# On a conjecture about the comparability of parallel systems with respect to convex transform order

## TL;DR

This paper demonstrates that two heterogeneous parallel systems with exponential components are comparable under the star transform order but not under the convex transform order, addressing an open problem in system comparison.

## Contribution

It proves the comparability of such systems under the star transform order and shows the failure of convex transform order comparison, resolving a decade-old open question.

## Key findings

- Systems are comparable via star transform order
- Comparison via convex transform order fails
- Addresses an open problem in system comparison

## Abstract

In this paper we prove that two heterogeneous parallel systems with independent exponentially distributed components are comparable via the star transform order while the comparison via the convex transform fails. The latter conclusion provides a partial answer to a problem that remained open for a decade.