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.


Introduction
Traffic engineering greatly benefits from models that are capable of accurately describing and predicting the performance of the system. The network nodes are usually modeled as queues, and queueing theory can be used to analyze the performance of these nodes. However, most studies address performance issues for single-node models. The single-node models can offer valuable insights, but are an oversimplification of reality, since traffic streams usually traverse concatenations of nodes (rather than just a single node). In this paper, we consider a tandem queueing model. Tandem queues consist of a very important type of queueing systems, which have numerous applications in many fields, including manufacturing, telecommunications, computer network management, supply network management, health care among others. For example, tandem queues are perfect models for manufacturing (product) assembly lines (say a car or aircraft product/assembly line), where intermediate inputs represent various parts or components arrived to different stages of the assembly line; in telecommunications, information (with the form of emails, documents, live conversations, videos, internet requests, and many others) is often partitioned into packets, which are transmitted, through routers, over telecommunication networks according to QoS criteria. The whole path is a model of a tandem queue, where intermediate inputs are traffic of other sources arrived to routers. More applications either in the above mentioned or other areas can Keywords and phrases: Brownian-driven tandem queue, stationary distribution, exact tail asymptotics, kernel method, extreme value theory. Before we conclude the introduction, it is worthwhile to point out that on one hand, we anticipate that the tools developed in this paper could be useful in analyzing a general d-dimensional SRBM (for example, for the rough decay rate), but on the other hand, one has to overcome more technical challenges for a more general case (with non-triangular reflection matrix, or with dimension d > 3). More detailed discussions are provided in the last section.

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 an 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 3 leaves the system (see Fig. 1). We assume that the exogenous inputs are independent Brownian processes of the form: where λ i > 0 is a positive constant, and B i (t) is a Brownian motion with variance σ 2 i and no drift. Denote the processing rate at node i by c i > 0. Let L i (t) be the buffer content at node i at time t ≥ 0 for i = 1, 2, 3, which are formally defined as L 1 (t) = L 1 (0) + X 1 (t) − c 1 t + Y 1 (t), (2.2) where the local time Y i (t) is a regulator at node i, that is, a minimal nondecreasing process for L i (t) to be nonnegative.
In this paper, all vectors are supposed to be column vectors. For a vector v, we write v to denote the transpose of it. To simplify the notation, let Namely, L(t) is a generated by a reflection mapping from net flow processesX(t) with reflection matrix R.
Remark 2.1. Here we point out that the reflection matrix R is triangular. In this sense, our model is degenerate. This triangular structure of the reflection matrix R is crucial for us to carry out analytic continuation and find dominant singularities in Section 3.
Without any difficulty, we can obtain that the tandem queue {L(t)} has a stationary distribution if and only if (2.7) Moreover, by Harrison and Williams [17], we can get that the stationary distribution of {L(t)} is unique. Throughout this paper, we denote this stationary distribution by π. In order to simplify the discussion, we refine the stability condition (2.7) to assume that λ 1 < c 1 , and λ i + c i−1 < c i , i = 2, 3. (2.8) Remark 2.2. From the proofs of the main results of this paper, it is clear that under the more general stability condition (2.7), we can use the same argument to discuss tail asymptotics. The only difference is that we need to discuss possible relationships between the parameters λ i and c i , i = 1, 2, 3, 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 the asymptotic tail behavior of the stationary distribution. Recall that a positive function g(x) is said to have exact tail asymptotic h(x), if lim x→∞ g(x) h(x) = 1.
Our main aim is to find exact tail asymptotics for various stationary distributions. Moment generating functions will play an important role in determining these exact tail asymptotics. We first introduce moment generating functions for stationary distributions. Let L = (L 1 , L 2 , L 3 ) be the stationary random vector with stationary distribution π. The moment generating function φ(·) for L is given by: φ(x, y, z) = E e xL1+yL2+zL3 , for any (x, y, z) ∈ R 3 . (2.9) We apply the kernel method to study tail asymptotics for marginal stationary distributions. In order to apply the kernel method, we need to establish a relationship between the moment generating function φ(·) for the stationary distribution π and the moment generating functions for the boundary measures defined below. For any Borel set A ⊂ B(R 3 ), we define the boundary measures V i (·), i = 1, 2, 3, by (2.10) Moreover, due to Harrison and Williams [17], we obtain that the density functions for V i , i = 1, 2, 3, exist. Then, their moment generating functions are defined by where w = (x, y, z) ∈ R 3 and w, θ denotes the inner product of vectors w and θ.
The following lemma is a particular case of Theorem 4 in Konstantopoulous, Last and Lin [18]. 14) Proof of Lemma 2.3. To make the paper self-contained, here we apply the Itô's formula to prove this lemma directly. Let C 2 (R 3 ) be the set of all functions from R 3 to R with continuous second-order partial derivatives. Let f (θ) ∈ C 2 (R 3 ), then by Itô's formula where r ji is the (j, i)-th entry of the reflection matrix R = (r ji ) 3×3 . Take the expectation at the both sides of (2.17) for t = 1, given that L(0) follows the stationary distribution, and denote this expectation by E π . Then, as long as all expectations are finite, we have (2.18) Therefore, taking f (θ) = exp{θ 1 x + θ 2 y + θ 3 z} with φ(x, y, z) < ∞ and φ i (x, y, z) < ∞, i = 1, 2, 3, in equation (2.18) completes the the proof of this lemma. From Lemma 2.3, we can prove the following lemma.
Before we prove this lemma, we first introduce the following notation for convenience: Proof of Lemma 2.4. From (2.11), we get that (2.12) makes sense for w ∈ (x, y, z) : Let j = 1. Then, one can easily get that By (2.22) and (2.23), the proof is completed.
In general, it is difficult or impossible to obtain the explicit expression for the stationary distribution π, or its moment generating function. Hence, our focus is on its tail asymptotics. There are a few available 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 P L 3 < z via the kernel method introduced by Li and Zhao [20] and asymptotic properties of the joint stationary distribution by extreme value theory and copula.
Remark 2.5. The kernel method introduced by Li and Zhao [20] is an analytic method for studying stationary tail behaviour of two-dimensional queueing systems. This method is a combination of analytic continuation and asymptotic analysis of complex functions. For more information about this method, refer the readers to the survey paper [37]. Here we also note that the term "kernel method" has also been used by others, for example, Banderier et al. [2], Bousquet-Mélou [3] among others. The kernel method in [2,3] continued the work of Fayolle et al. [10] and is used to solve one-dimensional unknown probability sequences (or functions) first through the kernel equation and then joint probability. Their methods focus on a complete determination of the unknown function and therefore involve much more work. The kernel method used here only requires the location of the dominant singularity of the unknown function and the asymptotic property at the dominant singularity.
At the end of this section, we recall the Tauberian-like Theorem for moment generating functions introduced in Dai, et al. [5]. Let g be the L-transformation of a nonnegative, continuous and integrable function f on [0, ∞), i.e., Then, g(s) is analytic on the left half-plane. Let C denote the complex plane. Moreover, for a point z 0 ∈ C and δ ≥ 0, define where arg(z) ∈ (−π, π) is the principal part of the argument of a complex number z. The following lemma comes from Dai et al. [5].

Kernel equation, dominant singularities and analytic continuation
In this paper, we apply the kernel method to study tail asymptotics for the marginal stationary measure P(L 3 < z). The original kernel method applies easily to one-dimensional problems, and for two-dimensional problems (or random walks in the quarter plane), we refer the readers to Li and Zhao [21]. The problem of interest in this paper is a three-dimensional problem. Significant efforts are required in order to apply the kernel method to our problem, which will be addressed in this and the next sections before we can use the Tauberian-like Theorem (see Lem. 2.6) to connect the asymptotic properties of the unknown function and the corresponding tail asymptotic properties of P(L 3 < z). Specifically, since φ(0, 0, z) is the transformation function for the marginal stationary measure for L 3 , we need to study analytic properties of the moment generating function φ(0, 0, z). In this section, we address analytic continuation and defer the singularity analysis in the next section.

Kernel equation and branch points
To study analytic 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 P(L 3 < z) is our focus, we first treat z in (x, y, z) ∈ R 3 as a variable. Inspired by the procedure of applying the kernel method, for example, see Li and Zhao [20,21], we first construct the relationship between z and x, y. The kernel equation in (3.1) defines an implicit function z in variables x and y when we only consider non-negative values for z.
In view of the kernel method for the bivariate case, we locate the maximum z max of z on H(x, y, z) = 0. In order to do it, taking the derivative with respect to x at the both sides of (3.1) yields and solve the system of equations (3.2) and (3.3), and then we have Similarly, take the derivative with respect to y, ∂z ∂y = 0, (3.5) to obtain It is easy to check that at the point x z max , y z max , z attains the maximum value z max . From (3.4) and (3.6), we can get that on the point (x z max , y z max , z max ), the coordinates x and y satisfy where Remark 3.1. Without loss of generality, we assume that k 1 = 1 in the rest of this paper. For the special case k 1 = 1, the discussion can be carried out by using the same idea, which is much simpler than the general case due to the fact that when k 1 = 1, the terms including k 1 − 1 in most equations will disappear.
From the above arguments, we obtain the maximum z max on the plane H(k 1 y, y, z) = 0. Now, we consider the new equation: From (2.8) and (3.1), we can easily know that (3.9) defines an ellipse. Thus, for fixed z, there are two solutions to (3.9) for y, which are given: and , Moreover, these two solutions are distinct except if ∆(z) = 0. We call a point z a branch point if ∆(z) = 0. For branch points, we have the following properties.
(i) ∆(z) has two real zeros, one of which is z max , and the other is denoted by z min . Moreover, they satisfy Proof. From (3.12), we obtain (3.14) where k 2 = 1 and c 0 = 0. On the other hand, From (3.14) and (3.15), we get (3.13). By properties of quadratic functions, we can get that (ii) holds. The proof of the lemma is completed now.
, (3.16) and . (3.17) In order to use the Tauberian-like Theorem, we consider the analytic continuation of the moment generating functions in the complex plane C. The function ∆(z) plays an important role in the process of the analytic continuation. Hence, we first study its analytic continuation. By Lemma 3.2, ∆(z) is well defined for z ∈ [z min , z max ]. Moreover, it is a multi-valued function in the complex plane. For convenience, in the sequel, ∆(z) denotes the principal branch, that is ∆(z) = ∆(Re(z)) for z ∈ (z min , z max ). In the following, we continue ∆(z) to the cut plane C \ (−∞, z min ] ∪ [z max , ∞) . In fact, we have The proof of Lemma 3.4 is standard. For example, see Dai and Miyazawa [7], and Dai, et al. [5]. Here we omit the proof.
Symmetrically, we can treat the kernel equation in (3.9) as a quadratic function in z, and obtain the parallel results to those in Lemmas 3.2 and 3.4, and Corollary 3.5, respectively. We list them below. Before stating them, we first introduce the following notation. Definē For fixed y, there are two solutions to (3.9), which are given by and (3.20) Similarly to Lemmas 3.2 and 3.4, and Corollary 3.5, we have: Lemma 3.6.
(i)∆(y) has two real zeros, denoted by y min and y max , respectively, satisfying y min < 0 < y max . In order to get the analytic continuation of the moment generating functions, we need some technical lemmas. For the function Y max,0 (z), we have the following properties.
. Proof. Since z min and z max are two zeros of ∆(z) = 0, we have It follows from (3.10) and (3.22) that By (3.23), in order to prove case (i), we only need to show We also note that (Re(z) − z min ) and (z max − Re(z)) are real parts of (z − z min ) and (z max − z), respectively, since z min and z max are real. Therefore, Thus, Hence, Since for Re(z) ∈ (z min , z max ), From (3.25) to (3.29), in order to prove (3.24), we only need to prove cos ω min (z) cos ω max (z) which directly follows from the proof of the inequality (6.2) in Dai and Miyazawa [7].
Next, we prove case (ii). We first assume that z min < Re(z) < z max . From (3.10) and Lemma 3.2, we have since ∆(z max ) = 0. From (3.31) and case (i), in order to prove case (ii), we only need to show It follows from (3.18) and Lemma 3.6 that Hence, (3.32) follows from (3.33). Finally, we assume that Re(z) ≥ z max . As δ → π 2 , we have that It follows from Lemma 3.6, (3.33) and (3.34) that we can find δ 0 ∈ [0, π 2 ) such that case (ii) holds. The proof of the lemma is completed.

Dominant singularities and analytic continuation
The analytic continuation of the moment generating function φ 2 (0, 0, z) 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.
Lemma 3.8. For the moment generating functions φ i (·), i = 1, 2, 3, we have Proof. We first prove case (i). In order to prove it, we first prove for some y > 0, and for some z > 0.
In fact, which suggests that we may restrict our analysis to the two-dimensional tandem queue L 1 (t), L 2 (t) with the two nodes 1 and 2. We note that L 1 (t), L 2 (t) is not affected by L 3 (t). Hence, (3.35) follows straightforwardly from Dai, et al. [5]. Next, we prove (3.36). Since Y 1 is a regulator, By (3.38) and (2.12), we get that the left-hand side of (3.36) satisfies Next, we study this system on the plane y = 0. We first consider the ellipse defined by For the point (x, z) on this ellipse, we have For fixed x, we can find two solutions to (3.40) for z. Denote one of these two solutions by (3.42) Using the same method as in the proof of Lemma 3.2, we can get that Hence, from (3.41) and (3.42), we have Hence, Z 0 (x)φ 3 (x, 0, 0) is finite if and only if the right-hand side of (3.44) is finite. On the other hand, from (3.42), we obtain that for x ∈ [x min , 0), and From (3.45) and (3.46), we obtain that On the other hand, noting that e yL2(u) ≥ 0 and e zL3(u) ≥ 0, we have Combining (3.35), (3.36) and (3.49), we get that for some y > 0 and z > 0 Next, we prove case (ii). Since we can consider the problem on the plane x = 0. It follows from (2.12) that Then, defines an ellipse. For every fixed y, definē (3.53) Then, (3.53) is a solution to equation (3.52). Similarly to Lemma 3.2,Z 0 (y) is well-defined on some region [a, b] with a < 0 and b > 0. It follows from (3.51) and (3.53) that Furthermore, from (3.53), we obtain that for y ∈ {y : a < y < 0} Hence, by case (i) and (3.55), we can choose y < 0 such that z =Z 0 (y) > 0 and It is also worthy noting that for y < 0, Case (ii) now follows from (3.54) to (3.57). Finally, we can use the same ideal as for cases (i) and (ii) to show cases (iii) and (iv). This completes the proof of the lemma.
For the continuation of the function φ 2 (0, 0, z), we need another technical tool.
being a bounded and continuously differentiable real function, and Then, the complex variable functionG(z) is analytic on {z ∈ C : Re(z) < τG}.
Proof. We use the Vitali's Theorem to prove it. In fact, we have is a density function, we can get that F (z, x 1 ) is analytic on the region {z ∈ C : Re(z) < τG} for any Now, it is obvious thatF (λ, x 1 ) satisfies the conditions of the Vitali's Theorem (see, for example, Markushevich [26]) on the region {z ∈ C : Re(z) < τG}. Then, the lemma holds. (ii) IfG(z) is singular at some z 0 ∈ C, then we must haveG(x) = ∞ for x ∈ Re(z 0 ), ∞ .
The next lemma enables us to express φ 2 (0, 0, z) in terms of the other moment generating functions.
Lemma 3.12. φ 2 (0, 0, z) can be analytically continued to the region z ∈ {z : Re(z) < } with > 0, and Proof. From Corollary 3.5 and (2.12), we get that On the other hand, equation (3.52) defines an ellipse. For fixed z, there are two solutions to (3.52) for y. Define Using the same method as in the proof of Lemma 3.4, we can get that Y 0 (z) is analytic in the cut plane By (3.51) and (3.64), we can find a region such that Next, we study the relationship between Y 0 (z) and Y max,0 (z) for z > 0. We note that both the two ellipses defined by (3.9) and (3.52), respectively, pass the origin (0, 0) and We should note that On the other hand, from (3.10) and (3.64), we have, for 0 < z < 2(c3−λ3−c2) Thus, It follows from Lemma 3.8, (3.63), (3.65) and (3.70) that where we use the principle of analytic continuation of several complex variables functions (see, for example, Narasimhan [28]). Therefore for Re (z) < with some > 0. The proof is completed.
We continue to address analytic continuation of the function φ 2 (0, 0, z). From Lemma 3.9, there exists only one dominant singularity. We denote it by z dom . We first characterize the dominant singularity z dom of φ 2 (0, 0, z).

For convenience, let
Moreover, let From Lemma 3.12, we have: Lemma 3.13. F (y) can be analytically continued to a region {y : Re(y) < } with > 0, and We introduce the following notation.
From the above arguments, we can get that the lemma holds.
The zero of Y max,0 (z) − z is critical for us to prove Lemma 3.17 below. Hence, we demonstrate how to evaluate (3.97) It follows from (3.8) and (3.10) that Hence, the non-zero root of Y max,0 (z) − z = 0 is Proof. From (3.62), we obtain that Next, we show that Finally, it follows from Lemma 3.16 thatḠ(y) is analytic at the point Y max,0 (z * ). From the above arguments and (3.73), we complete the proof of the lemma. Lemma 3.19. If the convergence parameter τ φ2 is less than z max , then To apply the Tauberian-like Theorem to connect the asymptotic property of φ 2 (0, 0, z) to the corresponding tail asymptotic property of the boundary measure V 2 , we need to continue the function φ 2 (0, 0, z) further to a larger domain except a neighbourhood of the dominant singularity z dom . By Lemma 3.9, there is exactly one dominant singularity for φ 2 (0, 0, z). By Lemma 3.17, there are two candidates for the dominant singularity z dom of φ 2 (0, 0, z): (1) A pole, i.e.., the zero z * of Y max,0 (z) − z; or (2) the branch point z max .
An illustration for z min , z * and z max is presented in Fig. 2.
For each of these two cases, we show that the unknown function φ 2 (0, 0, z) satisfies the analytic continuation condition required by the Tauberian-like Theorem.
Lemma 3.20. If z dom < z max , then there exists an > 0 such that φ 2 (0, 0, z) is analytic for Re(z) < z dom + except for z = z dom and for each a > 0 where B a (z dom ) = {z ∈ C : |z − z dom | < a}.
Proof. From Lemma 3.17, we see that if z dom < z max , then z dom is a pole of the function φ 2 (0, 0, z). Hence, φ 2 (0, 0, z) is analytic for Re(z) < z dom + except for z = z dom . It remains to show (3.104) for each a > 0. In such a case, z dom is a pole of φ 2 (0, 0, z). It follows from Lemma 3.17 that z dom is a zero of Y max,0 (z) − z. So On the other hand, from (3.10), we get that From (3.99) and (3.107), we obtain that From (3.102) and (3.108), we have that for Re(z) < z dom + , Finally, we can easily get that Lemma 3.21. If z dom = z max , then φ 2 (0, 0, z) is analytic in G δ0 (z max ), where δ 0 is chosen in Lemma 3.7 and G δ is defined by (2.24). Moreover, for each a > 0, Proof. We first show that φ 2 (0, 0, z) is analytic on z ∈ G δ0 (z max ). It follows from Lemma 3.9 that φ 2 (0, 0, z) is analytic for Re(z) < z dom . Furthermore, by (3.81), we have z dom > 0. Hence, in order to prove the lemma, it suffices to show that φ 2 (0, 0, z) is analytic on z ∈ G δ0 (z max ) ∩ {z ∈ C : Re(z) > 0}. Since z dom = z max , from Lemma 3.17, we must have z * ≥ z max . We first assume that Combining (3.92) and Lemma 3.16, we have that F (Y max,0 (z)) is analytic at G δ0 (z max ) . Hence, from (3.99) and Corollary 3.5, we can get the lemma. Next, we assume that z max = z * . The proof of this case is the combination of the proof of Lemma 3.20 and that of the case (3.111). So, we omit the details of the proof here.

Singularity Analysis of φ(0, 0, z) and Exact Tail Asymptotics for Marginal Distributions
From the arguments in the previous section, we are ready to present asymptotic properties of the function φ 2 (0, 0, z). Before stating these properties, we first present a technical lemma, which plays an important role in finding the tail asymptotics of the marginal distribution P(L 3 < z).
Based on Lemma 4.1, the asymptotic properties of φ 2 (0, 0, z) lead to corresponding properties of the function φ(0, 0, z). Then, the application of the Tauberian-like Theorem gives the exact tail asymptotics for the marginal distribution of L 3 .
From Lemmas 3.14 and 3.17, we can get that z dom is either z * or z max . In order to obtain tail asymptotics for the marginal L 3 , we need to study asymptotic properties of the moment generating function φ 2 at the point z dom . We first present asymptotic properties of G(z) defined in (3.72) at the point z dom . .
It is clear that the asymptotic behavior of the function φ(0, 0, z) at its dominant singularity z dom depends on the value of z dom , which is equal to z * or/and z max . In practice, it is important to compare the values of z * and z max . In fact, we have the following lemma. Proof. If z max = z * , one can easily see that the lemma holds. Next, we assume z max = z * . From (3.10), we obtain that Y max,0 (z) is increasing on ( 2(c3−λ3−c2) σ 2 3 , z max ]. We first assume that z * exists in (0, z max ). Since (4.45) On the other hand, we note that the line H 2 (y, z) = z − y = 0 intersects the ellipse H(ky, y, z) = 0 at one point except for the point (0, 0) . From (3.98), we know that the point (Y max,0 (z * ), z * ) is the other intersection point of H 2 (y, z) = 0 and H(k 1 y, y, z) = 0. Hence, we must havẽ Next, we assumeỹ We prove that z * belongs to (0, z max ). From (4.48), we obtain that the point (ỹ max , z max ) is above the line , z max . By the above arguments, one can get that the lemma holds. Theorem 4.7. For the marginal stationary distribution P{L 3 > z}, we have the following tail asymptotic properties for large z: Case 3: If z dom = z * = z max , then whereK i , i = 1, 2, 3, are non-zero constants.
Proof. Cases (1) and (3) are direct consequences of Lemmas 3.20, 3.21, 4.1, 4.4 and 2.6. Next, we prove case (2). From (4.43), we have On the other hand, it follows from Dai and Harrison [6] that the density function f (x) of the marginal distribution P(L 3 < x) exists. From Dai and Miyazawa [7], we get that is the moment generating function of the density function Therefore, from Lemma 2.6 and (4.49), we havē whereǨ is a constant depending on z dom . From (4.50) and (4.51), we obtain that as By (4.53), (4.54) and L'Hôspital's Rule, we obtain that That is where K 1 is a constant. From (4.56), we conclude that case (2) holds.

Tail behaviours of joint stationary distributions
In this section, we study the tail behavior of the joint stationary distribution π. It should be pointed out that the extension of the kernel method presented in previous sections for tail asymptotics of the marginal distribution of L 3 is not valid for the tail asympotics of the joint distribution. Instead, we propose a new idea for the main result in this section based on extreme value theory and copula. Before stating the main result, we first introduce the domain of attraction of some extreme value distribution function G DA (·).
where the maximum M k are the componentwise maxima, then we call the distribution function G DA (·) a multivariate extreme value distribution function, andF is in the domain of attraction of G DA (·). We denote this byF ∈ D(G DA ).
For convenience, we let F π (x, y, z) denote the joint stationary distribution function of {L(t)} and F i , i = 1, 2, 3, denote the stationary distribution function of the i-th buffer content process. From Dai and Miyazawa ([7], Thms. 2.2 and 2.3), and Theorem 4.7, we can easily get the following lemma.
where α i is the dominant singularity of the moment generating function of the marginal distribution F i , C i is a non-zero constant, and µ i ∈ {0, − 1 2 , − 3 2 }. From Lemma 5.2, we can get that Lemma 5.3. For any i ∈ {1, 2, 3}, we have Proof. From Dai and Harrison [6], F i , i = 1, 2, 3, have continuous densities. It straightforwardly follows from (5.1) and L'Hôspital's Rule that as x → ∞, Moreover, it is obvious that For large enough x > 0, due to (5.3), we have where o(·) denotes small oh as x → ∞. Now, we consider the existence of the second-order derivative F i (x) of the function F i (x) and the asymptotic equivalence of F i (x), as x → ∞. Let g i (x) = α i C i exp{−α i x}x µi for convenience. Then we can rewrite the equation (5.5) as To reach our aim, we first discuss some properties of the function g i (x). For any large enough u > R + fixed, we first show that where o u (·) denotes the small oh as x → u. Note that It is obvious that as x → u Moreover, since F i (x) and g i (x) are both continuous, from (5.6) and (5.8), as We first assume that x > u. From (5.9) and (5.10), for any > 0, there exists δ > 0 such that for all x − u < δ and o(g i (x)) − o(g i (u)) < 2 g i (u).
Below we assume that x < u, then from (5.9), for any small enough > 0, there exists δ > 0 such that for all u − x < δ and Hence, from (5.15) and (5.14), It follows from the asymptotic equivalence (5.3) and (5.19) that Then, it follows from Proposition 1.1 in Resnick [30] that F i ∈ D(G 1 ).
In the previous section, we obtained exact tail asymptotic properties of the marginal distributions. Now, based on these results, we can study the upper tail dependence for the joint stationary distribution. Before stating the tail dependent result for the joint stationary distribution, we introduce a technical lemma. Lemma 5.4. Suppose that X n = X (1) n ,X n ,X Then, the following are equivalent: (1)F is in the domain of attraction of a product measure, that is, where q ∈ {i, j}.
By a slight modification of the proof of Proposition 5.27 in Resnick [30], we can prove the above lemma. Hence, we omit the detail here.
Remark 5.5. By Proposition 5.24 in Resnick [30], the asymptotic independence in (5.21) can be reduced to two-dimensional case. Moreover, the bivariate asymptotic independence could be seen from the tail behaviour ofX (i) andX (j) . Hence, one can see the equivalence between (5.21) and (5.22) intuitively.
For the joint stationary distribution function F π (·), we have the following tail dependence.
To prove Lemma 5.6, we first introduce the inverse (left continuous) H ← of a function H by Proof. Here, we will use (5.22), an equivalent statement, to prove this lemma. Without loss of generality, we assume that L(0) follows the stationary distribution π. Hence for any t ∈ R + and (x, y, z) ∈ R 3 + , At the same time, we note that, to prove (5.22), it suffices to show the following upper tail dependence: Below, we let i = 1 and j = 2 for simplicity. Other cases can be proved in the same fashion. At the same time, from (5.23), we get that for any t ∈ R + Next, we introduce the last exit time τ t before t, out of the boundaries by τ t = inf s : L i (u) > 0 for all i = 1, 2, 3, s ≤ u ≤ t .
Remark 5.7. For a n (µ i , α i ) and b n (µ i , α i ), i = 1, 2, 3, in Lemma 5.6, we can use tail equivalence to obtain their explicit expressions. Since they are not the focus of this paper, we will not elaborate them here. Now, we present the main result of this section.
SRBM with triangular reflection matrix. A natural extension of our model is the d-dimensional Brownian-driven tandem queue with intermediate inputs, which is an SRBMZ = (Z 1 , . . . ,Z d ) , whose reflection matrixR = r ij d×d satisfiesr An interesting problem is to obtain exact tail asymptotics for the marginal stationary distributions ofZ. For this model, the kernel equation in (3.1) becomes a d-dimensional ellipsoid H(z 1 , . . . , z d ) = 0. The exact asymptotic analysis seems to be analogous to that for the 3-dimensional case. However, there are still some new technical challenges we need to address. For example, to study exact tail asymptotics for the marginal stationary distribution ofZ d , the main challenge comes from the construction of the ellipses which locate the maximum point z max d and the interlace between the moment generating functions of boundary measures and marginal stationary distributions. The detailed analysis is beyond the scope of the current paper.
SRBM with general reflection matrix. An immediate question is: Can we generalize our study to a general d-dimensional SRBMZ with d ≥ 3? To answer this question, we first recall the key components in our analysis for the 3-dimensional model L: (1) The fundamental form, or the functional equation satisfied by the (unknown) moment generating functions of the joint stationary distribution and boundary measures, see the equation in (2.12). Similar to Lemma 2.3, by using the Itô's formula, such a relationship can be obtained for a d-dimensional SRBMZ. (2) The kernel method, including analytic continuation of the unknown moment generating functions and asymptotic analysis. We first briefly review why the kernel method can be applied to our case. Inspired by the kernel method for 2-dimensional queueing systems, a binary alternative equation in (3.9) is constructed based on the ternary kernel equation in (3.1), and then the kernel method is employed to study the binary alternative equation in (3.9) instead of the kernel equation in (3.1). Finally the analytic continuation of the unknown moment generating functions and asymptotic analysis are obtained. However, for the general d-dimensional SRBM Z, it is not possible to find binary alternatives for applying the kernel method. Hence, analytic continuation could not be carried out by the kernel method at this moment. This looks to be the main challenge for studying a general modelZ. It is our conjecture that the counterpart analytic continuation property (to Lemma 3.12) still holds forZ. If this is true, the asymptotic analysis on the dominant singularity should prevail. (3) Based on the asymptotic analysis of the unknown moment generating functions, the Tauberian-like Theorem leads to exact tail asymptotic properties for the boundary measures and marginal stationary distributions, see Theorem 4.7. If Step (2) works well for a generalZ, then we can get exact tail asymptotic properties for the boundary measures and marginal stationary distributions ofZ, the counterpart to Theorem 4.7, by using the same Tauberian-like Theorem (see Lemma 2.6). (4) Furthermore, by extreme value theory and copula, asymptotic independence for the joint stationary distribution can be obtained, see Theorem 5.8. If we can obtain exact tail asymptotics for marginal stationary distributions of a generalZ, then, similar to Theorem 5.8, we can study exact tail asymptotics and dependence structure of the joint stationary distribution ofZ.
From above discussions, we know that, due to the challenge arisen in Step (2), we do not have a complete study on exact tail asymptotics for a general modelZ by using our method at this moment. However, the methods developed in this paper could be applied to discuss rough tail asymptotics for a general d-dimensional SRBM Z. The large deviations forZ have been studied intensively. Under some mild conditions (see, for example, Conditions 2.1 and 2.5 in Dupuis and Ramanan [9]), the large deviations principle forZ has been established, see, for example, Avram, Dai and Hasenbein [1], Dupuis and Ramanan [9] and Majewski [25]. Therefore, based on the large deviations forZ, we expect to establish rough asymptotitcs for the marginal stationary distributions ofZ, and we can then discuss rough asymptotics and dependence structure of the joint stationary distribution ofZ by the method developed in Section 5. In our ongoing work, we discuss this topic in detail.