scholarly journals Preservation properties of stochastic orders by transformation to Harris family

2018 ◽  
Vol 38 (2) ◽  
pp. 441-458
Author(s):  
Somayeh Abbasi ◽  
Mohammad Hossein Alamatsaz
Keyword(s):  
Random Number ◽  
Stochastic Orders ◽  
Stochastic Comparisons ◽  
Number Of Components ◽  
Lifetime Analysis ◽  
Reliability Systems ◽  
Series Systems

Stochastic comparisons of lifetime characteristics of reliability systems and their components are of common use in lifetime analysis. In this paper, using Harris family distributions, we compare lifetimes of two series systems with random number of components, with respect to several types of stochastic orders. Our results happen to enfold several previous findings in this connection. We shall also show that several stochastic orders and ageing characteristics, such as IHRA, DHRA, NBU, and NWU, are inherited by transformation to Harris family. Finally, some refinements are made concerning related existing results in the literature.

Some aging properties of parallel and series systems with a random number of components

2014 ◽  
Vol 61 (3) ◽  
pp. 238-243 ◽  
Author(s):  
Nil Kamal Hazra ◽  
Asok K. Nanda ◽  
Moshe Shaked
Keyword(s):  
Random Number ◽  
Number Of Components ◽  
Aging Properties ◽  
Series Systems

scholarly journals Optimization Problem in Series Systems with Random Number of Components from the Family of Power Series Distributions

2022 ◽  
Vol 15 (2) ◽  
pp. 481-504
Author(s):  
Motahare ZaeamZadeh ◽  
Jafar Ahmadi ◽  
Bahareh Khatib Astaneh ◽  
◽  
◽  
...  
Keyword(s):  
Power Series ◽  
Optimization Problem ◽  
Random Number ◽  
Number Of Components ◽  
The Family ◽  
In Series ◽  
Power Series Distributions ◽  
Series Systems

scholarly journals Stochastic Comparisons of Parallel and Series Systems Comprising Multiple-Outlier Scale Components

2022 ◽  
Vol 15 (2) ◽  
pp. 341-362
Author(s):  
Ebrahim Amini Seresht ◽  
Ghobad Barmalzan ◽  
◽  
Keyword(s):  
Stochastic Comparisons ◽  
Series Systems

The distribution of the maximum of a random number of random variables with applications

1973 ◽  
Vol 10 (01) ◽  
pp. 122-129 ◽  
Author(s):  
Janos Galambos
Keyword(s):  
Service Time ◽  
Joint Distribution ◽  
Random Number ◽  
Random Variables ◽  
Stochastic Independence ◽  
Fixed Integer ◽  
Number Of Components ◽  
Distribution Of The Maximum ◽  
Distribution Of Elements ◽  
Mild Assumption

The asymptotic distribution of the maximum of a random number of random variables taken from the model below is shown to be the same as when their number is a fixed integer. Applications are indicated to determine the service time of a system of a large number of components, when the number of components to be serviced is not known in advance. A much slighter assumption is made than the stochastic independence of the periods of time needed for servicing the different components. In our model we assume that the random variables can be grouped into a number of subcollections with the following properties: (i) the random variables taken from different groups are asymptotically independent, (ii) the largest number of elements in a subgroup is of smaller order than the overall number of random variables. In addition, a very mild assumption is made for the joint distribution of elements from the same group.

scholarly journals Stochastic Comparisons of Symmetric Supermodular Functions of Heterogeneous Random Vectors

2013 ◽  
Vol 50 (02) ◽  
pp. 464-474
Author(s):  
Antonio Di Crescenzo ◽  
Esther Frostig ◽  
Franco Pellerey
Keyword(s):  
Queueing Theory ◽  
Queueing Networks ◽  
Random Variables ◽  
Random Vectors ◽  
Convex Order ◽  
Stochastic Comparisons ◽  
Increasing Convex Order ◽  
Supermodular Functions ◽  
Stop Loss ◽  
Series Systems

Consider random vectors formed by a finite number of independent groups of independent and identically distributed random variables, where those of the last group are stochastically smaller than those of the other groups. Conditions are given such that certain functions, defined as suitable means of supermodular functions of the random variables of the vectors, are supermodular or increasing directionally convex. Comparisons based on the increasing convex order of supermodular functions of such random vectors are also investigated. Applications of the above results are then provided in risk theory, queueing theory, and reliability theory, with reference to (i) net stop-loss reinsurance premiums of portfolios from different groups of insureds, (ii) closed cyclic multiclass Gordon-Newell queueing networks, and (iii) reliability of series systems formed by units selected from different batches.

COMBINED SERIES AND PARALLEL SYSTEMS SUBJECT TO INDIVIDUAL VERSUS OVERARCHING DEFENSE AND ATTACK

2013 ◽  
Vol 30 (02) ◽  
pp. 1250056 ◽  
Author(s):  
KJELL HAUSKEN
Keyword(s):  
Parallel Systems ◽  
Series System ◽  
Number Of Components ◽  
Unit Costs ◽  
Special Cases ◽  
In Series ◽  
Individual Protection ◽  
Overarching Protection ◽  
Series Systems ◽  
Analytical Expressions

A system of components can be in series, parallel, or combined series/parallel. The components and system are protected individually and overarchingly by a defender, and attacked individually and overarchingly by an attacker. Both layers of protection have to be breached for an attack to be successful. Each component, and the system as a whole, have vulnerabilities determined by individual and overarching protection and attack. The agents choose their effort variables simultaneously and independently to maximize their utilities. Each component and the system have unit costs of protection and attack, and a contest intensity. We show for both the parallel and series systems that the defender always prefers overarching and individual protection and attack, while the attacker always prefers individual protection and attack. Analytical expressions are developed for the agents' effort variables, each individual component's vulnerability, and the system vulnerability. The expenditure ratio, between individual protection and attack, and overarching protection and attack, is shown to increase in the number of components for the parallel system, and decrease in the number of components for the series system. Special cases are considered and interpreted. Comparisons are made with only individual protection and attack. The model is applicable to determine how the defender and attacker should strike the balance between choosing efforts to protect and attack components individually versus overarchingly.

scholarly journals Extension of the past lifetime and its connection to the cumulative entropy

2015 ◽  
Vol 52 (04) ◽  
pp. 1156-1174 ◽  
Author(s):  
Antonio Di Crescenzo ◽  
Abdolsaeed Toomaj
Keyword(s):  
Random Variables ◽  
Independent Random Variables ◽  
Stochastic Orders ◽  
Absolutely Continuous ◽  
The Past ◽  
Probability Densities ◽  
The Mean ◽  
Weighted Probability ◽  
Series Systems ◽  
Cumulative Entropy

Given two absolutely continuous nonnegative independent random variables, we define the reversed relevation transform as dual to the relevation transform. We first apply such transforms to the lifetimes of the components of parallel and series systems under suitably proportionality assumptions on the hazard rates. Furthermore, we prove that the (reversed) relevation transform is commutative if and only if the proportional (reversed) hazard rate model holds. By repeated application of the reversed relevation transform we construct a decreasing sequence of random variables which leads to new weighted probability densities. We obtain various relations involving ageing notions and stochastic orders. We also exploit the connection of such a sequence to the cumulative entropy and to an operator that is dual to the Dickson-Hipp operator. Iterative formulae for computing the mean and the cumulative entropy of the random variables of the sequence are finally investigated.

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