Search

A single-server queue with renewal arrivals and generally distributed independent and identically distributed service times is considered. Customers are served using the longest remaining time first scheduling algorithm. In case of a tie, processor sharing is utilized. We introduce a fluid model for the evolution of a measure-valued state descriptor of this queue, and we investigate its properties. We also prove a fluid limit theorem justifying our fluid model as the first-order approximation of the queueing system under consideration. [ABSTRACT FROM AUTHOR]

This study considers a multi-class single-server retrial queue with constant retrial rates in which service times are generally distributed with different service rates for each class. Avrachenkov (Queueing Systems 100:177–179, 2022) formulated the conjecture on the stability of this queueing system. We completely resolve the conjecture by using the fluid limit approach. [ABSTRACT FROM AUTHOR]

In this paper, we analyze a model called the k-nearest neighbor queue with the possibility of having delayed queue length feedback. We prove fluid limits for the stochastic queueing model and show that the fluid limit converges to a system of delay differential equations. Using the properties of circulant matrices, we derive a closed form expression for the value of the critical delay, which governs whether the delayed information will induce oscillations or a Hopf bifurcation in our queueing system. [ABSTRACT FROM AUTHOR]

We prove a fluid limit describing coarsening for zero-range processes on a finite number of sites, with asymptotically constant jump rates. When time and occupation per site are linearly rescaled by the total number of particles, the evolution of the process is described by a piecewise linear trajectory in the simplex indexed by the sites. The linear coefficients are determined by the trace process of the underlying random walk on the subset of non-empty sites, and the trajectory reaches an absorbing configuration in finite time. A boundary of the simplex is called absorbing for the fluid limit if a trajectory started at a configuration in the boundary remains in it for all times. We identify the set of absorbing configurations and characterize the absorbing boundaries. [ABSTRACT FROM AUTHOR]

In this paper, we study an N server fork-join queue with nearly deterministic arrival and service times. Specifically, we present a fluid limit for the maximum queue length as N → ∞. This fluid limit depends on the initial number of tasks. In order to prove these results, we develop extreme value theory and diffusion approximations for the queue lengths. [ABSTRACT FROM AUTHOR]

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 a stochastic model for the shared ledger known as the tangle, which was devised 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 first prove ergodicity of the stochastic process, and then 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. [ABSTRACT FROM AUTHOR]

We consider a multihop wireless network. There are multiple source destination pairs. We propose a scheduling algorithm to provide end-to-end mean delay guarantees to the flows. We demonstrate that the algorithm is also throughput optimal. Under heavy traffic, we obtain a Brownian approximation of the network. Furthermore, we also show that the stationary distribution of the scaled process of the network converges to that of the Brownian limit, providing an approximation to the stationary distribution under heavy traffic. Finally simulations further verify our claims. [ABSTRACT FROM AUTHOR]

We consider a system of N identical server pools and a single dispatcher in which tasks with unit-exponential service requirements arrive at rate λ (N) . In order to optimize the experienced performance, the dispatcher aims to evenly distribute the tasks across the various server pools. Specifically, when a task arrives, the dispatcher assigns it to the server pool with the minimum number of tasks among d(N) randomly selected server pools. We construct a stochastic coupling to bound the difference in the system occupancy processes between the join-the-shortest-queue (JSQ) policy and a scheme with an arbitrary value of d(N). We use the coupling to derive the fluid limit in case d (N) → ∞ and λ (Ν) / Ν → λ as N → ∞ along with the associated fixed point. The fluid limit turns out to be insensitive to the exact growth rate of d(N) and coincides with that for the JSQ policy. We further establish that the diffusion limit corresponds to that for the JSQ policy as well, as long as d (N) / N log (N) → ∞ , and characterize the common limiting diffusion process. These results indicate that the JSQ optimality can be preserved at the fluid and diffusion levels while reducing the overhead by nearly a factor O(N) and O( N / log (N) ), respectively. [ABSTRACT FROM AUTHOR]

In multiserver distributed queueing systems, the access of stochastically arriving jobs to resources is often regulated by a dispatcher, also known as a 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 because of 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 λ is less than 1 − 1 d . If this condition is not satisfied, we show that queue lengths are bounded by ⌈ − log (1 − λ) log (λd + 1) ⌉ . This is in contrast with the classic version of the power-of-d-choice algorithm, in which, at the fluid scale, a strictly positive proportion of servers containing i jobs exists for all i ≥ 0 in equilibrium. Our results quantify and highlight the importance of using memory as a means to enhance performance in randomized load balancing. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Zhengzhou University (Natural Science Edition) is the property of Journal of Zhengzhou University (Natural Science Edition) Editorial Office and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

We consider load balancing in service systems with affinity relations between jobs and servers. Specifically, an arriving job can be assigned to a fast, primary server from a particular selection associated with this job or to a secondary server to be processed at a slower rate. Such job–server affinity relations can model network topologies based on geographical proximity, or data locality in cloud scenarios. We introduce load balancing schemes that assign jobs to primary servers if available, and otherwise to secondary servers. A novel coupling construction is developed to obtain stability conditions and performance bounds. We also conduct a fluid limit analysis for symmetric model instances, which reveals a delicate interplay between the model parameters and load balancing performance. [ABSTRACT FROM AUTHOR]

Understanding the Fundamentals of Empty-Car Routing in Ridesharing Systems How to efficiently route empty-cars in ridesharing systems? In this paper "Empty-car Routing in Ridesharing Systems," A. Braverman, J.G. Dai, X. Liu, and L. Ying introduce a novel model based on closed queueing networks and propose an optimization framework to optimize empty-car routing for maximizing system-wide utility functions. We propose a fluid-based optimal routing policy by solving the optimization problem in a large market regime. We establish both process-level and steady-state convergence of the closed queueing network to the fluid-limit and prove 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 under which the closed queueing network has a stationary distribution. This upper bound is achieved asymptotically under the fluid-based optimal routing policy. 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. [ABSTRACT FROM AUTHOR]

Abstract 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. [ABSTRACT FROM AUTHOR]

The synchronization process inherent to the Bitcoin network gives rise to an infinite-server model with the unusual feature that customers interact. Among the closed-form characteristics that we derive for this model is the busy period distribution which, counterintuitively, does not depend on the arrival rate. We explain this by exploiting the equivalence between two specific service disciplines, which is also used to derive the model's stationary distribution. Next to these closed-form results, the second major contribution concerns an asymptotic result: a fluid limit in the presence of service delays. Since fluid limits arise under scalings of the law-of-large-numbers type, they are usually deterministic, but in the setting of the model discussed in this paper the fluid limit is random (more specifically, of growth-collapse type). [ABSTRACT FROM AUTHOR]

Fractional-order epidemic models have become widely studied in the literature. Here, we consider the generalization of a simple SIR model in the context of generalized fractional calculus and we study the main features of such model. Moreover, we construct semi-Markov stochastic epidemic models by using time changed continuous time Markov chains, where the parent process is the stochastic analog of a simple SIR epidemic. In particular, we show that, differently from what happens in the classic case, the deterministic model does not coincide with the large population limit of the stochastic one. This loss of fluid limit is then stressed in terms of numerical examples.

We study a system where a random flow of customers is served by servers (called agents) invited on-demand. Each invited agent arrives into the system after a random time; after each service completion, an agent returns to the system or leaves it with some fixed probabilities. Customers and/or agents may be impatient, that is, while waiting in queue, they leave the system at a certain rate (which may be zero). We consider the queue-length-based feedback scheme, which controls the number of pending agent invitations, depending on the customer and agent queue lengths and their changes. The basic objective is to minimize both customer and agent waiting times. We establish the system process fluid limits in the asymptotic regime where the customer arrival rate goes to infinity. We use the machinery of switched linear systems and common quadratic Lyapunov functions to approach the stability of fluid limits at the desired equilibrium point and derive a variety of sufficient local stability conditions. For our model, we conjecture that local stability is in fact sufficient for global stability of fluid limits; the validity of this conjecture is supported by numerical and simulation experiments. When local stability conditions do hold, simulations show good overall performance of the scheme. [ABSTRACT FROM AUTHOR]

With the improvement of the Web service composition systems' size and complexity, the periodism and generalization of invoking requirements, the modeling and analyzing of systems become more difficult. To tackle this problem, this paper established a set of mapping rules to transform the interaction relations and factors including the processing capabilities and the load of Web servers into open queueing network. It developed a new analytic technique based on approximation theories and queueing networks, overcame the restriction of arrival processes and service processes for invoking requirements in past studies and got the fluid and diffusion limit theorems of some key performance indexes. At last, this paper analyzed the performance and its change rules of systems. The proposed results and methodology provide new highlight in the study of more general large-scale Web service composition systems. [ABSTRACT FROM AUTHOR]

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 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. [ABSTRACT FROM AUTHOR]

A parallel server system with n identical servers is considered. The service time distribution has a finite mean 1 / μ, but otherwise is arbitrary. Arriving customers are routed to one of the servers immediately upon arrival. The join-idle-queue routeing algorithm is studied, under which an arriving customer is sent to an idle server, if such is available, and to a randomly uniformly chosen server, otherwise. We consider the asymptotic regime where n → ∞ and the customer input flow rate is λn. Under the condition λ / μ < ½, we prove that, as n → ∞, the sequence of (appropriately scaled) stationary distributions concentrates at the natural equilibrium point, with the fraction of occupied servers being constant at λ / μ. In particular, this implies that the steady-state probability of an arriving customer waiting for service vanishes. [ABSTRACT FROM AUTHOR]

During the last decade we have seen an explosive development of wireless technologies. Consequently the demand for electromagnetic spectrum has been growing dramatically resulting in the spectrum scarcity problem. In spite of this, spectrum utilization measurements have shown that licensed bands are vastly underutilized while unlicensed bands are too crowded. In this context, Cognitive Radio Network emerges as an auspicious paradigm in order to solve those problems. The main question that motivates this work is: what are the possibilities offered by cognitive radio to improve the effectiveness of spectrum utilization? With this in mind, we propose a methodology, based on configuration models for random graphs, to estimate the medium access probability of secondary users. We perform simulations to illustrate the accuracy of our results and we also make a performance comparison between our estimation and one obtained by a stochastic geometry approach. [ABSTRACT FROM AUTHOR]

A service system with multiple types of customers, arriving according to Poisson processes, is considered. The system is heterogeneous in that the servers can also be of multiple types. Each customer has an independent, exponentially distributed service time, with the mean determined by its type. Multiple customers (possibly of different types) can be placed for service into one server, subject to `packing' constraints, which depend on the server type. Service times of different customers are independent, even if served simultaneously by the same server. The large-scale asymptotic regime is considered such that the customer arrival rates grow to ∞. We consider two variants of the model. For the infinite-server model, we prove asymptotic optimality of the greedy random (GRAND) algorithm in the sense of minimizing the weighted (by type) number of occupied servers in steady state. (This version of GRAND generalizes that introduced by Stolyar and Zhong (2015) for homogeneous systems, with all servers of the same type.) We then introduce a natural extension of the GRAND algorithm for finite-server systems with blocking. Assuming subcritical system load, we prove existence, uniqueness, and local stability of the large-scale system equilibrium point such that no blocking occurs. This result strongly suggests a conjecture that the steady-state blocking probability under the algorithm vanishes in the large-scale limit. [ABSTRACT FROM PUBLISHER]

We consider the asymptotics of the invariant measure for the process of the empirical spatial distribution of N coupled Markov chains in the limit of a large number of chains. Each chain reflects the stochastic evolution of one particle. The chains are coupled through the dependence of the transition rates on this spatial distribution of particles in the various states. Our model is a caricature for medium access interactions in wireless local area networks. It is also applicable to the study of spread of epidemics in a network. The limiting process satisfies a deterministic ordinary differential equation called the McKean-Vlasov equation. When this differential equation has a unique globally asymptotically stable equilibrium, the spatial distribution asymptotically concentrates on this equilibrium.More generally, its limit points are supported on a subset of the ω-limit sets of the McKean-Vlasov equation. Using a control-theoretic approach, we examine the question of large deviations of the invariant measure from this limit.

We study a mean-field approximation of the M/M/ $$\infty $$ queueing system. The problem we deal is quite different from standard games of congestion as we consider the case in which higher congestion results in smaller costs per user. This is motivated by a situation in which some TV show is broadcast so that the same cost is needed no matter how many users follow the show. Using a mean-field approximation, we show that this results in multiple equilibria of threshold type which we explicitly compute. We further derive the social optimal policy and compute the price of anarchy. We then study the game with partial information and show that by appropriate limitation of the queue-state information obtained by the players, we can obtain the same performance as when all the information is available to the players. We show that the mean-field approximation becomes tight as the workload increases, thus the results obtained for the mean-field model well approximate the discrete one. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We consider a general class of preferential attachment schemes evolving by a reinforcement rule with respect to certain sublinear weights. In these schemes, which grow a random network, the sequence of degree distributions is an object of interest which sheds light on the evolving structures. In this article, we use a fluid limit approach to prove a functional law of large numbers for the degree structure in this class, starting from a variety of initial conditions. The method appears robust and applies in particular to 'non-tree' evolutions where cycles may develop in the network. A main part of the argument is to show that there is a unique nonnegative solution to an infinite system of coupled ODEs, corresponding to a rate formulation of the law of large numbers limit, through C0-semigroup/dynamical systems methods. These results also resolve a question in Chung, Handjani and Jungreis (2003). © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 48, 703-731, 2016 [ABSTRACT FROM AUTHOR]

This paper considers the problem of scheduling real-time traffic in wireless networks. We consider ad hoc wireless networks with general conflict graph-based interference model and single-hop traffic. Each packet is associated with a deadline and will be dropped if it is not transmitted before the deadline. The number of packet arrivals in each time-slot and the maximum delay before the deadline are independent and identically distributed across time. We require a minimum fraction of packets to be delivered. At each link, we assume the link keeps track of the difference between the minimum number of packets that need to be delivered so far and the number of packets that are actually delivered, which we call the deficit. The largest-deficit-first (LDF) policy schedules links in descending order according to their deficit values, which is a variation of the longest-queue-first (LQF) policy for non-real-time traffic. We prove that the efficiency ratio of LDF, which is the fraction of the throughput region that LDF can achieve for given traffic distributions, can be lower-bounded by a quantity that we call the real-time local-pooling factor (R-LPF). We further prove that a lower bound on the R-LPF can be related to the weighted sum of the service rates, with a special case of 1/(\beta+1) by considering the uniform weight, where \beta is the interference degree of the conflict graph. We also propose a heuristic consensus algorithm that can be used to obtain a good weight vector for such lower bounds for given network topology. [ABSTRACT FROM AUTHOR]