may also share a set of semi-orthogonal codes denoted by $\varpi $ with cardinality $|\varpi|$, which reduces the interference from other devices reusing the same radio resource with a different code by factor of $\mathcal Q$. Examples of such codes are semi-orthogonal spreading codes in LoRa technology [@mag_all].

Analytical Modeling of KPIs
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Modeling of Reliability
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In the grant-free radio access system, transmitting devices are asynchronous in time and frequency domains, and hence, the received packets at the receiver could have partial overlaps in time-frequency. To model reliability in communications, we first derive analytical models for interference in subsection \[siI\], and for probability of success in subsection \[su1\]. These models are then employed in deriving reliability of communications in subsection \[rels\].

### Interference Analysis {#siI}

We assume a type-$i$ device has been located at point $\bf z$ in a 2D plane, and its respective AP has been located at the origin. In order to derive probability of success in data transmission from the device to the AP, we need to characterize the received interfere at the AP. A common practice in interference analysis is to determine its moments, which is possible by finding its generating function, i.e. the Laplace 