  To support ultra-reliable and low-latency services for mission-critical
applications, transmissions are usually carried via short blocklength codes,
i.e., in the so-called finite blocklength (FBL) regime. Different from the
infinite blocklength regime where transmissions are assumed to be arbitrarily
reliable at the Shannon's capacity, the reliability and capacity performances
of an FBL transmission are impacted by the coding blocklength. The relationship
among reliability, coding rate, blocklength and channel quality has recently
been characterized in the literature, considering the FBL performance model. In
this paper, we follow this model, and prove the joint convexity of the FBL
error probability with respect to blocklength and transmit power within a
region of interest, as a key enabler for designing systems to achieve globally
optimal performance levels. Moreover, we apply the joint convexity to general
use cases and efficiently solve the joint optimization problem in the setting
with multiple users. We also extend the applicability of the proposed approach
by proving that the joint convexity still holds in fading channels, as well as
in relaying networks. Via simulations, we validate our analytical results and
demonstrate the advantage of leveraging the joint convexity compared to other
commonly-applied approaches.
