2.1. Signal-interference ratio and coverage probability

We consider the classical signal-to-noise-plus-interference ratio (SINR) of a typical transmitter-receiver pair in slot $n$ connected by a link between origin and location $x$,

(3)\begin{equation} \mathrm{SINR}_n(x)= \frac{Q_n S_n a(x)}{\mathcal I_n(x) +\sigma^2}, \qquad \mathcal I_n(x) = \sum_{y\in \Phi_{x}} a(y) S^{(y)}_n Q^{(y)}_n, \quad n \ge 1, \end{equation}

where $S_n$ is the exponential signal power of the sender, $a(x)$ is the attenuation of the signal introduced in (1), $\mathcal I_n(x)$ is the interference experienced by the link at a distance $x$ from the origin generated by active nodes at the positions $\Phi_x$ of neighboring cell emitters, and $\sigma^2$ is additional noise. Moreover, $\{S_n^{(y)}\}$ is an i.i.d. sequence of exponentially distributed random variables that represents the signal power of interference transmitters at locations $y$. All signals are independent over time. Furthermore, $Q_n$ is the indicator function for the event that the BS in position $x$ is active, and $Q_n^{(y)}$, $y\in \Phi_x$, are indicator functions for the activity of the interference BSs.

The main simplifying assumption in our model framework is that, for all $n\ge 1$,

(4)\begin{equation} Q_n\sim\mathrm{Bin}(1,q_n),\quad Q_n^{(y)} \sim \mathrm{Bin}(1,q_n) \end{equation}

and all indicator functions are independent in a given slot $n$. We note that in a realistic buffered cellular network, transmitter activity is determined by the evolution of local queues, hence depends on past transmissions and interference conditions. However, the primary goal of the present work is to develop a tractable capacity model that captures the behavior of an average transmitter rather than that of each BS individually. Based on our previous assumptions of simplistic but natural properties of single-tier networks, all BSs are subjected to the same traffic conditions (same input rate $p$, same average signal strength, etc.), so it is reasonable that all servers are busy with the same probability $q_n$. Assumption (4), that activity indicators have the same distribution for each BS should be understood as a mean-field approximation that replaces the interacting queue dynamics with a system of network cells with the same average activity level $q_n$.

We now derive the coverage probability, essentially the probability of successful transmission, in the following steps. First, given $x$ and $\Phi_x$, using that all signals have an exponential distribution,

\begin{align*} V_T(q_n\,; x,\Phi_x) &=\mathbb P(\mbox{SINR}_n(x) \gt T|\,\Phi_x) \\ & = q_n \exp\Big\{\int_{\mathbb R^d} \ln \Big(1-\frac{q_n\,T a(y)}{a(x)+T a(y)} \Big)\,\Phi_x(dy)-\frac{\mu T\sigma^2}{a(x)}\Big\}. \end{align*}

Next, averaging over the Poisson locations in $ \Phi_x $, the associated transmission success probability, conditional on the link position $x$, $V_T(q_n\,;x)=\mathbb E[V_T(q_n\,; x,\Phi_x)]$, is obtained as a radial function of $|x|$,

\begin{equation*} V_T(q\,; x)= q \exp\Big\{-\int_{|y|\ge |x|} \frac{\lambda_0 q\,T a(y)}{a(x)+T a(y)}\,dy-\frac{\mu T\sigma^2}{a(x)}\Big\}. \end{equation*}

Let us also apply a change to polar coordinates $u=|y/\ell_0|^d$, and re-write as

\begin{align*} V_T(q\,;x)= q e^{-\mu T\sigma^2/a(x)}\exp\Big\{- \frac{\lambda_0 q}{\lambda_1} \int_{|x/\ell_0|^d}^\infty \frac{Ta(u^{1/d}\ell_0)}{a(x)+T a(u^{1/d}\ell_0)}\,du\Big\}. \end{align*}

In a final step, we take the expected value over the known random link position in a Voronoi cell,

\begin{align*} V_T(q) &= \lambda_1 \int_{\mathbb R^d} V_T(q_n\,; x)e^{-\lambda_1|B(0,1)||x|^d}\,dx\\ &= \int_0^\infty V_T(q\,; v^{1/d}\ell_0)\,e^{-v}\,dv, \end{align*}

which yields the probability of coverage as a function of threshold $T$, activity probability $q$, and the remaining parameters $p$, $\lambda_0$, $\lambda_1$, $\kappa$, $\sigma^2$, $\beta$, and $d$. To reduce the number of parameters, we use the dimension-scaled parameter $\delta = \beta/d \gt 1$ and the ratio $w=\lambda_0/\lambda_1$, which can be interpreted as the cell bandwidth. Integrating over $v$, we obtain the stationary downlink coverage probability

(5)\begin{equation} V_T(q) = \int_0^\infty q\, \exp\Big\{-\frac{\mu T\sigma^2(\kappa+v^\delta)}{1+\kappa} - q\int_v^\infty \frac{wT(\kappa+v^\delta)} {\kappa+u^\delta+T(\kappa+v^\delta)}\,du\Big\}e^{-v}\,dv. \end{equation}

2.2. Buffer dynamics and activity probability

Since $\{1-V_T(q)\}_{T\ge 0}$ are probability distribution functions, we can associate the coverage probabilities in (5) with a sequence of random variables $\{\mbox{SINR}_n\}_{n\ge 1}$, such that

(6)\begin{equation} \mathbb P(\mbox{SNIR}_n \gt T) = V_T(q_n). \end{equation}

Here, $\mathbb P(\mbox{SNIR}_n \gt 0)=V_0(q_n)=q_n$ and hence $\mbox{SNIR}_n$ is identically zero with probability $1-q_n$. Let us define the ratio

(7)\begin{equation} U(q_n)=V_T(q_n)/q_n, \end{equation}

which is the conditional probability of successful transmission in a cell where the buffer is active in the slot $n$. Let also $\{B_n\}$ be the sequence of buffer sizes in a typical cell and $K$, $0\le B_n\le K$, be the maximal buffer capacity. That is, any failed signal that arrived when $K$ signals are already stored in the buffer is permanently lost. Note that because only one transmission per cell per slot is allowed, $\{B_n\}$ is a finite birth-and-death process. Using (2) and (7), the jump probabilities are defined as

(8)\begin{equation} \begin{split} \mathbb P(B_{n+1}-B_n=1|0\le B_n\le K-1)&=p(1-U(q_n)),\\ \mathbb P(B_{n+1}-B_n=-1|1\le B_n\le K-1)&=(1-p)U(q_n), \\ \mathbb P(B_{n+1}-B_n=-1|B_n=K)&=U(q_n),\\ \mathbb P(B_n=0)&=(1-q_n)/(1-p). \end{split} \end{equation}

The equilibrium model of a cellular network, where all the dependence between cells is channeled through the coverage probability and the buffer sequences in separate cells are independent and satisfy the system of equations (8), is analyzed in [Reference Ingemar and Taisiia15]. It is shown that $\{B_n\}_{n\ge 0}$ with $B_0=0$ is a nonhomogeneous Markov chain, which is strongly ergodic with a unique asymptotic distribution $\pi=\{\pi_k\}_{0\le k\le K}$. This result relies on an iterative technique to determine the busy-link probability $q^*=q^*(p,w,T)$, as the limit of the increasing sequence

\begin{equation*} p=q_0\le q_1\le \dots,\le q_n \nearrow q^*, \quad \mbox{as} \quad n\to \infty, \end{equation*}

for $T$ less than the maximal SINR threshold that can be achieved in the network, that is,

(9)\begin{equation} T \lt T^*_\mathrm{max}(p,w) =\sup\{t \gt 0: \{q_n\}\ \mbox{has a limit}\ q^*(p,w,t),\ \mbox{such that}\ p \lt q^*\le 1\}. \end{equation}

It turns out that for limited loss systems, $1\le K \lt \infty$, the limit $q^*$ exists for all $(p,w,T)$, so $T^*_\mathrm{max}(p,w)=\infty$, and $q^*$ is the minimal solution of the non-linear equation

(10)\begin{equation} q=F_K(q),\qquad F_K(q)=1-\frac{U(q)-p}{U(q)-p\,A(q)^K}, \quad A(q)=\frac{p(1-U(q))}{(1-p)U(q)} \gt 0. \end{equation}

It is a crucial feature for the proof of $(10)$, that $U(q)$ defined in (7) is decreasing (non-increasing) in $q$. The function $F_K$ is well defined in the special case $U(q)=p$, and has the equivalent representation

(11)\begin{equation} F_K(q)={1-(1-p)\pi_0} =1-\frac{1-p}{1-p +\frac{p}{U(q)} \Big(1+\dots +A(q)^{K- 1}\Big)}, \end{equation}

In fact, re-writing (10) as

\begin{equation*} 1-F_K(q)=\frac{(U(q)-p)(1-p)^KU(q)^K}{U(q)^{K+1}(1-p)^K-p^{K+1}(1-U(q))^K}, \end{equation*}

the relation (11) follows by a polynomial division of the denominator with $U(q)-p$. As an alternative, this result can be derived from the probabilistic representation of the buffer size distribution as in [Reference Ingemar and Taisiia15, Corollary 1]. The resulting system is in a steady state with throughput $V_T(q^*) \lt p$ and loss probability given by $\mathbb{P}_{\text{loss}} = 1 - V_T(q^*)/p$. The loss probability specifically quantifies the likelihood of transmission failure for newly arrived signals. In contrast, the conventional outage probability is $1 - V_T(q^*)$, which represents the probability of transmission failure for an arbitrary signal, regardless of its origin. The no-loss system $K = \infty$, on the other hand, has $T^*_\mathrm{max}(p,w) \lt \infty$. The defining relation for $\{q_n\}$ with $q_0=p$ is $q_n=p/U(q_{n-1})$, $n\ge 1$, and $(p,w,t)$ is restricted to $t \lt T^*_\mathrm{max}(p,w)$, where there exists a minimal solution $q^*$ of the recursion $q_n=(p/V_t(q_{n-1}))q_{n-1}$, which satisfies the balance equation

(12)\begin{equation} V_t(q^*)=p \end{equation}

([Reference Ingemar and Taisiia15, Lemma 4]). The equilibrium buffer size in the no-loss model is the geometric distribution

(13)\begin{equation} \pi_k=\frac{1-q^*}{1-p}\Big(1-\frac{1-q^*}{1-p}\Big)^k,\quad k\ge 0. \end{equation}

We also observe that the reference case of pure loss, $K=0$, has $B_n=0$ and $q_n=p$ for all $n$, hence $q^*=p$.

To complement the previous results, for maximum buffer size $K$, $1\le K \lt \infty$, and given $(p,w,t)$, we let $q_K^*=q_K^*(p,w,t)$ be the minimal solution of the balance equation  $q=F_K(q)$. It follows from (10) or (11) that $F_K$ increases in $K$; that is, $F_K(q)\le F_{K+1}(q)$, $0 \lt q \lt 1$. Moreover, we know that $p \lt F_K(q) \lt 1$ for each $q$, $p \lt q \lt 1$. Thus, the solutions $q_K^*=F_K(q_K^*)$ and $q_{K+1}^*=F_{K+1}(q_{K+1}^*)$ must satisfy $q_K^*\le q_{K+1}^*$, and we have the monotonicity in $K$, that is

(14)\begin{equation} q_K^*\le q_{K+1}^*,\quad K\ge 1. \end{equation}

Restricting to $t \lt T^*_\mathrm{max}(p,w)$, these properties extend to $K=\infty$.

2.3. Coverage probability as a function of the $\mbox{SINR}$ threshold

To further develop the approach described above, we state upper bounds for the coverage probability, which will ensure suitable integrability properties of $\{V_t(q)\}_{t\ge 0}$. For convenience, we introduce the additional notation

(15)\begin{equation} G_\delta(t)= \int_1^\infty \frac{du}{1+(1+u^\delta)/t},\quad t \gt 0. \end{equation}

Lemma 1. The family of downlink coverage probability functions $V_t(q)$ in (5) for any $\sigma^2, \kappa \ge 0$ and $t\ge t_0 \gt 0$ satisfies the upper bound

\begin{equation*} V_t(q)\le \frac{q}{1+G_\delta\!(t_0)\,(t/t_0)^{1/\delta}\,qw }. \end{equation*}

Proof. For downlink traffic with $\sigma^2\ge 0$, $\kappa\ge 0$, using straightforward estimates in (5) and the auxiliary function $G_\delta(t)$ in (15),

\begin{align*} V_t(q) &\le \int_0^\infty q\, \exp\Big\{-v \int_1^\infty \frac{qw} {1+\frac{\kappa+v^\delta u^\delta}{t(\kappa +v^\delta)}}\,du\Big\}e^{-v}\,dv \nonumber \\ &\le \int_0^\infty q\, \exp\Big\{-v\int_1^\infty \frac{qw} {1+(1+u^\delta)/t}\,du\Big\}e^{-v}\,dv =\frac{q}{1+G_\delta\!(t)\, wq}. \end{align*}

Moreover, using $t\ge t_0$,

(16)\begin{equation} G_\delta(t)=\int_1^\infty \frac{du} {1+(1+u^\delta)/t} = \int_{(t_0/t)^{1/\delta}}^\infty \frac{(t/t_0)^{1/\delta}\,du } {1+1/t+u^\delta/t_0} \ge G_\delta(t_0)\, (t/t_0)^{1/\delta}, \end{equation}

which gives the stated upper bound.

We note that in the simplest case when $\kappa=0$ and $\sigma^2 = 0$, the coverage probabilities in (5) simplify considerably into the closed-form formula for downlink traffic

(17)\begin{equation} V_T(q) = \frac{q}{1+H_\delta(T)\, w\,q}, \end{equation}

where

(18)\begin{equation} H_\delta(t)=\int_1^\infty \frac{du}{1+u^\delta/t} ,\quad t \gt 0. \end{equation}

As an example, the no-loss cellular downlink network with unbounded attenuation and no noise ( $K=\infty$, $\kappa=0$, $\sigma^2=0$), has a traffic distribution in equilibrium satisfying (12) if $p\le 1/(1+w\,H_\delta(T))$, in which case the probability of activity $q^*$ and the geometric buffer distribution $\pi$ in (13) are determined by

(19)\begin{equation} q^*=\frac{p}{1-pw H_\delta(T)},\quad \pi_0=\frac{1-p-pwH_\delta(T)}{(1-p)(1-pwH_\delta(T))}. \end{equation}

3.1. Admissible capacity regions

To relate the Shannon capacity more directly to the traffic handled by the network, we introduce a notion of admissible operating parameters. Intuitively, the system should be considered in working order whenever the average transmission capacity of the channel exceeds the traffic demand. To address overload, we apply the Shannon capacity to distinguish between two types of network states, specifically oversaturated and non-oversaturated systems, which represent “non-workable” and “workable” networks, respectively. This motivates the definition of a region of admissible data rate-bandwidth pairs.

Definition 3.2. A wireless system with buffer size $K$, data rate $p$ and bandwidth $w$ is said to be capacity-admissible, if $(p,w)\in R_K$, where $R_K$ is the capacity-admissible region of data rate and bandwidth defined as

(27)\begin{equation} R_K=\{(p,w): \mathrm{Cap}_K(p,w)\ge pw\}. \end{equation}

From an engineering perspective, the capacity-admissible region $R_K$ describes the set of operating parameters $(p,w)$ for which the wireless system is able to maintain the incoming traffic. The quantity $pw$ represents the average traffic demand per cell, while the Shannon capacity $Cap_K(p,w)$ characterizes the average transmission capacity of the channel in the presence of interference and retransmissions. The condition $Cap_K(p,w)\ge pw$ therefore identifies a regime such that the available transmission resources are sufficient to process the incoming signals without long-term overload of the buffers. Parameters outside this region correspond to an over-saturated system where the traffic demand exceeds the achievable capability. In other words, the system is capacity-admissible if, for fixed bandwidth, the data rate is small enough so that the capacity exceeds the external traffic intensity. Equivalently, for a fixed data rate, the condition (27) holds if the bandwidth is small enough for the traffic intensity to stay below the Shannon capacity. For example, for $K=\infty$ we see from (24) that the system is capacity-admissible if $\log_2(1 + T_\mathrm{max}(p, w)) \ge 1$, that is, $T_\mathrm{max}(p,w)\ge 1$.

3.1.1. Special case $\kappa=\sigma^2=0$

Toexplicitly compute the capacity-admissible region, we consider the coverage probability in (17) for downlink traffic without noise and with unbounded attenuation, $\kappa=\sigma^2=0$. It is seen immediately in Theorem 3.1 i) that the Shannon capacity of the pure-loss system with $K=0$ is

(28)\begin{equation} \mbox{Cap}_0(p,w) = \frac{pw}{\ln2} \int_0^{\infty} \frac{1}{1+H_\delta(t)\,w p}\frac{1}{1 + t}\,dt. \end{equation}

The associated capacity-admissible region is

(29)\begin{equation} R_0=\Big\{(p,w): \frac{pw}{\ln 2}\int_0^\infty \frac{1}{1+w p H_\delta(t)}\frac{1}{1+t}\,dt \ge pw \Big\}, \end{equation}

which is the set $\{(p,w): pw\le \alpha\}$, with $\alpha$ the constant determined by

\begin{equation*} \int_0^\infty \frac{1}{1+\alpha H_\delta(t)}\frac{1}{1+t}\,dt=\ln 2. \end{equation*}

Turning to the other extrema case of the no-loss system $K = \infty$, we noticed in (19) that the limiting busy-link probability $q_\infty^*(p,w,t)$ is

\begin{equation*} q_\infty^*(p,w,t)=\frac{p}{1-pw H_\delta(t)},\quad t \lt T_\mathrm{max}(p,w), \end{equation*}

where $T_\mathrm{max}(p,w)$ is such that $q_\infty^*(p,w,t) \le 1$. Hence, using (18), $T_\mathrm{max}(p,w)$ is the solution of the equation

(30)\begin{equation} H_\delta(T_\mathrm{max}(p,w)) =\int_1^\infty \frac{1}{1+u^\delta/T_\mathrm{max}(p,w)}\,du=\frac{1}{w} \big(\frac{1}{p}-1\big). \end{equation}

By Theorem 3.1 iii) and the Definition 3.2, the no-loss system is capacity-admissible if $T_\mathrm{max}(p,w)\ge 1$, that is, $H_\delta(T_\mathrm{max}(p,w))\ge H_\delta(1)$, which implies that the capacity-admissible region is

(31)\begin{equation} R_\infty =\Big\{(p,w): p \le \frac{1}{1+w H_{\delta}(1)}\Big\}. \end{equation}

In the opposite situation of $T_\mathrm{max}(p,w) \lt 1$, the capacity of the no-loss system no longer covers the input rate intensity, in the sense made precise in Definition 3.2. A possible interpretation is degradation in delay and network performance to the extent that the system becomes non-workable.

To continue with finite buffer sizes $K\ge 1$, by (14), since $V_T(q)$ in (17) is an increasing function of $q$, it follows that $\mathrm{Cap}_K(p,w)$ defined in (23) also is increasing in $K$. Thus, we have

(32)\begin{equation} \mathrm{Cap}_0(p,w)\le\dots\le \mathrm{Cap}_K(p,w)\le \mathrm{Cap}_{K+1}(p,w)\le \dots \end{equation}

For $K=1$ the probability $q^*_1$ can be found explicitly as a function of $p$, $w$, and $t$ by solving the balance equation (10). The capacity $\mathrm{Cap}_1(p,w)$ and the region of admissible traffic $R_1$ are then obtained from (23). In principle, analogous numerical results can be derived for any finite buffer size $K \geq 1$. However, both theoretical and numerical analyses are increasingly challenging due to the growing complexity of the solutions $q^*_K$, which in turn affects the resulting capacity function $\mathrm{Cap}_K(p, w)$. To address this shortcoming, we propose an alternative approach to studying the Shannon capacity. Instead of computing the capacity for a specific value of $K$, we derive an upper bound as a generic measure of the Shannon capacity valid for any $K \lt \infty$. For this, we use $T_\mathrm{max}(p,w)$ defined in (30) and split the Shannon capacity in Theorem 2, ii) into two parts:

(33)\begin{equation} \mathrm{Cap}_K(p,w) = \frac{w}{\ln 2}\Big( \int_0^{T_\mathrm{max}(p,w)} \frac{V_t(q^*)}{1+t}\,dt +\int_{T_\mathrm{max}(p,w)}^\infty \frac{V_t(q^*)}{1+t}\,dt\Big). \end{equation}

Since $V_t(q^*)\le p$, the first integral in (33) is bounded by the no-loss capacity $wp\log_2(1+ T_\mathrm{max}(p,w))$ obtained in (24). For the second integral, since the function $V_t(q)$ for the special case (17) is monotone in $q$, the inequality $V_t(q^*)\le V_t(1)$ yields

(34)\begin{equation} \mathrm{Cap}_K(p,w)\le C(p,w), \end{equation}

where

(35)\begin{equation} C(p,w)=\frac{wp}{\ln 2} \ln(1+T_\mathrm{max}(p,w)) +\frac{w}{\ln 2}\int_{T_\mathrm{max}(p,w)}^\infty \frac{1}{1+H_\delta(t)w}\frac{1}{1+t}\,dt. \end{equation}

Next, we define

(36)\begin{equation} R =\{(p,w): C(p,w)\ge pw\} \end{equation}

to be the region of data rate-bandwidth pairs that are admissible for the capacity upper bound $C(p,w)$. Then it follows from (34) that the admissible regions $R_K$ are contained within the region admissible for the upper bound, $R_K\subset R$, for every finite $K$.

The numerical results are demonstrated in Figure 1, where plot (a) shows Shannon capacities $\mathrm{Cap}_0(p,w), \mathrm{Cap}_1(p,w)$ and $\mathrm{Cap}_\infty(p,w)$ together with the traffic intensity $\rho = pw$ as functions of $p$ for the choice $\delta=2$ and $w=10$. The plot (b) shows the admissible regions for capacity (cases $K = 0, 1, \infty$) as well as the admissible region $R$ from (36) for the upper limit of capacity for general $K$. The contour set $\{(p,w): C(p,w)=pw\}$ for varying $p$ and $w$ is displayed as a dashed line. It follows from (32) that the admissible regions $R_K$ are increasing sets with increasing $K \lt \infty$.

Figure 1.

Shannon capacity for different buffer sizes $K$ and associated admissible regions. The parameters are $\kappa=0$, $\sigma^2=0$, $\delta=2$, $w=10$. Left panel (a): Shannon capacity $\mathrm{Cap}_K(p,10)$ for $K=0,1,\infty$, and traffic intensity $\rho=wp$. Right panel (b): The admissible regions $R_0$, $R_1$, and the set $R$ introduced in (36), as well as $R_\infty$, which for these parameters satisfy $R_\infty\subset R_0$.

1 Long description

Line graph with legend entries: K equals 0, K equals 1, K equals infinity and rho. The horizontal axis label is p, with tick labels 0, 0.1, 0.2, 0.3, 0.4. The vertical axis label is Cap(p), with tick labels 0, 1, 2, 3, 4. A solid curve labeled K equals 0 starts near (0, 0) and rises to about (0.1, 2.1), (0.2, 2.8), (0.3, 3.3) and about (0.45, 3.8). A dashed curve labeled K equals 1 starts near (0, 0) and rises above the solid curve to about (0.1, 2.7), (0.2, 3.5), (0.3, 3.9) and about (0.45, 4.3). A curve marked with open circle points labeled K equals infinity starts near (0, 0), rises to about (0.05, 1.0), reaches about (0.1, 1.1), stays near about 1.1 around p from 0.1 to 0.2, then decreases to about (0.3, 1.0) and about (0.45, 0.8). A dashed straight line labeled rho runs upward from near (0, 0) through about (0.1, 1.0), (0.2, 2.0), (0.3, 3.0) and about (0.45, 4.5). Plot with a horizontal axis label p and tick labels 0, 0.2, 0.4, 0.6, 0.8, 1 and a vertical axis label w with tick labels 0, 10, 20, 30, 40, 50, 60. The legend entries are K equals 0, K equals 1, K equals infinity and K less than infinity. Three filled regions are bounded by curves that decrease as p increases. The boundary curve for K equals infinity is the lowest, passing near about (0.1, 10), (0.2, 5), (0.4, 2), (0.6, 1.2), (0.8, 0.8) and (1, 0.6). The boundary curve for K equals 0 lies above it, passing near about (0.1, 20), (0.2, 12), (0.4, 6), (0.6, 4), (0.8, 3) and (1, 2.5). The boundary curve for K equals 1 lies above the K equals 0 curve, passing near about (0.1, 25), (0.2, 15), (0.4, 7), (0.6, 4.5), (0.8, 3.2) and (1, 2.7). A dashed curve labeled K less than infinity lies above the filled regions, passing near about (0.1, 35), (0.2, 20), (0.4, 10), (0.6, 7), (0.8, 5) and (1, 4).

3.1.2. General case

The observations discussed in the previous subsection demonstrate that pure loss and finite buffer networks possess a potential gain in capacity relative to no-loss networks in the scenario where $\kappa = \sigma^2 = 0$. This increase arises from the fact that loss systems retain access to higher transmission rates at $t \geq T_{\mathrm{max}}(p, w)$. To briefly discuss the capacity behavior in a more general network, we keep $\kappa=0$ but allow $\sigma^2\ge 0$ in (5). The associated coverage probability $V_t^{(\sigma^2)}(q)$, which in this case also depends on the parameter $\mu$, satisfies $V_t^{(\sigma^2)}(q)\le V_t^{(0)}(q)$ for all $t \gt 0$ and $q$. Moreover, the maximum transmission rate is reduced, $T_\mathrm{max}^{(\sigma^2)}(p,w)\le T_\mathrm{max}^{(0)}(p,w)$ and therefore

\begin{equation*} \mbox{Cap}^{(\sigma^2)}_0(p,w)\le \mbox{Cap}_0(p,w), \qquad \mbox{Cap}^{(\sigma^2)}_\infty(p,w)\le \mbox{Cap}_\infty(p,w). \end{equation*}

4.1. Frequency reuse

To meet realistic coverage targets with a sufficiently high probability, it is common to apply frequency-reuse methods. In this approach, the reuse factor $k\ge 1$ is the number of frequency bands available in the network. The higher the number of frequency bands, the lower the number of cells that share the same frequency ([Reference Andrews, Baccelli and Ganti4]). Since each BS can pick one of $k$ different frequency bands, the interference field is reduced and now only occurs between the BSs that use the same frequency. Therefore, a simplified approach to implement frequency reuse is to reduce the bandwidth from $w$ to $w/k$, and hence obtain the relevant performance measures under frequency reuse as functions of $k$. For the pure-loss case, using (22), we implement the capacity frequency reuse as

\begin{equation*} \mathrm{Cap}_0(p,w/k) = \frac{1}{\ln2} \int_0^{\infty}\frac{wp} {(k+H_\delta(t)\,wp)(1 + t)}\,dt. \end{equation*}

We see that the above function decreases with increasing reuse factor $k$, and vanishes in the limit as $k\to\infty$. This is again consistent with the results presented in [Reference Andrews, Baccelli and Ganti4] in which a normalization $w=1$ applies. The function $\tau(4,k)$ in [Reference Andrews, Baccelli and Ganti4] is the same as our $\mathrm{Cap}_0(1,1/k)$ for $\delta=2$, compare Figure 2.

Figure 2.

Comparison of the Shannon capacity $Cap_0(1, w/k)$ for different bandwidths $w$ with $\tau(4, k)$ in [Reference Andrews, Baccelli and Ganti4, VI.B], as a function of the reuse factor $k$.

4.2. Densification

Earlier work on the performance analysis of wireless pure-loss systems also considered limit schemes in the opposite direction of increasing bandwidth. This is a densification setting, widely used in modern communication systems in light of ultra-dense 5G networks designed to handle vast amounts of data from a large number of users ([Reference Bhushan, Junyi, Malladi, Gilmore, Brenner, Damnjanovic, Sukhavasi, Patel and Geirhofer7]). In terms of our model, the increase in traffic demand can be implemented by letting the number of users per cell grow, $\lambda_0\to\infty$, formalized as the limit relation $w=\lambda_0/\lambda_1 \to \infty$. In [Reference AlAmmouri, Andrews and Baccelli2, Theorem 1], under assumptions related to our pure-loss model, the authors study the expected average rate and obtain in the many-user limit the existence of a densification plateau, in practice a saturation limit of performance. In parallel, we obtain a similar capacity limit as the user density increases. By (23),

(37)\begin{align} \mathrm{Cap}_0&(p, \infty) := \lim_{w \to \infty} \mathrm{Cap}_0(p, w) \nonumber \\ &=\frac{1}{\ln2} \int_0^\infty \lim_{w \to \infty} \frac{wp} {(1+H_\delta(t)\,wp)(1 + t)}\,dt \nonumber \\ &= \frac{1}{\ln2} \int_0^{\infty} \frac{dt}{H_\delta(t)(1 + t)} \lt \infty, \end{align}

where we use dominated convergence to perform the limit-integral interchange. It is noted that the limit does not depend on the individual data rate $p$.

To extend the analysis beyond the model investigated in [Reference AlAmmouri, Andrews and Baccelli2], we consider the same limit of densification for the no-loss system. Using (24) for $K = \infty$ we compute the limiting capacity

(38)\begin{equation} \mathrm{Cap}_{\infty}(p, \infty) := \lim_{w\to\infty} wp \log_2(1 + T_\mathrm{max}(p,w)), \end{equation}

recalling from (30) that $T_\mathrm{max} := T_\mathrm{max}(p,w)$ is the unique solution of

\begin{equation*} H_\delta(T_\mathrm{max})=(1-p)/pw. \end{equation*}

Using this relation, it first follows that $T_\mathrm{max} \to 0$ as $w\to\infty$ for a fixed $p$. Moreover, in this limit,

\begin{equation*} w\,T_\mathrm{max} = \frac{1 - p}{p} \frac{T_\mathrm{max}}{H_\delta(T_\mathrm{max})} =\frac{1 - p}{p} \Big(\int_1^{\infty} \frac{du}{T_\mathrm{max} + u^{\delta}}\Big)^{-1} \to \frac{(1 - p)(\delta-1)}{p}. \end{equation*}

By expanding the logarithm in (38) this yields

(39)\begin{equation} \mathrm{Cap}_{\infty}(p, \infty) =\frac{(1 - p)(\delta-1)}{\ln 2}. \end{equation}

Furthermore, it can be checked for $K=1$ that $\mathrm{Cap}_1(p, \infty)$ coincides with the limit obtained in (37). As a consequence, by (34) and (35), for any $K$,

\begin{equation*} \frac{1}{\ln2} \int_0^{\infty} \frac{dt}{H_\delta(t)(1 + t)} \le \mathrm{Cap}_K(p, \infty) \le C(p, \infty) = \frac{(1 - p)(\delta-1)}{\ln 2} +\frac{1}{\ln2} \int_0^{\infty} \frac{dt}{H_\delta(t)(1 + t)}, \end{equation*}

where $C(p, \infty) = \lim_{w \to \infty} C(p, w)$ with $C(p,w)$ defined in (35). The limit result for the system with an unlimited buffer may seem counterintuitive. On the one hand, network densification forces the admissible maximal SINR threshold $T_\mathrm{max}$ to approach zero, indicating that the network cannot handle increasing traffic demand, but on the other hand, network performance improves as the average capacity reaches the plateau level in (39). To clarify, we let $p_\mathrm{max}$ be the maximal $p$, such that $(p,w)$ belongs to $R_\infty$, defined in (31). We then note that $p_\mathrm{max}$ also tends to zero as $w \to \infty$. The limiting network state, therefore, lies outside the admissible region $R_\infty$. Figure 3(a) illustrates the effect of densification on the no-loss capacity. The plot shows $\mathrm{Cap}_\infty(p,w)$ for increasing values of $w$ (dashed red lines), beginning from $w = 1$ (black line). For each $w$, the maximum rate $p_\mathrm{max}$ is displayed with dashed blue lines. The plot shows that the critical threshold $p_\mathrm{max}$ approaches $1$ when the limiting capacity approaches $0$, which is natural and consistent with the previous discussion.

Figure 3.

Capacity limits in different representations for $\delta=2$. (a) Convergence of the channel capacity $\mathrm{Cap}_\infty(p,w)$ as $w \to \infty$, starting from $w = 1$ (solid black line). The capacity increases for larger $w$, leading to $p_{max}$ approaching zero. (b) Capacity limit as a function of $c$ for $K = 0, K = 1$, and $K = \infty$.

Figure 3 Long description

A) Left plot with x-axis label p, ranging from 0.1 to 0.9 with ticks at 0.1 intervals. The y-axis label is Cap infinity (w, p), ranging from 0.0 to 1.4 with ticks at 0.2 intervals. A legend lists: Cap infinity (p, 1) as a solid line; Cap infinity (p, w) as dashed lines; lim w arrow infinity Cap infinity (p, w) as a dashed line; and p max as dotted markers with vertical dashed guide lines. The solid curve starts near about 0.55 at p equals 0.1, rises to a broad peak near about 0.7 around p between 0.25 and 0.35, then decreases, reaching about 0.5 near p equals 0.6 and approaching 0 near p equals 0.9. Multiple dashed curves form a band above the solid curve, with values near about 1.3 around p equals 0.1 and decreasing roughly linearly toward 0 as p approaches 0.9. A dashed limiting curve follows the upper envelope and also trends down to 0 near p equals 0.9. Along the bottom, dotted markers indicate p max positions and many vertical dashed guide lines rise from these markers up into the plot. B) Right plot with x-axis label c, ranging from 0 to 6 with ticks at 1 intervals. The y-axis label is Cap (c), ranging from 0 to 6 with ticks at 1 intervals. The legend lists Cap 1 (c), Cap 0 (c), Cap infinity (c), c and c max. Three smooth curves start near 0 at c equals 0 and increase with c. The top curve rises steeply at small c and then bends toward a plateau, reaching a little above 5 at c near 5 and about 5.4 near c equals 6. A middle curve increases more slowly, reaching about 4.0 near c equals 6. A lower curve increases and then flattens near about 1.4 by c around 2 to 6. A diagonal dashed line labeled c runs from near (0, 0) upward to near (6, 6). Three vertical dashed lines labeled c max are drawn at approximately c equals 1.3, c equals 3.4 and c equals 5.2, each with a circular marker at the top where it meets the top curve and a circular marker near the bottom on the x-axis.

4.3. Shannon capacity under balanced densification

In this section, we propose a new approach to studying capacity performance under densification. To better understand the limiting effects discussed in previous subsections, we also consider a balanced densification regime in which the bandwidth (or equivalently the number of users per cell) increases while the input rate per transmitter decreases in such a way that the overall traffic intensity remains constant. In this scaling limit, the capacity approaches a well-defined limit, and the analysis reveals a maximal throughput level. Such a scaling is motivated by dense cellular networks where the number of connected devices grows, while the traffic generated by each device is typically smaller. In this situation, the total load per cell may remain approximately constant even as the number of users increases. To exemplify, we restrict the analysis to the case $\kappa=\sigma^2=0$, for which the coverage probability is given by (17).

Mathematically, we consider the joint convergence of increasing bandwidth $w$ and decreasing data rate $p$, such that the system has a limiting traffic intensity $c \gt 0$, in the sense of

(40)\begin{equation} wp\to c,\quad w\to \infty, \quad p\to 0. \end{equation}

For convenience, we implement the limit procedure by setting $p = c/w$ and allowing $w \to \infty$. It is easy to see that the maximal transmission rate sequence $T_{\mathrm{max}}(c/w,w)$ in (30) is non-decreasing in $w$. The limit, denoted as

\begin{equation*} T_{\mathrm{max}}(c) = \lim_{w\to\infty} T_{\mathrm{max}}(c/w,w), \end{equation*}

is the solution to the equation

(41)\begin{equation} H_\delta(T_{\mathrm{max}}(c)) = 1/c. \end{equation}

Next, we define the limiting capacity under balanced densification for $0 \leq K \leq \infty$, again with some abuse of notation, as

(42)\begin{equation} \mathrm{Cap}_K(c) = \lim_{w\to\infty} \mathrm{Cap}_K(c/w,w), \end{equation}

provided the relevant limit exists. Analogously to Definition 3.2, we say that the traffic intensity $c$ falls within the balanced Shannon capacity if $c \in I_K = [0, c_{\mathrm{max}}]$, where

(43)\begin{equation} I_K = \{c \gt 0 : \mathrm{Cap}_K(c) \geq c \}, \end{equation}

denotes the admissible interval of traffic intensities supported within the balanced Shannon capacity. The upper bound $c_{\mathrm{max}}$ of this interval is the maximum throughput that can be sustained by the maximum average rate $T_{\mathrm{max}}(c)$, which therefore theoretically could be transmitted in the network.

Theorem 4.1. Under the assumption of balanced densification (40), the limiting Shannon capacity $\mathrm{Cap}_K(c)$ and the maximal admissible intensity $c_\mathrm{max}$ are

i) for the pure-loss model $K=0$,

\begin{equation*} \mathrm{Cap}_0(c) = \frac{1}{\ln 2} \int_0^\infty \frac{c}{1 + c H_{\delta}(t)}\frac{dt}{1 + t}, \end{equation*}

with $c_\mathrm{max}$ the unique solution of

\begin{equation*} \frac{1}{\ln 2}\int_0^\infty \frac{1}{1 + c_\mathrm{max} H_{\delta}(t)}\frac{dt}{1 + t}=1; \end{equation*}

ii) for the loss and delay models with any finite buffer size $K\ge 1$,

\begin{equation*} \mathrm{Cap}_K(c) = c\,\log_2\big( 1 + T_\mathrm{max}(c) \big) + \frac{1}{\ln 2} \int_{T_\mathrm{max}(c)}^{\infty} \frac{dt}{H_{\delta}(t)(1 + t)} , \end{equation*}

with $c_\mathrm{max}$ the solution $c$ of

\begin{equation*} \log_2\big( 1 + T_\mathrm{max}(c) \big) + \frac{1}{c\,\ln 2} \int_{T_\mathrm{max}(c)}^{\infty} \frac{dt}{H_{\delta}(t)(1 + t)}=1; \end{equation*}

iii) and for the no-loss model $K=\infty$,

(44)\begin{equation} \mathrm{Cap}_{\infty}(c) = c\,\log_2\big( 1 + T_\mathrm{max}(c) \big), \end{equation}

with $c_\mathrm{max}=1/H_\delta(1)$.

Proof. In the pure-loss case, $\mathrm{Cap}_0(c/w,w)$ derived in (28) does not depend on $w$ and we obtain immediately

(45)\begin{equation} \mathrm{Cap}_0(c)=\frac{1}{\ln2} \int_0^\infty \frac{c} {(1+H_\delta(t)\,c)(1 + t)}\, dt. \end{equation}

By Definition 3.2, the limiting traffic intensity $c$ is within balanced Shannon capacity whenever $\mathrm{Cap}_{0}(c) \geq c$. Hence, as stated, $c_\mathrm{max}$ is the solution of $\mathrm{Cap}_0(c) = c$. Turning to case ii), finite maximal buffer size $1\le K \lt \infty$, we first observe that the scaled coverage probability for any $K$ has the upper bound

\begin{equation*} V_t\left(q^*\left(\frac{c}{w},w,t \right)\right)\le \frac{c}{w} \cdot \mathbf 1_{\{t \lt T_\mathrm{max}(c)\}} + \frac{\mathbf 1_{\{t \gt T_\mathrm{max}(c)\}}}{1+wH_\delta(t)}. \end{equation*}

Hence $wV_t(q^*(c/w,w,t))$ has the dominating function

(46)\begin{equation} wV_t(q^*(c/w,w,t)) \le c \cdot \mathbf 1_{\{t \lt T_\mathrm{max}(c)\}} + \frac{\mathbf 1_{\{t \gt T_\mathrm{max}(c)\}}}{H_\delta(t)}, \end{equation}

which is independent of $w$ and integrable in $t$, such that

\begin{equation*} \int_0^\infty \Big(c \cdot \mathbf 1_{\{t \lt T_\mathrm{max}(c)\}} + \frac{1}{H_\delta(t)}\cdot \mathbf 1_{\{t \gt T_\mathrm{max}(c)\}}\Big)\frac{dt}{1+t} \lt \infty. \end{equation*}

Thus,

(47)\begin{equation} \mathrm{Cap}_K(c) =\frac{1}{\ln 2} \int_0^\infty \lim_{w\to\infty} wV_t(q^*(c/w,w,t)) \frac{dt}{1+t}. \end{equation}

Moreover, it follows from (46),(47) that

(48)\begin{equation} \mathrm{Cap}_K(c) \le \frac{c}{\ln 2}\int_0^{T_\mathrm{max}(c)} \frac{dt}{1+t}+\frac{1}{\ln 2}\int_{T_\mathrm{max}(c)}^\infty \frac{1}{H_\delta(t)}\frac{dt}{1+t}. \end{equation}

Now we return to the explicit result for $K=1$ mentioned in Section 3.1. The defining relation for $q^* = q^*(p,w,t)$ is found by solving (10) with $K = 1$, and is obtained explicitly as

\begin{equation*} q^*=\frac{\sqrt{(1-w H_\delta(t) p)^2+4w H_\delta(t) p^2}-(1-w H_\delta(t) p)}{2w H_\delta(t) p}. \end{equation*}

For this case

\begin{equation*} \mathrm{Cap}_1(c/w,w)\cdot \ln 2 = \int_0^\infty \frac{1+cH_\delta(t)-\sqrt{(1-cH_\delta(t))^2+4c^2 H_\delta(t)/w} }{2H_\delta(t)(1-c/w) }\frac{dt}{1+t}. \end{equation*}

By (47),

\begin{align*} \mathrm{Cap}_1(c) &= \frac{1}{\ln 2} \int_0^\infty \frac{1+cH_\delta(t)-|1-cH_\delta(t)|} {2H_\delta(t)}\frac{dt}{1+t}\\ &= \frac{1}{\ln 2}\int_0^\infty c\cdot \mathbf 1_{\{cH_\delta(t) \lt 1\}} +\frac{1}{H_\delta(t)} \mathbf 1_{\{cH_\delta(t) \gt 1\}} \, \frac{dt}{1+t}. \end{align*}

Now, $cH_\delta(t) \lt 1$ iff $t \lt T_\mathrm{max}(c)$ and, therefore, the derived expression for $\mathrm{Cap}_1(c)$ coincides with the upper bound of $\mathrm{Cap}_K(c)$ in (48). The conclusion is that all capacities $\mathrm{Cap}_K(c)$ for finite $K\ge 1$ are equal. The corresponding $c_{max}$ can be read as stated in claim ii).

Finally, for the no-loss system iii), we have

\begin{equation*} \mathrm{Cap}_{\infty}(c/w,w) = c \log_2(1 + T_\mathrm{max}(c/w,w)). \end{equation*}

By monotone convergence, $\mathrm{Cap}_{\infty}(c/w,w) \to c \log_2(1 + T_\mathrm{max}(c))$ as $w \to \infty$. In this case, the maximum admissible intensity $c_{max}$ is obtained by solving $c \log_2(1 + T_\mathrm{max}(c)) = c$, or equivalently, $T_\mathrm{max}(c) = 1$. Then it follows from (41) that $c_\mathrm{max} = 1/H_{\delta}(1)$.

The results presented in this section demonstrate that channel capacity remains an informative performance measure in the balanced scaling regime of small input rate and large UE density. In particular, channel capacity is an explicit, relatively simple, increasing function of scaled traffic intensity $c$. A new aspect of the analysis is that, under balanced densification, the retransmission mechanism does not influence the channel capacity when the buffer size $K$ is finite. Thus, physically, irrespective of the coding scheme employed, the channel capacity remains consistent, distinguishing only between systems with no buffer, a finite buffer, and an infinite buffer. An additional conclusion from Theorem 4.1 is that only scaled traffic intensities within the admissible intervals $[0, c_\mathrm{max}]$ are covered by the balanced Shannon capacity, expressed as

\begin{equation*} \mathrm{Cap}_K(c) \geq c. \end{equation*}

As $c$ grows larger, the channel capacities approach asymptotic plateau values in the limit $c \to \infty$, given by

(49)\begin{equation} \lim_{c\to\infty}\mathrm{Cap}_K(c) =\frac{1}{\ln2}\Big((\delta-1)\mathbf 1_{\{1\le K\le \infty\}} +\int_0^\infty \frac{1}{H_\delta(t)} \frac{dt}{1 + t} \,\mathbf 1_{\{0\le K \lt \infty\}}\Big). \end{equation}

Figure 3(b) illustrates the three cases of channel capacity limits under balanced densification obtained in Theorem 4.1, for $K=0$, finite $K\geq 1$, and $K=\infty$. The limit functions shown in the figure indicate that an increase in bandwidth combined with a reduction in input rate leads to an improvement in scaled capacity up to a certain threshold. Beyond this threshold, any further increase in $c$ no longer improves network performance. The dashed red lines represent the maximum values $c_\mathrm{max}$ for each type of network (finite buffer, infinite buffer, and no buffer). The interpretation of these values is similar to the capacity-admissible regions introduced in Definition 3.2. That is, the system is deemed workable whenever $c \le c_\mathrm{max}$, and non-workable otherwise.