Stationary Distributions for Two-Dimensional Sticky Brownian Motions: Exact Tail Asymptotics and Extreme Value Distributions

In this paper, we consider a two-dimensional sticky Brownian motion. Sticky Brownian motions can be viewed as time-changed semimartingale reflecting Brownian motions, which find applications in many areas including queueing theory and mathematical finance. For example, a sticky Brownian motion can be used to model a storage system.with exceptional services. In this paper, we focus on stationary distributions for sticky Brownian motions. The main results obtained here include tail asymptotic properties in boundary stationary distributions, marginal distributions, and joint distributions. The kernel method, copula concept and extreme value theory are main tools used in our analysis.Comment: 32 page

Tail Asymptotics for a Retrial Queue with Bernoulli Schedule

In this paper, we study the asymptotic behavior of the tail probability of the number of customers in the steady-state $M/G/1$ retrial queue with Bernoulli schedule, under the assumption that the service time distribution has a regularly varying tail. Detailed tail asymptotic properties are obtained for the (conditional and unconditional) probability of the number of customers in the (priority) queue, orbit and system, respectively.Comment: 18 pages; revised version: 20 page

GTH Algorithm, Censored Markov Chains, and RGRG-Factorization

In this paper, we provide a review on the GTH algorithm, which is a numerically stable algorithm for computing stationary probabilities of a Markov chain. Mathematically the GTH algorithm is an rearrangement of Gaussian elimination, and therefore they are mathematically equivalent. All components in the GTH algorithm can be interpreted probabilistically based on the censoring concept and each elimination in the GTH algorithm leads to a censored Markov chain. The $RG$-factorization is a counterpart to the LU-decomposition for Gaussian elimination. The censored Markov chain can also be treated as an extended version of the GTH algorithm for a system consisting of infinitely many linear equations. The censored Markov chain produces a minimal error for approximating the original chain under the $l_1$-norm.Comment: 13 pages; a few editorial edits in the revised versio

Join-the-Shortest-Queue Model as a Mean Field Approximation to a Local Balancing Network

In this paper, we consider a queueing network with $N$ nodes, each of which has a fixed number $k$ of neighboring nodes, referred to as the $N$ node network with local balancing. We assume that to each of the $N$ nodes, an incoming job (or task) chooses the shortest queue from this node and its neighboring nodes. We construct an appropriate Markov process for this network and find a mean field approximation to this network as $N\rightarrow\infty$, which turns out to be the standard join-the-shortest-queue model.Comment: This draft has some major issues to be fixed

Exact Tail Asymptotics --- Revisit of a Retrial Queue with Two Input Streams and Two Orbits

We revisit a single-server retrial queue with two independent Poisson streams (corresponding to two types of customers) and two orbits. The size of each orbit is infinite. The exponential server (with a rate independent of the type of customers) can hold at most one customer at a time and there is no waiting room. Upon arrival, if a type $i$ customer $(i=1,2)$ finds a busy server, it will join the type $i$ orbit. After an exponential time with a constant (retrial) rate $\mu_i$, an type $i$ customer attempts to get service. This model has been recently studied by Avrachenkov, Nain and Yechiali~\cite{ANY2014} by solving a Riemann-Hilbert boundary value problem. One may notice that, this model is not a random walk in the quarter plane. Instead, it can be viewed as a random walk in the quarter plane modulated by a two-state Markov chain, or a two-dimensional quasi-birth-and-death (QBD) process. The special structure of this chain allows us to deal with the fundamental form corresponding to one state of the chain at a time, and therefore it can be studied through a boundary value problem. Inspired by this fact, in this paper, we focus on the tail asymptotic behaviour of the stationary joint probability distribution of the two orbits with either an idle or busy server by using the kernel method, a different one that does not require a full determination of the unknown generating function. To take advantage of existing literature results on the kernel method, we identify a censored random walk, which is an usual walk in the quarter plane. This technique can also be used for other random walks modulated by a finite-state Markov chain with a similar structure property.Comment: 23 pages; 2 figure

Light-tailed behavior of stationary distribution for state-dependent random walks on a strip

In this paper, we consider the state-dependent reflecting random walk on a half-strip. We provide explicit criteria for (positive) recurrence, and an explicit expression for the stationary distribution. As a consequence, the light-tailed behavior of the stationary distribution is proved under appropriate conditions. The key idea of the method employed here is the decomposition of the trajectory of the random walk and the main tool is the intrinsic branching structure buried in the random walk on a strip, which is different from the matrix-analytic method

Explicit stationary distribution of the (L,1)(L,1)-reflecting random walk on the half line

In this paper, we consider the $(L,1)$ state-dependent reflecting random walk (RW) on the half line, which is a RW allowing jumps to the left at a maxial size $L$. For this model, we provide an explicit criterion for (positive) recurrence and an explicit expression for the stationary distribution.As an application, we prove the geometric tail asymptotic behavior of the stationary distribution under certain conditions. The main tool employed in the paper is the intrinsic branching structure within the $(L,1)$-random walk

Equilibrium customer and socially optimal balking strategies in a constant retrial queue with multiple vacations and NN-policy

In this paper, equilibrium strategies and optimal balking strategies of customers in a constant retrial queue with multiple vacations and the $N$-policy under two information levels, respectively, are investigated. We assume that there is no waiting area in front of the server and an arriving customer is served immediately if the server is idle; otherwise (the server is either busy or on a vacation) it has to leave the system to join a virtual retrial orbit waiting for retrials according to the FCFS rules. After a service completion, if the system is not empty, the server becomes idle, available for serving the next customer, either a new arrival or a retried customer from the virtual retrial orbit; otherwise (if the system is empty), the server starts a vacation. Upon the completion of a vacation, the server is reactivated only if it finds at least $N$ customers in the virtual orbit; otherwise, the server continues another vacation. We study this model at two levels of information, respectively. For each level of information, we obtain both equilibrium and optimal balking strategies of customers, and make corresponding numerical comparisons. Through Particle Swarm Optimization (PSO) algorithm, we explore the impact of parameters on the equilibrium and social optimal thresholds, and obtain the trend in changes, as a function of system parameters, for the optimal social welfare, which provides guiding significance for social planners. Finally, by comparing the social welfare under two information levels, we find that whether the system information should be disclosed to customers depends on how to maintain the growth of social welfare.Comment: 27 page

Mean Field Approximations to a Queueing System with Threshold-Based Workload Control Scheme

In this paper, motivated by considerations of server utilization and energy consumptions in cloud computing, we investigate a homogeneous queueing system with a threshold-based workload control scheme. In this system, a virtual machine will be turned off when there are no tasks in its buffer upon the completion of a service by the machine, and turned on when the number of tasks in its buffer reaches a pre-set threshold value. Due to complexity of this system, we propose approximations to system performance measures by mean field limits. An iterative algorithm is suggested for the solution to the mean field limit equations. In addition, numerical and simulation results are presented to justify the proposed approximation method and to provide a numerical analysis on the impact of the system performances by system parameters.Comment: Revised version, 26 pages, 5 figure

Exact tail asymptotics for a three dimensional Brownian-driven tandem queue with intermediate inputs

The semimartingale reflecting Brownian motion (SRBM) can be a heavy traffic limit for many server queueing networks. Asymptotic properties for stationary probabilities of the SRBM have attracted a lot of attention recently. However, many results are obtained only for the two-dimensional SRBM. There is only little work related to higher dimensional ($\geq 3$) SRBMs. In this paper, we consider a three dimensional SRBM: A three dimensional Brownian-driven tandem queue with intermediate inputs. We are interested in tail asymptotics for stationary distributions. By generalizing the kernel method and using copula, we obtain exact tail asymptotics for the marginal stationary distribution of the buffer content in the third buffer and the joint stationary distribution.Comment: 35 page