A quantitative analysis of the throughput gains and the energy efficiency of multi-radio transmission diversity in dense access networks

Abstract

Densification of mobile network infrastructure and integration of multiple radio access technologies are important approaches to support the increasing demand for mobile data traffic and to reduce energy consumption in future 5G networks. In this paper, the benefits of multi-radio transmission diversity (MRTD) are investigated by modelling the radio access link throughputs as uniform- and Rayleigh-distributed random variables and evaluating different user schedulers and resource allocation strategies. We examine different strategies for the allocation of radio accesses to individual users ranging from independent utilisation of the radio accesses to MRTD-enabled schemes. The schemes are compared by considering the statistics of the system throughput and energy consumption of the mobile devices. It is shown that MRTD can increase the throughput significantly through two types of diversity gain: Firstly by having multiple radio accesses to choose from for each user and secondly by having more available users to choose from for each radio access. The increased throughput also helps to reduce the energy consumption per bit, but this comes at a cost of increased energy consumption for channel measurement and reporting.

Appendix: Theorem proof

Proposition 4.1

Given definition 4.1, $P(R_{g[m]} {\vert }K)$, the probability that the maximum allocation process transmits with link throughput equal to $R_{g[m]}$ is

$$\begin{aligned} P\left( {R_{g[m]} \left| K \right. } \right) =P_{g[m]} ^{K}-P_{g\left[ {m-1} \right] } ^{K} \end{aligned}$$

(8)

Proof

We start by deriving an expression for $P(R_{g[M]}{\vert } K)$. Following our notation, $p_{g[M]}$ denotes the probability that a user can transmit over the $g$th RA with the maximum link throughput $R_{g[M]}$, while $P_{g[M]}$ denotes the cumulative probability that a user can transmit at any of the rates up to $R_{g[M]}$. Consequently, the probability that a user will send at rates lower than the maximum link throughput equals $(1-p_{g[M]})= P_{g[M-1]}$. Since users’ link throughputs at any instant of time $t$ are assumed to be independent events, the outcome that all $K$ users may send at lower link thoughput values than maximum is then given by $(1-p_{g[M]})^{K}=P(r<R_{g[M]})^{K}= Pr\le R_{g[M-1]})^{K}$. The relevant probabilities are reproduced in Table 2.

Table 2 Probabilities for the system to transmit at some rates

The derivation of the probability that at least one user among $K$ users can send with $r=R_{g[M-1]}$ equals $1-P(r\le R_{g[M-1]})^{K}$ is summarized in (48) below:

$$\begin{aligned}&P\left( {R_{g[M]} |K } \right) \nonumber \\&=P\left( {\hbox {at least one among }K\hbox { users can send with }r=R_{g[M]} } \right) \nonumber \\&=1-P\left( {\hbox {no one among }K\hbox { users can send with }r=R_{g[M]} } \right) \nonumber \\&=1-(1-p_{g[M]} )^{K} \nonumber \\&=1-P\left( {\hbox {all }K\hbox { users have maximum rate }r\le R_{g\left[ {M-1} \right] } } \right) \nonumber \\&=1-P(r\le R_{g\left[ {M-1} \right] })^{K} =1-P_{g\left[ {M-1} \right] }^{K} \end{aligned}$$

(48)

Next we will derive the probability of having the system transmitting at the next highest rate $P(R_{g[M-1]}{\vert } K)$. According to the maximum rate allocation scheme the system transmits at the next highest link throughput, $R_{g[M-1]}$, if and only if there is no user that can transmit at $R_{g[M]}$ and there is at least one user that can transmit at $R_{g[M-1]}$. A user will transmit at rates lower than $R_{g[M]}$ and $R_{g[M-1]}$ with probability $(1-p_{g[M]} -p_{g\left[ {M-1} \right] } )=P\left( {r\le R_{g\left[ {M-2} \right] } } \right) =P_{g\left[ {M-2} \right] }$. This is also shown in Table 2.

For $K$ users with independent rates this becomes $(1-p_{g[M]} -p_{g\left[ {M-1} \right] } )^{K}=P\left( {r\le R_{g\left[ {M-2} \right] } } \right) ^{K}$ and equals the probability that the system will neither transmit with $R_{g[M]}$ nor $R_{g[M-1]}$. Thus the probability of having the system transmitting at the next highest rate $P(R_{g[M-1]} {\vert }K)$ is given by

$$\begin{aligned}&P\left( {R_{g\left[ {M-1} \right] } |K} \right) \nonumber \\&\quad =P\left( {\hbox {at least one among }K\hbox { users can send with }r\ge R_{g\left[ {M-1} \right] } } \right) \nonumber \\&\qquad -P\left( {\hbox {at least one among }K\hbox { users can send at }r\ge R_{g[M]} } \right) \nonumber \\&\quad =1-P\left( {\hbox {no one among }K\hbox { users can send with }r\ge R_{g\left[ {M-1} \right] } } \right) \nonumber \\&\qquad -\left( {1-P\left( {\hbox {no one among }K\hbox { users can send with }r\ge R_{g[M]} } \right) } \right) \nonumber \\&\quad =1-(1-p_{g[M-1]} -p_{g[M]} )^{K}-(1-(1-p_{g[M]} )^{K}) \nonumber \\&\quad =P\left( {\hbox {all }K\hbox { users have maximum rates }r\le R_{g\left[ {M-1} \right] } } \right) \nonumber \\&\qquad -P\left( {\hbox {all }K\hbox { users have maximum rates }r\le R_{g\left[ {M-2} \right] } } \right) \nonumber \\&\quad =P_{g\left[ {M-1} \right] } ^{K}-P_{g\left[ {M-2} \right] } ^{K} \end{aligned}$$

(49)

Similarly we derive the probability that the system will transmit at rate $R_{g[M-m]}$ as follows

$$\begin{aligned}&P\left( {R_{g\left[ {M-m} \right] } |K} \right) \nonumber \\&\quad =P\left( {\hbox {at least one among }K\hbox { users can send with }r\ge R_{g\left[ {M-m} \right] } } \right) \nonumber \\&\qquad -P\left( {\hbox {at least one among }K\hbox { users can send at }r\ge R_{g\left[ {M-m+1} \right] } } \right) \nonumber \\&\quad =1-P\left( {\hbox {no one among }K\hbox { users can send with }r\ge R_{g\left[ {M-m} \right] } } \right) \nonumber \\&\qquad -\left( {1-P\left( {\hbox {no one among }K\hbox { users can send with }r\ge R_{g\left[ {M-m+1} \right] } } \right) } \right) \nonumber \\&\quad =P\left( {\hbox {all }K\hbox { users have maximum rates }r\le R_{g\left[ {M-m} \right] } } \right) \nonumber \\&\qquad -P\left( {\hbox {all }K\hbox { users have maximum rates }r\le R_{g\left[ {M-m-1} \right] } } \right) \nonumber \\&\quad =P_{g\left[ {M-m} \right] } ^{K}-P_{g\left[ {M-m-1} \right] } ^{K} \end{aligned}$$

(50)

Generally, by replacing $M-m$ in (13) by $m$ the probability that a system will transmit at rate $R_{g[m]}$ is given by

The above proposition provides an expression to calculate the probability of the definition 4.1 as expressed in (7). $\square$