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-diemnsional SRBM. There is only little work related to higher dimensional () 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 coupla, we obtain exact tail asymptotics for the marginal stationary distribution of the buffer content in the third buffer and the joint stationary distribution.
MSC(2000): 60K25, 60J10.
Keywords: Brownian-driven tandem queue, stationary distribution, kernel method, exact tail asymptotics
Since Harrison and Reiman [10, 11], Varadhan and Williams, and Williams[29, 30] introduced the semimartingale reflecting Brownian motion, SRBM has received a lot of attention. Stationary properties of stationary distributions of SRBM when they exist, are important, especially in applications. However, except for a very limited number of special cases, a simple closed expression for the stationary distribution is not available. Hence, exact tail behaviour of stationary distributions becomes most important. Recently, many results about two-dimensional SRBM have been obtained. Harrison and Hasenbein  presented sufficient and necessary conditions for the existence of a stationary distribution. Dai and Miyazawa  studied exact tail asymptotics for the marginal distributions of SRBM by using a geometric method. Dai, Dawson and Zhao  applied the kernel method to obtain exact tail asymptotics for the boundary measures of SRBM. Franceschi and Kurkova  studied exact tail asymptotics of the stationary distribution along some path by analytic methods. Franceschi and Raschel  studied exact tail behaviour of the boundary stationary distributions of SRBM by using the boundary value problems. However, we note that all aforementioned results are only for two-dimensional SRBM. In this paper, we will consider a three dimensional SRBM.
Miyazawa and Rolski  generalized the result of Lieshout and Mandjes [19, 20], and studied a two-dimensional Lévy-driven tandem queue with an intermediate input. They obtained exact tail asymptotics for the Brownian inputs, while weaker tail asymptotic results were obtained for the general Lévy input. They also tried to discuss higher-dimensional cases. However only the stationary equation was obtained in terms of moment generating functions, and tail asymptotic properties for the marginal distributions were left for a future work. In this paper, we consider a three-dimensional Brownian-driven tandem queue with intermediate inputs. We derive exact tail aysmptotics for the marginal stationary distribution of the third buffer content, since exact tail asymptotic results for the first two buffer content can be obtained directly from results for two-dimensional SRBM. We also note that all results related to exact tail asymptotics for stationary distributions of SRBM are only for marginal stationary distributions and boundary stationary distributions. There are no results referred in the literature on asymptotic properties for the joint stationary distribution of SRBM, which is also considered in this paper.
In this paper, we apply both the Kernel method and extreme value theory to study tail asymptotics. The kernel method has been systematically applied to study random walks in the quarter plane by Li and Zhao  and references therein. Key steps in applying the kernel method for random walks in the quarter plane are:(i) Establishing the fundamental form:
where , and are unknown generating functions for joint and two boundary probabilities, respectively. (ii) Finding a branch such that , which leads to a relationship between the two unknown boundary generating functions:
(iii) Based on (1.1), carrying out a singularity analysis for and , which leads to not only a decay rate, but also exact tail asymptotic properties of the boundary probabilities through a Tauberian-like theorem. In this paper, we will extend this method to study a three-dimensional SRBM. By using the kernel method, we can get exact tail asymptotics for the marginal stationary distributions.
In this paper, we also study asymptotic properties for the joint stationary distributions. However, we cannot use the kernel method to study tail behaviours of the joint stationary distribution, since the kernel method relies on the Tauberian-like Theorem, which is valid only for univariate functions. By using the kernel method, we can get tail equivalence for the marginal distributions, from which we will further study the tail dependence of the joint stationary distribution. Tail dependence describes the amount of dependence in the upper tail or lower tail of a multivariate distributions and has been widely used in extreme value analysis and in quantitative risk management. Once we get the dependence, we can study multivariate extreme value distribution of the joint stationary distribution. The extreme value distribution is very useful since from a sample of vectors of maximum, one can make inferences about the upper tail of the stationary distribution using multivariate extreme value theory. Based on the multivariate extreme distribution, by using copula, we can get tail behaviour of the joint stationary distributions.
In this paper, we study a three dimensional SRBM and anticipate the tools developed in this paper will be useful in analyzing the general -dimensional case. The rest of this paper is organized as follows: In Section 2, a three-dimensional Brownian-driven tandem queue with intermediate inputs is introduced. To apply the kernel method for asymptotic properties for the marginal , we study the kernel equation and the analytic continuation of moment generating functions in Section 3. We study some asymptotic properties of moment generating functions in Section 4. Asymptotic results for the marginal distributions are present in Section 5. In Section 6, we study asymptotic properties of the joint stationary distribution.
2 Model and Preliminaries
In this section, we introduce a three dimensional Brownian-driven tandem queue with intermediate inputs and establish a stationary equation satisfied by stationary probabilities. This tandem queue has three nodes, numbered as 1,2, 3, each of which has exogenous input process and a constant processing rate. Outflow from the node 1 goes to node 2, and the outflow from node 2 goes to node 3. Finally, outflow from node leaves the system, see Fig.1 below.
Fig. 1 A tandem queue with 3 nodes.
We assume that the exogenous inputs are Brownian processes of the form:
where is a nonnegative constant, and is a Brownian motion with variance and no drift. Without loss of generality, we assume that the correlation coefficients , , where . Denote the processing rate at node by . Let be the buffer content at node at time for , which are formally defined as
where is a regulator at node , that is, a minimal nondecreasing process for to be nonnegative. In fact, we can regard as a reflection mapping from the net flow processes with the reflection matrix
Let and . Then
where and .
Without any difficulty, we can obtain that the tandem queue has the stationary distribution if and only if
Moreover, by Harrison and Williams , we can get that the stationary distribution of is unique. Throughout this paper, we denote this stationary distribution by . In order to simplify the discussion, in this paper, we refine the stability condition (2.5) to assume that
From the proofs of the main results of this paper, it is clear that under the more general stability condition (2.5), we can use the same argument to discuss tail asymptotics. The only difference is that we need to discuss possible relationships between the parameters and , , before we use the arguments in the proofs in this paper. For each of the possible relationships, we repeat the method applied in this paper to study tail asymptotics.
We are interested in asymptotic tail behaviour of the stationary distribution. Recall that a positive function is said to have exact tail asymptotic , if
In this paper, our main aim is to find exact tail asymptotics for various stationary distributions. Moment generating function will play an important role in determining these exact tail asymptotics. We first introduce moment generating functions for stationary distributions. Let be the stationary random vector with the stationary distribution . The moment generating function for is given by:
We apply the kernel method to study tail asymptotics for stationary distributions. In order to apply the kernel method, we need establish a relationship between the moment generating function for the stationary distribution and the moment generating functions for the boundary measures defined below. For any Borel set , we define the boundary measures , , by
Moreover, due to Harrison and Williams , we obtain that the density functions for , exist. Then, their moment generating functions are defined by
Next, we establish the relationship between these moment generating functions. In fact, there is a nice connection during them. The following lemma is due to Konstantopoulos, Last and Lin .
[14, Theorem 4] For each with and , , we have
From Lemma 2.1, we can prove the following lemma.
For , we have
since . Hence,
Let . Then, one can easily get that
In general, it is difficult or impossible to obtain the explicit expression for the stationary distribution or its moment generating function. However in some special cases, it becomes possible. For example, if there are no intermediate inputs, that is , , Miyazawa and Rolski  obtained an explicit expression of . For a general case, our focus is on its tail asymptotics. There are a few aviable methods for studying tail asymptotics, for example, in terms of large deviations and boundary value problems. In this paper, we study tail asymptotics of the marginal distribution via the kernel method introduced by Li and Zhao  and asymptotic properties of the joint stationary distribution by extreme value theory and copula.
At the end of this section, we present a technical lemma, which plays an important role in finding the tail asymptotics of the marginal distribution .
and have the same singularities.
Proof: Let . Then,
Note that for any , since only if , we have
From (4.11) below, we get that
Letting in (2.20), we obtain
3 Kernel Equation and Analytic Continuation
In this paper, we apply the kernel method to study tail asymptotics for the marginal stationary measure . In order to do it, we need the Tauberian-like Theorem (Theorem 5.1). For applying this theorem, we need to study the analytic properties of the moment generating function .
3.1 Kernel Equation and Branch Points
To study analyitic properties of the moment generating functions, we first focus on the kernel equation and the corresponding branch points. For this purpose, we consider the kernel equation:
which is critical in our analysis.
Since tail asymptotics for is our focus, we first treat in as a variable. Inspired by the procedure of applying the kernel method, for example, see Li and Zhao , we first construct the relationship between and , . The kernel equation in (3.1) defines an implicit function in variables and when we only consider non-negative values for . For convenience, let .
In view of the kernel method for the bivariate case, we locate the maximum of on . In order to do it, taking the derivative with respect to at the both side of (3.1) yields
Similarly, take the derivative with respect to ,
Without loss of generality, we assume that in the rest of this paper. For the special case , the discussion can be carried out by using the same ideal which is much simpler than the general case due to the fact that when , the term including in most equations will disappear.
From the above arguments, we obtain the maximum on the plane . Now, we consider the new equation:
Moreover, these two solution are distinct except . We call a point a branch point if . For branch points, we have the following property.
has two real zeros, one of which is , and the other is denoted by . Moreover, they satisfy
in and in .
Proof: From (3.12), we obtain
where . On the other hand,
By properties of quadratic functions, we can get that (ii) holds. The proof of the lemma is completed now.
In order to use the Tauberian-like Theorem below, we consider the analytic continuation of the moment generating functions in the complex plane . The function plays an important role in the procedure of the analytic continuation. Hence, we first study its analytic continuation. By Lemma 3.1, is well defined for . Moreover, it is a multi-valued function in the complex plane. For convenience, in the sequel, denotes the principle branch, that is for . In the follow, we continue to the cut plane . In fact, we have
is analytic in the cut plane .
The proof of Lemma 3.2 is standard. For example, see Dai and Miyazawa , or Dai, Dawson and Zhao . However, for the completeness of the paper, we provide a proof following the ideal used by Dai and Miyazawa .
Proof: Since and are two zeros of , we have
Next, we rewrite (3.16) in the polar form. Let and denote the principal arguments of and , respectively. Therefore,
Hence, (3.16) can be rewritten as
Moreover for , the functions and are analytic. Since is the principle part, we have
Both and are analytic in the cut plane .
Symmetrically, we can treat the kernel equation (3.9) as a quadratic function in , and obtain parallel results to those in Lemmas 3.1 and 3.2, and Corollary 3.1. We list them below. Before stating them, we first introduce the following notation. Define
For fixed , there are two solutions to (3.9), which are given by
has two real zeros, denoted by and , respectively, satisfying
in and in .
are analytic in the cut plane .
In order to get the analytic continuation of the moment generating functions, we need some technical lemmas. Before we introduce these lemmas, we first present an important notation. Define
where , and .
For the function , we have the following properties.
For , we have
for with some .
By (3.1), in order to prove case (i), we only need to show
We also note that and are real parts of and , respectively, since and are real. Therefore,
Similarly, we have
Since for ,
which directly follows from Dai and Miyazawa .
since . From (3.32) and case (i), in order to prove the case (ii), we only need to show
3.2 Analytic Continuation
The analytic continuation of the moment generating function plays an important role in our analysis, which is the focus in this subsection. In order to carry out this, we need the following technical lemma.
For the moment generating functions ,, we have
is finite on some region with .
is finite on some region with .
is finite on some region with .
is finite on some region with .
Proof: We first prove case (i). In order to prove it, we first prove
for some , and
for some .
which suggests that we may restrict our analysis to the two-dimensional tandem queue