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March  2022, 18(2): 969-984. doi: 10.3934/jimo.2021004

## Stochastic comparisons of parallel systems with scale proportional hazards components equipped with starting devices

 1 School of Mathematics and Statistics, Anhui Normal University, Wuhu 241002, China 2 Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada L8S 4K1

* Corresponding author: Longxiang Fang

Received  June 2019 Revised  November 2020 Published  March 2022 Early access  December 2020

Fund Project: The first author is supported by the Anhui Provincial Natural Science Foundation (No.1808085MA03), and the PhD research startup foundation of Anhui Normal University (No.2014bsqdjj34). The second author thanks the National Sciences and Engineering Research Council of Canada for supporting this research

In this paper, we discuss stochastic comparisons of parallel systems with scale proportional hazards components equipped with starting devices. To begin with, we present the hazard rate order of parallel systems with two scale proportional hazards components equipped with starting devices for two different cases: first when the starting devices with different probability have the same scale proportional hazards components, and the second when the different scale proportional hazards components have the same starting devices probability. Next, we present the usual stochastic order of parallel systems with $n$ scale proportional hazards components equipped with starting devices. Finally, we provide some numerical examples to illustrate all the results established here.

Citation: Longxiang Fang, Narayanaswamy Balakrishnan, Wenyu Huang. Stochastic comparisons of parallel systems with scale proportional hazards components equipped with starting devices. Journal of Industrial and Management Optimization, 2022, 18 (2) : 969-984. doi: 10.3934/jimo.2021004
##### References:
 [1] R. Fang, C. Li and X. Li, Ordering results on extremes of scaled random variables with dependence and proportional hazards, Statistics, 52 (2018), 458-478.  doi: 10.1080/02331888.2018.1425998. [2] N. K. Hazra, A. K. Nanda and M. Shaked, Some aging properties of parallel and series systems with a random number of components, Naval Research Logistics, 61 (2014), 238-243.  doi: 10.1002/nav.21580. [3] H. Jin, L. Hai and X. Tang, An optimal maintenance strategy for multi-state systems based on a system linear integral equation and dynamic programming, Journal of Industrial & Management Optimization, 16 (2020), 965-990.  doi: 10.3934/jimo.2018188. [4] X. Li and M. J. Zuo, Preservation of stochastic orders for random minima and maxima with applications, Naval Research Logistics, 51 (2004), 332-344.  doi: 10.1002/nav.10122. [5] X. Li, P. Parker and S. Xu, A stochastic model for quantitative security analyses of networked systems, IEEE Transactions on Dependable and Secure Computing, 8 (2011), 28-43.  doi: 10.1109/TDSC.2008.75. [6] C. Li and X. Li, Stochastic comparisons of parallel and series systems of dependent components equipped with starting devices, Communications in Statistics-Theory and Methods, 48 (2019), 694-708.  doi: 10.1080/03610926.2018.1435806. [7] A. W. Marshall, I. Olkin and B. C. Arnold, Inequalities: Theory of Majorization and Its Applications, Springer, New York, 2011. doi: 10.1007/978-0-387-68276-1. [8] A. K. Nanda and M. Shaked, Partial ordering and aging properties of order statistics when the sample size is random: A brief review, Communications in Statistics-Theory and Methods, 37 (2008), 1710-1720.  doi: 10.1080/03610920701826195. [9] M. Shaked and J. G. Shanthikumar, Stochastic Orders, Springer, New York, 2007. doi: 10.1007/978-0-387-34675-5.

show all references

##### References:
 [1] R. Fang, C. Li and X. Li, Ordering results on extremes of scaled random variables with dependence and proportional hazards, Statistics, 52 (2018), 458-478.  doi: 10.1080/02331888.2018.1425998. [2] N. K. Hazra, A. K. Nanda and M. Shaked, Some aging properties of parallel and series systems with a random number of components, Naval Research Logistics, 61 (2014), 238-243.  doi: 10.1002/nav.21580. [3] H. Jin, L. Hai and X. Tang, An optimal maintenance strategy for multi-state systems based on a system linear integral equation and dynamic programming, Journal of Industrial & Management Optimization, 16 (2020), 965-990.  doi: 10.3934/jimo.2018188. [4] X. Li and M. J. Zuo, Preservation of stochastic orders for random minima and maxima with applications, Naval Research Logistics, 51 (2004), 332-344.  doi: 10.1002/nav.10122. [5] X. Li, P. Parker and S. Xu, A stochastic model for quantitative security analyses of networked systems, IEEE Transactions on Dependable and Secure Computing, 8 (2011), 28-43.  doi: 10.1109/TDSC.2008.75. [6] C. Li and X. Li, Stochastic comparisons of parallel and series systems of dependent components equipped with starting devices, Communications in Statistics-Theory and Methods, 48 (2019), 694-708.  doi: 10.1080/03610926.2018.1435806. [7] A. W. Marshall, I. Olkin and B. C. Arnold, Inequalities: Theory of Majorization and Its Applications, Springer, New York, 2011. doi: 10.1007/978-0-387-68276-1. [8] A. K. Nanda and M. Shaked, Partial ordering and aging properties of order statistics when the sample size is random: A brief review, Communications in Statistics-Theory and Methods, 37 (2008), 1710-1720.  doi: 10.1080/03610920701826195. [9] M. Shaked and J. G. Shanthikumar, Stochastic Orders, Springer, New York, 2007. doi: 10.1007/978-0-387-34675-5.
Plot of $P(V_{2:2}>x)/ P(W_{2:2}>x),$ for $x\geq 0$, for Example 3.3
plot of $P(V_{2:2}>x)/ P(W_{2:2}>x),$ for $x\geq 0$, for Example 3.6 (1)
Plot of $P(V_{2:2}>x)/ P(W_{2:2}>x),$ for $x\geq 0$, for Example 3.6 (2)
Plot of $P(V_{2:2}>x)/ P(W_{2:2}>x),$ for $x\geq 0$, for Example 3.8
Plot of $P(V_{2:2}>x)/ P(W_{2:2}>x),$ for $x\geq 0$, for Example 3.10
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