Bounds on the rate on convergence for Markovian queueing models with catastrophes

Transitory Queueing Networks

Queueing networks are notoriously difficult to analyze sans both Markovian and stationarity assumptions. Much of the theoretical contribution towards performance analysis of time-inhomogeneous single class queueing networks has focused on Markovian networks, with the recent exception of work in Liu and Whitt (2011) and Mandelbaum and Ramanan (2010). In this paper, we introduce transitory queueing networks as a model of inhomogeneous queueing networks, where a large, but finite, number of jobs arrive at queues in the network over a fixed time horizon. The queues offer FIFO service, and we assum…

Corrected phase-type approximations of heavy-tailed queueing models in a Markovian environment

We develop accurate approximations of the delay distribution of the MArP/G/1 queue that cap- ture the exact tail behavior and provide bounded relative errors. Motivated by statistical analysis, we consider the service times as a mixture of a phase-type and a heavy-tailed distribution. With the aid of perturbation analysis, we derive corrected phase-type approximations as a sum of the delay in an MArP/PH/1 queue and a heavy-tailed component depending on the perturbation parameter. We exhibit their performance with numerical examples.

Corrected phase-type approximations of heavy-tailed queueing models in a Markovian environment

Significant correlations between arrivals of load-generating events make the numerical evaluation of the workload of a system a challenging problem. In this paper, we construct highly accurate approximations of the workload distribution of the MAP/G/1 queue that capture the tail behavior of the exact workload distribution and provide a bounded relative error. Motivated by statistical analysis, we consider the service times as a mixture of a phase-type and a heavy-tailed distribution. With the aid of perturbation analysis, we derive our approximations as a sum of the workload distribution of th…

On the ergodicity bounds for a constant retrial rate queueing model

We consider a Markovian single-server retrial queueing system with a constant retrial rate. Conditions of null ergodicity and exponential ergodicity for the correspondent process, as well as bounds on the rate of convergence are obtained.

Stochastic Monotonicity of Markovian Multi-class Queueing Networks

Multi-class queueing networks (McQNs) extend the classical concept of Jackson network by allowing jobs of different classes to visit the same server. While such a generalization seems rather natural, from a structural perspective there is a significant gap between the two concepts. Nice analytical features of Jackson networks, such as stability conditions, product-form equilibrium distributions, and stochastic monotonicity do not immediately carry over to the multi-class framework. The aim of this paper is to shed some light on this structural gap, focusing on monotonicity properties. To this …

具有随机重置的排队模型

Queueing models with random resetting

本文研究了具有随机重置的排队模型，通过新的方法推导了平稳分布并改进了现有分析。

Robust bounds and optimization at the large deviations scale for queueing models via Rényi divergence

This paper develops tools to obtain robust probabilistic estimates for queueing models at the large deviations (LD) scale. These tools are based on the recently introduced robust Rényi bounds, which provide LD estimates (and more generally risk-sensitive (RS) cost estimates) that hold uniformly over an uncertainty class of models, provided that the class is defined in terms of Rényi divergence with respect to a reference model and that estimates are available for the reference model. One very attractive quality of the approach is that the class to which the estimates apply may consist of hard …

Tail asymptotics for cumulative processes sampled at heavy-tailed random times with applications to queueing models in Markovian environments

This paper considers the tail asymptotics for a cumulative process $\{B(t); t \ge 0\}$ sampled at a heavy-tailed random time $T$. The main contribution of this paper is to establish several sufficient conditions for the asymptotic equality ${\sf P}(B(T) > bx) \sim {\sf P}(M(T) > bx) \sim {\sf P}(T>x)$ as $x \to \infty$, where $M(t) = \sup_{0 \le u \le t}B(u)$ and $b$ is a certain positive constant. The main results of this paper can be used to obtain the subexponential asymptotics for various queueing models in Markovian environments. As an example, using the main results, we derive subexponen…

Analysis of an infinite-buffer batch-size-dependent service queue with discrete-time Markovian arrival process: D-$MAP/G_n^{(a,b)}/1$

Discrete-time queueing models find huge applications as they are used in modeling queueing systems arising in digital platforms like telecommunication systems, computer networks, etc. In this paper, we analyze an infinite-buffer queueing model with discrete Markovian arrival process. The units on arrival are served in batches by a single server according to the general bulk-service rule, and the service time follows general distribution with service rate depending on the size of the batch being served. We mathematically formulate the model using the supplementary variable technique and obtain …

Non-Markovian state dependent networks in critical loading

We establish heavy traffic limit theorems for queue-length processes in critically loaded single class queueing networks with state dependent arrival and service rates. A distinguishing feature of our model is non-Markovian state dependence. The limit stochastic process is a continuous-path reflected process on the nonnegative orthant. We give an application to generalised Jackson networks with state-dependent rates.

Performance analysis of Markovian queue with impatience and vacation times

We consider an $M/M/1$ queueing system with impatient customers with multiple and single vacations. It is assumed that customers are impatient whenever the state of the server. We derive the probability generating functions of the number of customers in the system and we obtain some performance measures.

Covert queueing problem with a Markovian statistic

Based on the covert communication framework, we consider a covert queueing problem that has a Markovian statistic. Willie jobs arrive according to a Poisson process and require service from server Bob. Bob does not have a queue for jobs to wait and hence when the server is busy, arriving Willie jobs are lost. Willie and Bob enter a contract under which Bob should only serve Willie jobs. As part of the usage statistic, for a sequence of N consecutive jobs that arrived, Bob informs Willie whether each job was served or lost (this is the Markovian statistic). Bob is assumed to be violating the co…

A sufficient condition for the subexponential asymptotics of GI/G/1-type Markov chains with queueing applications

The main contribution of this paper is to present a new sufficient condition for the subexponential asymptotics of the stationary distribution of a GI/GI/1-type Markov chain without jumps from level "infinity" to level zero. For simplicity, we call such Markov chains {\it GI/GI/1-type Markov chains without disasters} because they are often used to analyze semi-Markovian queues without "disasters", which are negative customers who remove all the customers in the system (including themselves) on their arrivals. In this paper, we demonstrate the application of our main result to the stationary qu…

基于欧拉近似的马尔可夫排队网络快速离散事件仿真

Fast Discrete-Event Simulation of Markovian Queueing Networks through Euler Approximation

本文针对大规模马尔可夫排队网络提出基于欧拉近似的后向与前向两种快速仿真方案，在保证随机上下界和误差有界的同时实现数万倍加速，显著降低计算复杂度。

A rate balance principle and its application to queueing models

We introduce a rate balance principle for general (not necessarily Markovian) stochastic processes. Special attention is given to processes with birth and death like transitions, for which it is shown that for any state $i$, the rate of two consecutive transitions from $i-1$ to $i+1$, coincides with the corresponding rate from $i+1$ to $i-1$. This observation appears to be useful in deriving well-known, as well as new, results for the Mn/Gn/1 and G/Mn/1 queueing systems, such as a recursion on the conditional distributions of the residual service times (in the former model) and of the residual…

Two-sided bounds on the rate of convergence for continuous-time finite inhomogeneous Markov chains

We suggest an approach to obtaining general two-sided bounds on the rate of convergence in terms of special "weighted" norms related to total variation. Some important classes of continuous-time Markov chains are considered: birth-death-catastrophes processes, queueing models with batch arrivals and group services, chains with absorption in zero.

A Nonlinear Solution to Closed Queueing Networks for Bike Sharing Systems with Markovian Arrival Processes and under an Irreducible Path Graph

As a favorite urban public transport mode, the bike sharing system is a large-scale and complicated system, and there exists a key requirement that a user and a bike should be matched sufficiently in time. Such matched behavior makes analysis of the bike sharing systems more difficult and challenging. This paper considers a more general large-scale bike sharing system from two important views: (a) Bikes move in an irreducible path graph, which is related to geographical structure of the bike sharing system; and (b) Markovian arrival processes (MAPs) are applied to describe the non-Poisson and …

马尔可夫负载均衡排队网络在平衡状态下的完美模拟

Perfect simulation of Markovian load balancing queueing networks in equilibrium

本文针对马尔可夫负载均衡排队网络平衡态模拟问题，提出三种基于支配过程和预序耦合的完美模拟算法，解决了状态空间无限且不变分布未知情况下的蒙特卡洛估计难题。

带弃单的马尔可夫多类多服务器队列的遍历风险敏感控制

Ergodic Risk Sensitive Control of Markovian Multiclass Many-Server Queues with Abandonment

本文研究 Halfin--Whitt 机制下带弃单的多类多服务器队列的遍历风险敏感最优调度问题，通过引入辅助控制并利用变分表示，证明了扩散尺度化排队过程的最优 ERSC 值渐近收敛到极限扩散的最优值。

基于学习的非再生到达过程在排队网络中的叠加算子

A Learning-Based Superposition Operator for Non-Renewal Arrival Processes in Queueing Networks

本文提出了一种基于深度学习的叠加算子，用于处理非再生到达过程在排队网络中的叠加问题，通过合成数据训练模型，实现对合并流的矩和依赖结构的准确重建，并在不同变异性及相关性条件下优于传统方法。

Hitting probabilities in a Markov additive process with linear movements and upward jumps: applications to risk and queueing processes

Motivated by a risk process with positive and negative premium rates, we consider a real-valued Markov additive process with finitely many background states. This additive process linearly increases or decreases while the background state is unchanged, and may have upward jumps at the transition instants of the background state. It is known that the hitting probabilities of this additive process at lower levels have a matrix exponential form. We here study the hitting probabilities at upper levels, which do not have a matrix exponential form in general. These probabilities give the ruin probab…

On limiting characteristics for a non-stationary two-processor heterogeneous system

In this paper we study a non-stationary Markovian queueing model of a two-processor heterogeneous system with time-varying arrival and service rates. We obtain the bounds on the rate of convergence and find the main limiting characteristics of the queue-length process.

Stability of multidimensional skip-free Markov modulated reflecting random walks: Revisit to Malyshev and Menshikov's results and application to queueing networks

Let $\{\boldsymbol{X}_n\}$ be a discrete-time $d$-dimensional process on $\mathbb{Z}_+^d$ with a supplemental (background) process $\{J_n\}$ on a finite set and assume the joint process $\{\boldsymbol{Y}_n\}=\{(\boldsymbol{X}_n,J_n)\}$ to be Markovian. Then, the process $\{\boldsymbol{X}_n\}$ can be regarded as a kind of reflecting random walk (RRW for short) in which the transition probabilities of the RRW are modulated according to the state of the background process $\{J_n\}$; we assume this modulation is space-homogeneous inside $\mathbb{Z}_+^d$ and on each boundary face of $\mathbb{Z}_+^d…

扩展的开放马尔可夫Jackson网络：流极限和无限维的斯科罗霍德问题

Growing open Markovian Jackson networks: Fluid limit and infinite-dimensional Skorokhod problem

本文研究了扩展的开放Jackson网络，其中每个站点是一个遵循先进先出服务纪律、具有泊松到达和指数服务时间的单服务器队列，其特征由节点特定的速率决定。通过应用流标度到队列长度过程，证明在某些条件下，排队系统可以被无限维的流极限近似，该极限过程由一个核函数代替转移矩阵。该极限过程可以由一个无限维的斯科罗霍德问题来描述，我们通过考虑更广泛的反射算子和一般无限维过程发展了新的理论。我们建立了解的存在性和唯一性以及Lipschitz连续性，前提是反射算子的谱半径小于1。通过引入一个中间过程，其中补偿泊松成分被移除，然后将其提升为无限维过程，我们利用无限维斯科罗霍德映射的新Lipschitz性质来证明中间过程的收敛性。然后通过使用鞅性质证明原始过程和中间过程之间的差值估计。最后，我们考虑排队过程的经验测度，证明其收敛到与无限维流极限路径相关的测度，从而扩展到特定性能相关函数的收敛性。

Stationary queue and server content distribution of a batch-size-dependent service queue with batch Markovian arrival process: BMAP/G^{(a,b)}_n/1

Queueing systems with batch Markovian arrival process (BMAP) have paramount applications in the domain of wireless communication. The BMAP has been used to model the superposition of video sources and to approximate the super-position of data, voice and video traffic. This paper analyzes an infinite-buffer generally distributed batch-service queue with BMAP, general bulk service ($a,b$) rule and batch-size-dependent service time. In this proposed analysis, we mainly focus on deriving the bivariate vector generating function of queue and server content distribution together at departure epoch u…

Asymptotic Analysis for Markovian Queues with Two Types of Nonpersistent Retrial Customers

We consider Markovian multiserver retrial queues where a blocked customer has two opportunities for abandonment: at the moment of blocking or at the departure epoch from the orbit. In this queueing system, the number of customers in the system (servers and buffer) and that in the orbit form a level-dependent quasi-birth-and-death (QBD) process whose stationary distribution is expressed in terms of a sequence of rate matrices. Using a simple perturbation technique and a matrix analytic method, we derive Taylor series expansion for nonzero elements of the rate matrices with respect to the number…

On the BMAP_1, BMAP_2/PH/g, c retrial queueing system

In this paper, we analyze a retrial queueing system with Batch Markovian Arrival Processes and two types of customers. The rate of individual repeated attempts from the orbit is modulated according to a Markov Modulated Poisson Process. Using the theory of multi-dimensional asymptotically quasi-Toeplitz Markov chain, we obtain the stability condition and the algorithm for calculating the stationary state distribution of the system. Main performance measures are presented. Furthermore, we investigate some optimization problems. The algorithm for determining the optimal number of guard servers a…

Stationary analysis of certain Markov-modulated reflected random walks in the quarter plane

In this work, we focus on the stationary analysis of a specific class of continuous time Markov-modulated reflected random walks in the quarter plane with applications in the modelling of two-node Markov-modulated queueing networks with coupled queues. The transition rates of the two-dimensional process depend on the state of a finite state Markovian background process. Such a modulation is space homogeneous in the set of inner states of the two-dimensional lattice but may be different in the set of states at its boundaries. To obtain the stationary distribution, we apply the power series appr…

关于在Halfin-Whitt regime中马尔可夫多类多服务器队列的统一指数遍历性的研究

On uniform exponential ergodicity of Markovian multiclass many-server queues in the Halfin-Whitt regime

本文研究了在Halfin-Whitt regime中重载的马尔可夫多类多服务器队列的遍历性质，通过Lyapunov函数方法统一建立了极限扩散和预极限扩散缩放队列过程的Foster-Lyapunov方程，证明了当备用容量参数为正时，扩散过程在所有平稳马尔可夫控制下均匀指数遍历，并且不变概率测度具有均匀指数尾部特性。

Ergodicity Bounds for the Markovian Queue With Time-Varying Transition Intensities, Batch Arrivals and One Queue Skipping Policy

In this paper we revisit the Markovian queueing system with a single server, infinite capacity queue and the special queue skipping policy. Customers arrive in batches, but are served one by one according to any conservative discipline. The size of the arriving batch becomes known upon its arrival and at any time instant the total number of customers in the system is also known. According to the adopted queue skipping policy if a batch, which size is greater than the current system size, arrives to the system, all current customers in the system are removed from it and the new batch is placed …

Processus aléatoires et applications

Lecture notes for a master-level applied mathematics course on stochastic processes and applications, held at the University of Orléans, France. Contents: Markov chains, Poisson point processes, Markovian jump processes, queueing theory. ----- Notes d'un cours de Master 2 professionnel en mathématiques appliquées, donné à l'Université d'Orléans. Contenu: Chaînes de Markov, processus ponctuel de Poisson, files d'attente.

Markovian Transition Counting Processes: An Alternative to Markov Modulated Poisson Processes

Stochastic models for performance analysis, optimization and control of queues hinge on a multitude of alternatives for input point processes. In case of bursty traffic, one very popular model is the \textit{Markov Modulated Poisson Process} (MMPP), however it is not the only option. Here, we introduce an alternative that we call \textit{Markovian transition counting process} (MTCP). The latter is a point process counting the number of transitions of a finite continuous-time Markov chain. For a given MTCP one can establish an MMPP with the same first and second moments of counts. In this paper…

马尔可夫到达与服务分析及其在间歇性过载中的应用

Analysis of Markovian Arrivals and Service with Applications to Intermittent Overload

本文针对到达率和服务率均按有限状态马尔可夫链变化的 MAMS 队列系统，提出基于相对到达与相对完成的新框架，推导出平均队列长度的显式边界，并在重流量及两层间歇性过载特例下证明边界的紧性与更强结果。

交互式随机环境中的指数型单服务器队列

Exponential single server queues in an interactive random environment

本文研究与随机环境双向交互的状态依赖指数单服务器队列，证明一大类系统稳态分布具乘积形式，并为不可分离系统建立遍历性判据及吞吐量界。

马尔可夫链的深度学习：Lyapunov函数、Poisson方程与平稳分布

Deep Learning for Markov Chains: Lyapunov Functions, Poisson's Equation, and Stationary Distributions

本文提出利用深度学习训练神经网络满足首达分析导出的积分方程，自动构造Lyapunov函数，并扩展用于求解Poisson方程和估计平稳分布，在非紧致状态空间上依然有效。

Approximations for the Moments of Nonstationary and State Dependent Birth-Death Queues

In this paper we propose a new method for approximating the nonstationary moment dynamics of one dimensional Markovian birth-death processes. By expanding the transition probabilities of the Markov process in terms of Poisson-Charlier polynomials, we are able to estimate any moment of the Markov process even though the system of moment equations may not be closed. Using new weighted discrete Sobolev spaces, we derive explicit error bounds of the transition probabilities and new weak a priori estimates for approximating the moments of the Markov processs using a truncated form of the expansion.…

Perfect Sampling of Markov Chains with Piecewise Homogeneous Events

Perfect sampling is a technique that uses coupling arguments to provide a sample from the stationary distribution of a Markov chain in a finite time without ever computing the distribution. This technique is very efficient if all the events in the system have monotonicity property. However, in the general (non-monotone) case, this technique needs to consider the whole state space, which limits its application only to chains with a state space of small cardinality. We propose here a new approach for the general case that only needs to consider two trajectories. Instead of the original chain, we…

四分之一平面中的扩散极限与非半鞅反射布朗运动

Diffusion limits in the quarter plane and non-semimartingale reflected Brownian motion

本文证明四分之一平面随机游走在扩散标度下收敛于 Varadhan–Williams 斜反射布朗运动，填补了非半鞅参数范围 $α\in(1,2)$ 的不变原理空白。