In this paper we consider the user association problem in heterogeneous networks where each user chooses to be associated to a base station based on the biased downlink received power. In contrast to previous studies where users are usually assumed to be uniformly distributed, and thereby a per-tier SINR biasing factor is used to balance the load of BSs among different tiers, we examine in this paper a scenario that one cell is overloaded, i.e., has a higher user intensity. In this case, the adjustment of the per-tier biasing factor becomes unreasonable, and thus we propose to adjust the biasing factor of the overloaded cell to offload the traffic to its surrounding cells. By maximizing the mean user utility in the area of this overloaded cell and its neighboring cells, the optimal biasing factor can be obtained. It is found that in the scenario where the overloaded cell is fully surrounded by a macro cell, the optimal biasing factor logarithmically decreases with the user’s intensity of the overloaded cell. Numerical results demonstrate that by using the optimal biasing factor of the overloaded cell in the considered scenario, the mean user rate in the overloaded cell increases from 23 to 87 % as the average number of cluster users increases from 10 to 80, and the overall mean user rate is also improved compared to the previous biased scheme without the adjustment of the overloaded cell in the literature. Our analysis provides guidance on the optimal tuning of the biasing factor of an overloaded cell and, is a step forward towards the goal of the adjustment of the biasing factor in a per-station fashion under heterogeneous spatial user distribution.

1. \({A_0}\cap {A_m}\) is the boundary of two cells, and they are adjacent to each other if it is not a void set.

This work was supported by National Basic Research Program of China (973 program: 2013CB329005), National Natural Science Foundation of China (61401224, 61271237), Natural Science Foundation of Jiangsu Province of China (BK20140882) and NUPTSF (NY213061).

Authors and Affiliations

1. College of Telecommunications & Information Engineering, Nanjing University of Posts and Telecommunications, Nanjing, China

   Fancheng Kong, Xinghua Sun & Hongbo Zhu

Corresponding author

Correspondence toHongbo Zhu.

Appendix 1: Proof of Eq. (21)

For a random user in Region\(A'_0\) where the undesired signal from\({BS}_1\) is the major source of the interference, we have

where\(pdf_{{g_{{k_1}}}}\left( {{g_{{k_1}}}} \right)\) is probability density function (PDF) of random variable\(g_{k_1}\). Conditioned on a given\(g_{k_1}\) we have

where\(pdf_{{d_{{k_0}},{d_{{k_1}}}}}\left( {{d_{{k_0}}},{d_{{k_1}}}} \right)\) denotes the joint PDF of\(d_{k_0}\) and\(d_{k_1}\). Hence, conditioned on\(g_{k_1}\),\(d_{k_0}\) and\(d_{k_1}\), we can obtain that

where (a) follows that\(g_{k_0}\) is an exponential random variable with unit mean.

By substituting (27) and (28) into (26), we have

where (b) follows that\(g_{k_1}\) is exponentially distributed with unit mean, i.e.,\({E_{{g_{{k_1}}}}}\left[ {{g_{{k_1}}}} \right] = 1\), and (c) follows that users are uniformly deployed within each association region of a cell, i.e.,\(pdf_{x,y}(x,y) = 1/\pi {{r^{\prime }}^2}\).

On the other hand,\(E\left[ {\log \left( {\Pr \left( {SI{R_{\Delta {A_0}}} \geqslant \tau } \right) } \right) } \right]\) can be derived by a similar approach and thus omitted here.

Appendix 2: Proof of Eq. (25)

With\({P_{{k_1}}} \gg {P_{{k_0}}}\), the radius of Region\({A_0}\) is much smaller than the distance between\({BS}_1\) and the overloaded\({BS}_0\), i.e.,\(r = \sqrt{\frac{{B{a^2}}}{{{{\left( {B - 1} \right) }^2}}}} \ll a\). We then have\(B \gg 1\), and thus\(\frac{{Ba}}{{B - 1}} \approx a\). In addition, as the biasing factor of\({BS}_0\) decreases to offload some users to\({BS}_1\), we have\(B^{\prime } > B \gg 1\), and thus\(\frac{{Ba}}{{B - 1}} = \frac{{B'a}}{{B^{\prime } - 1}} \approx a\). Therefore, both\(A_0\) and\(A^{\prime }_0\) can be regarded as circular areas with the same center\(\left( {a,0} \right)\) but different radius. Meanwhile, since\(r^{\prime } < r \ll a\), the distance from\({BS}_1\) to a random user in region\({A_0}\) or\({A^{\prime }_0}\) can be approximately regarded as a constant, i.e.,\({d_{{k_1}}} \approx a\). We then have

and

where (a) and (b) follow the property of uniform distribution of a random user inside a circular area.

As regular users’ intensity is usually much lower than that of cluster users, i.e.,\({\lambda _r} \ll {\lambda _c}\), and the association area of the macro cell is much larger than that of other cells, (19) and (20) can be further written as

and

Finally, by substituting (18), (30)–(32) into (13), the objective function (25) can be obtained.

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