Poisson Input Flow Research Articles

Transient Behavior of a Two-Phase Queuing System with a Limitation on the Total Queue Size

The transient mode of a two-phase queuing system with a Poisson input flow, exponential distribution of service time in each phase, and a limitation on the total buffer size of the two phases is considered. Nonstationary probabilities of system states are found using the Laplace transform. A numerical calculation and analysis of the system performance characteristics in transient mode with parameters corresponding to new-generation optical networks were carried out.

Methodology for calculating the delay time in the management system infocommunication networks based on the theory of mass service

The article is devoted to the consideration of delay time calculation methods in the M/M/m service system, the main components of the model, such as information flows and the service device, are investigated, and the delay time distribution formula based on Little’s law is improved. The importance of this indicator in determining the effectiveness of information communication systems and optimizing their functioning is noted. The article provides examples of delay time calculations and considers factors that affect this indicator, such as the workload of the telecommunications system and the characteristics of the service device. In addition, possible ways to improve the system are explored, including the use of buffers and more complex models. M/M/m is a system with m service lines, a Poisson input flow, an indicative service time distribution, and a First-Come-First-Serve (FCFS) discipline, which means that requests are served in the order in which they are received. A detailed analysis of the M/M/m model and the study of its main components allow a deeper understanding of the processes in mass service systems. The delay time in the M/M/m system is reflected in its direct impact on the quality of service and satisfaction of users of such networks

Властивості моделі обслуговування паралельної структури

The present article is devoted to research the multi-channelk model with the parallel structure. It means that we consider the model which consists of two infinite-server queues. The service time in the each system has general function of distribution. In this case the stochastic dynamic of our model cannot be defined by Markov chain. As a result, analysis of such models is much more difficult than that of the corresponding Markovian queueing models. Besides we assume that customers arrive to our model according a bivariate Poisson input flow. This input process is characterized by the fact that customers arrive according to a bivariate Poisson flow simultaneously. We consider the number of customers in the systems at time t. This stochastic process describes the state of our model. In present paper we find the limit joint distribution of the number of customers in the systems. In a general way (by differentiating the corresponding generating function.) we obtain the main characteristics of this distribution, such as the expected number of customers in the nodes, its variance and correlation. In the case when parameters of our model dependent on the parameter n (number of series) the limit normal distribution was obtained for the service process in the stationary regime.

Analysis of a system with exponential and hyper-Erlang distributions by the method of spectral decomposition of the solution the Lindley integral equation

In this work, we obtain the spectral decomposition of the solution of the Lindley integral equation for a queuing system with a Poisson input flow of requirements and a hyper-Erlang distribution of the service time. Based on it, a calculation formula is derived for the average queue waiting time for this system in a closed form. As you know, all other characteristics of the queuing systems are derived from the average waiting time. The resulting calculation formula complements and extends the well-known Polyachek-Khinchin formula in queuing theory for M/G/1 systems. In the queueing theory, studies of private systems of the M/G/1 type are relevant due to the fact that they are still actively used in the modern theory of teletraffic.

On Gaussian Approximation of Multi-Channel Networks with Input Flows of General Structure

In this paper, a multi-channel queueing network with input flow of a general structure is considered. The multi-dimensional service process is introduced as the number of customers at network nodes. In the heavy-traffic regime, a functional limit theorem of diffusion approximation type is proved under the condition that the input flows converge to their limits in the uniform topology. A limit Gaussian process is constructed and its correlation characteristics are represented explicitly via the network parameters. A network with nonhomogeneous Poisson input flow is studied as a particular case of the general model, and a correspondent Gaussian limit process is built.

СЛОТОВЫИ ALOHA С ИТЕРАЦИОННОМ ПРОЦЕДУРОЙ РАЗРЕШЕНИЯ КОЛЛИЗИЙ. СТАБИЛЬНОСТЬ И НЕСТАБИЛЬНОСТЬ

Introduction: Cellular networks of the new generation consider Massive Machine Type Communication scenarios and a class of multiple random access algorithms Slotted ALOHA with coded random access. The algorithms of this class allow you to maintain a large number of devices, but they are unstable. Their non-stability leads to a longer time of delivering a message from a subscriber to the base station during the operation of large-scale systems of inter-machine communication. Purpose: Substantiation of non-stability of Slotted ALOHA algorithms with coded random access at any intensity of the input stream; proposal of a method for its stabilization; determination of the intensity at which the system will be stable. Results: A model of a random multiple access system is introduced for coded random access and a Poisson input flow. The functioning of the model is described with the use of a Markov chain with a countable number of states. It is proved that the Markov chain is non-returnable for any non-zero intensity of the input stream. Thus, the non-stability of a multiple access system for any coded random access algorithm is proved, and a modification of these algorithms is proposed. The operation of the model for the proposed modification is described using a two-dimensional Markov chain with a countable number of states. If the intensity of the input stream does not exceed a certain value limit, the two-dimensional Markov chain is ergodic. This suggests that the proposed modification of the algorithms ensures stable operation of the system. For any algorithm from the considered class, a method is proposed for determining the numerical value of the limiting intensity of the input stream at which the system is stable. Practical relevance: The proposed modification of the algorithms can be used to develop protocols oriented to a scenario with a large number of devices, a low message rate per device and a large total input intensity in the whole system.

Engineering Approach to Calculating QoS of Server with Self-Similar Incoming Traffic Based on Recursive Scalable Poisson Model

To date the prospects for using the accumulated over many years mathematical and software for the modeling of telecommunications with the Poisson input flow are under a big question. The matter is that a new fractal queuing theory is already on the threshhold. This article formulates and solves the problem of application of a queuing system model with a Poisson incoming flow for the purposes of server modeling described by QS with self-similar incoming traffic of the "fractal Brownian motion" type (according to Norros). Based on the results of the morphological analysis, the Norros model was decomposed into Poisson components connected by a scalable recurrence scheme. The variance of the number of packets in the server, raised to the power determined by the Hurst parameter acts as the similarity coefficient of fractal and Poisson QSs. The method for rescaling Poisson solutions into fractal solutions was constructed on the basis of the similarity coefficient. According to this method in order to find the fractal delay of access, the Poisson delay should be multiplied by the similarity coefficient, and to estimate the probability of packet loss, it is necessary to extract a root of degree equal to the similarity coefficient from classical exponential losses. The scope of the re-scaling method focuses on the pre-project stages of creating telecommunications, where there is no need for high accuracy of simulation results.

Two-phase queueing system optimization in applications to data transmission control

We consider the controlled tandem queuing network with two single-server finite-length queues with non-stationary Poisson input flow of packets. The first station controls the service rate and drops any incoming packet if its buffer is full. The second station controls the packet acceptance probability. The tandem is described by a controlled finite-state Markov process and optimized on a fixed time interval by minimizing average number of dropped packets with constraints on the average sojourn time and energy consumption.

Gaussian and Diffusion Limits for Multi-Channel Stochastic Networks

In the paper the multichannel stochastic networks are considered. A non-homogeneous Poisson input flow of calls arrives at each node. An approximate method of investigation of the service process in heavy traffic is developed. For the multichannel Markovian networks the limit process is represented as a multidimensional diffusion.

Queueing System with Preemptive Resume Service Discipline and Unreliable Server*

We consider a single server queue with an unreliable server and two Poisson input flows. For this system we find stability conditions and obtain the limit distribution of the number of customers in the system.

Limit theorems for an infinite-server queuing system

We consider an infinite-server queuing system with a doubly stochastic Poisson input flow. Assuming that the service time does not have expectation, we prove limit theorems for the number of occupied servers. As a consequence, we obtain limit theorems for systems in which the input flow intensity is a regenerative process.

Stationary characteristics of M 2|G|1|r system with hysteretic policy for arrival rate control

A finite queuing system with one server, Poisson input flow, arbitrary service time distribution, and hysteretic policy for arrival rate control is considered. An analytical method is proposed for determination of the stationary distribution of the number of customers in the system. Several numerical examples are given.

B.A. Sevastyanov’s famous theorem

In 1956, at the Third All-Union Mathematical Congress, Boris Aleksandrovich Sevastyanov gave a talk on the ergodic theorem proved by him for Markov processes and on its application to queueing systems. In 1957, this result was published in the journal Teoriya Veroyatnostei i Ee Primeneniya (Theory of Probability and Its Applications). An important corollary to the ergodic theorem is a generalization of Erlang’s well-known formula to a queueing system with a Poisson input flow and an arbitrary distribution of the service time. This result of Sevastyanov has served as a starting point for numerous studies on the problem, which was later called the insensitivity (invariance) problem for queueing systems with losses. There are hundreds of references to this result of Sevastyanov.

Configuration of overloaded servers with dynamic routing

We consider overload of servers in a network with dynamic routing of messages. The system consists of k servers and ? independent Poisson input flows. Messages from each flow are directed to m servers, and each message is directed to a server that is the least loaded at the moment of its arrival. In such a system, configuration of overloaded servers depends on the intensity of input flows. A similar effect was considered in [1] for a system with another geometry.

Product Theorem in a Jackson Network with Switching Centers

1. Classical Jackson Product Theorem A network G is called a classical Jackson network if it has Poisson input flow with intensity λ and m multi-server centers. Suppose that the ith center has ri ≤ ∞ servers (each server has exponentially distributed serving times with intensity μi) and allows an infinite queue, i = 1, 2, . . . ,m. Take an external source as a center with index 0. Determine a customer route in the network G by the route matrix Θ = (θij)i,j=0, where θij is the probability of transiting from the ith serving center to the jth serving center, θ00 = 0. Suppose that the route matrix is indivisible: ∀i, j ∈ {0, 1, . . . ,m} ∃i1, i2, . . . , ir ∈ {0, 1, ...,m}: θii1 > 0, θi1i2 > 0, . . . , θirj > 0. Thus, there is a single solution (λ, λ1, λ2, . . . , λm) of the system λ = λ, (λ, λ1, λ2, . . . , λm) = (λ, λ1, λ2, . . . , λm)Θ. Separate in the network G the ith serving center and consider it as an isolated ri-server queuing system with Poisson input flow which has intensity λi. Describe the number of customers in this system by a birth-and-death process ni(t). If ρi = λi riμi < 1, (1)

Limit Theorems for One Class of Polling Models

The asymptotic independence of any finite set of random processes describing states of nodes is proved for an open transportation network with N nodes and M servers moving between them provided $N\to\infty$ and $M/N\to r$ ($0 < r <\infty$). We determine that the limit flow of servers to a fixed node is a nonstationary Poisson process and describe behavior of the system in the thermodynamic limit. A system with Poisson input flow and a special kind of travel time distribution is examined as an example of such networks. The convergence to a limit dynamic system is proved for it. Also the steady point is specified and it is discovered that this point depends on the shape of travel time distribution only through its mean.

On the Invariance of Stationary State Probabilities of a Non-Product-Form Single-Line Queueing System

We consider a single-line queueing system (QS) with Poisson input flow of varying intensity, which depends on the number of demands in the system. The job size (length) distribution for a demand depends on the number of demands in the system at the arrival moment. The service rate also depends on the number of calls in the QS. If the job size for a new arrival is larger than the remaining job size for the currently processed demand, then the arrival is put at the beginning of the queue with a certain probability, which depends on the total number of demands in the system. Otherwise, it occupies the server and displaces the currently processed demand, which is put at the beginning of the queue. The probability distribution of stationary states of the QS is found and necessary and sufficient conditions for this distribution to be invariant with respect to the job size distribution with a fixed mean are obtained.

An M/D/1 Queueing System with Partial Synchronization of Its Incoming Flow and Demands Repeating at Constant Intervals

A one-channel queueing system with Poisson input flow, a constant service time, and a constant time of being in orbit is considered. The system admits of synchronization of its input flow. Some indicators of efficiency of operation of the system are deduced and numerical calculations are presented.

Asymptotical properties of erlang queueing systems with a dependent Poisson input flow

The purpose of this paper is to obtain stationary characteristics of the mathematical model, which consists of two Erlang queueing systems with a bivariate Poisson input flow of customers. Functional theorems of diffusion approximation type are proved.

Averaging methods for transient regimes in overloading retrial queueing systems

A new approach is suggested to study transient and stable regimes in overloading retrial queueing systems. This approach is based on limit theorems of averaging principle and diffusion approximation types for so-called switching processes. Two models of retrial queueing systems of the types M ̄ / G ̄ / 1 ̄ /w.r (multidimensional Poisson input flow, one server with general service times, retrial system) and M/M/m/w.r ( m servers with exponential service) are considered in the case when the intensity of calls that reapply for the service tends to zero. For the number of re-applying calls, functional limit theorems of averaging principle and diffusion approximation types are proved.