Spatial fluid limits for stochastic mobile networks
Introduction
This paper is concerned with the modeling and analysis of large-scale computer and communication networks with mobile nodes. We consider an expressive model of a node which is assumed to evolve over a set of local states. These may represent, for instance, distinct phases of a node’s behavior that distinguish between periods of activity and inactivity in a wireless network or, with a richer granularity, they may be related to different states in a communication protocol. We are interested in understanding how the interplay between the node’s local behavior and its mobility pattern has an impact on the overall system’s performance.
In this paper, we consider Markov population processes as the starting point of our analysis. These represent a general class of continuous-time Markov chains (CTMCs), characterized by a state descriptor which is given by a vector where each element gives the population of nodes in a specific local state. In general, such stochastic models do not scale well with increasing system sizes because they are based on an explicit discrete-state representation. This makes the analysis very difficult computationally and poses a detriment to model-assisted parameter-space exploration and capacity planning. However, under mild assumptions, a Markov population process may yield a fluid approximation [1], [2], [3], [4], [5], [6], [7] as a system of ordinary differential equations (ODEs). This can be shown to be the deterministic limit behavior that holds when the number of nodes goes to infinity; see, e.g., [8]. Fluid limits have been developed for a wide range of models of distributed and networked systems, including, for instance, load balancing [9], [10], optical switches [11], virtualized environments [12], and peer-to-peer networks [13]. In [14] Benaïm and Le Boudec offer a general framework for a Markov model of interacting agents evolving through a set of local states. In all cases, while the Markov process has a state space with cardinality that is exponential in the number of agents (in the worst case), the fluid limit only depends linearly on the number of local states. This procedure is very attractive in practice because it enables computationally efficient solutions, when the local state space is small, that are typically very accurate for large populations.
In the aforementioned models, however, spatial effects are not explicitly present because the system is inherently static, or because it is a convenient simplification to abstract away from them for the purposes of model tractability [15]. Unfortunately, explicitly taking into account mobility and space in a fluid model generally leads to a rapid growth of the local state space. In this paper, for instance, we deal with population processes defined by a CTMC with a random walk (RW) model over a two-dimensional topology. We consider a typical partitioning of the spatial domain into regions (or patches, as usually called in theoretical biology [16]). Within each region an assumption of homogeneity is made, and spatial effects are incorporated as transitions across regions [17]. If the dynamics of each region admits a fluid limit, then the overall system behavior is given by ODEs, where, according to the notation of this paper, is the local state space size and is the total number of regions. Such dependence is simply due to the fact that the model must keep track of the total population of nodes in each of the local states, at each region.
We consider this problem in the context of a generic framework for mobile reaction networks, a concise model to describe Markov population processes. This is inspired by stochastic reaction networks, which are frequently used for the analysis of systems of (bio-)chemical reactions (e.g., [18]). We assume that nodes may interact with each other within the same region, and undergo unbiased RW with a parameter, the migration rate, which can be dependent on their local state. We also consider absorbing and reflective boundary conditions for the spatial domain. In the presence of the former, each node that hits the boundary exits the domain without the possibility of ever re-entering it. Instead, if the latter is considered, the boundaries build a barrier of the domain that cannot be crossed by nodes. Those assumptions do not exclude the possibility of having exogenous arrivals into the system, e.g., by means of a Poisson process in each region.
The purpose of this paper is to develop a technique to effectively analyze models when and the total population of nodes are large. It is natural to consider an approach with two dimensions of scaling: the first one is the celebrated density-dependent form that allows us to obtain a continuous deterministic limit for the concentrations (or densities) of the node populations [8], while keeping the regions discrete. This is achieved by constructing a family of population processes indexed by a parameter, hereafter denoted by , such that the initial state of each element of the family is a population of nodes that grows linearly with . Then, the result of Kurtz [8] ensures that the sequence converges, as , to the solution of an ODE system of size . The second scaling occurs on the spatial dimension. We define a suitable sequence of migration rates such that increasing means considering regions which are closer and closer to each other on a regular mesh in the unit square in (i.e., the regions are at a distance from each other). After setting the scene, we show that the sequence of ODE systems converges, as , to a system of partial differential equations (PDEs) of reaction–diffusion type. In these PDEs, the diffusive terms model the continuous migration across regions, whereas the reactive terms describe the local interactions between nodes. To the best of our knowledge, such a convergence result has not been proven before. We also argue that our limit may allow for a more efficient numerical solution. This may seem surprising because analytical solutions of PDE systems are scarce and numerical solutions of PDE systems rely on a discretization of space. However while in the case of the limit ODE system of size the discretization is dictated by the stochastic model, in our limit PDE system the coarseness of the discrete mesh depends on the PDE solver. In other words, the PDE solver may in effect give rise to a coarsening of the original spatial domain.
A typical situation of practical interest to which our framework can be applied is that of large-scale mobile networks such as personal communication services (e.g., [19]): there are many base stations (e.g., in a wide-area cellular network) and the area served by a base station can be modeled as a region, which can contain potentially many mobile nodes that may migrate across regions. Here, the modeler may wish to predict how nodes distribute across the network over a given time horizon [20]. In this paper, we demonstrate the applicability of our method by studying a network of mobile nodes with connectivity provided by an 802.11 access point located in each region. First, we successfully validate our CTMC model against discrete-event simulation using the JiST/SWANS discrete-event simulation framework [21]. Then, we show that the PDE solution provides an excellent estimate of the network’s performance for relatively small population sizes and moderate lattice granularity.
Paper outline. The remainder of the paper is organized as follows. Section 2 overviews related work. In order to fix notation and provide intuition on the scaling limits considered in the paper, Section 3 introduces stationary reaction networks, which describe static networks with no mobility, whilst allowing nodes to be described by a local state space. Here we discuss how such models admit a classic fluid limit as a system of ODEs. Section 4 presents mobile reaction networks, which are defined as a conservative extension of stationary networks with an explicit mobility model. We introduce a straightforward spatial ODE limit that depends upon the lattice granularity. In Section 5 we discuss the main contribution of this paper, namely the convergence of the spatial ODE limit to a system of reaction–diffusion PDEs by assuming that nodes undergo unbiased random walk during their evolution. Section 6 discusses the numerical tests on our validation model. Finally, Section 7 concludes the paper.
Section snippets
Related work
Reaction–diffusion PDEs. PDEs of reaction–diffusion type are very well understood in many disciplines, such as biology [22], ecology [23], and chemistry [24]. It is beyond the scope of this paper to provide a general overview of the literature. Instead, we focus on related work that, similarly to ours, considers PDEs as the macroscopic deterministic behavior of a stochastic process.
In physical chemistry, one such approach is to consider the so-called reaction–diffusion master equation, which
Stationary reaction networks
In this section we provide a definition of a classic stochastic reaction network which does not consider an explicit mobility model. For this reason, this is called a stationary reaction network. For illustrative purposes, we also show how this notation allows us to recover two different fluid models of networked computing systems already studied in the literature. This sets the stage for Section 4, which, instead, discusses the novel contribution of this paper, consisting of extending a
Mobile reaction networks
A mobile reaction network is a Markov population process with an explicit notion of locality and mobility. Space is partitioned in a number of regions. Nodes within the same region may communicate with each other using the interaction functions (1). Additionally, nodes may move to neighboring regions by performing an unbiased RW. The purpose of this section is to show that a straightforward spatial ODE limit result for such a CTMC where the explicit location of nodes must be kept track of leads
Spatial reaction–diffusion limit
The spatial ODE fluid limit of Theorem 2 holds for any arbitrary but fixed . However, for large , the analysis may become infeasible because the ODE system size grows with . The purpose of this section is to consider a limit behavior that allows the analysis to be independent from . To do so, we study a limit PDE of reaction–diffusion type.
Space scaling. In addition to a suitable scaling of the population sizes with , we require a proper scaling of the migration rates with .
Case study
The purpose of this section is to show how to use our PDE limit result for the performance modeling and analysis of a communication network with mobile nodes. In the following, we study in detail the mobile version of Example 3 with its PDE limit (19) using absorbing boundary conditions.
System description. Our mobile network consists of a lattice of regions where each region represents an area offering Internet connectivity to all nodes therein located by means of an 802.11 access point.
Conclusion
In this paper we have proposed an approach to analyze stochastic models of mobile networks by means of a deterministic approximation represented by a system of partial differential equations. These offer a macroscopic, continuous view of the network dynamics which holds in the limit of infinite populations of nodes in the network and infinite number of regions in a regular two-dimensional spatial lattice. In practice, in our numerical tests we observed good accuracy in the case of moderate
Acknowledgment
This work was partially supported by the EU project QUANTICOL, 600708.
Max Tschaikowski is an Assistant Professor at IMT Lucca, Italy. Before he was a Research Fellow at the University of Southampton, UK, and a Research Assistant at the LMU in Munich, Germany. He was awarded a diploma in Mathematics and a Ph.D. in Computer Science by LMU Munich in 2010 and 2014, respectively. His research focusses on formal methods in quantitative modeling of biochemical and concurrent systems.
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Max Tschaikowski is an Assistant Professor at IMT Lucca, Italy. Before he was a Research Fellow at the University of Southampton, UK, and a Research Assistant at the LMU in Munich, Germany. He was awarded a diploma in Mathematics and a Ph.D. in Computer Science by LMU Munich in 2010 and 2014, respectively. His research focusses on formal methods in quantitative modeling of biochemical and concurrent systems.
Mirco Tribastone is Associate Professor at IMT Lucca, Italy. Prior to joining IMT Lucca he was an Associate Professor at the School of Electronics and Computer Science of Southampton University, United Kingdom, and an Assistant Professor (Juniorprofessor) at the Institute for Informatics of the Ludwig-Maximilians University of Munich, Germany. He received his Ph.D. in Computer Science from the School of Informatics of the University of Edinburgh in 2010. His research interests are in the analysis and reduction of large-scale models, with applications to computing systems and computational systems biology.