Let us consider the downlink of a cellular network where base stations form a stationary and ergodic point process Φ and define the SIR at each location *x* ∈ **R**^{2} as

Here *N(x)* is the nucleus of the Voronoi cell that *x* belongs to, *h _{x,y}* is the fading coefficient between

*x*and

*y*, and ℓ is the path loss function. Due to the stationarity of Φ, the SIR statistics do not depend on the location

*x*. In other words, any arbitrary location can be taken to be the typical location that the analysis focuses on.

**Example result**

*: If Φ is Poisson, the fading is Rayleigh, and ℓ is a power-law function with exponent 2/δ, it is known that for all x ∈*

**R**^{2},*where _{2}F_{1} is the Gauss hypergeometric function.*

Since Φ is ergodic, the probability that the SIR exceeds

*θ*is the fraction of the plane that achieves an SIR of at least

*θ*for all realizations of Φ. This means that the probability (ensemble average) can be replaced by a spatial average over an increasingly large region. Sometimes this probability (or spatial average) is questionably called “coverage probability” (see this post), and the area fraction is termed “covered area fraction”.

It is important to note that results such as (1) do not require any specification of a point process of users. This answers the question in the title: No, users are not necessary in the downlink SIR analysis.

That said, in the literature we observe that in many cases, a point process of users is defined before such downlink SIR results are derived. The reason could be that it may seem overly abstract to consider a cellular network model devoid of any users and view the SIR as a random field on the plane. Specializing the location *x* to the points of a user point process (assumed independent of Φ), we observe that (1) is the SIR distribution at the typical user for *any stationary point process of users*. So there is nothing wrong in introducing a point process of users, focus on the typical user, and state a result such as (1). It would, however, be potentially misleading to specify the user point process to be a Poisson process, since the reader may then believe that the result only holds for Poisson distributed users.

There is one caveat when introducing a point process of users to formulate downlink results: The interpretation of the SIR distribution as the fraction of users who achieve SIR>*θ* in each realization of the user and base station point processes may no longer be correct, even if the two point processes are independent and stationary and ergodic. For instance, consider the case where both are stationary (i.e., randomly translated) lattices of the same intensity. Then, given the point processes, the SIR distribution at each user is the same and depends on the relative shift of the lattices. For example, if a user is very close to its serving base station, then all users are close to their serving base station, and the SIR at all users is likely to exceed *θ* even when *θ* is, say, 20 dB. In contrast, if one user is equidistant to two base stations, then all users are, and it is unlikely that the SIR (at any or all of them) exceeds 1. So averaging over the users in one realization cannot yield the same result as averaging over the point processes (ensemble averaging). But doesn’t ergodicity imply that the two results are the same? The answer is yes, it does, but individual ergodicity of the two point processes is not sufficient. Since the SIR depends on both of them jointly, they need to be *jointly ergodic*. This is the condition that is not met in this example scenario of two lattices.

## 3 thoughts on “Are users needed in cellular networks?”