Category: Cellular networks

Q cell analysis

Coverage is a key metric in wireless networks but usually only analyzed in terms of the covered area fraction, which is a scalar in [0,1]. Here we discuss a method to analytically bound the coverage manifold 𝒞, formed by the locations y on the plane where users are covered in the sense that the probability that the signal-to-interference ratio (SIR) exceeds a threshold θ is at least u (as argued here, this is a sensible definition of coverage):

$\displaystyle\mathcal{C}\triangleq\{y\in\mathbb{R}^2\colon \mathbb{P}(\text{SIR}(y)>\theta)>u\}.$

SIR(y) is the SIR measured at y for a given set of transmitters 𝒳={x₁,x₂,…} where the nearest transmitter provides the desired signal while all others interfere. It is given by

$\displaystyle\text{SIR}(y)=\frac{h_1r_1^{-\alpha}}{\sum_{i>1}h_ir_i^{-\alpha}}=\frac{1}{\sum_{i>1}\frac{1}{H_i}\left(\frac{r_1}{r_i}\right)^{\alpha}},$

where r_i is the distance to the i-th nearest transmitter to y and the coefficients H_i=h₁/h_i are identically distributed with cdf F_H. The coverage cell when x is the desired transmitter is denoted as C_x, so 𝒞 is the union of all C_x, x∈𝒳. To find an outer bound to the C_x, we are defining a cell consisting of locations at which there is no single interferer that causes them to be uncovered. In that cell, which we call the Q cell Q_x, the ratio of the distance to the nearest interferer to the distance of the serving transmitter has to be larger than a suitably chosen parameter ρ, i.e.,

$\displaystyle Q_x(\rho)\triangleq\left\{y\in\mathbb{R}^2\colon \frac{\|\mathcal{X}\setminus\{x\}-y\|}{\|x-y\|}>\rho\right\}.$

The numerator is the distance to the nearest interferer to y. The shape of the Q cell is governed by a first key insight (which is 2200 years old):
Key Insight 1: Given two points, the locations where the ratio of the distances to the two points equals ρ form a circle whose center and radius depend on the two points and ρ.
For ρ>1, which is the regime of practical interest, the Q cell is the intersection of a number of disks, one for each interferer. For the more distant interferers, the radii of the disks get large that they are not restricting the Q cell. As a result, a Q cell consists of either just one disk, the intersection of two disks, or a polygon plus a small number of disk segments. In the example in Figure 1, only the four disks with black boundaries are relevant. Their centers and radii are given by the locations of the four strongest interferers, marked by red crosses.

Fig. 1. *The boundary of the Q cell for ρ*=`√`2 *of the transmitter at the origin is shown in red. It is formed by four black circles.* The square is colored according to ℙ(SIR>1) for Rayleigh fading.

The remaining question is the connection between the distance ratio parameter ρ and the quality-of-service (QoS) parameters θ and u. It is given as

$\displaystyle \rho=\sigma(\theta,u)^{1/\alpha},$

where

$\displaystyle \sigma(\theta,u)\triangleq\frac{\theta}{F_H^{-1}(1-u)}$

is called the stringency and F_H^-1 is the quantile function of H (inverse of F_H). For Rayleigh fading,

$\displaystyle \sigma(\theta,u)=\theta\frac{u}{1-u}.$

Figure 1 shows a Q cell for θ=1 and u=0.8, hence (with Rayleigh fading) ρ^α=4, and ρ=√2 for α=4. It is apparent from the coloring that outside the Q cell, the reliability is strictly smaller than 0.8.
The second key important insight is that the Q cells only depend on ρ, so there is no need to explore the three-dimensional parameter space (θ, u, α) in analysis or visualization.
Key Insight 2: The Q cells are the same for all combinations of θ, u, and α that have the same ρ.
Accordingly, Figure 2, which shows Q cells for four values of ρ, encapsulates many combinations of θ, u, and α.

Fig. 2. *Q cells for ρ=1.25, 1.50, 2.00, 2.50 for 25 transmitters.*

If ρ=1, the Q cells equal the Voronoi cells, and for ρ<1, they span the entire plane minus an exclusion disk for each interferer. This case is less relevant in practice since it corresponds to very lax QoS requirements (small rate and/or low reliability, resulting in a stringency less than 1).

Since C_x ⊂ Q_x for each transmitter x, the union of Q cells is an outer bound to the coverage manifold.

If 𝒳 is a stationary and ergodic point process, the area fraction of the coverage manifold corresponds to the meta distribution (MD) of the SIR, and the area fraction covered by the Q cells is an upper bound to the MD. For the Poisson point process (PPP), the Q cells cover an area fraction of ρ^-2, and for any stationary point process of transmitters, they cover less than 4/(1+ρ)². This implies that for any stationary point process of users, at least a fraction 1-4/(1+ρ)² of users is uncovered. This shows how quickly users become uncovered if the stringency increases.

Details on the calculation of the stringency for different fading models and the proofs of the properties of the Q cells can be found here. This paper also shows a method to refine the Q cells by “cutting corners”, which is possible since the corners of the Q cells have two interferers are equal distance but only one is taken into account. Further, if the MD is known, the refined Q cells can be shrunk so that their covered area fraction matches the MD; this way, accurate estimates of the coverage manifolds are obtained. Figure 3 shows an example of refined and scaled refined Q cells where the transmitters form a perturbed (noisy) square lattice, in comparison with the exact coverage cells C_x.

Fig. 3. *Refined Q cells (blue) and estimated coverage cells (yellow) for ρ=√2. The boundaries of the exact coverage cells are shown in red.*

Due to the simple geometric construction, bounding property, and general applicability to all fading and path loss models, Q cell analysis is useful for rapid yet accurate explorations of base station placements, resource allocation schemes, and advanced transmission techniques including base station cooperation (CoMP) and non-orthogonal multiple access (NOMA).

Is the IRS interfering?

There is considerable interest in wireless networks equipped with intelligent reflecting surfaces (IRSs), aka reconfigurable intelligent surfaces or smart reflecting surfaces. While everyone agrees that IRSs can enhance the signal over a desired link, there are conflicting views about whether IRSs matched to a certain receiver causes interference at other receivers. The purpose of this blog is to clarify this point.

To this end, we focus on a downlink cellular setting where BSs form a stationary point process Φ and a user and a passive IRS with N elements are placed in each (Voronoi) cell. The exact distribution is irrelevant for our purposes. The desired signal at each user consists of the direct signal from its BS plus the intelligently reflected signal at its IRS. The SIR of a user at the origin served by the BS at x₀ can be expressed as

$\displaystyle \text{SIR}=\frac{G \ell(x_0)}{I},$

where ℓ(x₀) is the path loss from the serving BS and

$\displaystyle G=\Big|g_0+\sqrt{\Delta}\sum_{i=1}^N g_{i,1}g_{i,2}\Big|^2.$

Here the random variables g₀ and g_i capture the (amplitude) fading over the direct link and reflected (indirect) link respectively, and the triangle parameter Δ captures the distances in the BS-IRS-user triangle. For details, please see this paper. The pertinent question is what constitutes the interference I. If all BSs are active, their interference is

$\displaystyle I_{\rm BS}=\sum_{x\in\Phi\setminus\{x_0\}} |g_x|^2\ell(x).$

But how about the IRSs? This is where it becomes contentious. Some argue that the IRSs emit signals and thus cause extra interference that is not captured in this sum over the BSs only. This would mean that we need to add a sum over the IRSs of the form

$\displaystyle Q=\sum_{z\in\Psi\setminus\{z_0\}} p_z |g_z|^2\ell(z),$

where Ψ is the point process of IRSs and p_z is the power emitted by the IRS located at z. To decide whether Q is real or fake, let us have a look at the underlying network model. With fading modeled as Rayleigh, it is understood that the propagation of all signals is subject to rich scattering, i.e., there is multi-path propagation. For the case without IRSs, the propagation of interfering signals is illustrated here:

Figure 1. *Propagation in a rich scattering environment without IRSs.*

Figure 1 shows the user at the origin, its serving BS (blue triangle), and the interfering BSs (red triangles). The small black lines indicate scattering and reflecting objects. Some of the paths from interfering BSs via scattering objects to the user are shown with 319 red lines.

Now, let’s add one IRS per interfering BS. The resulting propagation environment is shown in Figure 2.

Figure 2. *Propagation in a rich scattering environment with IRSs.*

From the perspective of the user at the origin, the IRSs are unmatched, which means they act just like any of the other scattering and reflecting objects. So the difference to the IRS-free case is that there is a tiny fraction of additional scatterers, as highlighted in Figure 3.

Figure 3. *Same as Figure 2, but with the IRSs encircled.*

In this case, the IRSs add 4% to the scattering objects. If we did raytracing, they could make a small difference. In analytical work where the fading model is based on the assumption of a large number of scatterers, with the number of propagation paths tending to infinity, they make no difference at all. So we conclude that there is no extra interference Q due to the presence of passive IRSs and thus I=I_BS. The signals reflected at unmatched IRSs are no different from signals reflected at any other object, and multi-path fading models already incorporate all such reflections.

Another argument that IRSs cannot cause extra interference is based on the fundamental principle of energy conservation. If Q did exist, its mean would be proportional to the density of IRSs deployed. This implies that there would be a density of IRSs beyond which the “interference” from the IRSs exceeds the total power transmitted by all BSs, which obviously is not physically possible.

Meta visualization

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.

Fig. 1: *The setup. Base stations are red crosses, and users are blue circles. They are served by the nearest base station. Cell boundaries are dashed.*

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.

Fig. 2: *Transmission success (green) and failure (red) at each user over 100 time slots.*

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).

Fig. 3: *Per-user reliabilities, obtained by averaging over fading. These are captured by the SIR meta distribution*. Near the top left is a user that almost never succeeds since it is equidistant to three base stations.

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:

Fig. 4: *The standard SIR distribution only reveals the overall reliability of 70%.*

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.

Realistic communication

Today’s blog is about realistic communication, i.e., what kind of performance can realistically be expected of a wireless network. To get started, let’s have a look at an excerpt from a recent workshop description:

“Future wireless networks will have to support many innovative vertical services, each with its own specific requirements, e.g.

End-to-end latency of 1 ns and reliability higher than 99.999% for URLLCs.
Terminal densities of 1 million of terminals per square kilometer for massive IoT applications.
Per-user data-rate of the order of Terabit/s for broadband applications.”

Let’s break this down, bullet by bullet.

First bullet: In 1 ns, light travels 30 cm in free space. So “end-to-end” here would mean a distance of at most 10 cm, to leave some fraction of a nanosecond for encoding, transmission, and decoding. But what useful wireless service is there where transceivers are within at most 10 cm? Next, a packet loss rate of 10^-5 means that the spectral efficiency must be very low. Together with a latency constraint of 1 ns, ultrahigh bandwidths must be used, which, in turn, makes the design of circuitry and antenna arrays extremely challenging. At least the channel can be expected to be benign (line-of-sight).

Where does stochastic geometry come in? Assuming that these ultrashort links live in a network and not in isolation, interference will play a role. Let us consider a Poisson bipolar network with normalized link distance 1, a path loss exponent α and Rayleigh fading. What is the maximum density of links that can be supported that have an outage of at most ε? This quantity is known as the spatial outage capacity (SOC). For small ε, which is our regime of interest here, we have

$\displaystyle\text{SOC}\sim\left(\frac{\varepsilon}{\rho}\right)^\delta c_\delta,\quad \varepsilon\to 0,$

where δ=2/α and c_δ is a constant that only depends on the path loss exponent 2/δ. ρ is the spectral efficiency (in bits/s/Hz or bps/Hz). This shows the fundamental tradeoff between outage and spectral efficiency: Reducing the outage by a factor of 10 reduces the rate of transmission by the same factor if the same link density is to be maintained. Compared to a more standard outage constraint of 5%, this means that the rate must be reduced by a factor 5,000 to accommodate the 99.999% reliability requirement. Now, say we have 0.5 ns for the transmission of a message of 50 bits, the rate is 100 Gbps. Assuming a very generous spectral efficiency of 100 bps/Hz for a system operating at 5% outage, this means that 100 Gbps must be achieved at a spectral efficiency of a mere 0.02 bps/Hz. So we would need 5 THz of bandwidth to communicate a few dozen bits over 10 cm.
Even relaxing the latency constraint to 1 μs still requires 5 GHz of bandwidth.

In cellular networks, the outage-rate relationship is governed by a very similar tradeoff.
For any stationary point process of base stations and Rayleigh fading, the SIR meta distribution asymptotically has the form

$\displaystyle \bar F(\rho,\varepsilon)\sim\left(\frac{\varepsilon}{\rho}\right)^\delta C_\delta,\quad \varepsilon\to 0,$

where C_δ again depends only on the path loss exponent. This is the fraction of users who achieve a spectral efficiency of ρ with an outage less than ε, remarkably similar to the bipolar result. To keep this fraction fixed at, say, 95%, again the spectral efficiency needs to be reduced in proportion to a reduction of the outage constraint ε.

Second bullet: Per the classification and nomenclature in a dense debate, this density falls squarely in the tremendously dense class, above super-high density and extremely high density. So what do the anticipated 100 devices in an average home or 10,000 devices in an average parking lot do? What kind of messages are they exchanging or reporting to a hub? How often? What limits the performance? These devices are often said to be “connected“, without any specification what that means. Only once this is clarified, a discussion can ensue whether such tremendous densities are realistic.

Third bullet: Terabit-per-second (Tbps) rates require at least 10 GHz of spectrum, optimistically. 5G in its most ambitious configuration, ignoring interference, has a spectral efficiency of about 50 bps/Hz, and, barring any revolutionary breakthrough, more than 100 bps/Hz does not appear feasible in the next decade. Similarly, handling a signal 10 GHz wide would be an order of magnitude beyond what is currently possible. Plus such large junks of spectrum are not even available at 60 GHz (the current mm-wave bands). At 100 GHz and above, link distances are even more limited and more strongly subject to blockages, and analog beamforming circuitry becomes much more challenging and power-hungry. Most importantly, though, peak rates are hardly achieved in reality. In the 5G standard, the user experienced data rate (the rate of the 5-th percentile user) is a mere 1% of the peak rate, and this fraction has steadily decreased over the cellular generations:

So even if 1 Tbps peak rates became a reality, users would likely experience between 1 Gbps to at most 10 Gbps – assuming their location is covered, which may vary over short spatial scales. Such user percentile performance can be analyzed using meta distributions.

In conclusion, while setting ambitious goals may trigger technological advances, it is important to be realistic of what is achievable and what performance the user actually experiences. For example, instead of focusing on 1 Tbps peak rates, we could focus on delivering 1 Gbps to 95% of the users, which may still be very challenging but probably achievable and more rewarding to the user. And speaking of billions of “connected devices” is just marketing unless it is clearly defined what being connected means.

For more information on the two analytical results above, please see this paper (Corollary 1) and this paper (Theorem 3).

What to expect (over)

In performance analyses of wireless networks, we frequently encounter expectations of the form

$\displaystyle\mathbb{E}\log(1+{\rm SIR}),\qquad\qquad\qquad\qquad\qquad\qquad (*)$

called average (ergodic) spectral efficiency (SE) or mean normalized rate or similar, in units of nats/s/Hz. For networks models with uncertainty, its evaluation requires the use stochastic geometry. Sometimes the metric is also normalized per area and called area spectral efficiency. The SIR is expressed in the form

$\displaystyle {\rm SIR}=\frac{h_y \|y\|^{-\alpha}}{\sum_{x\in\Phi} h_x \|x\|^{-\alpha}},$

with Φ being the point process of interferers.
There are several underlying assumption made when claiming that (*) is a relevant metric:

It is assumed that codewords are long enough and arranged in a way (interspersed in time, frequency, or across antennas) such that fading is effectively averaged out. This is reasonable for several current networks.
It is assumed that desired signal and interference amplitudes are Gaussian. This is sensible since if a decoder is intended for Gaussian interference, then the SE is as if the interference amplitude were indeed Gaussian, regardless of its actual distribution.
Most importantly and questionably, taking the expectation over all fading random variables h_x implies that the receiver has knowledge of all of them. Gifting the receiver with all the information of the channels from all interferers is unrealistic and thus, not surprisingly, leads to (*) being a loose upper bound on what is actually achievable.

So what is a more realistic and accurate approach? It turns out that if the fading in the interferers’ channels is ignored, i.e., by considering

$\displaystyle {\rm SIR}^\prime=\frac{h_y \|y\|^{-\alpha}}{\sum_{x\in\Phi} \|x\|^{-\alpha}},$

we can obtain a tight lower bound on the SE instead of a loose upper bound. A second key advantage is that this formulation permits a separation of temporal and spatial scales, in the sense of the meta distribution. We can write

$\displaystyle {\rm SIR}^\prime=h_y\rho,\qquad\text{where }\; \rho=\frac{\|y\|^{-\alpha}}{\sum_{x\in\Phi} \|x\|^{-\alpha}}$

is a purely geometric quantity that is fixed over time and cleanly separated from the time-varying fading term h_y. Averaging locally (over the fading), the SE follows as

$\displaystyle C(\rho)=\mathbb{E}_h \log(1+h\rho),$

which is a function of (conditioned on) the point process. For instance, with Rayleigh fading,

$\displaystyle C(\rho)=e^{1/\rho} {\rm Ei}_1(1/\rho),$

where Ei₁ is an exponential integral. The next step is to find the distribution of ρ to calculate the spatial distribution of the SE – which would not be possible from (*) since it is an “overall average” that lumps all randomness together. In the case of Poisson cellular networks with nearest-base station association and path loss exponent 2/δ, a good approximation is

Here s* is given by

$\displaystyle s^{*\delta}\gamma(-\delta,s^*)=0,$

and γ is the lower incomplete gamma function. This approach lends itself to extensions to MIMO. It turns out that the resulting distribution of the SE is approximately lognormal, as illustrated in Fig. 1.

Figure 1: Spatial distribution of ergodic spectral efficiency for 2×2 MIMO in a Poisson cellular network and its lognormal approximation (red).

For SISO and δ=1/2 (a path loss exponent of 4), this (approximative) analysis shows that the SE achieved in 99% of the network is 0.22 bits/s/Hz, while a (tedious) simulation gives 0.24 bits/s/Hz. Generally, for small ξ, 1/ln(1/ξ) is achieved by a fraction 1-ξ of the network. As expected from the discussion above, this is a good lower bound.

In contrast, using the SIR distribution directly (and disregarding the separation of temporal and spatial scales), from

$\displaystyle \bar F_{\rm SIR}(\theta)=0.99 \quad\Longrightarrow\quad \theta=-20\text{ dB},$

we would obtain an SE of only log₂(1.01)=0.014 bits/s/Hz for 99% “coverage”, which is off by a factor of 16! So it is important that coverage be gleaned from the ergodic SE rather than a quantity subject to the small-scale variations of fading. See also this post.

The take-aways for the ergodic spectral efficiency are:

Avoid mixing time and spatial scales by expecting first over the fading and separately over the point process; this way, the spatial distribution of the SE can be obtained, instead of merely its average.
Avoid gifting the receiver with information it cannot possibly have; this way, tight lower bounds can be obtained instead of loose upper bounds.

The details can be found here.

Signal-to-interference, reversed

Interference is the key performance-limiting factor in wireless networks. Due to the many unknown parts in a large network (transceiver locations, activity patterns, transmit power levels, fading), it is naturally modeled as a random variable, and the (only) theoretical tool to characterize its distribution is stochastic geometry. Accordingly, many stochastic geometry-based works focus on interference characterization, and some closed-form expressions have been obtained in the Poisson case.

If the path loss law exhibits a singularity at 0, such as the popular power-law r^-α, the interference (power) may not have a finite mean if an interferer can be arbitrarily close to the receiver. For instance, if the interferers form an arbitrary stationary point process, the mean interference (at an arbitrary fixed location) is infinite irrespective of the path loss exponent. If α≤2, the interference is infinite in an almost sure sense.

This triggered questions about the validity of the singular path loss law and prompted some to argue that a bounded (capped) path loss law should be used, with α>2, to avoid such divergence of the mean. Of course the singular path loss law becomes unrealistic at some small distance, but is it really necessary to use a more complicated model that introduces a characteristic distance and destroys the elegant scale-free property of the singular (homogeneous) law?

The relevant question is to which extent the performance characterization of the wireless network suffers when using the singular model.

First, for practical densities, there are very few realizations where an interferer is within the near-field, and if it is, the link will be in outage irrespective of whether a bounded or singular model is used. This is because the performance is determined by the SIR, where the interference is in the denominator. Hence whether the interference is merely large or almost infinite makes no difference – for any reasonable threshold, the SIR will be too small for communication.
Second, there is nothing wrong with a distribution with infinite mean. While standard undergraduate and graduate-level courses rarely discuss such distributions, they are quite natural to handle and pose no significant extra difficulty.

That said, there is a quantity that is very useful when it has a finite mean: the interference-to-(average)-signal ratio ISR, defined as

$\displaystyle {\rm ISR}=\frac{\sum_{x\in\Phi\setminus\{x_0\}} h_x \|x\|^{-\alpha}}{\|x_0\|^{-\alpha}},$

where x₀ is the desired transmitter and the other points of Φ are interferers. The h_x are the fading random variables (assumed to have mean 1), only present in the numerator (interference), since the signal power here is averaged over the fading.
Taking the expectation of the ISR eliminates the fading, and we arrive at the mean ISR

$\displaystyle {\rm MISR}=\mathbb{E} \sum_{x\in\Phi\setminus\{x_0\}}\left(\frac{\|x_0\|}{\|x\|}\right)^{\alpha},$

which only depends on the network geometry. It follows that the SIR distribution is

$\displaystyle \mathbb{P}({\rm SIR}<\theta)=\mathbb{P}(h<\theta\, {\rm ISR})=\mathbb{E} F_h(\theta\,{\rm ISR}),$

where h is a generic fading random variable. If h is exponential (Rayleigh fading) and the MISR is finite,

$\displaystyle F_h(x)\sim x\qquad\Longrightarrow\qquad \mathbb{P}({\rm SIR}<\theta)\sim \theta\,{\rm MISR}.$

Hence for small θ, the outage probability is proportional to θ with proportionality factor MISR. This simple fact becomes powerful in conjunction with the observation that in cellular networks, the SIR distributions (in dB) are essentially just shifted versions of the basic SIR distribution of the PPP (and of each other).

Figure 1: Examples for SIR distributions in cellular networks that are essentially shifted versions of each other.

In Fig. 1, the blue curve is the standard SIR ccdf of the Poisson cellular network, the red one is that of the triangular lattice, which has the same shape but shifted by about 3 dB, with very little dependence on the path loss exponent. The other two curves may be obtained using base station silencing and cooperation, for instance. Since the shift is almost constant, it can be determined by calculating the ratios of the MISRs of the different deployments or schemes. The asymptotic gain relative to the standard Poisson network (as θ→0) is

$\displaystyle G_0=\frac{\rm MISR_{PPP}}{{\rm MISR}},\quad \text{where }\; {\rm MISR_{PPP}}=\frac{2}{\alpha-2},\quad \alpha>2,$

The MISR in this expression is the MISR for an alternative deployment or architecture. The MISR for the PPP is not hard to calculate. Extrapolating to the entire distribution by applying the gain everywhere, we have

$\displaystyle \bar F_{\rm SIR}(\theta) \approx \bar F_{\rm SIR}^{\rm PPP}(\theta/G_0).$

This approach of shifting a baseline SIR distribution was proposed here and here. It is surprisingly accurate (as long as the diversity order of the transmission scheme is unchanged), and it can be extended to other types of fading. Details can be found here.

Hence there are good reasons to focus on the reversed SIR, i.e., the ISR.

Path loss point processes

Naturally the locations of wireless transceivers are modeled as a point process on the plane or perhaps in the three-dimensional space. However, key quantities that determine the performance of a network do not directly nor exclusively depend on the locations but on the received powers. For instance, a typical SIR expression (at the origin) looks like

$\displaystyle {\rm SIR}=\frac{P_y h_y \|y\|^{-\alpha}}{\sum_{x\in\Phi} P_x h_x \|x\|^{-\alpha}},$

where y is the location of the intended transmitter and Φ is the point process of interferers. P_x and h_x are the transmit powers and fading coefficients of x, respectively. It is apparent that what matters are the distances raised to some power, not the locations themselves. So instead of working with Φ⊂ℝ², we can focus on the one-dimensional process

$\displaystyle \Psi\triangleq\{x\in\Phi\colon \|x\|^{\alpha}/h_x\},$

called the path loss point process (PLPP) (with fading). The reason why the positive exponent α is preferred over -α is that otherwise the resulting point process is no longer locally finite (assuming Φ is stationary) since infinitely many points would fall in the interval [0,ε] for any ε>0. Transmit power levels could be included as displacements, either deterministically or randomly.

Path loss processes are particularly useful when Φ is a PPP. By the mapping and displacement theorems, the PLPPs are also PPPs whose intensity function is easy to calculate. For a stationary PPP Φ of intensity λ and iid fading, the intensity function of Ψ is

$\displaystyle \mu(r)=\pi\lambda\delta r^{\delta-1}\mathbb{E}(h^\delta),\quad r\geq 0, \delta\triangleq2/\alpha.$

where h is a generic fading random variable. If h has mean 1, then for δ<1, which is necessary to keep the interference finite, 𝔼(h^δ)<1 from Jensen’s inequality, hence the effect of fading is a reduction of the intensity function by a fixed factor.

As an immediate application we observe that fading reduces the expected number of connected nodes, defined as those whose received power is above a certain threshold, by the δ-th moment of the fading coefficients.

More importantly, PLPPs lead to two key insights for Poisson cellular networks. Let us assume the elements of Ψ are ordered and denoted as ξ₁<ξ₂<… . Then the SIR with instantaneously-strongest base station association (ISBA) is

$\displaystyle {\rm SIR}=\frac{\xi_1^{-1}}{\sum_{i=2}^\infty \xi_i^{-1}}.$

First, it is not hard to show that for ISBA with Rayleigh fading, the SIR distribution does not depend on the density of the underlying PPP. But since the effect of fading is but a scaling of the density, it follows that the SIR distribution does not depend on the fading statistics, either. In particular, the result for Rayleigh fading also applies to the non-fading case (where ISBA corresponds to nearest-base station association, NBA), which is often hard to analyze in stochastic geometry models.

Second, the intensity function of the PLPP also shows that the SIR performance of the heterogeneous independent Poisson (HIP) model is the same as that of the simple PPP model. The HIP model consists of an arbitrary number n of tiers of base stations, each modeled as an independent PPP of arbitrary densities λ_k and transmitting at arbitrary (deterministic) power levels P_k. The point process of inverse received powers (i.e., the PLPP with transmit powers included) from tier k has intensity

$\displaystyle \mu_k(r)=\pi\lambda_k\delta r^{\delta-1}\mathbb{E}(h^\delta)P_k^\delta,\quad r\geq 0.$

Since the superposition of n PPPs is again a PPP, the overall intensity is just the sum of the μ_k, which is still proportional to r^δ-1. This shows that the SIR performance (with ISBA or NBA) of any HIP model is the same as that of just a single PPP.

For further reading, please refer to A Geometric Interpretation of Fading in Wireless Networks: Theory and Applications and The Performance of Successive Interference Cancellation in Random Wireless Networks.

The typical user and her malfunctioning base station

Let us consider a cellular network with Poisson distributed base stations (BSs). We assume that in each Voronoi cell, one user is located uniformly at random (and independently across cells), and, naturally, the user is connected to the nucleus of that cell. This is the user point process of type I defined in this article. In this case, the typical user, named Alice, does reside in the typical cell since there is no size-biased sampling involved in defining the typical user. The downlink SIR performance of this network has been analyzed here (SIR distribution) and here (SIR meta distribution).

Suddenly and unfortunately, Alice’s serving BS is malfunctioning. Her downlink is, well, down, and she gets reconnected to the next-nearest one. How does that affect her SIR performance?
In another network, also with Poisson BSs, lives another type of typical user, namely that of a stationary point process of users that is independent of the base station process. This typical user’s name is Bob. Bob’s SIR performance is the same as the one measured at an arbitrary deterministic location on the plane, as discussed in this post. He resides in the 0-cell, not the typical cell. So in his case, the typical user does not reside in the typical cell.

Noticing that Alice’s original BS ceased to operate, Bob says: “I think now your SIR performance is the same as mine. After all, your cell was formed by adding a BS at the origin, while my network has no such BS. With that added BS removed, we are in the same situation.”

Alice responds: “You may have a point, but I am not sure that my location is uniform in the 0-cell, as yours is.”

Bob: “Good point, but wouldn’t that be the natural conjecture?”

Alice: “I am not sure. How about we verify? Let’s look at the distances.”

Bob: “Ok. For me, if B_n is the distance to my n-th nearest BS, we have

$\displaystyle \pi\mathbb{E}(B_n^2)=n.$

Alice: “For me, the distance A₁ to the malfunctioning BS satisfies π𝔼(A₁²)≈10/13, by the properties of the typical cell. If you are correct, then the distribution of A₂ should be the same as that of B₁. But that’s not the case, see this figure.”

Mean squared distances times π. For Alice, the distances are A_n, for Bob, B_n.

Bob: “I see – your new serving BS at distance A₂ is quite a bit further away than mine at B₁. So my conjecture was wrong.“

Alice: “Yes, but the question of how resilient a cellular network is to BS outages is an interesting one. How about we compare the SIR performance with and without BS outage in different networks, say Poisson networks and lattice networks? I bet Poisson networks are more robust, in the sense that the downlink SIR statistics change less when there is an outage and users need to be handed off to the next-nearest BS.“

Bob: “Hmm… that would make sense. But would that mean we should build clustered networks, to achieve even higher robustness?“

Alice: “Possibly – if all we worry about is a small loss when a user is offloaded. But we should take into account the absolute performance also, and clustered networks are worse in this regard. If the starting performance is much higher, it is acceptable to have a somewhat bigger loss due to outage and handover.“

Bob: “Makes sense. Sorry, I have to go. My SIR is so high that I just got a phone call.“

Percentage games

Let us consider a hypothetical situation where the authors of a paper promise the following:

“In the next figure, we compare our Quantum Ultra-Enhanced Superior-Throughput Intelligent Objectively-Novel Adaptive Beamformed Low-Latency Emission (QUESTIONABLE) scheme with the Arbitrarily-Massive Antenna Zero-Innovation Neural Grandiloquence (AMAZING) scheme, which is the best previously known scheme. As the figure shows, the QUESTIONABLE scheme outperforms the AMAZING scheme by 6122% at a threshold of 2 dB.”

They refer to this figure:

Fig. 1: *Fair comparison of our QUESTIONABLE scheme with the AMAZING scheme reported elsewhere. As is clearly apparent, the reliability gain at 2 dB is very high, unquestionably.*

A gain of more than 6,000% sounds, well, amazing. But it also questionable: Is it good practice to state this figure of 6122%? Is it good marketing? Is it acceptable? Should it be discouraged? Is it even unethical?
If claiming 6122% at 2 dB is acceptable, how about a 25119% gain at 10 dB? Here the absolute reliability of AMAZING is 6.3e-9, and that of QUESTIONABLE is 1.60e-6. Not a practical regime, but a more than 25000% gain nonetheless.

Conversely, at -1.5 dB, the improvement is from 0.996 (which is AMAZING) to 0.9999999 (which is QUESTIONABLE). This is a modest 0.4% improvement both in absolute and relative terms. However, focusing on outage instead of reliability, the factor between the two outage probabilities is 62763, or 6,276,267%. So the 0.4% gap becomes a 6M% gap (that’s mega-percent, admittedly an usual combination of a metric prefix with a unit).

Independently of such amazing and questionable transmission schemes, is an improvement from 10% to 13% a 3% improvement or a 30% improvement? Frequently we see such a gain reported as a 30% improvement. Is it fair to use the ratio to measure improvement, or should it be the gap (difference)? Equivalently, should the percentage improvement be measure in percent of the underlying quantity (in which case it is 3%) or in percent of the percentages (in which case it is 30%)? The problem is that terms like “gain”, “increase”, “improvement”, “boost”, “raise” are imprecise as they may refer to differences or ratios.

How about an improvement from 10 dB to 13 dB? Here, most people would agree that the gain is 3 dB. The reason why there is no ambiguity here is that 3 dB is both the difference 13 dB-10 dB and also the ratio of the underlying quantities expressed in dB, 20/10. Thanks to the logarithm in the dB scale, both lead to the same result. With percentages, this is not the case.

The ambiguity also manifests itself when we compare percentages with physical units such as meters or seconds. The difference between 10 m and 13 m is 3 m, and 13 m is 30% longer than 10 m. Percentage gains always measure ratios in this case. But if “meter” is replaced by “percent”, the situation becomes murky. By analogy then the difference between 10% and 13% is 3%, so 13% is 3% more than 10%, in the same way that 13 m is 3 m more than 10 m. But at the same time, 13% is 30% more than 10%, in the same way that 13 m is 30% more than 10 m.

So how do we resolve this? Here are some suggestions:

Always state whether we refer to ratios or difference. For instance, instead of saying “The increase is 30%” say the “increase is a factor of 1.3 or 30%”.
Avoid ambiguity by using fractions instead of percentages. For instance, the difference between 0.1 and 0.13 is 0.03, and the ratio is 1.3.
Avoid aggressive marketing when quantities are in very impractical regimes. For instance, in the example above, advertising huge gains at reliabilities below 1e-5 is not good practice. Conversely, in regimes of interest, such as the high-reliablity regime, it is perfectly acceptable to report a gain of 100 in the outage performance if the outage probability is reduced from 0.01 to 0.0001. (However, talking about 10,000% is not recommended.)
Consider using horizontal distances (gaps) instead of vertical ones. Especially with steep (almost step-like) curves like the one in Fig. 1, the vertical gap strongly depends on where it is measured along the x axis. Conversely, the horizontal gap may be fairly constant. In the case of Fig. 1, the horizontal gap is, in fact, constant. It is exactly 3 dB at all reliabilities. A quasi-constant gap is often observed in SIR distributions (cdfs) of cellular networks when different transmission schemes are compared. The theory behind is explained in this letter (here on IEEE Xplore) and in this paper (here on IEEE Xplore). In this case, the gains are either factors (in linear scale) or gaps (in the dB) scale. This method of reporting gains is the one adopted by the coding theory community – coding gains are reported in dB (horizontally), not as gains in the bit error performance (vertically).

On cell slicing

Network slicing is a warm topic these days. Here we discuss cell slicing, where a polygon is cut in three pieces (sub-polygons) by two lines through its nucleus and a random point, respectively. First, as a sequel to this post, we focus on the 0-cell in the Poisson-Voronoi tessellation, which is the Voronoi cell of a PPP that contains the origin.
As in that earlier post, we rotate and shift the 0-cell so that its nucleus is at the origin and the point located uniformly randomly in the cell is to the right (on the positive x axis). In a cellular network application, the nucleus (now at the origin) is the location of the base station, while the random point is the typical user. Then we draw vertical lines through the origin and the random point and calculate the areas of the polygons to the left of the origin and to the left of the random point. This movie shows a number of realizations of this setup:

Movie 1. Realizations of 0-cells in Poisson-Voronoi tessellation sliced into three sub-polygons.

For intensity λ=1, the 0-cell in the Poisson-Voronoi tessellation has a mean area of 1.280176, obtained by the numerical evaluation of an integral. The polygon to the left of the nucleus o, shown in green in the movie, has a mean area of 0.517649, also obtained by numerical integration. The red polygon, to the right of o and to the left of x, has a mean area of 0.529111, obtained by simulation. Relative to the total mean area, we thus have:

Mean area to the left of o: 40.4%
Mean area to the left of x: 81.8%
Mean area to the right of x: 18.2%

Hence, on average, almost 60% of the (other) users in the cell are on the same side of the base station as the typical user, and the mean area to the left of the typical user is larger than the entire mean area of the typical cell (which is 1 for λ=1). Also, more than 18% of the users are “behind” the typical user.

Not surprisingly, the area fractions are quite similar in the typical Poisson-Voronoi cell. (Note that the uniformly random point in the typical cell does not correspond to the typical user of a user point process that is independent of the base station process – see this post). These percentages (area fractions) for the typical cell are all simulated:

Mean area to the left of o: 39.8%
Mean area to the left of x: 81.3%
Mean area to the right of x: 18.7%

How unusual are these area fractions? Let us compare them with those in the disk, which is, in some sense, the ideal cell shape. In the disk, with the nucleus o at the center, the mean area fraction to the left of o is trivially 1/2, while the area to the left of a uniformly random point is easily determined to be 7/8. This shows that the (roughly) 2/5 – 2/5 – 1/5 split in the typical Poisson-Voronoi cells is relatively far from the 1/2 – 3/8 – 1/8 split of the disk. How about regular polygons with a finite number of sides? The second movie shows some realizations for 3,4, … ,10,12,15,20,32 sides.

Movie 2. Realizations of regular polygons sliced into three sub-polygons.

As expected, for a larger number of sides the area fractions approach those of the disk. Here is a plot of the relative deviations to the disk of regular polygons, where o is the centroid:

Figure 1. Percentage deviation of mean areas left of o and left of x in regular polygons relative to the disk.

For instance, for the triangle, the area fraction to the left of o is 0.516, which is 3.2% larger than for the disk. Hence the first blue point in Fig. 1 (top left), corresponding to 3 sides, is at 3.2. To better see how the deviations behave for 5 or more sides, here is another version of the figure that shows only pentagons and higher.

Figure 2. Same as Fig. 1 but starting at pentagons.

From the blue curve it is apparent that the area to the left of the center is always exactly one half if the number of sides is even. The reason is that with an even number of sides, the polygons to the left of o and to the right of o are always congruent irrespective of the location of the random point. We also see that the hexagon is quite close to the disk already, with a mere 0.15% deviation from the 7/8 area fraction (to the left of x) of the disk.