We consider a two-dimensional semimartingale reflecting Brownian motion (SRBM) in the nonnegative quadrant. The data of the SRBM consists of a two-dimensional drift vector, a 2 ×2 positive definite covariance matrix, and a 2×2 reflection matrix. Assuming the SRBM is positive recurrent, we are interested in tail asymptotic of its marginal stationary distribution along each direction in the quadrant. For a given direction, the marginal tail distribution has the exact asymptotic of the form bx κ exp(−αx) as x goes to infinity, where α and b are positive constants and κ takes one of the values −3/2, −1/2, 0, or 1; both the decay rate α and the power κ can be computed explicitly from the given direction and the SRBM data. A key tool in our proof is a relationship governing the moment generating function of the two-dimensional stationary distribution and two moment generating functions of the associated one-dimensional boundary measures. This relationship allows us to characterize the convergence domain of the two-dimensional moment generating function. For a given direction c, the line in this direction intersects the boundary of the convergence domain at one point, and that point uniquely determines the decay rate α. The one-dimensional moment generating function of the marginal distribution along direction c has a singularity at α. Using analytic extension in complex analysis, we characterize the precise nature of the singularity there. Using that characterization and complex inversion techniques, we obtain the exact asymptotic of the marginal tail distribution.

A double quasi-birth-and-death (QBD) process is the QBD process whose background process is a homogeneous birth-and-death process, which is a synonym of a skip-free random walk in the two-dimensional positive quadrant with homogeneous reflecting transitions at each boundary face. It is also a special case of a 0-partially homogenous chain introduced by Borovkov and Mogul'skii. Our main interest is in the tail decay behavior of the stationary distribution of the double QBD process in the coordinate directions and that of its marginal distributions. In particular, our problem is to get their rough and exact asymptotics from primitive modeling data. We first solve this problem using the matrix analytic method. We then revisit the problem for the 0-partially homogenous chain, refining existing results. We exemplify the decay rates for Jackson networks and their modifications.

Queueing network, Reflecting random walk, Semi-martingale reflecting Brownian motion, Stationary distribution, Tail asymptotic, Tail decay rate, Large deviations, Light tail, Stability, Stationary inequality, Server collaboration, Join the shortest queue, Markov additive process, Convergence domain, Multidimensional moment generating function, 60K27, 90B15, 60G50, 60J70, 60F10,

A geometric tail decay of the stationary distribution has been recently studied for the GI /G /1 type Markov chain with both countable level and background states. This method is essentially the matrix analytic approach, and simplicity is an obvious advantage of this method. However, so far it can be only applied to the -positive case (or the jittered case, as referred to in the literature). In this paper, we specialize the GI /G /1 type to a quasi-birth-and-death process. This not only refines some expressions because of the matrix geometric form for the stationary distribution, but also allows us to extend the study, in terms of the matrix analytic method, to non--positive cases. We apply the result to a generalized join-the-shortest-queue model, which only requires elementary computations. The obtained results enable us to discuss when the two queues are balanced in the generalized join-the-shortest-queue model, and establish the geometric tail asymptotics along the direction of the difference between the two queues.

We consider the asymptotic behaviour of the stationary tail probabilities in the discrete-time GI/G/1-type queue with countable background state space. These probabilities are presented in matrix form with respect to the background state space, and shown to be the solution of a Markov renewal equation. Using this fact, we consider their decay rates. Applying the Markov renewal theorem, it is shown that certain reasonable conditions lead to the geometric decay of the tail probabilities as the level goes to infinity. We exemplify this result using a discrete-time priority queue with a single server and two types of customer.

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We present three sets of results for the stationary distribution of a two-dimensional semimartingale reflecting Brownian motion (SRBM) that lives in the nonnegative quadrant. The SRBM data can equivalently be specified by three geometric objects, an ellipse and two lines, in the two-dimensional Euclidean space. First, we revisit the variational problem (VP) associated with the SRBM. Building on Avram, Dai and Hasenbein (2001), we show that the value of the VP at a point in the quadrant is equal to the optimal value of a linear function over a convex domain. Depending on the location of the point, the convex domain is either D (1) or D (2) or D (1) ∩ D (2) , where each D (i) , i = 1, 2, can easily be described by the three geometric objects. Our results provide a geometric interpretation for the value function of the VP and allow one to see geometrically when one edge of the quadrant has influence on the optimal path traveling from the origin to a destination point. Second, we provide a geometric condition that characterizes the existence of a product form stationary distribution. Third, we establish exact tail asymptotics of two boundary measures that are associated with the stationary distribution; a key step in our proof is to sharpen two asymptotic inversion lemmas in Dai and Miyazawa (2011) that allow one to infer the exact tail asymptotic of a boundary measure from the singularity of its moment generating function.

In the seminal paper of Gamarnik and Zeevi [17], the authors justify the steady-state diffusion approximation of a generalized Jackson network (GJN) in heavy traffic. Their approach involves the so-called limit interchange argument, which has since become a popular tool employed by many others who study diffusion approximations. In this paper we illustrate a novel approach by using it to justify the steady-state approximation of a GJN in heavy traffic. Our approach involves working directly with the basic adjoint relationship (BAR), an integral equation that characterizes the stationary distribution of a Markov process. As we will show, the BAR approach is a more natural choice than the limit interchange approach for justifying steadystate approximations, and can potentially be applied to the study of other stochastic processing networks such as multiclass queueing networks. Introduction.This paper considers open single-class queueing networks that have d service stations. Each station has a single server operating under the first-in-first-out (FIFO) service discipline. Upon completing service at a particular station, customers are either routed to another station, or exit the network. There is a single class of customers at each station, meaning that all customers are homogenous in terms of service times and routing. A customer entering the network will exit in finite time with probability one, hence the term open network. For each station, the external interarrival times (possibly null), service times, and routing decisions are assumed to follow three separate i.i.d. sequences of random variables; the three sequences are assumed to be independent. Furthermore, the interarrival times, service times and routing decisions are assumed to be independent between different stations. Such a network is hereafter referred to as a generalized Jackson network (GJN).