text: In heterogeneous networks (HetNets), load balancing among different tiers can be effectively achieved by a biased user association scheme with which each user chooses to associate with one base station (BS) based on the biased received power. In contrast to previous studies where a BS always has packets to transmit, we assume in this paper that incoming packets intended for all the associated users form a queue in the BS. In order to find the delay limit of the network to support real-time service, we focus on the delay optimization problem by properly tuning the biasing factor of each tier. By adopting a thinned Poisson point process (PPP) model to characterize the locations of BSs in the busy state, an explicit expression of the average traffic intensity of each tier is obtained. On that basis, an optimization problem is formulated to minimize a lower bound of the network mean queuing delay. By showing that the optimization problem is convex, the optimal biasing factor of each tier can be obtained numerically. When the mean packet arrival rate of each user is small, a closed-form solution is derived. The simulation results demonstrate that the network queuing performance can be significantly improved by properly tuning the biasing factor. It is further shown that the network mean queuing delay might be improved at the cost of a deterioration of the network signal-to-interference ratio (SIR) coverage, which indicates a performance tradeoff between real-time and non-real-time traffic in HetNets.
  @cite_24: 
  @cite_18: Several queueing processes may be modeled as random walks on a multidimensional grid. In this paper the equilibrium distribution for the case of a two-dimensional grid is considered. In previous research it has been shown that for some two-dimensional random walks the equilibrium distribution has the form of an infinite series of products of powers which can be constructed with a compensation procedure. The object of the present paper is to investigate under which conditions such an elegant solution exists and may be found with a compensation approach. The conditions can be easily formulated in terms of the random behaviour in the inner area and the drift on the boundaries.
  @cite_8: Delay-sensitive traffic services, such as live streaming video, voice over IP and multimedia teleconferencing, requires low end-to-end delay in order to maintain its interactive and streaming nature. In recent years, the popularity of delay-sensitive applications has been rapidly growing. In this study, the authors consider the delay-sensitive user scheduling and power control design for heterogeneous networks (HetNets). They model the problem as an infinite-horizon average cost Markov decision process MDP. Based on the dynamics of channel state information (CSI) and queue state information (QSI), they focus on finding an adaptive user scheduling and power control policy to minimise the average delay of network subjected to the power constraints of heterogeneous base stations. In order to reduce the computational complexity and facilitate the distributive implementation, the authors derive a distributive stochastic learning algorithm which only requires local CSI and local QSI to determine the optimal user scheduling and power control policy. The simulation results confirm that significant delay performance gain can be achieved compared with various baseline schemes.
summary: The analysis of coupled queues is a long-standing open problem, and even solving a special case of two interacting queues is challenging @cite_18 . Zhuang @cite_24 modeled multiple interacting queues as a continuous-time Markov chain (CTMC) with fixed BS locations in the network. By minimizing the average packet queuing delay, the optimal spectrum allocation pattern was obtained for each BS. Similarly, Cheng @cite_8 optimized the average queuing delay of network subjected to the BS power constraints. They formulated a Markov decision process (MDP) problem based on the instantaneous channel state information and queue state information, and proposed an adaptive user scheduling and power control policy. However, the state space of the Markov process may become huge as the network scales up, and the analysis would become intractable. Hence, this motivates us to deal with the coupled queue problem with the tool of stochastic geometry to account for the random BS deployment, and derive the mean performance metrics analytically such that some insights can be gained for system design.