Performance metrics of interest in our system model are the outage probability and average delivery rate. We start by defining these metrics for the downlink. From now on, without loss of generality, we refer to the user o as typical user, which is located at the origin on the plane.
We know that the downlink rate depends on the signaltointerferenceplusnoise ratio. The signaltointerferenceplusnoise ratio of user o which is located at a random distance r far away from its small base station b
_{
o
} is given by:
$$\begin{array}{@{}rcl@{}} \textrm{SINR} &\triangleq \frac{hr^{\alpha}}{\sigma^{2} + I_{r}}, \end{array} $$
((1))
where
$$\begin{array}{@{}rcl@{}} I_{r} &\triangleq \sum_{i \in \Phi / b_{o}}{g_{i}{R^{\alpha}_{i}}}, \end{array} $$
((2))
is the total interference experienced from all other small base station at a distance R
_{
i
} from the typical user (except the connected small base station b
_{
o
}) which have fading value g
_{
i
}. Assume that the success probability is the probability of the downlink rate exceeding the file bitrate T and the probability of requested file being in the local cache. Then, the outage probability can be given as the complementary of the success probability as follows:
$$\begin{array}{*{20}l}{} p_{\text{out}}(\lambda,T,\alpha,S, L, \gamma) &\triangleq 1  \underbrace{\mathbb{P}\Big[\text{ln}(1 \,+\, \text{SINR}) >\! T, f_{o} \!\in\! \Delta_{b_{o}}\Big]}_{\text{success probability}}, \end{array} $$
((3))
where f
_{
o
} is the requested file by the typical user, and \(\Delta _{b_{o}}\) is the local cache of the serving small base station b
_{
o
}. Indeed, such a definition of the outage probability comes from a simple observation. Ideally, if a requested file is in the cache of the serving small base station (thus, the limited backhaul is not used) and if the downlink rate is higher than the file bitrate T (thus, the user does not observe any interruption during the playback of the file), we then expect the outage probability to be close to zero. Given this explanation and the assumptions made in the previous section, we state the following theorem for outage probability.
Theorem 1 (Outage probability).
The typical user has an outage probability from its tagged base station which can be expressed as:
$$ \begin{aligned} p_{\text{out}}(\lambda,T,\alpha,S, L, \gamma) &=\!1 \pi\lambda \int^{\infty}_{0} \int^{S/L}_{0}\\ &\quad\times e^{\pi\lambda v\beta(T,\alpha)  \mu(e^{T}  1)\sigma^{2}v^{\alpha/2}} f_{\text{pop}}(\,f,\gamma\!) \mathrm{d}\,f \mathrm{d}v, \end{aligned} $$
((4))
where β(T,α) is given by:
$$ \begin{aligned} \beta(T,\alpha) &= \frac{2\left(\mu(e^{T}  1)\right)}{\alpha} \mathbb{E}_{g}\left[ g^{\frac{2}{\alpha}} \left(\Gamma\left(\frac{2}{\alpha},\mu\left(e^{T}  1\right)g\right) \right.\right.\\ &\quad\left.\left.\Gamma\left(\frac{2}{\alpha}\right)\right) \right], \end{aligned} $$
((5))
where \(\Gamma (a,x) = \int ^{\infty }_{x}{t^{a1}e^{t}\mathrm {d}t}\) is the upper incomplete Gamma function and \(\Gamma (x) = \int ^{\infty }_{0}{t^{x1}e^{t}\mathrm {d}t}\) is the Gamma function.
Proof.
The proof is provided in Appendix A Proof of Theorem 1.
Yet another useful metric in our system model is the delivery rate, which we define as follows:
$${} \begin{aligned} \tau\triangleq\left\{ \begin{aligned} &T,\qquad \text{if ln}(1 + \text{SINR}) > T \mathrm{\;and\;} f_{o} \in \Delta_{b_{o}}, \\ &C(\lambda),\quad\!\! \text{if ln}(1 + \text{SINR}) > T \mathrm{\;and\;} f_{o} \not\in \Delta_{b_{o}},\quad \text{nats/s/Hz}\\ &0,\qquad\, \text{otherwise}, \end{aligned}\right. \end{aligned} $$
((6))
where C(λ) is the backhaul capacity provided to the small base station for single frequency in the downlink^{c}. The definition above can be explained as follows. If the downlink rate is higher than the threshold T (namely, the bitrate of the requested file) and the requested file is available in the local cache, the rate T is dedicated to the user by the tagged small base station, which in turn is sufficient for qualityofexperience. On the other hand, if the downlink rate is higher than T but the requested file does not exist in the local cache of the tagged small base station, the delivery rate will be limited by the backhaul link capacity C(λ), for which we assume that C(λ)<T. Given this definition for the delivery rate, we state the following theorem.
Theorem 2 (Average delivery rate).
The typical user has an average delivery rate from its tagged base station which can be expressed as:
$$ \begin{aligned} {\bar \tau}(\lambda,T,\alpha,S, L, \gamma) &= \pi\lambda \int^{\infty}_{0} e^{\pi\lambda v\beta(T,\alpha)  \mu(e^{T}  1)\sigma^{2}v^{\alpha/2}}\mathrm{d}v \\ &\quad\times\left(C(\lambda)+(T  C(\lambda))\int^{S/L}_{0}{f_{\text{pop}}(\,f,\gamma)\mathrm{d}\,f} \right), \end{aligned} $$
((7))
where β(T,α) has the same definition as in Theorem 1.
Proof.
The proof is deferred to Appendix B Proof of Theorem 2.
What we provided above are the general results. The exact values of outage probability and average delivery rate can be obtained by specifying the distribution of the interference, the backhaul link capacity C(λ), and the file popularity distribution f
_{pop}(f,γ). If this treatment does not yield closed form expressions, numerical integration can be done as a last resort for evaluating the functions. In the next section, as an example, we derive special cases of these results after some specific assumptions, which in turn yield much simpler expressions.
Special Cases
Assumption 1.
The following assumptions are given for the system model:

1.
The noise power σ
^{2} is higher than 0, and the pathloss component α is 4.

2.
Interference is Rayleigh fading, which in turn g
_{
i
}∼Exponential(μ).

3.
The capacity of backhaul links is given by:
$$ C\left(\lambda\right) \triangleq \frac{C_{1}}{\lambda} + C_{2}, $$
((8))
where C
_{1}>0 and C
_{2}≥0 are some arbitrary coefficients such that C(λ)<T holds.

4.
The file popularity distribution of users is characterized by a power law [25] such as:
$$ f_{\text{pop}}\left(f,\gamma\right) \triangleq\left\{ \begin{aligned} & \left(\gamma  1\right)f^{\gamma},\quad f \geq 1, \\ &0,\qquad\qquad\quad\,\,\, f < 1, \end{aligned}\right. $$
((9))
where γ>1 is the shape parameter of the distribution.
The assumption C(λ)<T comes from the observation that the highspeed fiberoptic backhaul links might be very costly in densely deployed small base station scenarios. Therefore, we assume that C(λ) is lower than the bitrate of the file. On the other hand, we characterize the file popularity distribution with a power law. Indeed, this comes from the observation that many realworld phenomena can be characterized by power laws (i.e., distribution of files in web proxies, distribution of word counts in natural languages) [25]. According to our system model and the specific assumptions made in Assumption 1, we state the following results.
Proposition 1 (Outage probability).
The typical user has an outage probability from its tagged base station which can be expressed as:
$$ \begin{aligned} p_{\text{out}}(\lambda,T,4,S, L,\gamma)&=1\frac{\pi^{\frac{3}{2}}\lambda}{\sqrt{\frac{e^{T}1}{\text{SNR}}}} \text{exp}\left(\!\frac{\left(\lambda\pi(1 + \rho(T,4))\right)^{2}}{4(e^{T}\,\,1)/\text{SNR}}\! \right) \\ &\quad\times Q\left(\!\frac{\lambda\pi(1 + \rho(T,4))}{\sqrt{2(e^{T}\,\,1)/\text{SNR}}}\! \right)\!\left(\! 1 \! \left(\!\frac{L}{L\,+\,S}\!\right)^{\gamma  1}\!\right), \end{aligned} $$
((10))
where \(\rho (T,4) = \sqrt {e^{T}  1}\left (\frac {\pi }{2}  \text {arctan}\left (\frac {1}{\sqrt {e^{T}1}}\right) \right)\) and the standard Gaussian tail probability is given as \(Q\left (x\right) = \frac {1}{\sqrt {2\pi }}\int _{x}^{\infty }{e^{y^{2}/2}\mathrm {d}y}\).
Proof.
The proof is given in Appendix C Proof of Proposition 1.
Proposition 2 (Average delivery rate).
The typical user has an average delivery rate from its tagged base station which can be expressed as:
$$ \begin{aligned} {\bar \tau}(\lambda,T,4,S, L, \gamma)&=\frac{\pi^{\frac{3}{2}}\lambda}{\sqrt{\frac{e^{T}1}{\text{SNR}}}} \text{exp}\left(\frac{\left(\lambda\pi(1 + \rho(T,4))\right)^{2}}{4(e^{T}1)/\text{SNR}} \right) \\ &\quad\times Q\left(\!\frac{\lambda\pi(1 + \rho(T,4))}{\sqrt{2(e^{T}1)/\text{SNR}}}\!\right)\!\left(\!T \,+\, \left(\!\frac{C_{1}}{\lambda} + C_{2} \! T\!\right)\right.\\ &\quad\left.\times\left(\!\frac{L}{L+S}\!\right)^{\gamma  1}\right), \end{aligned} $$
((11))
where ρ(T,4) and Q(x) have the same definition as in Proposition 1.
Proof.
The proof is given in Appendix D Proof of Proposition 2.
The expressions obtained for special cases are cumbersome but fairly easy to compute and do not require any integration. Note that the Q(x) function given in the expressions is a wellknown function and can be computed by using lookup tables or standard numerical packages.