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

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.

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 .

We consider a Levy-driven tandem queue with an intermediate input assuming that its buffer content process obtained by a reflection mapping has the stationary distribution. For this queue, no closed form formula is known, not only for its distribution but also for the corresponding transform. In this paper, we consider only light-tailed inputs. For the Brownian input case, we derive exact tail asymptotics for the marginal stationary distribution of the second buffer content, while weaker asymptotic results are obtained for the general Levy input case. The results generalize those of Lieshout and Mandjes from the recent papers (Lieshout and Mandjes in Math. Methods Oper. Res. 66:275---298, 2007 and Queueing Syst. 60:203---226, 2008) for the corresponding tandem queue without an intermediate input.

This paper studies a fluid queueing model driven by an M / M / 1 queue subject to working vacation. The underlying system of differential difference equations that governs the process is solved using continued fraction and generating function methodologies. Explicit expressions for the joint steady-state probabilities of the state of the background queueing model and the content of the buffer are obtained in terms of modified Bessel function of the first kind. Numerical illustrations are added to depict the variations of the state probabilities and the buffer content distributions against varying values of the parameters subject to the stability conditions.

The existence of maximal and minimal positive solutions for singular infinite-point p-Laplacian fractional differential equation is investigated in this paper. Green's function is derived, and some properties of Green's function are obtained. Based upon these properties of Green's function, the existence of maximal and minimal positive solutions is obtained, and iterative schemes are established for approximating the maximal and minimal positive solutions.

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.

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].

More study has been performed on the steady state analysis of fluid models. However, the transient analysis of fluid models has not been extensively carried due to complexity of the problem. This is because the system of conservation laws for which no closed form solution is available. The motivation behind this paper is to provide a new methodology to find transient distribution of buffer content for fluid models. In this paper, the time-dependent solution of a fluid model driven by an M/M/1 queue is derived using a simple probability approach. Finally, numerical results are illustration for the proposed approach.

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.

The King–Werner method for solving nonsymmetric algebraic Riccati equation

In the Covid-19 pandemic process that broke out in Wuhan city, China and affected the whole world, the active use of social media by healthcare professionals is considered as significantly important in preventing information pollution and transferring scientific knowledge to the society. In this context, this study, which examined the 10 instagram account of health scientists in Turkey, aims to reveal the message format usage preferences of healthcare professional opinion leaders, distribution of themed message using rates and preferences based on the type of sharing. Moreover, in scope of this study which is based on user content collected using the Instagram application (API) between March 11, 2020 and May 1, 2020, the structure of the visual used concerning the visual content (inactive-animated), the distribution of the animated content based on the structure of the visual and the time distributions were examined. In the verbal qualities related to sharing, the theme of the messages in terms of sharing content, the message format (rational-emotional orientation) coding units were formed at a nominal scale. According to the findings of the research, it is seen that healthcare professional opinion leaders prefer to use the rationally oriented message format more. In the posts shared by the " kayihanpala" and "akinkaya4677" named Instagram accounts, chosen among the healthcare professionals examined within the framework of the research unit, the rates of 70% and 66.7%, respectively, stand out in the use of emotionally oriented messages. In the Instagram account named “dr.fahrettinkoca”, which defines the institutional representation of the Minister of Health, the usage rate of rationally oriented messages and emotionally oriented messages can be considered equal to each other. When the data obtained regarding the use of themed messages and the data on message formats are evaluated, it is seen that the "dr.fahrettinkoca", "kayihanpala", and "akinkaya4677", named accounts, where emotionally oriented message sharing occurs at high rates, also focus on themed messages for the measures to be taken in the process. When the posts shared by the healthcare professional opinion leaders are evaluated regarding format, it is seen that the inactive sharing type is preferred. In the presentation of animated visuals, based on real life animated image sharing patterns is preferred. It is one of the findings that IGTV posts longer than 1 minute come to the fore in posts where animation, videography and photo clip use are not preferred in content design.

Positive maximal and minimal solutions for non-homogeneous elliptic equations depending on the gradient

On the reversibility of manufacturing networks

A numerical solution of the flow equation for water in soil was obtained for a flow system consisting of a vertical column of soil which had been drained from saturation to equilibrium with a water table. A thin layer of ponded water was assumed to be applied to the top end of the column. This meant that each point in the column wetted up along a different scanning curve of the moisture characteristic curves. The solution of the equation depicts the time and depth distributions of water content and pressure head during the resulting infiltration. The results show that for soils that have water content‐pressure head curves with regions of steep slopes, the wetting front advances as a steep, well‐defined front. In those soils with more gently sloping water content‐pressure head curves, the wetting front was more diffuse. If hysteresis was ignored, the position of the front at any one time was overestimated or underestimated—depending upon whether the drainage or wetting curve was used.