How Typical is "Typical"? Characterizing Deviations Using the Meta Distribution Martin Haenggi University of Notre Dame, IN, USA SpaSWiN 2017 Keynote May 19, 2017 Available at http://www.nd.edu/~mhaenggi/talks/spaswin17.pdf M. Haenggi (ND) How Typical is "Typical"? 05/19/2017 1 / 55
Growth of stochastic geometry for wireless networks Growth of articles on IEEE Xplore over 1.5 decades IEEE Xplore Articles on "Stochastic Geometry" & "Wireless" 30 linear fit: f(x)=10*(x/6+1/3) 25 Growth: 16 dB/decade (factor 40) number of publications [dB] 20 15 10 5 0 0 5 10 15 year-2000 M. Haenggi (ND) How Typical is "Typical"? 05/19/2017 1 / 55
Overview Menu Overview What is "typical"? From spatial averages to the meta distribution Poisson bipolar networks Poisson cellular networks ◮ Downlink ◮ Uplink ◮ D2D ◮ What is coverage? Spatial outage capacity Ergodic spectral efficiency Conclusions M. Haenggi (ND) How Typical is "Typical"? 05/19/2017 2 / 55
Typicality The typical person The typical French person (in numbers) Lives 82 years. Makes USD 29,759 per year (disposable income). Lives in a household whose wealth is USD 53,851. Lives in 1.8 rooms. The typical American person (in numbers) Lives 79 years. Makes USD 41,071 per year (disposable income). Lives in a household whose wealth is USD 163,268. Lives in 2.4 rooms. M. Haenggi (ND) How Typical is "Typical"? 05/19/2017 3 / 55
Typicality The (stereo)typical French person M. Haenggi (ND) How Typical is "Typical"? 05/19/2017 4 / 55
Typicality The (stereo)typical American tourist M. Haenggi (ND) How Typical is "Typical"? 05/19/2017 5 / 55
Typicality The globally typical person The typical user! M. Haenggi (ND) How Typical is "Typical"? 05/19/2017 6 / 55
Typicality Inequality Inequality Inequality in France Social inequality (household income and wealth): 4.67. 1% income: $242,000. 0.1% income: $720,000. 0.01% income: $2,252,000. Top 20% earns 5 × as much as the bottom 20%. Inequality in the USA Social inequality (household income and wealth): 8.19. 1% income: $465,000. 0.1% income: $1,695,000. 0.01% income: $9,141,000. Top 20% earns 8 × as much as the bottom 20%. M. Haenggi (ND) How Typical is "Typical"? 05/19/2017 7 / 55
Typicality Inequality Typicality and inequality In many situations, merely considering the typical entity reveals only limited information. The satisfaction of people in a country depends as much (or more) on the inequality than the absolute level of income, wealth, number of rooms, etc. Similarly, in a cellular network, whether a user is happy or not with 1 Mb/s strongly depends on whether other users get 100 kb/s or 10 Mb/s. Industry often focuses on the performance of the "5% user", which is the performance that the top 95% of the users experience. Increasing the performance of the typical user may decrease the performance of the 5% user. M. Haenggi (ND) How Typical is "Typical"? 05/19/2017 8 / 55
From spatial averages to the meta distribution Spatial or ensemble averages Stochastic geometry: From spatial averages to the meta distribution Spatial/ensemble average Let Φ be a point process and f : N → R + a performance function. Ensemble average: ¯ f = E o f (Φ) User at o is the typical user. In an ergodic setting, its performance is the performance averaged over all users (spatial average). But in a realization of Φ , no user is typical. All users have (much) better/worse performance than ¯ f . To quantify this, we need to calculate other properties of the random variable f (Φ) than just its mean. M. Haenggi (ND) How Typical is "Typical"? 05/19/2017 9 / 55
From spatial averages to the meta distribution Spatial distribution Spatial distribution Refined analysis: ¯ F ( x ) = E o 1 ( f (Φ) > x ) ≡ P o ( f (Φ) > x ) In an ergodic setting, this yields the fraction of users that achieve performance at least x (spatial distribution). This is more informative. This is still an average (but so is any distribution of any random variable), and it is again evaluated at the typical user. � ¯ Of course, ¯ f = F ( x ) d x . Key example in wireless networks: The SIR distribution. M. Haenggi (ND) How Typical is "Typical"? 05/19/2017 10 / 55
From spatial averages to the meta distribution SIR distribution SIR distribution Let f θ (Φ) = 1 ( SIR (Φ) > θ ) . The SIR distribution at the typical user is P o ( SIR (Φ) > θ ) ≡ E o f θ (Φ) , which is a family of spatial averages since the performance function has an extra parameter θ . It yields the fraction of users that achieve an SIR of θ . It is also interpreted as the success probability of the typical user. Does this give us complete information on the network performance, such as the performance of the 5% user? M. Haenggi (ND) How Typical is "Typical"? 05/19/2017 11 / 55
From spatial averages to the meta distribution SIR distribution Interpretation of the SIR distribution Let us assume that P o ( SIR (Φ) > 1 / 10 ) = 0 . 95 and consider a realization of Φ representing the node locations for a certain period of time. 95% of the users achieve an SIR of -10 dB, at any given time. However, the set of users that achieve this SIR changes in each coherence interval. Hence each user is likely to belong to the bottom 5% and to the top 95% many times in a short period. (This is why the SIR distribution is not the coverage—details to follow.) Moreover, this does not mean that an individual user achieves -10 dB SIR 95% of the time. In fact, nothing can be said about the SIR at an individual user. It could be that for each group of 100 users, 5 never achieve -10 dB (100% outage for 5 users, 0% outage for the rest). Or it could be that the 5 who do not achieve -10 dB are picked uniformly at random every 10ms (5% outage for all users.) M. Haenggi (ND) How Typical is "Typical"? 05/19/2017 12 / 55
From spatial averages to the meta distribution SIR distribution How to get more information? To capture user performance, we 3 0.68 0.27 0.25 0.44 0.28 0.20 0.36 0.81 need to adopt a longer-term 2 viewpoint. This way, we can talk 0.50 0.42 0.88 0.15 0.98 0.46 0.86 0.69 consistently about the 5% user. 1.00 0.30 0.99 0.25 0.70 0.93 0.30 0.58 1 This means we need to average over 0.44 0.96 0.34 0.06 0.97 0.92 0.12 0.81 0 the fading (and channel access). 0.74 0.98 0.48 0.56 0.66 0.44 0.70 1.00 So lets assign to each user a −1 0.66 0.76 0.98 0.56 0.99 0.52 0.49 0.87 personal SIR distribution (success −2 probability): 0.32 0.35 0.50 0.78 0.50 0.18 0.04 0.35 P ( SIR u (Φ) > θ | Φ) −3 −3 −2 −1 0 1 2 3 E o ( P ( SIR (Φ) > θ | Φ)) SIR distribution: M. Haenggi (ND) How Typical is "Typical"? 05/19/2017 13 / 55
From spatial averages to the meta distribution Meta distribution The meta distribution of the SIR E o ( P ( SIR (Φ) > θ | Φ)) SIR distribution: So the SIR distribution is just the spatial average of the conditional success probability random variable P s ( θ ) � P ( SIR o (Φ) > θ | Φ o ) . But, as before, instead of considering only the average, let’s consider the distribution! The meta distribution of the SIR is the ccdf � Haenggi , 2016 � F ( θ, x ) = ¯ ¯ F P s ( θ ) ( x ) � P o ( P s ( θ ) > x ) , θ ∈ R + , x ∈ [ 0 , 1 ] . ¯ F ( θ, x ) is the fraction of users that achieve an SIR of θ with probability at least x , in each realization of Φ . Those users do not change over time. M. Haenggi (ND) How Typical is "Typical"? 05/19/2017 14 / 55
From spatial averages to the meta distribution Meta distribution Meta distribution of the SIR Using the previous notation, we have f θ (Φ) = P ( SIR o > θ | Φ) and � � ¯ F ( θ, x ) = E o 1 ( f θ (Φ) > x ) = E o 1 E 1 ( SIR o (Φ) > θ | Φ o ) > x . It is the distribution of the conditional SIR distribution, hence the term "meta". The standard SIR distribution (mean success probability) is � 1 ¯ p s ( θ ) = P o ( SIR > θ ) = F ( θ, x ) d x . 0 Performance of the 5% user: Rate (spectral efficiency) is determined by θ , e.g., through log ( 1 + θ ) . The 5% user achieves the rate-reliability trade-off pairs ( θ, x ) given by ¯ F ( θ, x ) = 0 . 95. M. Haenggi (ND) How Typical is "Typical"? 05/19/2017 15 / 55
From spatial averages to the meta distribution Meta distribution Example contour plot of ¯ F ( θ, x ) : Trading off rate and reliability 1 0.8 u=0.5 0.6 x u=0.95 0.4 0.2 0 −20 −15 −10 −5 0 5 θ [dB] Contours ¯ F ( θ, x ) = u for u ∈ { 0 . 5 , 0 . 6 , 0 . 7 , 0 . 8 , 0 . 9 , 0 . 95 } . The bottom curve u = 0 . 95 gives the performance of the 5% user. For example, this user achieves an SIR of − 10 dB with reliability 0 . 72 or an SIR of − 4 . 3 dB with reliability 0 . 3. M. Haenggi (ND) How Typical is "Typical"? 05/19/2017 16 / 55
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