In this section, we introduce the meta distribution, overview the most common approaches for computing it, and show that it can be efficiently computed, within a given and bounded error, by using the trapezoidal integration rule and the Euler sum method. In what follows, we consider the equivalent network model based on the inhomogeneous PPPs ΦBS(F) and ΦBS(K). For ease of writing, we employ the notation ΦBS={ΦBS(F),ΦBS(K)}.
Definition
According to [8], the meta distribution is defined as follows:
$$ \begin{aligned} &{\overline {\mathrm{F}}_{{{\mathrm{P}}_{{\text{cov}}}}}}\left({{\gamma_{\mathrm{D}}},{\gamma_{\mathrm{A}}},z} \right) = {\text{Pr}_{{\Phi_{{\text{BS}}}}}}\left\{ {{{\mathrm{P}}_{{\text{cov}} }}\left({\left. {{\gamma_{\mathrm{D}}},{\gamma_{\mathrm{A}}}} \right|{\Phi_{{\text{BS}}}}} \right) \ge z} \right\}\\ &\quad {\text{with}} \quad z \in \left[ {0,1} \right] \end{aligned} $$
(8)
where Pcov(γD,γA|ΦBS) is the coverage probability conditioned upon ΦBS (and by assuming that the typical MT is at the origin), which is defined as follows:
$$ {}{{\mathrm{P}}_{\text{cov} }}\left({\left. {{\gamma_{\mathrm{D}}},{\gamma_{\mathrm{A}}}} \right|{\Phi_{{\text{BS}}}}} \right) = \Pr \left\{ {\left. {{\text{SIR}} \ge {\gamma_{\mathrm{D}}},\overline {{\text{SNR}}} \ge {\gamma_{\mathrm{A}}}} \right|{\Phi_{{\text{BS}}}}} \right\} $$
(9)
It is worth mentioning that, in (8), we have emphasized that the probability is computed only with respect to ΦBS.
The spatially averaged coverage probability in (2) can be retrieved from the meta distribution in (7) directly from its definition, as follows:
$$ {}{{\mathrm{P}}_{\text{cov}} }\left({{\gamma_{\mathrm{D}}},{\gamma_{\mathrm{A}}}} \right) \,=\, {{\mathbb{E}}_{{\Phi_{{\text{BS}}}}}}\left\{ {{{\mathrm{P}}_{\text{cov} }}\left({\left. {{\gamma_{\mathrm{D}}},{\gamma_{\mathrm{A}}}} \right|{\Phi_{{\text{BS}}}}} \right)} \right\} \,=\, {\int\nolimits}_{0}^{1} {{{\overline {\mathrm{F}} }_{{{\mathrm{P}}_{{\text{cov}}}}}}\left(z \right)dz} $$
(10)
In practical terms, the meta distribution provides one with the fraction of links whose SIR is greater than γD and whose average SNR is greater than γA with probability at least equal to z in each network realization. Therefore, it yields a more general statistical characterization of the performance of cellular networks beyond spatial averages.
Computation: Gil-Pelaez method
As discussed in [8], the direct computation of the meta distribution in (8) is not straightforward. A general approach to overcome this issue is to capitalize on the Gil-Pelaez inversion theorem [15], which allows one to formulate the meta distribution as a function of the moments of the (conditional) coverage probability in (9).
In particular, the following holds [8]:
$$ \begin{aligned} {}{\overline {\mathrm{F}}_{{{\mathrm{P}}_{{\text{cov}}}}}}& \left({{\gamma_{\mathrm{D}}},{\gamma_{\mathrm{A}}},z} \right) = \\ &\frac{1}{2} + \frac{1}{\pi }{\int\nolimits}_{0}^{+ \infty} {\frac{{{\text{Im}} \left\{ {{{\mathcal{M}}_{jt}}\left({{\gamma_{\mathrm{D}}},{\gamma_{\mathrm{A}}}} \right)\exp \left({ - jt\ln \left(z \right)} \right)} \right\}}}{t}dt} \end{aligned} $$
(11)
where \(j = \sqrt { - 1}\) is the imaginary unit, Im{·} denotes the imaginary part operator, and \({{\mathcal {M}}_{b}}\left ({{\gamma _{\mathrm {D}}},{\gamma _{\mathrm {A}}}} \right)\) is the bth moment of the (conditional) coverage probability Pcov(γD,γA|ΦBS), which is defined as follows:
$$ {}\begin{aligned} {{\mathcal{M}}_{b}}\left({{\gamma_{\mathrm{D}}},{\gamma_{\mathrm{A}}}} \right) &= {{\mathbb{E}}_{{\Phi_{{\text{BS}}}}}}\left\{ {{{\left({{{\mathrm{P}}_{\text{cov}} }\left({\left. {{\gamma_{\mathrm{D}}},{\gamma_{\mathrm{A}}}} \right|{\Phi_{{\text{BS}}}}} \right)} \right)}^{b}}} \right\}\\ &= {\int\nolimits}_{0}^{1} {b{z^{b - 1}}{{\overline {\mathrm{F}} }_{{{\mathrm{P}}_{{\text{cov}}}}}}\left(z \right)dz} \end{aligned} $$
(12)
Therefore, the meta distribution can be obtained by first computing the moments of the (conditional) coverage probability Pcov(γD,γA|ΦBS) in (12) and by then computing the integral in (11). As remarked in [12–14], however, the computation of (11) is not always straightforward. Other methods need, in general, to be used instead, e.g., the Fourier-Jacobi expansion [12], and the Mnatsakanov’s theorem [13, 14].
An alternative approach relies on approximating the meta distribution with another distribution. A nota- ble example is using the beta distribution for approximating it over the entire range of values z∈[0,1]. As remarked in [14], however, this approach cannot be applied if the actual distribution does not fulfill the class of the beta distribution. The following lemma shows, e.g., that this is the case if the coverage probability is defined in terms of SIR and \({\overline {{\text {SNR}}} }\).
Lemma 1
Let (aF,bF,cF) be the generic triplet of parameters introduced in (
6
) and (
7
), and denote dF=(cF−bF)/aF≥0. The meta distribution in (
11
) satisfies the following properties:
$$ {}\begin{aligned} {\overline {\mathrm{F}}_{{{\mathrm{P}}_{{\text{cov}}}}}}&\left({{\gamma_{\mathrm{D}}},{\gamma_{\mathrm{A}}},z \to 0} \right) =\\ &1 \,-\, \exp \!\left(\!{ - 2 \pi \lambda_{{\text{BS}}} \Psi \!\left(\! \!{{{\left({\frac{{{{\mathrm{P}}_{{\text{tx}}}}}}{{\kappa \sigma_{\mathrm{N}}^{2}{\gamma_{\mathrm{A}}}}}} \right)}^{{1 / \gamma }}};{{\mathrm{a}}_{\mathrm{F}}},{{\mathrm{b}}_{\mathrm{F}}},{{\mathrm{c}}_{\mathrm{F}}}} \!\right)} \!\right) \end{aligned} $$
(13)
$$ {\overline {\mathrm{F}}_{{{\mathrm{P}}_{{\text{cov}}}}}}\left({{\gamma_{\mathrm{D}}},{\gamma_{\mathrm{A}}},z \to 1} \right) = 0 $$
(14)
$$ {}\begin{aligned} 0 &\!\le\! {\overline {\mathrm{F}}_{{{\mathrm{P}}_{{\text{cov}}}}}}\left({{\gamma_{\mathrm{D}}},{\gamma_{\mathrm{A}}},z} \right) \le 1\\ &\quad- \!\exp\! \left({ - 2 \pi \lambda_{{\text{BS}}} \Psi \left({{{\left({\frac{{{{\mathrm{P}}_{{\text{tx}}}}}}{{\kappa \sigma_{\mathrm{N}}^{2}{\gamma_{\mathrm{A}}}}}} \right)}^{{1 / \gamma }}};{{\mathrm{a}}_{\mathrm{F}}},{{\mathrm{b}}_{\mathrm{F}}},{{\mathrm{c}}_{\mathrm{F}}}} \right)} \right) \end{aligned} $$
(15)
where \({\Lambda _{\Phi _{{\text {BS}}}^{\left (F \right)}}}\left ({{\mathcal {B}}\left ({0,r} \right)} \right) = 2 \pi \lambda _{{\text {BS}}} \Psi \left ({r;{{\mathrm {a}}_{\mathrm {F}}},{{\mathrm {b}}_{\mathrm {F}}},{{\mathrm {c}}_{\mathrm {F}}}} \right)\), and:
$$ {}\begin{aligned} \Psi \left({r;{{\mathrm{a}}_{\mathrm{F}}},{{\mathrm{b}}_{\mathrm{F}}},{{\mathrm{c}}_{\mathrm{F}}}} \right) \!&=\! \left({\frac{{{{\mathrm{a}}_{\mathrm{F}}}}}{3}{r^{3}} + \frac{{{{\mathrm{b}}_{\mathrm{F}}}}}{2}{r^{2}}} \right){\mathbbm{1}}\left({r \le {{\mathrm{d}}_{\mathrm{F}}}} \right)\\&\quad\,+\, \left(\!{\frac{{{{\left({{{\mathrm{b}}_{\mathrm{F}}} - {{\mathrm{c}}_{\mathrm{F}}}} \right)}^{3}}}}{{6{\mathrm{a}}_{\mathrm{F}}^{2}}} + \frac{{{{\mathrm{c}}_{\mathrm{F}}}}}{2}{r^{2}}} \right)\!{\mathbbm{1}}\!\left({r > {{\mathrm{d}}_{\mathrm{F}}}} \right) \end{aligned} $$
(16)
Proof
See Appendix A. □
From Lemma 1, we evince that the meta distribution lies in the range [0,1] only if γA=0, i.e., the conventional definition of coverage probability based only on the SIR is used [11]. If γA≠0, on the other hand, the meta distribution lies in the range \(\left [ {0,1 - \exp \left ({ - 2 \pi \lambda _{{\text {BS}}} \Psi \left ({{{\left ({{{{{\mathrm {P}}_{{\text {tx}}}}} / {\left ({\kappa \sigma _{\mathrm {N}}^{2}{\gamma _{\mathrm {A}}}} \right)}}} \right)}^{{1 / \gamma }}};{{\mathrm {a}}_{\mathrm {F}}},{{\mathrm {b}}_{\mathrm {F}}},{{\mathrm {c}}_{\mathrm {F}}}} \right)} \right)} \right ]\). This implies that the beta distribution is not necessarily a good approximation for the meta distribution, since the former distribution lies always in the range [0,1].
Computation: Euler sum method
In this section, motivated by the considerations just made, we show that the meta distribution can be efficiently computed, with a known and bounded approximation error, by using the trapezoidal integration rule and the Euler sum method as originally proposed in [16] and recently used, e.g., in [17]. The following proposition states the result in rigorous terms.
Proposition 1
Let A, N, and Q be three positive integer numbers. Let us define the following functions:
$$ \begin{array}{l} {\beta_{0}} = 2\\ {\beta_{n}} = 1\quad {\text{for}}\quad n = 1,2, \ldots,N\\ {s_{n}} = \frac{{A + 2\pi jn}}{2}\quad {\text{for}}\quad n = 0,1, \ldots,N \end{array} $$
(17)
The meta distribution can be formulated as follows:
$$ {}\begin{aligned} {\overline {\mathrm{F}}_{{{\mathrm{P}}_{{\text{cov}}}}}}\left({{\gamma_{\mathrm{D}}},{\gamma_{\mathrm{A}}},z} \right) & \approx \frac{{{2^{- Q}}\exp \left({{A / 2}} \right)}}{{{{\ln }^{2}}\left(z \right)}}\sum\limits_{q = 0}^{Q} \left({\begin{array}{*{20}{c}} Q\\ q \end{array}} \right)\\ &\quad\sum\limits_{n = 0}^{N + q} {\frac{{{{\left({ - 1} \right)}^{n}}}}{{{\beta_{n}}}}{\text{Re}} \left\{ {\frac{{{{\mathcal{M}}_{- {{{s_{n}}} / {\ln \left(z \right)}}}}\left({{\gamma_{\mathrm{D}}},{\gamma_{\mathrm{A}}}} \right)}}{{{s_{n}}}}} \right\}} \\ &\quad+ \left| {{\mathcal{E}}\left({A,N,Q} \right)} \right| \end{aligned} $$
(18)
where Re{·} is the real part operator, \({{\mathcal {M}}_{b}}\left ({{\gamma _{\mathrm {D}}},{\gamma _{\mathrm {A}}}} \right)\) is the bth moment in (
12
), and \({{\mathcal {E}}\left ({A,N,Q} \right)}\) is the approximation error as follows:
$$ {}\begin{aligned} \left| {{\mathcal{E}}\left({A,N,Q} \right)} \right| & \approx \frac{1}{{\exp \left(A \right) - 1}}\\ & \quad+ \left| \frac{{{2^{- Q}}\exp \left({{A / 2}} \right)}}{{{{\ln }^{2}}\left(z \right)}}\sum\limits_{q = 0}^{Q} \left({\begin{array}{*{20}{c}} Q\\ q \end{array}} \right){{\left({ - 1} \right)}^{N + 1 + q}}\right.\\ &\qquad\left.{\text{Re}} \left\{ {\frac{{{{\mathcal{M}}_{- {{{s_{N + 1 + q}}} / {\ln \left(z \right)}}}}\left({{\gamma_{\mathrm{D}}},{\gamma_{\mathrm{A}}}} \right)}}{{{s_{N + 1 + q}}}}} \right\} \right| \end{aligned} $$
(19)
Proof
See Appendix B. □
There are three main advantages in favor of using the Euler sum method instead of the Gil-Pelaez method: (1) Eq. 18 does not need the explicit computation of an integral, which makes the numerical estimation of the meta distribution easier; (2) the method can be applied to any family of non-negative meta distributions; and (3) the approximation error in (19) is known in closed-form and the accuracy of the numerical computation can be controlled by using the triplet of parameters (A,N,Q). As discussed in [16], in particular, typical values of these parameters are A=10 ln(10), which guarantees a discretization error of the order of 10−10, and N and Q of the order of 10 or 20.