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

• * Corresponding author: Longxiang Fang

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.

Mathematics Subject Classification: Primary: 60E15, 62G30; Secondary: 62N05.

 Citation: • • Figure 1.  Plot of $P(V_{2:2}>x)/ P(W_{2:2}>x),$ for $x\geq 0$, for Example 3.3

Figure 2.  plot of $P(V_{2:2}>x)/ P(W_{2:2}>x),$ for $x\geq 0$, for Example 3.6 (1)

Figure 3.  Plot of $P(V_{2:2}>x)/ P(W_{2:2}>x),$ for $x\geq 0$, for Example 3.6 (2)

Figure 4.  Plot of $P(V_{2:2}>x)/ P(W_{2:2}>x),$ for $x\geq 0$, for Example 3.8

Figure 5.  Plot of $P(V_{2:2}>x)/ P(W_{2:2}>x),$ for $x\geq 0$, for Example 3.10

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