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This letter analyses both static and dynamic resource allocation methodologies of the Multi-User Orthogonal Frequency Division Multiple Access (MU-OFDMA) feature of the Wi-Fi 6 network. In contrast to static MU-OFDMA, in the case of dynamic MU-OFDMA system, resource units (RUs) are allocated dynamically between the schedule and random access (RA) RUs based on the heterogeneous traffic requirements of the stations (STAs). In this letter, we first analyze the static MU-OFDMA system using the Fluid limit approach and derive the transmission probability of a STA in a time slot denoted as τ as a function of the number of RA RUs (NRA). Later the dynamic MU-OFDMA system is modeled as a Discrete-Time Markov Chain (DTMC) wherein the NRA changes with every time slot. The DTMC is solved by using the parameter τ as a function of NRA that is derived previously using the Fluid limit approach. The trade-off between throughput and RA capability (rate of new users accessing the system) of a dynamic system is studied and validated against simulation.

We study a stylized queueing model motivated by paid express lanes on highways. There are two parallel, observable first-come, first-served queues with finitely many servers: one queue has a faster service rate, but charges a fee to join, and the other is free but slow. Upon arrival, customers see the state of each queue and choose between them by comparing the respective disutility of time spent waiting, subject to random shocks. This framework encompasses both the multinomial logit and exponomial customer choice models. Using a fluid limit approximation, we give a detailed characterization of the equilibrium in this system. We show that social welfare is optimized when the express queue is exactly at (but not over) full capacity; however, in some cases, revenue is maximized by artificially creating congestion in the free queue. The latter behavior is caused by changes in the price elasticity of demand as the service capacity of the free queue fills up.

Pender, Rand and Wesson recently established a delay differential equation limit for a parallel service system with routing based on delayed information. We provide an interpretation of their scaling under which their limit can be regarded as an instance of a law of large numbers in the familiar many-server heavy-traffic scaling. It requires scaling in the probabilistic routing function. We also obtain related many-server heavy-traffic delay-differential-equation limits for more general models.

A shared ledger is a record of transactions that can be updated by any member of a group of users. The notion of independent and consistent record-keeping in a shared ledger is important for blockchain and more generally for distributed ledger technologies. In this paper we analyze the growth of a model for the tangle, which is the shared ledger protocol used as the basis for the IOTA cryptocurrency. The model is a random directed acyclic graph, and its growth is described by a non-Markovian stochastic process. We derive a delay differential equation for the fluid model which describes the tangle at high arrival rate. We prove convergence in probability of the tangle process to the fluid model, and also prove global stability of the fluid model. The convergence proof relies on martingale techniques., 21 pages

We consider a multi-class multi-server retrial queueing system. Customers of each class arrive from outside the system according to a Poisson process. The service times for each class of customers are assumed to be exponentially distributed with service rates depending on both the customers’ class and the servers. We define the offered load and provide stability and instability conditions for this retrial queueing system. The stability result can be obtained by introducing artificial primitive processes and using the fluid limit approach.

This paper considers a closed queueing network model of ridesharing systems, such as Didi Chuxing, Lyft, and Uber. We focus on empty-car routing, a mechanism by which we control car flow in the network to optimize system-wide utility functions, for example, the availability of empty cars when a passenger arrives. We establish both process-level and steady-state convergence of the queueing network to a fluid limit in a large market regime where demand for rides and supply of cars tend to infinity and use this limit to study a fluid-based optimization problem. We prove that the optimal network utility obtained from the fluid-based optimization is an upper bound on the utility in the finite car system for any routing policy, both static and dynamic, under which the closed queueing network has a stationary distribution. This upper bound is achieved asymptotically under the fluid-based optimal routing policy. Simulation results with real-world data released by Didi Chuxing demonstrate the benefit of using the fluid-based optimal routing policy compared with various other policies.

International audience; We propose a Markovian queueing model for computing travel times at a macroscopic scale during rush hour. The service rates of the queueing model are state-dependent, reflecting the speed/density relation of the fundamental diagram of traffic flow. In the fluid limit, the dynamics of the transient queue size and travel time processes are governed by a set of differential equations. As an application of the model, we consider the rational time-dependent choice between public and private transport, assuming that there is a congestion-free public alternative to private transportation. Numerical examples reveal that a small reduction in peak traffic can significantly reduce the average travel times.

In this paper, we adopt the fluid limits to analyze Age of Information (AoI) in a wireless multiaccess network with many users. We consider the case wherein users have heterogeneous i.i.d. channel conditions and the statuses are generate-at-will. Convergence of the AoI occupancy measure to the fluid limit, represented by a Partial Derivative Equation (PDE), is proved within an approximation error inversely proportional to the number of users. Global convergence to the equilibrium of the PDE, i.e., stationary AoI distribution, is also proved. Based on this framework, it is shown that an existing AoI lower bound in the literature is in fact asymptotically tight, and a simple threshold policy, with the thresholds explicitly derived, achieves the optimum asymptotically. The proposed threshold-based policy is also much easier to decentralize than the widely-known index-based policies which require comparing user indices. To showcase the usability of the framework, we also use it to analyze the average non-linear AoI functions (with power and logarithm forms) in wireless networks. Again, explicit optimal threshold-based policies are derived, and average age functions proven. Simulation results show that even when the number of users is limited, e.g., $10$, the proposed policy and analysis are still effective., Comment: Extended version of a conference paper accepted to INFOCOM 2021

We propose a novel dynamic car relocation policy for a car-sharing network with centralized control and uncertain, unbalanced demand. The policy is derived from a reformulation of the linear programming fluid model approximation of the dynamic problem. We project the full-dimensional fluid approximation onto the lower-dimensional space of relocation decisions only. This projection results in a characterization of the problem as n+1 linear programs, where n is the number of nodes in the network. The reformulation uncovers structural properties that are interpretable using absorbing Markov chain concepts and allows us to write the gradient with respect to the relocation decisions in closed form. Our policy exploits these gradients to make dynamic car relocation decisions. We provide extensive numerical results where our dynamic car relocation policy consistently outperforms the standard static policy in realistic instances from the literature. In fact, it reduces the static policy's optimality gap by more than 13% in all of these instances and up to 35% in some.

The purpose of this paper is to bridge kinetic plasma descriptions and low-frequency single-fluid models. More specifically, the asymptotics leading to magnetohydrodynamic regimes starting from the Vlasov–Maxwell system are investigated. The analogy with the derivation, from the Vlasov–Poisson system, of a fluid representation for ions coupled to the Boltzmann relation for electrons is also outlined. The aim is to identify asymptotic parameters explaining the transitions from one microscopic description to a macroscopic low-frequency model. These investigations provide groundwork for the derivation of multi-scale numerical methods, model coupling or physics-based preconditioning.

In past studies, the reorientation or true polar wander (TPW) of visco-elastic bodies has been studied with approximated solutions. Two types of methods are commonly adopted: those based on the quasi-fluid approximation e.g. Ricard et al., 1993 and those based on the fluid limit approximation which only consider the final orientation (e.g. Matsuyama and Nimmo, 2007). Recently, Hu et al., (2017b) established a method which provides a dynamic solution for calculating the reorientation of tidally deformed bodies. However, they did not provide the links between the complete solution and the fluid limit solution. This paper provides a semi-analytical method for calculating the reorientation of tidally deformed bodies and shows the relation between the complete and the approximated solutions. Furthermore, we provide a criterion, the fluid limit process number F , to test for a given model and estimated TPW speed if the quasi-fluid approximation or fluid limit solution is valid. This number is a quantitative description of how close the body stays in hydrostatic equilibrium during a reorientation process. We use this number to obtain the largest allowed TPW speed of Mars as a function of viscosity, for which the approximated solutions may be used.

We derive the exact relation for the energy transfer in three-dimensional compressible two-fluid plasma turbulence. In the long-time limit, we obtain an exact law which expresses the scale-to-scale average energy flux rate in terms of two point increments of the fluid variables of each species, electric and magnetic field and current density, and puts a strong constraint on the turbulent dynamics. The incompressible single fluid and two-fluid limits and the compressible single fluid limit are recovered under appropriate assumption. In the single fluid limits, analyses are done with and without neglecting the electron mass thereby making the exact relation suitable for a broader range of application. In the compressible two-fluid regime, the total energy flux rate, unlike the single fluid case, is found to be unaltered by the presence of a background magnetic field. The exact relation provides a way to test whether a range of scales in a plasma is inertial or dissipative and is essential to understand the nonlinear nature of both space and dilute astrophysical plasmas. Fil: Banerjee, Supratik. Indian Institute of Technology Kanpur; India Fil: Andrés, Nahuel. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Astronomía y Física del Espacio. - Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Astronomía y Física del Espacio; Argentina

This paper focuses on the mechanism of Blocking Before Service (BBS), in time-varying many-server queues in tandem. BBS arises in telecommunication networks, production lines and healthcare systems. We model a stochastic tandem network under BBS and develop its corresponding fluid limit, which includes reflection due to jobs lost. Comparing our fluid model against simulation shows that the model is accurate and effective. This gives rise to design/operational insights regarding network throughput, under both BBS and BAS (Blocking After Service).

In this paper, we develop time-varying fluid models for tandem networks with blocking. Beyond having their own intrinsic value, these mathematical models are also limits of corresponding many-server stochastic systems. We begin by analyzing a two-station tandem network with a general time-varying arrival rate, a finite waiting room before the first station, and no waiting room between the stations. In this model, customers that are referred from the first station to the second when the latter is saturated (blocked) are forced to wait in the first station while occupying a server there. The finite waiting room before the first station causes customer loss and, therefore, requires reflection analysis. We then specialize our model to a single station (many-server fluid limit of the $$G_t/M/N/(N +H)$$ queue), generalize it to k stations in tandem, and allow finite internal waiting rooms. Our models yield operational insights into network performance, specifically on the effects of line length, bottleneck location, waiting room size, and the interaction among these effects.

We consider a queueing system with heterogeneous customers. One class of customers is eager; these customers are impatient and leave the system if service does not commence immediately upon arrival. Customers of the second class are tolerant; these customers have larger service requirements and can wait for service. In this paper, we establish pseudo-conservation laws relating the performance of the eager class (measured in terms of the long run fraction of customers blocked) and the tolerant class (measured in terms of the steady state performance, e.g., sojourn time, number in the system, workload) under a certain partial fluid limit. This fluid limit involves scaling the arrival rate as well as the service rate of the eager class proportionately to infinity, such that the offered load corresponding to the eager class remains constant. The workload of the tolerant class remains unscaled. Interestingly, our pseudo-conservation laws hold for a broad class of admission control and scheduling policies. This means that under the aforementioned fluid limit, the performance of the tolerant class depends only on the blocking probability of the eager class, and not on the specific admission control policy that produced that blocking probability. Our psuedo-conservation laws also characterize the achievable region for our system, which captures the space of feasible tradeoffs between the performance experienced by the two classes. We also provide two families of complete scheduling policies, which span the achievable region over their parameter space. Finally, we show that our pseudo-conservation laws also apply in several scenarios where eager customers have a limited waiting area and exhibit balking and/or reneging behaviour.

We analyze a tractable model of a limit order book on short time scales, where the dynamics are driven by stochastic fluctuations between supply and demand. We establish the existence of a limiting distribution for the highest bid, and for the lowest ask, where the limiting distributions are confined between two thresholds. We make extensive use of fluid limits in order to establish recurrence properties of the model. We use the model to analyze various high-frequency trading strategies, and comment on the Nash equilibria that emerge between high-frequency traders when a market in continuous time is replaced by frequent batch auctions., Revision of the submitted version: correcting an error (caught by Jan Swart) in the statement of Proposition 4.1 (1), propagating to Theorem 2.1 (1)

Stochastic encounter-mating (SEM) models describe monogamous permanent pair formation in finite zoological populations of multitype females and males. In this article, we study SEM with Poisson firing times. First, we prove that the model enjoys a fluid limit as the population size diverges, i.e., the stochastic dynamics converges to a deterministic system governed by coupled ODEs. Then, we convert these ODEs to the well-known Lotka-Volterra and replicator equations from population dynamics. Next, under the so-called fine balance condition which characterizes panmixia, we solve the corresponding replicator equations and give an exact expression for the fluid limit. Finally, we consider the case with two types of females and males. Without the fine balance assumption, but under certain symmetry conditions, we give an explicit formula for the limiting mating pattern, and then use it to characterize assortative mating., 23 pages, 1 figure. We resolved a technical issue in the beginning of Section 2.1, included a self-contained proof of Theorem 2.1, removed the part about the diffusion limit which did not fit the rest of the paper (hence the revised title), corrected/completed the proof of Theorem 4.1, and made various other (minor) changes

We consider two players, starting with $m$ and $n$ units, respectively. In each round, the winner is decided with probability proportional to each player's fortune, and the opponent loses one unit. We prove an explicit formula for the probability $p(m,n)$ that the first player wins. When $m\sim Nx_{0}$, $n\sim N y_{0}$, we prove the fluid limit as $N\to \infty$. When $x_{0}=y_{0}$, then $z\to p(N,N+z\sqrt{N})$ converges to the standard normal CDF and the difference in fortunes scales diffusively. The exact limit of the time of ruin $\tau_{N}$ is established as $(T-\tau_N) \sim N^{-\beta}W^{\frac{1}{\beta}}$, $\beta=\frac{1}{4}$, $T=x_{0}+y_{0}$. Modulo a constant, $W \sim \chi^{2}_{1}(z_{0}^{2}/T^{2})$., Comment: 34 pages

We consider the model of a token-based joint auto-scaling and load balancing strategy, proposed in a recent paper by Mukherjee, Dhara, Borst, and van Leeuwaarden [4] (SIGMETRICS '17), which offers an efficient scalable implementation and yet achieves asymptotically optimal steady-state delay performance and energy consumption as the number of servers N ! 1. In the above work, the asymptotic results are obtained under the assumption that the queues have fixed-size finite buffers, and therefore the fundamental question of stability of the proposed scheme with infinite buffers was left open. In this paper, we address this fundamental stability question. The system stability under the usual subcritical load assumption is not automatic. Moreover, the stability may not even hold for all N. The key challenge stems from the fact that the process lacks monotonicity, which has been the powerful primary tool for establishing stability in load balancing models. We develop a novel method to prove that the subcritically loaded system is stable for large enough N, and establish convergence of steady-state distributions to the optimal one, as N ! 1. The method goes beyond the state of the art techniques " it uses an induction-based idea and a "weak monotonicity" property of the model; this technique is of independent interest and may have broader applicability.

We consider a parallel-queue system in which each queue is served by a dedicated service provider. The arrival process is driven by a discrete choice model, that is, customers observe the queue length for each service provider and choose one to join upon arrival. We assume that a customer's utility is difference between the service reward and the waiting cost, both of which are heterogeneous. Empirical analysis of the vehicle queues at the U.S.-Canada border-crossing port of entry supports our model setting. We show that with such a choice model, the arrival rate function satisfies certain properties, which allow us to characterizes the fluid and diffusion limit of the queue-length process. In particular, we show that even without the well-used Lipschitz-continuity assumption, the fluid limit process is unique and is attracted to a unique equilibrium. The diffusion limit process is a reflected multi-dimensional Ornstein-Uhlenbeck process centered at that equilibrium. We prove that the stationary distribution of the diffusion limit is a truncated multivariant Gaussian and interchange of limits holds.

Inaccuracies in the initial conditions for cosmological N-body simulations could easily be the largest source of systematic error in predicting the non-linear large-scale structure. From the theory side, initial conditions are usually provided by using low-order truncations of the displacement field from Lagrangian perturbation theory, with the first and second-order approximations being the most common ones. Here we investigate the improvement brought by using initial conditions based on third-order Lagrangian perturbation theory (3LPT). We show that with 3LPT, truncation errors are vastly suppressed, thereby opening the portal to initializing simulations accurately as late as z=12 (for the resolution we consider). We analyse the competing effects of perturbative truncation and particle discreteness on various summary statistics. Discreteness errors are essentially decaying modes and thus get strongly amplified for earlier initialization times. We show that late starting times with 3LPT provide the most accurate configuration, which we find to coincide with the continuum fluid limit within 1 per cent for the power- and bispectrum at z=0 up to the particle Nyquist wave number of our simulations (k ~ 3h/Mpc). In conclusion, to suppress non-fluid artefacts, we recommend initializing simulations as late as possible with 3LPT. We make our 3LPT initial condition generator publicly available., Comment: 22 pages, 13 figures, minor revisions (matches published version), code available from https://bitbucket.org/ohahn/monofonic

We introduce a new framework for the analysis of large-scale load balancing networks with general service time distributions, motivated by applications in server farms, distributed memory machines, cloud computing and communication systems. For a parallel server network using the so-called $SQ(d)$ load balancing routing policy, we use a novel representation for the state of the system and identify its fluid limit, when the number of servers goes to infinity and the arrival rate per server tends to a constant. The fluid limit is characterized as the unique solution to a countable system of coupled partial differential equations (PDE), which serve to approximate transient Quality of Service parameters such as the expected virtual waiting time and queue length distribution. In the special case when the service time distribution is exponential, our method recovers the well-known ordinary differential equation characterization of the fluid limit. Furthermore, we develop a numerical scheme to solve the PDE, and demonstrate the efficacy of the PDE approximation by comparing it with Monte Carlo simulations. We also illustrate how the PDE can be used to gain insight into the performance of large networks in practical scenarios by analyzing relaxation times in a backlogged network. In particular, our numerical approximation of the PDE uncovers two interesting properties of relaxation times under the SQ(2) algorithm. Firstly, when the service time distribution is Pareto with unit mean, the relaxation time decreases as the tail becomes heavier. This is a priori counterintuitive given that for the Pareto distribution, heavier tails have been shown to lead to worse tail behavior in equilibrium. Secondly, for unit mean light-tailed service distributions such as the Weibull and lognormal, the relaxation time decreases as the variance increases. This is in contrast to the behavior observed under random routing, where the relaxation time increases with increase in variance.

We consider a single server queueing system with two classes of jobs: eager jobs with small sizes that require service to begin almost immediately upon arrival, and tolerant jobs with larger sizes that can wait for service. While blocking probability is the relevant performance metric for the eager class, the tolerant class seeks to minimize its mean sojourn time. In this article, we analyse the performance of each class under dynamic scheduling policies, where the scheduling of both classes depends on the instantaneous state of the system. This analysis is carried out under a certain fluid limit, where the arrival rate and service rate of the eager class are scaled to infinity, holding the offered load constant. Our performance characterizations reveal a (dynamic) pseudo-conservation law that ties the performance of both classes to the standalone blocking probabilities associated with the scheduling policies for the eager class. Furthermore, the performance is robust to other specifics of the scheduling policies. We also characterize the Pareto frontier of the achievable region of performance vectors under the same fluid limit, and identify a (two-parameter) class of Pareto-complete scheduling policies.

This thesis is about coping with variability in outcomes for complex stochastic systems. We focus on systems where jobs arrive randomly throughout time to utilise resources for a random amount of time before departure. The systems we investigate are primarily concerned with the communication and storage of data. The thesis is partitioned into two parts. The first part studies systems where congestion leads to jobs waiting for service (queueing systems) and the second part considers systems where congestion leads to losses due to departures before provision of service (loss systems).For queueing systems, we are mainly interested in the management objective of ensuring that the expected time a job must wait before entering is finite --- a property known as stability. Finite waiting times occur naturally for loss systems due to the balking behaviour of jobs in response to congestion and so our attention in this case turns to the more ambitious goal of managing systems in such a way that the number of lost jobs is minimised.Each part consists of an introductory chapter providing background knowledge, which is followed by three chapters containing original research. In both parts we progress through these chapters by first applying traditional analytical approaches to novel models and then developing novel simulation-based approaches for models which are out of reach of traditional approaches.We begin our research on queueing networks in Part 1 by considering a network of infinite-server queues with the special feature that, triggered by specific events, the network population vector may undergo a linear transformation. We use moment generating functions to obtain expressions for transient and stationary moments of the queue size vector and characterise the set of parameters for which the system is stable. A variety of systems fit in the framework developed, such as networks of retrial queues, networks in which jobs can be rerouted when links fail, and storage systems.In the next chapter of Part 1 we study the recently introduced Queue-Proportional Rate Allocation scheduling algorithm for multihop radio networks. The main contribution is a proof using fluid limit techniques to show that a natural generalisation of this policy to allow weighting of packets at each link, to reflect nonhomogeneous priorities, retains the maximal stability property. We also state a conjecture that in heavy traffic the diffusion-scaled workload process of the network converges weakly to a reflected Brownian motion and that in this weak limit the vector of queue lengths is always proportional to the traffic arrival rate vector.We conclude Part 1 by devising a simulation-based method for detecting whether a non-negative Markov chain is unstable for a given set of parameter values. More precisely, for a given subset of the parameter space, we develop an algorithm that can decide whether the set has a subset of positive Lebesgue measure for which the Markov chain is unstable. The approach is based on a variant of simulated annealing, and consequently only mild assumptions are needed to obtain rigorous performance guarantees. Our framework leads to a procedure that can perform statistically rigorous tests for instability, which has been extensively tested using several examples of standard and non-standard queueing networks.We begin our investigation of loss systems in Part 2 by considering a finite-capacity Erlang B model that alternates between active and inactive states according to a two-state modulating Markov process. Jobs arrives to the system as a Poisson process but are blocked from entry when the system is at capacity or inactive. We use Laplace transforms to derive expressions for the revenue lost during short term planning horizons. These expressions can be used to assess alternative system designs.In the next chapter of Part 2 we develop a sophisticated loss system type model for cloud computing systems. User demand on the computational resources of cloud computing platforms varies over time. These variations in the arrival process can be predictable or unpredictable, resulting in time-varying and `bursty' demand fluctuations. Furthermore, jobs can arrive in batches, and users whose demands are not met can be impatient. We demonstrate how to compute the expected revenue loss over a finite time horizon in the presence of all these model characteristics using matrix analytic methods. It is seen that taking these characteristics of fluctuating user demand into account can result in a substantial reduction of losses.We conclude Part 2 by developing an optimisation framework for a model applicable to mobile cloud edge computing systems. Our model is a stochastic network with blocking: jobs attempt to be processed sequentially at nodes in a network but are lost when they attempt to access a node that is at capacity. The problem is mathematically intractable in general and time consuming to solve using standard simulation methods. Our novel method combines simulation with analytical approximations to quickly obtain high quality solutions. We extensively test our approach using several complex models.

This study is concerned with a throughput maximization problem of prioritized customer orders. Customer orders with different priorities arrive at a server station dynamically. Each order consists of multiple product types with random workloads. These workloads will be assigned to and processed by a set of unrelated servers. Two commonly applied assignment schemes, named Workload Assignment Scheme (WAS) and Server Assignment Scheme (SAS) are considered. The objective is to determine the optimal assignments under the two assignment schemes to maximize the long-run throughput. Mathematical programming models with relaxed stability constraints are developed for the two assignment schemes, and the adequacy of the programes is guaranteed through fluid limit model analysis. It is shown that these two mathematical programes share the same optimal value, and that there exists a one-to-one correspondence between the optimal assignments. Numerical experiment verifies that the two proposed mathematical programes yield the same optimal throughput, and demonstrates that the corresponding optimal assignments under the two assignment schemes can be transformed into each other.

In this paper, we propose a new model for binary opinion dynamics in a (fully connected) structurally balanced network. In a structurally balanced network, agents are classified into two clusters and two agents in the same cluster (resp. different clusters) are connected with a positive (resp. negative) edge. Initially, every agent is assigned with one of the two opinions randomly. In every time slot, three agents are randomly selected to have their opinions updated. If the three agents belong to the same cluster, the majority rule (MR) is used to update their opinions. On the other hand, if the three agents belong to two different clusters, with probability $p$ , a consensus is reached by the MR, and with probability $1-p$ , a polarization (in line with the signs of the three edges) is reached. The probability $p$ , called the rationality probability, plays a significant role for measuring how rational the agents in a network behave when they encounter different opinions. By applying a fluid limit theorem for jump Markov processes, we derive a system of differential equations for the density functions of opinions for large networks. We show that the equilibrium points corresponding to consensus and polarization are the only stable equilibrium points. All other equilibrium points are all unstable. As such, as time goes on, the network eventually reaches a consensus or a polarization, depending on the rationality probability and the initial state of the network.

In this paper, we establish a fluid limit for a two-sided Markov order book model. The main result states that in a certain asymptotic regime, a pair of measure-valued processes representing the “sell-side shape” and “buy-side shape” of an order book converges to a pair of deterministic measure-valued processes in a certain sense. We also test the fluid approximation on data. The empirical results suggest that the approximation is reasonably good for liquidly traded stocks in certain time periods.

A single queueing station serving K input streams with renewal arrivals and generally distributed independent and identically distributed service times is considered. Customers are served by the Shortest Remaining Processing Time policy. In the case of a tie, the first-in, first-out policy is utilized. We analyze a fluid model for the evolution of a measure-valued state descriptor of this system, with particular emphasis on its limiting behavior in the critical case as time gets large. We also prove a fluid limit theorem justifying our fluid model as the first-order approximation of the queueing system under consideration. Along the way, we establish fluid limits for the corresponding state-dependent response times.

The backpressure scheduling algorithm for multihop wireless networks is known to be throughput optimal, but it requires each node to maintain per-destination queues. Recently, a clever generalization of processor sharing has been proposed which is also throughput optimal, but which only uses per-link queues. Here, we propose another algorithm, called Queue-Proportional Rate Allocation (QPRA), which also only uses per-link queues and allocates service rates to links in proportion to their queue lengths, and employs the Serve-In-Random-Order queueing discipline within each link. Through fluid limit techniques and using a novel Lyapunov function, we show that the QPRA algorithm achieves the maximum throughput. We demonstrate an advantage of QPRA by showing that, for the so-called primary interference model, it is able to develop a low-complexity scheduling scheme which approximates QPRA and achieves a constant fraction of the maximum throughput region, independent of network size.

We consider the following distributed service model: jobs with unit mean, exponentially distributed, and independent processing times arrive as a Poisson process of rate $\lambda n$, with $00$ (no matter how small), if our policy only uses a linear message rate $\alpha n$, the resulting asymptotic delay is upper bounded, uniformly over all $\lambda

The purpose of this note is to show that it is possible to establish a many-server functional strong law of large numbers (FSLLN) for the fraction of occupied servers (i.e., the scaled number-in-system) without explicitly tracking through a measure valued process either the age or the residual service times of the jobs in a non-Markovian, non-stationary loss model. This considerable analytical simplification is achieved by exploiting a semimartingale representation. The fluid limit is shown to be the unique solution of a Volterra integral equation.

We develop diffusion approximations for parallel-queueing systems with the randomized longest-queue-first scheduling (LQF) algorithm by establishing new mean-field limit theorems as the number of buffers n → ∞. We achieve this by allowing the number of sampled buffers d = d(n) to depend on the number of buffers n, which yields an asymptotic 'decoupling' of the queue length processes. We show through simulation experiments that the resulting approximation is accurate even for moderate values of n and d(n). To the best of the authors' knowledge, this is the first derivation of diffusion approximations for a queueing system in the large-buffer mean-field regime. Another noteworthy feature of our scaling idea is that the randomized LQF algorithm emulates the LQF algorithm, yet is computationally more attractive. The analysis of the system performance as a function of d(n) is facilitated by the multi-scale nature in our limit theorems: the various processes we study have different space scalings. This allows us to show the trade-off between performance and complexity of the randomized LQF scheduling algorithm.

In this paper we analyze the dynamics of a P2P file exchange swarm from a queueing standpoint. In such systems, the service rate a peer receives depends on one mostly fixed component (servers or seeders), and another that scales with the number of peers present. We analyze a class of M / G Processor Sharing queues that describe populations and residual workloads in this situation, characterizing its stationary regime in the case of a fixed population of servers; the result behaves like a combination of M / G / 1 and M / G / ∞ queues. We apply scaling limits to this queue and identify two limiting regimes, depending on whether the server or peer contribution becomes dominant. For the latter, more important case we refine the fluid limit description of the download profile with a suitable functional Gaussian approximation. We also analyze the case of a slowly varying population of servers, extending the fixed case through a quasi-stationary analysis. For practical validation we offer comparisons with detailed packet simulations.

A single-server queueing model is considered with customers that have deadlines. If a customer’s deadline elapses before service is offered, the customer abandons the system (customers do not abandon while being served). When the server becomes available, it offers service to the customer having the earliest deadline among those that are in the queue. We obtain a fluid limit of the queue length and abandonment processes and for the occupation measure of deadlines, in the form of measure-valued processes. We characterize the limit by means of a Skorohod problem in a time-varying domain that has an explicit solution. The fluid limits also describe a certain process called the frontier that is well known to play a key role in systems operating under this scheduling policy.

The backpressure scheduling algorithm for multihop wireless networks is known to be throughput optimal, but it requires each node to maintain per-destination queues. Recently, a clever generalization of processor sharing has been proposed which is also throughput optimal, but which only uses per-link queues. Here we propose another algorithm called Queue Proportional Rate Allocation (QPRA) which also only uses per-link queues, and allocates service rates to links in proportion to their queue-lengths and employs the Serve-In-Random-Order (SIRO) discipline within each link. Through fluid limit techniques and using a novel Lyapunov function, we show that the QPRA achieves the maximum throughput. We demonstrate an advantage of QPRA by showing that, for the so-called primary interference model, it is able to develop a low-complexity scheduling scheme which approximates QPRA and achieves a constant fraction of the maximum throughput region, independent of network size.

In multi-server distributed queueing systems, the access of stochastically arriving jobs to resources is often regulated by a dispatcher, also known as load balancer. A fundamental problem consists in designing a load balancing algorithm that minimizes the delays experienced by jobs. During the last two decades, the power-of-$d$-choice algorithm, based on the idea of dispatching each job to the least loaded server out of $d$ servers randomly sampled at the arrival of the job itself, has emerged as a breakthrough in the foundations of this area due to its versatility and appealing asymptotic properties. In this paper, we consider the power-of-$d$-choice algorithm with the addition of a local memory that keeps track of the latest observations collected over time on the sampled servers. Then, each job is sent to a server with the lowest observation. We show that this algorithm is asymptotically optimal in the sense that the load balancer can always assign each job to an idle server in the large-system limit. This holds true if and only if the system load $\lambda$ is less than $1-\frac{1}{d}$. If this condition is not satisfied, we show that queue lengths are tightly bounded by $\left\lceil - \frac{ \log (1-\lambda)}{\log (\lambda d +1)} \right\rceil$. This is in contrast with the classic version of the power-of-$d$-choice algorithm, where at the fluid scale a strictly positive proportion of servers containing $i$ jobs exists for all $i\ge 0$, in equilibrium. Our results quantify and highlight the importance of using memory as a means to enhance performance in randomized load balancing.

We consider a simple stochastic model for the spread of a disease caused by two virus strains in a closed homogeneously mixing population of size N. The spread of each strain in the absence of the other one is described by the stochastic logistic SIS epidemic process, and we assume that there is perfect cross-immunity between the two strains, that is, individuals infected by one are temporarily immune to re-infections and infections by the other. For the case where one strain has a strictly larger basic reproductive ratio than the other, and the stronger strain on its own is supercritical (that is, its basic reproductive ratio is larger than 1), we derive precise asymptotic results for the distribution of the time when the weaker strain disappears from the population, that is, its extinction time. We further extend our results to certain parameter values where the difference between the two reproductive ratios may tend to 0 as $N \to \infty$. In proving our results, we illustrate a new approach to a fluid limit approximation for a sequence of Markov chains in the vicinity of a stable fixed point of the limit., Comment: 53 pages

In this paper, we are concerned with threshold voter models on tori. Assuming that the initial distribution of the process is product measure with density p, we obtain a fluid limit of the proportion of vertices in state 1 as the dimension of the torus grows to infinity. The fluid limit performs a phase transition phenomenon from p < 1/2 to p > 1/2., 18 pages

We justify the global-in-time diffusive limit of the Boltzmann equation inside a periodic domain $\mathbb{T}^3$. We only assume that in the initial expansion $F^{\epsilon}=\mu+\sqrt{\mu} f^{\epsilon}$ at $t=0$, the kinetic parts are well-prepared, but the fluid parts could be general, i.e., the fluids parts are not required to satisfy the incompressibility and Boussinesq relations. For this case, the fluid initial layers are created and preserved in the periodic domain. Employing the method of time averaging, we analyze the propagation of the initial layers, and thus extend Guo's results in [Comm. Pure Appl. Math., 59 (2006), pp. 626--687] in which both fluid and kinetic parts are required to be well-prepared.

Fluid approximations have seen great success in approximating the macro-scale behaviour of Markov systems with a large number of discrete states. However, these methods rely on the continuous-time Markov chain (CTMC) having a particular population structure which suggests a natural continuous state-space endowed with a dynamics for the approximating process. We construct here a general method based on spectral analysis of the transition matrix of the CTMC, without the need for a population structure. Specifically, we use the popular manifold learning method of diffusion maps to analyse the transition matrix as the operator of a hidden continuous process. An embedding of states in a continuous space is recovered, and the space is endowed with a drift vector field inferred via Gaussian process regression. In this manner, we construct an ordinary differential equation whose solution approximates the evolution of the CTMC mean, mapped onto the continuous space (known as the fluid limit).

The object of this research in the queueing theory is a theorem about the Strong-Law-of-Large-Numbers (SLLN) under the conditions of heavy traffic in a multiserver open queueing network. SLLN is known as a fluid limit or fluid approximation. In this work, we prove that the long-term average rate of growth of the queue length process of a multiserver open queueing network under heavy traffic strongly converges to a particular vector of rates. SLLN is proved for the values of an important probabilistic characteristic of the multiserver open queueing network investigated as well as the queue length of jobs.

This paper extends the works of Kang and Ramanan (2010) and Kaspi and Ramanan (2011), removing the hypothesis of absolute continuity of the service requirement and patience time distributions. We consider a many-server queueing system in which customers enter service in the order of arrival in a non-idling manner and where reneging is considerate. Similarly to Kang and Ramanan (2010), the dynamics of the system are represented in terms of a process that describes the total number of customers in the system as well as two measure-valued processes that record the age in service of each of the customers being served and the “potential” waiting times. When the number of servers goes to infinity, fluid limit is established for this triple of processes. The convergence is in the sense of probability and the limit is characterized by an integral equation.

Fluid or mean-field methods are approximate analytical techniques which have proven effective in tackling the infamous state-space explosion problem which typically arises when modelling large-scale concurrent systems based on interleaving semantics. These methods are particularly suitable in situations which present large populations of simple interacting objects characterised by small local state spaces, since they require the analysis of a problem which is insensitive to the population sizes but is dependent only on the size of the local state spaces. This paper studies the case when the replicated objects are best described as composites which consist of smaller simple objects. A congenial formal modelling framework for situations of this kind may be given by stochastic process algebra. Using PEPA as a representative case, we find that fluid models with replicated copies of composite processes do not scale well with increasing population sizes, thus rendering intractable the analysis of the underlying system of ordinary differential equations (ODEs). We call this problem continuous state-space explosion, by analogy with its counterpart phenomenon in discrete state spaces. The main contribution of this paper is a result of equivalence that simplifies, in an exact way, the potentially massive ODE system arising in those circumstances to one whose size is independent from all the multiplicities in the model. As a byproduct, we find that these simplified ODEs turn out to characterise the fluid behaviour of a family of PEPA models whose elements cannot be related to each other through any known equivalence relation. A substantial numerical assessment investigates the relationship between the different underlying Markov chains and their unique fluid limit, demonstrating its generally good accuracy for all practical purposes.

We consider a service system model primarily motivated by the problem of efficient assignment of virtual machines to physical host machines in a network cloud, so that the number of occupied hosts is minimized. There are multiple input flows of different type customers, with a customer mean service time depending on its type. There is infinite number of servers. A server packing {\em configuration} is the vector $k=\{k_i\}$, where $k_i$ is the number of type $i$ customers the server "contains". Packing constraints must be observed, namely there is a fixed finite set of configurations $k$ that are allowed. Service times of different customers are independent; after a service completion, each customer leaves its server and the system. Each new arriving customer is placed for service immediately; it can be placed into a server already serving other customers (as long as packing constraints are not violated), or into an idle server. We consider a simple parsimonious real-time algorithm, called {\em Greedy}, which attempts to minimize the increment of the objective function $\sum_k X_k^{1+\alpha}$, $\alpha>0$, caused by each new assignment; here $X_k$ is the number of servers in configuration $k$. (When $\alpha$ is small, $\sum_k X_k^{1+\alpha}$ approximates the total number $\sum_k X_k$ of occupied servers.) Our main results show that certain versions of the Greedy algorithm are {\em asymptotically optimal}, in the sense of minimizing $\sum_k X_k^{1+\alpha}$ in stationary regime, as the input flow rates grow to infinity. We also show that in the special case when the set of allowed configurations is determined by {\em vector-packing} constraints, Greedy algorithm can work with {\em aggregate configurations} as opposed to exact configurations $k$, thus reducing computational complexity while preserving the asymptotic optimality., Comment: 22 pages

Stiffness manifests in stochastic dynamic systems in a more complex manner than in deterministic systems; it is not only important for a time-stepping method to remain stable but it is also important for the method to capture the asymptotic variances accurately. In the context of stochastic chemical systems, time stepping methods are known as tau leaping. Well known existing tau leaping methods have shortcomings in this regard. The implicit tau method is far more stable than the trapezoidal tau method but underestimates the asymptotic variance. On the other hand, the trapezoidal tau method which estimates the asymptotic variance exactly for linear systems suffers from the fact that the transients of the method do not decay fast enough in the context of very stiff systems. We propose a tau leaping method that possesses the same stability properties as the implicit method while it also captures the asymptotic variance with reasonable accuracy at least for the test system S"[email protected]?S"2. The proposed method uses a central limit approximation (CLA) locally over the tau leaping interval and is referred to as the [email protected] The CLA predicts the mean and covariance as solutions of certain differential equations (ODEs) and for efficiency we solve these using a single time step of a suitable low order method. We perform a mean/covariance stability analysis of various possible low order schemes to determine the best scheme. Numerical experiments presented show that [email protected] performs favorably for stiff systems and that the [email protected] is also able to capture bimodal distributions unlike the CLA itself. The proposed [email protected] method uses a split implicit step to compute the mean update. We also prove that any tau leaping method employing a split implicit step converges in the fluid limit to the implicit Euler method as applied to the fluid limit differential equation.

We prove a many-server heavy-traffic fluid limit for an overloaded Markovian queueing system having two customer classes and two service pools, known in the call-center literature as the X model. The system uses the fixed-queue-ratio-with-thresholds (FQR-T) control, which we proposed in a recent paper as a way for one service system to help another in face of an unexpected overload. Under FQR-T, customers are served by their own service pool until a threshold is exceeded. Then, one-way sharing is activated with customers from one class allowed to be served in both pools. After the control is activated, it aims to keep the two queues at a prespecified fixed ratio. For large systems that fixed ratio is achieved approximately. For the fluid limit, or FWLLN (functional weak law of large numbers), we consider a sequence of properly scaled X models in overload operating under FQR-T. Our proof of the FWLLN follows the compactness approach, i.e., we show that the sequence of scaled processes is tight and then show that all converging subsequences have the specified limit. The characterization step is complicated because the queue-difference processes, which determine the customer-server assignments, need to be considered without spatial scaling. Asymptotically, these queue-difference processes operate on a faster time scale than the fluid-scaled processes. In the limit, because of a separation of time scales, the driving processes converge to a time-dependent steady state (or local average) of a time-varying fast-time-scale process (FTSP). This averaging principle allows us to replace the driving processes with the long-run average behavior of the FTSP.

In this paper, we consider the problem of link scheduling in multihop wireless networks under general interference constraints. Our goal is to design scheduling schemes that do not use per-flow or per-destination information, maintain a single data queue for each link, and exploit only local information, while guaranteeing throughput optimality. Although the celebrated back-pressure algorithm maximizes throughput, it requires per-flow or per-destination information. It is usually difficult to obtain and maintain this type of information, especially in large networks, where there are numerous flows. Also, the back-pressure algorithm maintains a complex data structure at each node, keeps exchanging queue-length information among neighboring nodes, and commonly results in poor delay performance. In this paper, we propose scheduling schemes that can circumvent these drawbacks and guarantee throughput optimality. These schemes use either the readily available hop-count information or only the local information for each link. We rigorously analyze the performance of the proposed schemes using fluid limit techniques via an inductive argument and show that they are throughput-optimal. We also conduct simulations to validate our theoretical results in various settings and show that the proposed schemes can substantially improve the delay performance in most scenarios.

Bike sharing systems have rapidly developed around the world, and they are served as a promising strategy to improve urban traffic congestion and to decrease polluting gas emissions. So far performance analysis of bike sharing systems always exists many difficulties and challenges under some more general factors. In this paper, a more general large-scale bike sharing system is discussed by means of heavy traffic approximation of multiclass closed queueing networks with non-exponential factors. Based on this, the fluid scaled equations and the diffusion scaled equations are established by means of the numbers of bikes both at the stations and on the roads, respectively. Furthermore, the scaling processes for the numbers of bikes both at the stations and on the roads are proved to converge in distribution to a semimartingale reflecting Brownian motion (SRBM) in a \(N^{2}\)-dimensional box, and also the fluid and diffusion limit theorems are obtained. Furthermore, performance analysis of the bike sharing system is provided. Thus the results and methodology of this paper provide new highlight in the study of more general large-scale bike sharing systems.

In this work, the fluid limit approach methodology is applied to find a sufficient and necessary stability condition for the Basic Collaboration (BC) system with feedback allowed, which is a generalization of the so-called W-model. In this queueing system, some customer classes need cooperation of a subset of (non-overlapping) servers. We assume that each customer class arrives to the system following a renewal input with general i.i.d. inter-arrival times, and general i.i.d. service times are also assumed. Priority is given to customer classes that can not be served by a single server but need a cooperation.