguments.

1 of 1

Citations

Cites

Flagged first, then by citations

Research works printed but not linked: 5

  1. Jelenković, 2007, 14th Informs Applied Probability Conference, 9
  2. Asmussen, 2014, Analytic and Stochastic Modelling Techniques and Applications, 253
  3. Jelenković, 2007, Proc. 26th IEEE INFOCOM'07, 892
  4. Tan, 2010, Proc. IEEE INFOCOM'10, 1334
  5. Nair, 2010, Proc. IEEE INFOCOM'10, 965

Cited by

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Related

Metadata

Authors
Jelenković, P. R., & Skiani, E. D.
Affiliations
Columbia University (US)
Corresponding author
  • Jelenković, Predrag R.
  • Skiani, Evangelia D.
Venue
Advances in Applied Probability
Publisher
Cambridge University Press (CUP)
Published
2015-06
Volume
47
Issue
2
Pages
425-449
ISSN
  • 0001-8678 (ISSN-L)
  • 1475-6064
DOI
10.1239/aap/1435236982
OpenAlex
W2963087972
Semantic Scholar
c85190ca45fe4702708740b243c5d6cb1cc6034d
Cited by
7
References
23
Citations by year
2017: 1 · 2016: 1 · 2015: 4
FWCI
2.30
Type
journal article
Language
en
Open access
closed
Licence
www.cambridge.org/core/terms (unspecified, 266 d)
Locations
  • Advances in Applied Probability (publishedVersion)
  • Project Euclid (Cornell University) (submittedVersion)
Funders
National Science Foundation (0915784)
Topics
  • Physical Sciences › Computer Science › Computer Networks and Communications › Network Traffic and Congestion Control
  • Social Sciences › Decision Sciences › Management Science and Operations Research › Advanced Queuing Theory Analysis
  • Social Sciences › Decision Sciences › Management Science and Operations Research › Probability and Risk Models
Keywords
retransmission, fault recovery, power law, gamma distribution, hazard function, restart mechanism, communication networks

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