In this post we contrast the meta distribution of the SIR with the standard SIR distribution. The model is the standard downlink Poisson cellular network with Rayleigh fading and path loss exponent 4. The base station density is 1, and the users form a square lattice of density 5. Hence there are 5 users per cell on average.
We assume the base stations transmit to the users in their cell at a rate such that the messages can be decoded if an SIR of θ =-3 dB is achieved. If the user succeeds in decoding, it is marked with a green square, otherwise red. We let this process run over many time slots, as shown next.
The SIR meta distribution captures the per-user statistics, obtained by averaging over the fading, i.e., over time. In the next figure, the per-user reliabilities are illustrated using a color map from pure red (0% success) to pure green (100% success).
The SIR meta distribution provides the fractions of users that are above (or below) a certain reliability threshold. For instance, the fraction of users that are at least dark green in the above figure.
In contrast, the standard SIR distribution, sometimes misleadingly called “coverage probability”, is just a single number, namely the average reliability, which is close to 70% in this scenario (see, e.g., Eqn. (1) in this post). Since it is obtained by averaging concurrently over fading and point process, the underlying network structure is lost, and the only information obtained is the average of the colors in Fig. 3:
The 70% reliability is that of the typical user (or typical location), which does not correspond to any user in our network realization. Instead, it is an abstract user whose statistics correspond to the average of all users.
Acknowledgment: The help of my Ph.D. student Xinyun Wang in writing the Matlab program for this post is greatly appreciated.
Stogblog by M. Haenggi 