Performance analysis of mobility management architectures in cellular networks

Abstract

Flattened cellular network architecture for the mobile internet is expected to meet the demands of rapidly increasing traffic from mobile users. Dynamic mobility anchoring (DMA) mechanism distributes the mobility management functions using such network architecture. In this paper, we compare DMA with a classical mobility protocol like proxy mobile IP (PMIP). A major cost factor of a mobility protocol is the management of contexts and tunnels. We propose an analytical model to compute the number of contexts and tunnels with DMA and with PMIP in a homogeneous network with random mobility of mobile nodes. The model is used under different configurations by varying the traffic loads and the capacities of access nodes in order to analyze the distributed and dynamic characteristics of DMA. The results show that the required number of contexts on an anchor node with DMA is significantly less than that required on an anchor node with PMIP and the required number of visitor contexts with DMA is significantly less in magnitude than that with PMIP for most of the configurations. The results also show that the number of required tunnels with DMA is less than those required with PMIP for most configurations.

Appendices

Appendix 1: A steady state numerical solution of CTMC

Solving a continuous time Markov chain (CTMC) with $n$ states corresponds to solving the set of steady-state equations of the form:

$$\begin{aligned}&\pi Q = 0 \end{aligned}$$

(26)

$$\begin{aligned}&\sum _{i=0}^{n-1} \pi _i = 1 \end{aligned}$$

(27)

where $Q$ is the $n \times n$ infinitesimal generator matrix and $\pi$ is the $n$-element steady-state solution vector. $Q$ is singular and it can be shown that $Q$ is of rank $n-1$ for any Markov chain of size $n$. The resulting set of equations is not linearly independent and one of the equations is redundant. To yield a unique solution, a normalization condition is imposed on Eq. (26). This can be done by substituting one of the columns (usually the last column) of $Q$ with the unit vector $[1,1,\dots ,1]^T$. The resulting linear system of non-homogenous equations can be rearranged as $Q^T\pi ^T = c^T$ with $c=[0,0,0,\dots ,0,1]$ which yields an expression in the form $Ax=b$. For this expression, a number of well known direct and iterative solution techniques exist. Let $A=Q^T$, $x=\pi ^T$, and $b=c^T$; the system Eq. (26) can be written as:

$$\begin{aligned} Ax=b \end{aligned}$$

(28)

We solve Eq. (28) by implementing a direct method in C++.

Appendix 2: Life-time of a Type-A context

We calculate $\frac{1}{\mu _A}$ as follows:

$$\begin{aligned}&\frac{1}{\mu _A} = \frac{1}{1-\pi _0} \displaystyle \sum _{k=0}^\infty \pi _k \int _0^\infty \left( 1- \left( 1-e^{- \mu _s t}\right) ^k \right) dt; \\&\text{ where } \text{ k }\; > \; 0 \end{aligned}$$

Using binomial expansion, it can be written as:

$$\begin{aligned} \frac{1}{\mu _A}&= \frac{1}{1-\pi _0} \displaystyle \sum _{k=1}^\infty \pi _k \\&\int _0^\infty { \left( 1 - \displaystyle \sum _{j=0}^k (-1)^j ({^k}\mathrm {C}_j) e^{-j\mu _s t} \right) dt}\!; \end{aligned}$$

where ${^k}\mathrm {C}_j = \frac{k!}{j!(k-j)!}$.

It can be further simplified as follows:

$$\begin{aligned} \frac{1}{\mu _A}&= \frac{1}{1-\pi _0} \displaystyle \sum _{k=1}^\infty \pi _k {\displaystyle \sum _{j=1}^k (-1)^{j-1} ({^k}\mathrm {C}_j) \int _0^\infty e^{-j\mu _s t} dt} \end{aligned}$$

By using the formula, $\displaystyle \int _0^\infty e^{-ax}dx=\frac{1}{a}$, we have:

$$\begin{aligned} \frac{1}{\mu _A}&= \frac{1}{1-\pi _0} \displaystyle \sum _{k=1}^\infty \pi _k \displaystyle \sum _{j=1}^k (-1)^{j-1} ({^k}\mathrm {C}_j) \frac{1}{j\mu _s} \end{aligned}$$

Appendix 3: Life-time of a Type-V context which is created by a set of k sessions

We calculate $\frac{1}{\mu _{V,k}}$ as follows:

$$\begin{aligned} \frac{1}{\mu _{V,k}}&= \int _0^\infty \left( 1- \left( 1-e^{- \mu _s t}\right) ^k \right) e^{-\alpha t} dt; k>0 \end{aligned}$$

Using binomial expansion, it can be written as:

$$\begin{aligned} \frac{1}{\mu _{V,k}} = \displaystyle \int _0^\infty \left( 1 - \displaystyle \sum _{j=0}^k (-1)^j ({^k}\mathrm {C}_j) e^{-j\mu _s t} \right) e^{-\alpha t} dt; \end{aligned}$$

where ${^k}\mathrm {C}_j = \frac{k!}{j!(k-j)!}$.

It can be further simplified as follows:

$$\begin{aligned} \frac{1}{\mu _{V,k}} = \displaystyle \sum _{j=1}^k (-1)^{j+1} ({^k}\mathrm {C}_j) \displaystyle \int _0^\infty e^{-(\alpha +j\mu _s)t} dt \end{aligned}$$

By using the formula, $\displaystyle \int _0^\infty e^{-ax}dx=\frac{1}{a}$, we have:

$$\begin{aligned} \frac{1}{\mu _{V,k}} = \displaystyle \sum _{j=1}^k (-1)^{j+1} ({^k}\mathrm {C}_j) \frac{1}{\alpha +j\mu _s}; k>0 \end{aligned}$$