Tail asymptotics for a Lévy-driven tandem queue with an intermediate input

This article examines an system with catastrophic server failures and repairs that feeds a fluid queue. The buffer content's stationary distribution function is expressed explicitly in terms of confluent queueing hyper geometric functions. The expression for the joint probability of the buffer's content and the number of customers in an queueing system has been obtained using continued fraction techniques. The stationary analyses of fluid flow queueing models in a stochastic environment can be effectively scrutinized and applied in the modern telecommunication systems and various advanced computer communication networks. Especially, the fluid queues are used in the performance analyses of data packet transmission networks, traffic flows at high-speed computer networks, and rechargeable batteries in smart mobile phones. A few key performance indicators are acquired. Finally, the numerical work is shown to show how the buffer content distribution behaves and how the performance metrics for the suggested fluid queueing system work.

We establish mean-field limits for large-scale random-access networks with buffer dynamics and arbitrary interference graphs. Although saturated buffer scenarios have been widely investigated and yield useful throughput estimates for persistent sessions, they fail to capture the fluctuations in buffer contents over time and provide no insight in the delay performance of flows with intermittent packet arrivals. Motivated by that issue, we explore in the present paper random-access networks with buffer dynamics, where flows with empty buffers refrain from competition for the medium. The occurrence of empty buffers thus results in a complex dynamic interaction between activity states and buffer contents, which severely complicates the performance analysis. Hence, we focus on a many-sources regime where the total number of nodes grows large, which not only offers mathematical tractability but is also highly relevant with the densification of wireless networks as the Internet of Things emerges. We exploit timescale separation properties to prove that the properly scaled buffer occupancy process converges to the solution of a deterministic initial value problem and establish the existence and uniqueness of the associated fixed point. This approach simplifies the performance analysis of networks with huge numbers of nodes to a low-dimensional fixed-point calculation. For the case of a complete interference graph, we demonstrate asymptotic stability, provide a simple closed form expression for the fixed point, and prove interchange of the mean-field and steady-state limits. This yields asymptotically exact approximations for key performance metrics, in particular the stationary buffer content and packet delay distributions.

We consider a tandem fluid system composed of multiple buffers connected in a series. The first buffer receives input from a number of independent homogeneous on-off sources and each buffer provides input to the next buffer. The active (on) periods and silent (off) periods follow general and exponential distribution, respectively. Furthermore, the generally distributed active periods are controlled by an exponential timer. Under this assumption, explicit expressions for the distribution of the buffer content for the first buffer fed by a single source is obtained for the fluid queue driven by discouraged arrivals queue and infinite server queue. The buffer content distribution of the subsequent buffers when the first buffer is fed by multiple sources are found in terms of confluent hypergeometric functions. Numerical results are illustrated to compare the trend of the average buffer content for the models under consideration.

Heavy-tailed asymptotics for a fluid model driven by an [formula omitted] queue

In this work we compute the exact tail asymptotics of the stationary sojourn time Z, associated to the discretetime tandem queue D/G/1/∞ → ∙/G/1/∞, with constant release rate, infinite buffer capacity, and with service time sequences generated by the busy server processes of the two M 1/G 1/∞ and M 2/G 2/∞independent queues exhibiting long-range dependence. We choose the same regularly varying distribution with parameter α > 1 for the general distributions G 1 and G 2. We show that the exact asymptotics of the sojourn time is a specific regularly varying function under some assumptions on the parameters.

We consider two parallel, infinite capacity, M/G/1 queues characterized by ( U_{1}(t), U_{2}(t) ) with U_{j}(t) denoting the unfinished work (buffer content) in queue j . A new arrival is assigned to the queue with the smaller buffer content. We construct formal (as opposed to rigorous) asymptotic approximations to the Joint stationary distribution of the Markov process ( U_{1}(t), U_{2}(t) ), treating separately the asymptotic limits of heavy traffic, light traffic, and large buffer contents. In heavy traffic, the stochastic processes U_{1}(t) + U_{2}(t) and U_{2}(t) - U_{1}(t) become independent, with the distribution of U_{1}(t) + U_{2}(t) identical to the heavy traffic waiting time distribution in the standard M/G/2 queue, and the distribution of U_{2}(t) - U_{1}(t) closely related to the tail of the service time density. In light traffic, we obtain a formal expansion of the stationary distribution in powers of the arrival rate.

A communication system of n buffered input terminals, numbered 1,2,3,..., n, connected to a computer by a single channel, is analyzed. The n terminals are serviced according to a SCAN policy: the terminals are polled in sequence 1,2,3,..., n, n, n-1,...,2,1. Data are removed one unit at a time from the terminals’ buffer. Data arriving to the terminal during this time are stored in an auxilliary buffer and are served in the next poll to the terminal. When the buffer has been emptied, the channel is used for system overhead for a randomly determined length of time. The stationary distribution of buffer contents, assumed to exist if the total normalized arrival rate is less than one, is used to find the average buffer contents and the average intervisit time. General, independent arrival rates are assumed for the n terminals. The analysis follows that in Swartz [2].

Distributed algorithms such as CSMA provide a popular mechanism for sharing the transmission medium among competing users in large-scale wireless networks. Conventional models for CSMA that are amenable for analysis assume that users always have packets to transmit. In contrast, when users do not compete for medium access when their buffers are empty, a complex interaction arises between the activity states and the buffer contents. We develop a meanfield approach to investigate this dynamic interaction for networks with many users. We identify a time-scale separation between the evolution of the activity states and the buffer contents, and obtain a deterministic dynamical system describing the network dynamics on a macroscopic scale. The fixed point of the dynamical system yields highly accurate approximations for the stationary distribution of the buffer contents and packet delay, even when the number of users is relatively moderate.

On the reversibility of manufacturing networks

short-paper Share on Tail asymptotics of the stationary distribution for a two-node generalized Jackson network Author: Masakiyo Miyazawa Tokyo University of Science, Noda, Chiba, Japan Tokyo University of Science, Noda, Chiba, JapanView Profile Authors Info & Claims ACM SIGMETRICS Performance Evaluation ReviewVolume 42Issue 2September 2014pp 70–72https://doi.org/10.1145/2667522.2667545Published:04 September 2014Publication History 1citation26DownloadsMetricsTotal Citations1Total Downloads26Last 12 Months0Last 6 weeks0 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteGet Access

We consider a tandem queue with coupled processors, which is subject to global breakdowns. When the network is in the operating mode and both queues are non empty, the total service capacity is shared among the stations according to fixed proportions. When one of the stations becomes empty, the total service capacity is assigned to the non-empty station. Moreover, arrival rates depend on the state of the network. The system is described by a Markov modulated random walk in the quarter plane representing the number of jobs in the two stations, and the state of the network. We first apply the power series approximation method to obtain power series expansions of the generating function of the stationary joint queue length distribution for both network states. We also provide a way to derive the generating function of the stationary joint queue length distribution for both network states in terms of the solution of a Riemann-Hilbert boundary value problem. Numerical results are obtained to show insights in the system performance.

A Fluid queue is a mathematical model used to describe fluid level in a reservoir subject to randomly determined periods of filling and emptying the system without interruption called a buffer, according to a randomly varying rate regulated by an external stochastic environment. Such fluid queues are used as a mathematical tool for modeling, for example, to approximate discrete models, model the spread of wildfires in ruin theory and to model high speed data networks, a router, computer networks including call admission control, traffic shaping and modeling of TCP and production and inventory systems. The fluid from the first phase (i.e, fluid output of the queue) goes to the second phase represents a buffer with a constant leak rate c. We always assume that service rate is greater than buffer, μ ˃ c. In this paper, a fluid queue driven by infinite queue with fixed –size batch arrivals. The generating function technique is used to obtain the expressions for steady-state distribution of both the buffer content and stationary state probabilities of background birth-death process. Hence, the performance measures are computed whereas the server utilization analysis and mean buffer content are investigated. . In addition, some numerical results are provided to illustrate the effect of various parameters on the distribution of the buffer content.

ABSTRACTWe consider a model consisting of two fluid queues driven by the same background continuous-time Markov chain, such that the rates of change of the fluid in the second queue depend on whether the first queue is empty or not: when the first queue is nonempty, the content of the second queue increases, and when the first queue is empty, the content of the second queue decreases.We analyze the stationary distribution of this tandem model using operator-analytic methods. The various densities (or Laplace–Stieltjes transforms thereof) and probability masses involved in this stationary distribution are expressed in terms of the stationary distribution of some embedded process. To find the latter from the (known) transition kernel, we propose a numerical procedure based on discretization and truncation. For some examples we show the method works well, although its performance is clearly affected by the quality of these approximations, both in terms of accuracy and run time.

This paper presents the large-buffer asymptotics for a multiplexer which serves N types of heterogeneous sessions which have long-tailed session lengths. Specifically, the model considered is that sessions of type i ∈ {1,…,N} arrive as a Poisson process with rate λ i . Each type of session (independently) remains active for a random duration, say τ i , where P(τ i > x) ~ α i x -(1 + β i ) for positive numbers α i and β i . While active, a session transmits at a rate r i . Under the assumption that the average load ρ = ∑ N i=1 r i λ i E[τ i ] < C, where C denotes the server capacity, we show that both the tail distribution of the stationary buffer content and the loss asymptotics in finite buffers of size z behave approximately as z -κ J 0 , where κ J 0 depends not only on the β i but also on the transmission rates r i ; it is the ratio of β i to r i which determines κ J 0 . When specialized to the homogeneous case, i.e., when r i =r and β i = β for all i, the result coincides with results reported in the literature which have been shown under more restrictive hypotheses. Finally, it is a simple observation that light-tailed sessions only have the effect of reducing the available capacity for long-tailed sessions, but do not contribute otherwise to the definition of κ J 0 .

BackgroundMolecular evolution is usually described assuming a neutral or weakly non-neutral substitution model. Recently, new data have become available on evolution of sequence regions under a selective pressure, e.g. transcription factor binding sites. To reconstruct the evolutionary history of such sequences, one needs evolutionary models that take into account a substantial constant selective pressure.ResultsWe present a simple evolutionary model with a single preferred (consensus) nucleotide and the neutral substitution model adopted for all other nucleotides. This evolutionary model has a rate matrix in which all substitutions that do not involve the consensus nucleotide occur with the same rate. The model has two time scales for achieving a stationary distribution; in the general case only one of the two rate parameters can be evaluated from the stationary distribution. In the middle-time zone, a counterintuitive behavior was observed for some parameter values, with a probability of conservation for a non-consensus nucleotide greater than that for the consensus nucleotide. Such an effect can be observed only in the case of weak preference for the consensus nucleotide, when the probability to observe the consensus nucleotide in the stationary distribution is less than 1/2. If the substitution rate is represented as a product of mutation and fixation, only the fixation can be calculated from the stationary distribution. The exhibited conservation of non-consensus nucleotides does not take place if the elements of mutation matrix are identical, and can be related to the reduced mutation rate between the non-consensus nucleotides. This bias can have no effect on the stationary distribution of nucleotide frequencies calculated over the ensemble of multiple alignments, e.g. transcription factor binding sites upstream of different sets of co-regulated orthologous genes.ConclusionThe derived model can be used as a null model when analyzing the evolution of orthologous transcription factor binding sites. In particular, our findings show that a nucleotide preferred at some position of a multiple alignment of binding sites for some transcription factor in the same genome is not necessarily the most conserved nucleotide in an alignment of orthologous sites from different species. However, this effect can take place only in the case of a mutation matrix whose elements are not identical.