Modelling the time-varying cell capacity in LTE networks

Abstract

In wireless orthogonal frequency-division multiple access (OFDMA) based networks like Long Term Evolution (LTE) or Worldwide Interoperability for Microwave Access (WiMAX) a technique called adaptive modulation and coding (AMC) is applied. With AMC, different modulation and coding schemes (MCSs) are used to serve different users in order to maximise the throughput and range. The used MCS depends on the quality of the radio link between the base station and the user. Data is sent towards users with a good radio link with a high MCS in order to utilise the radio resources more efficiently while a low MCS is used for users with a bad radio link. Using AMC however has an impact on the cell capacity as the quality of a radio link varies when users move around; this can even lead to situations where the cell capacity drops to a point where there are too little radio resources to serve all users. AMC and the resulting varying cell capacity notably has an influence on admission control (AC). AC is the algorithm that decides whether new sessions are allowed to a cell or not and bases its decisions on, amongst others, the cell capacity. The analytical model that is developed in this paper models a cell with varying capacity caused by user mobility using a continuous -time Markov chain (CTMC). The cell is divided into multiple zones, each corresponding to the area in which data is sent towards users using a certain MCS and transitions of users between these zones are considered. The accuracy of the analytical model is verified by comparing the results obtained with it to results obtained from simulations that model the user mobility more realistically. This comparison shows that the analytical model models the varying cell capacity very accurately; only under extreme conditions differences between the results are noticed.

The developed analytical and simulation models are then used to investigate the effects of a varying cell capacity on AC. Also, an optimisation algorithm that adapts the parameter of the AC algorithm which determines the amount of resources that are reserved in order to mitigate the effects of the varying cell capacity is studied using the models. Updating the parameter of the AC algorithm is done by reacting to certain triggers that indicate good or bad performance and adapt the parameters of the AC algorithm accordingly. Results show that using this optimisation algorithm improves the quality of service (QoS) that is experienced by the users.

Appendices

Appendix A: Static AC algorithm

In this appendix the matrices of the system with the static AC algorithm where N=3 are described. The whole analytical model for this case is described in Sect. 4.1. Remember that the function a _i (x) denotes whether a session that arrives in zone i when the system is in state x is accepted by the AC algorithm or not, a _i (x)=1 means that the session is accepted and a _i (x)=0 means that the session is blocked. The notation has been simplified as a _i (x)=a _i . The block matrices that were not described in Sect. 4.1 are listed below. In these matrices p=h+l and the values of δ _i equal the opposite of the sum of the other elements in the same row to make the elements of each row of the transition rate matrix Q sum to 0.

$$ \boldsymbol{A_1^{h,l}}=\left [ \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} \delta_0 & a_3\epsilon_3 & 0 & 0 &\cdots \cr \gamma& \delta_1 & a_3\epsilon_3 & 0 & \cdots \cr 0 & 2\gamma& \delta_2 & a_3\epsilon_3 & \cdots \cr &\ddots& \ddots& \ddots & \cr \cdots& 0 & 0 & (M-p)\gamma& \delta_{M-p} \cr \end{array} \right ] $$

where the size of $\boldsymbol{A_{1}^{h,l}}$ is (M+1−p)×(M+1−p).

$$ \boldsymbol{A_0^{h,l}}=\left [ \begin{array}{c@{\quad }c@{\quad }c@{\quad }c} a_2\epsilon_2 & 0 & 0 &\cdots \cr \lambda_3 &a_2\epsilon_2 & 0 & \cdots \cr 0 & 2\lambda_3 & a_2\epsilon_2 &\cdots \cr & \ddots& \ddots& \cr \cdots& 0 & 0 & (M-p)\lambda_3 \end{array} \right ] $$

where the size of $\boldsymbol{A_{0}^{h,l}}$ is (M+1−p)×(M−p).

$$ \boldsymbol{A_2^{h,l}}=\left [ \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} l\gamma& l\mu_2 & 0 & 0 &\cdots \cr0 & l\gamma & l\mu_2 & 0 & \cdots \cr & \ddots& \ddots& \ddots & \cr\cdots & 0 & 0 & l\gamma& l\mu_2 \cr \end{array} \right ] $$

where the size of $\boldsymbol{A_{2}^{h,l}}$ is (M+1−p)×(M+2−p).

$$ \boldsymbol{B_1^{h,l}}=\left [ \begin{array}{c@{\quad }c@{\quad }c@{\quad }c} a_1\epsilon_1 & 0 & 0 &\cdots\cr0 & a_1\epsilon _1 & 0 & \cdots\cr & \ddots&\ddots & \cr\cdots& 0 & 0 & a_1\epsilon_1 \cr\cdots& 0 & 0 & 0 \cr \end{array} \right ] $$

(15)

where the size of $\boldsymbol{B_{1}^{h,l}}$ is (M+1−p)×(M−p).

The matrix $\boldsymbol{B_{2}^{h,l}}$ is a diagonal matrix where the values of the diagonal are equal to lλ ₂ and its size is (M+1−p)×(M+1−p).

$$ \boldsymbol{C_1^{h,l}}=\left [ \begin{array}{c@{\quad }c@{\quad }c@{\quad }c} h\gamma& 0 & 0 &\cdots\cr0 & h\gamma& 0 & \cdots\cr & \ddots&\ddots & \cr \cdots& 0 & h\gamma& 0 \cr \end{array} \right ] $$

(16)

where the size of $\boldsymbol{C_{1}^{h,l}}$ is (M+1−p)×(M+2−p).

The matrix $\boldsymbol{C_{0}^{h,l}}$ is a diagonal matrix where the values of the diagonal are equal to hμ ₁ and its size is (M+1−p)×(M+1−p).

Appendix B: AC optimisation algorithm

When the AC optimisation algorithm described in Sect. 3.2 is considered, the analytical model has one more level and hence one more block level than the analytical model for the static AC algorithm with the same number of zones N, see Sect. 4.2. The functions t ₁(x) and t ₂(x) denote whether the parameter f must be changed or not when the system is in state x according to the AC optimisation algorithm in Eq. (2). t ₁(x)=1 and t ₂(x)=0 means that the parameter f must be increased, t ₁(x)=0 and t ₂(x)=1 means that the parameter f must be decreased and t ₁(x)=0 and t ₂(x)=0 means that f does not change. Note that t ₁(x) and t ₂(x) cannot be 1 at the same time. The notation has been simplified as t ₁(x)=t ₁ and t ₂(x)=t ₂. Remember that n _f is the number of the different discrete values that f can take and the intervals after which the optimisation is performed are exponentially distributed with mean 1/η. Also remember that p=h+m. The block matrices for N=3 that were not described in Sect. 4.2 are listed below. Again, the values of δ _i equal the opposite of the sum of the other elements in the same row to make the elements of each row of the transition rate matrix Q sum to 0.

$$ \boldsymbol{D_1^{h,m,l}}=\left [ \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} \delta_0 & t_1 \eta& 0 & 0 &\cdots\cr t_2 \eta& \delta_1 & t_1 \eta& 0 & \cdots\cr0 & t_2\eta& \delta_2 & t_1 \eta & \cdots\cr & \ddots&\ddots & \ddots& \cr \cdots& 0 & 0 & t_2 \eta& \delta_{n_f} \cr \end{array} \right ] $$

where the size of $\boldsymbol{D_{1}^{h,m,l}}$ is n _f ×n _f . Note that $\boldsymbol{D_{1}^{h,m,l}}$ does not depend on the levels, i.e. it is equal for all the (h,m,l) levels.

The matrices $\boldsymbol{D_{0}^{h,m,l}}$, $\boldsymbol{D_{2}^{h,m,l}}$, $\boldsymbol{E_{1}^{h,m,l}}$, $\boldsymbol{E_{2}^{h,m,l}}$, $\boldsymbol {F_{1}^{h,m,l}}$ and $\boldsymbol{F_{0}^{h,m,l}}$ are diagonal matrices with size n _f ×n _f ; the values on the diagonal are a ₃ ϵ ₃, lγ, a ₂ ϵ ₂, lλ ₃, mγ and mμ ₂ respectively.

$$ \boldsymbol{B_1^{h,m}}=\left [ \begin{array}{c@{\quad }c@{\quad }c@{\quad }c} a_1\epsilon_1 & 0 & 0 &\cdots\cr0 & a_1\epsilon _1 & 0 & \cdots\cr & \ddots&\ddots & \cr\cdots& 0 & 0 & a_1\epsilon_1 \cr\cdots& 0 & 0 & 0 \cr\cdots& \vdots& \vdots& \vdots\cr\cdots& 0 & 0 & 0 \cr \end{array} \right ] $$

where the number of rows with all elements 0 is n _f and the size of $\boldsymbol{B_{1}^{h,m}}$ is (n _f (M+1−p))×(n _f (M−p)).

The Matrix $\boldsymbol{B_{2}^{h,m}}$ is a diagonal matrix where the values of the diagonal are mλ ₂ and its size is (n _f (M+1−p))×(n _f (M+1−p)).

$$ \boldsymbol{C_1^{h,m}}=\left [ \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} h\gamma& 0 & 0 & 0 & 0 &\cdots& 0 \cr0 & h\gamma& 0 & 0 & 0 & \cdots& 0 \cr & \ddots& \ddots & \ddots& \ddots \cr \cdots& 0 & h\gamma& 0 & 0 & \cdots& 0 \cr \cdots& 0 & 0 & h\gamma& 0 & \cdots& 0 \cr \end{array} \right ] $$

where the number of columns with all elements 0 is n _f and the total size of $\boldsymbol{C_{1}^{h,m}}$ is (n _f (M+1−p))×(n _f (M+2−p)).

The matrix $\boldsymbol{C_{0}^{h,m}}$ is a diagonal matrix where the values of the diagonal are hμ ₁ and its size is (n _f (M+1−p))×(n _f (M+1−p)).