Limit theorems for general shock models with infinite mean intershock times

Anderson, Kevin K.

doi:10.1017/s0021900200031089

Cited by 18 publications

(39 citation statements)

References 7 publications

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“…where random variable Θv represents node v's global pull-based attack environment ( e.g., the extent of malicious websites presence in cyberspace), random variable X (2) i (Θv) represents the magnitude (i.e., sophistication or power) of the ith pull-based attack against v with X Specifying when a node is compromised. Having specified when an attack is launched against a secure node v, we now specify when v becomes compromised.…”

Section: The New Approachmentioning

confidence: 99%

“…For example, a greater Jv (in a stochastic sense) would imply a severer local environment, namely more intense attacks (i.e., smaller attack inter-arrival times) and/or greater attack magnitudes (i.e., more powerful attacks). Similarly, random variables X (2) i (Θv) for i = 1, 2, . .…”

Section: The New Approachmentioning

confidence: 99%

“…Characterizing distribution function qv,c(t) under Assumption 1. Denote by F (1) i,r (·) and F (2) i,θ (·) respectively the distribution functions of X…”

Section: Assumption 1 (A) For Any V ∈ V and For Given Local Environmentmentioning

confidence: 99%

“…i (θ) , i ≥ 1 are independently and identically distributed samples of (X(2) (θ), Y (2) (θ)) .Now we present an upper bound for qv,c(t), denoted by q + v,c (t). Denote by F…”

mentioning

confidence: 99%

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A new approach to modeling and analyzing security of networked systems

Da¹,

Xu³

2014

Proceedings of the 2014 Symposium and Bootcamp on the Science of Security

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Modeling and analyzing security of networked systems is an important problem in the emerging Science of Security and has been under active investigation. In this paper, we propose a new approach towards tackling the problem. Our approach is inspired by the shock model and random environment techniques in the Theory of Reliability, while accommodating security ingredients. To the best of our knowledge, our model is the first that can accommodate a certain degree of adaptiveness of attacks, which substantially weakens the often-made independence and exponential attack inter-arrival time assumptions. The approach leads to a stochastic process model with two security metrics, and we attain some analytic results in terms of the security metrics.

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Section: The New Approachmentioning

confidence: 99%

Section: The New Approachmentioning

confidence: 99%

“…Characterizing distribution function qv,c(t) under Assumption 1. Denote by F (1) i,r (·) and F (2) i,θ (·) respectively the distribution functions of X…”

Section: Assumption 1 (A) For Any V ∈ V and For Given Local Environmentmentioning

confidence: 99%

“…i (θ) , i ≥ 1 are independently and identically distributed samples of (X(2) (θ), Y (2) (θ)) .Now we present an upper bound for qv,c(t), denoted by q + v,c (t). Denote by F…”

mentioning

confidence: 99%

See 2 more Smart Citations

A new approach to modeling and analyzing security of networked systems

Da¹,

Xu³

2014

Proceedings of the 2014 Symposium and Bootcamp on the Science of Security

View full text Add to dashboard Cite

show abstract

“…Shanthikumar and Sumita (1983), Abdel-Hameed (1986), Aven and Gaader (1987), Anderson (1987Anderson ( , 1988, Rangan and Grace (1988), Gut (1990), Lai and Yuan (1993), Gut and Husler (1999), and Li et al (1999) are just a few. Shanthikumar and Sumita (1983), Abdel-Hameed (1986), Aven and Gaader (1987), Anderson (1987Anderson ( , 1988, Rangan and Grace (1988), Gut (1990), Lai and Yuan (1993), Gut and Husler (1999), and Li et al (1999) are just a few.…”

Section: Introductionmentioning

confidence: 99%

Preventive replacement policy with cumulative repair-cost limit for a system under increasing failure rate shock model

Lai

Hung

2012

IJPQM

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A system is subject to shocks and these shocks are divided into two types: non-lethal and lethal shocks. Non-lethal shocks increase the failure rate of the system along with the age of the system and make the system eventually into a minor failure. Lethal shock whenever occurs makes the system critical failure. This shock process is called an increasing failure rate shock model. A minor failure is corrected by minimal repair, while a critical failure is rectified by replacement. In this paper, we propose a periodical replacement policy for a system, incorporating with the concept of cumulative repair-cost limit. While carrying out minimal repair, the repair cost and the accumulated repair cost are calculated. The system is replaced at age T or a minor failure at which the accumulated repair cost exceeds a predetermined threshold L or at a critical failure. The minimum-cost replacement policy is studied and a sensitivity analysis is conducted to evaluate the effect of model parameters on the optimal T*.Keywords: periodical replacement policy; increasing failure rate shock model; cumulative repair-cost limit; minimal repair.Reference to this paper should be made as follows: Lai, M-T. and Hung, C-W. (2012) 'Preventive replacement policy with cumulative repair-cost limit for a system under increasing failure rate shock model', Int.

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System Reliability with Multiple Failure Modes and Time Scales

Noorossana

Sabri-Laghaie

2015

Quality & Reliability Eng

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Lifetime of some systems can be measured based on multiple time scales. For instance, the lifetime of an airplane may be affected by its mileage or number of landings. Furthermore, most systems are exposed to competing risks. In this regard, time scales can accelerate the failure mechanism of these systems. In this paper, the behavior of systems is investigated under competing risks and multiple time scales. The time scales follow independent Poisson processes. As it is not straight forward obtaining closed-form relations for the system reliability, we have provided a parametric upper bound. Also, the upper bound can be tightened by considering an error function. The error function can be built by regression on a sample containing real values of system reliability for given time units. Performance of the upper bound is studied in two numerical examples and a case study. Results show that the obtained upper bound is very tight.

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Limit theorems for general shock models with infinite mean intershock times

Abstract: A general shock model in which the time intervals between shocks have infinite expectation is considered. Limit theorems for the first time the magnitude of a shock exceeds a and the historical maximum magnitude are given.

Cited by 18 publications

References 7 publications

A new approach to modeling and analyzing security of networked systems

A new approach to modeling and analyzing security of networked systems

Preventive replacement policy with cumulative repair-cost limit for a system under increasing failure rate shock model

System Reliability with Multiple Failure Modes and Time Scales

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