HetHetNets: Heterogeneous Traffic Distribution in Heterogeneous Wireless Cellular Networks

A recent approach in modeling and analysis of the supply and demand in heterogeneous wireless cellular networks has been the use of two independent Poisson point processes (PPPs) for the locations of base stations (BSs) and user equipments (UEs). This popular approach has two major shortcomings. First, although the PPP model may be a fitting one for the BS locations, it is less adequate for the UE locations mainly due to the fact that the model is not adjustable (tunable) to represent the severity of the heterogeneity (non-uniformity) in the UE locations. Besides, the independence assumption between the two PPPs does not capture the often-observed correlation between the UE and BS locations. This paper presents a novel heterogeneous spatial traffic modeling which allows statistical adjustment. Simple and non-parameterized, yet sufficiently accurate, measures for capturing the traffic characteristics in space are introduced. Only two statistical parameters related to the UE distribution, namely, the coefficient of variation (the normalized second-moment), of an appropriately defined inter-UE distance measure, and correlation coefficient (the normalized cross-moment) between UE and BS locations, are adjusted to control the degree of heterogeneity and the bias towards the BS locations, respectively. This model is used in heterogeneous wireless cellular networks (HetNets) to demonstrate the impact of heterogeneous and BS-correlated traffic on the network performance. This network is called HetHetNet since it has two types of heterogeneity: heterogeneity in the infrastructure (supply), and heterogeneity in the spatial traffic distribution (demand).Comment: JSA

Point Process Models for Heterogeneous Event Time Data

Interaction event times observed on a social network provide valuable information for social scientists to gain insight into complex social dynamics that are challenging to understand. However, it can be difficult to accurately represent the heterogeneity in the data and to model the dependence structure in the network system. This requires flexible models that can capture the complicated dynamics and complex patterns. Point process models offer an elegant framework for modeling event time data. This dissertation concentrates on developing point process models and related diagnostic tools, with a real data application involving an animal behavior network. In this dissertation, we first propose a Markov-modulated Hawkes process (MMHP) model to capture the sporadic and bursty patterns often observed in event time data. A Bayesian inference procedure is developed to evaluate the likelihood by using a variational approximation and the forward-backward algorithm. The validity of the proposed model and associated estimation algorithms is demonstrated using synthetic data and the animal behavior data. Facilitated by the power of the MMHP model, we construct network point process models that can capture a social hierarchy structure by embedding nodes in a latent space that can represent the underlying social ranks. Our model provides a ranking method for social hierarchy studies and describes the dynamics of social hierarchy formation from a novel perspective – taking advantage of the detailed information available in event time data. We show that the network point process models appropriately captures the temporal dynamics and heterogeneity in the network event time data, by providing meaningful inferred rankings and by calibrating the accuracy of predictions with relevant measures of uncertainty. In addition to developing a sensible and flexible model for network event time data, the last part of this dissertation provides essential tools for diagnosing lack of fit issues for such models. We develop a systematic set of diagnostic tools and visualizations for point process models fitted to data in the dynamic network setting. By inspecting the structure of the residual process and Pearson residual on the network, we can validate whether a model adequately captures the temporal and network dependence structures in the observed data

Fast MCMC sampling for Markov jump processes and extensions

Markov jump processes (or continuous-time Markov chains) are a simple and important class of continuous-time dynamical systems. In this paper, we tackle the problem of simulating from the posterior distribution over paths in these models, given partial and noisy observations. Our approach is an auxiliary variable Gibbs sampler, and is based on the idea of uniformization. This sets up a Markov chain over paths by alternately sampling a finite set of virtual jump times given the current path and then sampling a new path given the set of extant and virtual jump times using a standard hidden Markov model forward filtering-backward sampling algorithm. Our method is exact and does not involve approximations like time-discretization. We demonstrate how our sampler extends naturally to MJP-based models like Markov-modulated Poisson processes and continuous-time Bayesian networks and show significant computational benefits over state-of-the-art MCMC samplers for these models.Comment: Accepted at the Journal of Machine Learning Research (JMLR

GPS queues with heterogeneous traffic classes

We consider a queue fed by a mixture of light-tailed and heavy-tailed traffic. The two traffic classes are served in accordance with the generalized processor sharing (GPS) discipline. GPS-based scheduling algorithms, such as weighted fair queueing (WFQ), have emerged as an important mechanism for achieving service differentiation in integrated networks. We derive the asymptotic workload behavior of the light-tailed class for the situation where its GPS weight is larger than its traffic intensity. The GPS mechanism ensures that the workload is bounded above by that in an isolated system with the light-tailed class served in isolation at a constant rate equal to its GPS weight. We show that the workload distribution is in fact asymptotically equivalent to that in the isolated system, multiplied with a certain pre-factor, which accounts for the interaction with the heavy-tailed class. Specifically, the pre-factor represents the probability that the heavy-tailed class is backlogged long enough for the light-tailed class to reach overflow. The results provide crucial qualitative insight in the typical overflow scenario

Analysis of Markov-modulated infinite-server queues in the central-limit regime

This paper focuses on an infinite-server queue modulated by an independently evolving finite-state Markovian background process, with transition rate matrix $Q\equiv(q_{ij})_{i,j=1}^d$. Both arrival rates and service rates are depending on the state of the background process. The main contribution concerns the derivation of central limit theorems for the number of customers in the system at time $t\ge 0$, in the asymptotic regime in which the arrival rates $\lambda_i$ are scaled by a factor $N$, and the transition rates $q_{ij}$ by a factor $N^\alpha$, with $\alpha \in \mathbb R^+$. The specific value of $\alpha$ has a crucial impact on the result: (i) for $\alpha>1$ the system essentially behaves as an M/M/$\infty$ queue, and in the central limit theorem the centered process has to be normalized by $\sqrt{N}$; (ii) for $\alpha<1$, the centered process has to be normalized by $N^{{1-}\alpha/2}$, with the deviation matrix appearing in the expression for the variance

Modulated Branching Processes, Origins of Power Laws and Queueing Duality

Power law distributions have been repeatedly observed in a wide variety of socioeconomic, biological and technological areas. In many of the observations, e.g., city populations and sizes of living organisms, the objects of interest evolve due to the replication of their many independent components, e.g., births-deaths of individuals and replications of cells. Furthermore, the rates of the replication are often controlled by exogenous parameters causing periods of expansion and contraction, e.g., baby booms and busts, economic booms and recessions, etc. In addition, the sizes of these objects often have reflective lower boundaries, e.g., cities do not fall bellow a certain size, low income individuals are subsidized by the government, companies are protected by bankruptcy laws, etc. Hence, it is natural to propose reflected modulated branching processes as generic models for many of the preceding observations. Indeed, our main results show that the proposed mathematical models result in power law distributions under quite general polynomial Gartner-Ellis conditions, the generality of which could explain the ubiquitous nature of power law distributions. In addition, on a logarithmic scale, we establish an asymptotic equivalence between the reflected branching processes and the corresponding multiplicative ones. The latter, as recognized by Goldie (1991), is known to be dual to queueing/additive processes. We emphasize this duality further in the generality of stationary and ergodic processes.Comment: 36 pages, 2 figures; added references; a new theorem in Subsection 4.