Queueing Syst (2011) 68:251–260
DOI 10.1007/s11134-011-9251-0
Conjectures on tail asymptotics of the marginal
stationary distribution for a multidimensional SRBM
Masakiyo Miyazawa · Masahiro Kobayashi
Received: 10 May 2011 / Revised: 10 May 2011 / Published online: 9 July 2011
© Springer Science+Business Media, LLC 2011
Abstract We are concerned with the stationary distribution of a d-dimensional semimartingale reflecting Brownian motion on a nonnegative orthant, provided it is stable,
and conjecture about the tail decay rate of its marginal distribution in an arbitrary direction. Due to recent studies, the conjecture is true for d = 2. We show its validity
for the skew symmetric case for a general d.
Keywords Queueing network · Semi-martingale reflecting Brownian motion ·
Stationary distribution · Tail asymptotic · Tail decay rate · Stationary inequality ·
Multidimensional moment generating function
Mathematics Subject Classification (2000) 60J65 · 60F10 · 90B15 · 60K25
1 Introduction
We consider the tail asymptotics of the stationary distribution of a semimartingale
reflecting Brownian motion, SRBM for short, provided it is stable. This process
takes values in the nonnegative orthant of the d-dimensional Euclid space. Its sample
path is identical with a Brownian motion inside this state space, and reflected at the
boundary of the orthant. This SRBM has been well studied for many years (see, e.g.,
[3, 7, 8]). Throughout the paper, we assume that the SRBM has a stationary distribution, that is, it is stable.
As for the stable SRBM, the large deviations principle has been established for the
tail asymptotic behavior of the stationary distribution under mild regularity conditions
(e.g., see [12] and Theorem 4.3 of [1]). Then, the problem is reduced to find the rate
function, which is formulated as a variational problem. However, this problem is
M. Miyazawa () · M. Kobayashi
Tokyo University of Science, Yamazaki 2641, Noda, Chiba 278-8510, Japan
e-mail: [email protected]
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Queueing Syst (2011) 68:251–260
known to be very hard to analytically solve, particularly to get the rate function in
closed form except for the two dimensional case. In the latter case, the rate function
has been obtained in [1, 6]. There are some discussions about why the more than
two dimensional case is so difficult (see, for example, [1]). Thus, it is challenging to
consider the more than two dimensional case.
In this paper we differently look at the problem for general d ≥ 2. In applications,
we may like to see the tail asymptotics of the stationary distribution of the SRBM in
a given direction, particularly in the directions of coordinate axes. They may change
according to a tail set to be considered. We choose the tail set which is a collection
of d-dimensional nonnegative vectors x satisfying c, x > u for each d-dimensional
nonnegative directional vector c and tail level u, where a, b is the inner product
of vectors a and b. We then consider the decay rate of the stationary probability
concerning this tail set as u goes to infinity. Here, the decay rate means the limiting
ratio of logarithm of the tail probability over the tail level. Thus, we are interested in
the decay rate of the marginal stationary distribution in an arbitrary direction. This
rate can be computed from the large deviations principle, so the problem is a special
case of the above variational problem. However, it is still a hard problem.
Thus, we here approach the decay rate problem, working directly with the moment
generating functions of the d-dimensional stationary distribution and its marginals on
boundary faces which constitute the boundary. Unfortunately this moment generating
function is not given in an explicit form. So, we need ideas to overcome this difficulty.
A basic idea is the decay rates are determined by the convergence domain of the moment generating function. For d = 2, this approach has been successfully performed
by solving a certain fixed point equation in [4]. Similar ideas have been used for the
special case of the two dimensional SRBM in [16] and the two dimensional skip free
reflecting random walk in [14]. Both mainly consider the decay rates in the directions
of coordinate axes, that is, those of the marginal stationary distributions. We extend
those ideas for the general d-dimensional case, and conjecture the decay rates in an
arbitrary direction. It may be notable that another analytic approach is studied for a
special case of the two dimensional reflecting random walk in [9] (see also [2, 5]).
The advantage of the present approach is its feasibility to compute the decay rates
from modeling parameters. Furthermore, we may find exact asymptotics including
polynomial prefactor to the exponential main term as studied in [4, 16]. Our conjectures may suggest how the optimal path looks like in the variational problem for the
large deviations principle. We hope it will be useful to solve the variational problem.
2 Semimartingale reflecting Brownian motion
For a positive integer d, let X(t) be a d-dimensional Brownian motion. We express
it as X(t) = tμ + B(t), where μ is the mean drift vector, and B(t) is the null drift
Brownian motion with positive definite covariance matrix Σ . Let R be d × d matrix.
We assume
(i) R is a completely S-matrix, that is, all principal sub-matrices of R are S matrices,
where a square matrix A is said to be an S matrix if there is a vector x ∈ Rd such
that x > 0 and Rx > 0.
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253
Then, it is known that, for each initial distribution of Z(0), there exists a Rd -valued
continuous process Z(t) such that
Z(t) = X(t) + RY (t),
t ≥ 0,
(2.1)
where Y (t) is a regulator, that is, a minimal continuous and nondecreasing process
such that Y (0) = 0 and its ith entry Yi (t) is increased only when Zi (t) = 0 for each
i = 1, 2, . . . , d (see [17]). In the literature, this reflecting process is referred to as
semi-martingale reflecting Brownian motion, SRBM for short (for example, see [6]
for its recent development). A typical application of SRBM is a Brownian fluid network (for example, see Sect. 7.5 of [3] and Sect. 6 of [16]).
Throughout the paper, we assume
(ii) Z(t) has a stationary distribution, which is denoted by π .
It is known that
R −1 μ < 0
(2.2)
is necessary for the stationary distribution to exist. This condition is also sufficient
for d = 2, but not the case for d ≥ 3. Hereafter, we assume that Z(t) is a stationary process with the initial distribution π and the stationary incremental regulator
Y (t). We first derive the stationary equation for Z(t) in terms of moment generating
functions. A more general stationary equation has already been obtained in the text
book [3] (see also Proposition 6.1 of [16]). We here repeat some of its derivation for
introducing notation.
We first note that Eπ (Y (1)) = −R −1 μ is a finite and positive vector. Let ϕ(θ ) be
the moment generating function of π . Similarly, let
1
ϕi (θ ) = Eπ
eθ,Z(u) dYi (u),
0
and let
T
ϕ(θ ) = ϕ1 (θ ), . . . , ϕd (θ ) ,
where “T” stands for transposing a vector.
Note that ϕi (θ ) does not depend on θi because Yi (u) is increased only when
Zi (u) = 0. We use Itô’s integral formula for deriving the stationary equation. For
a twice continuously differentiable function f of d variables, let
∂ 2f
1 σij
(x),
2
∂xi ∂xj
d
∇f = (f1 , . . . , fd )T ,
Lf (x) =
d
i=1 j =1
where σij is (i, j ) entry of the covariance matrix Σ . Then, Itô’s integral formula reads
1
T f Z(u) − f Z(0) =
∇f Z(u)
μ du + dB(u)
0
+
0
1
T
∇f Z(u) dY (t) +
1
0
Lf Z(u) du.
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Let f (x) = exp(θ , x) for θ = (θ1 , . . . , θd )T and x = (x1 , . . . , xd )T , where μ, c is
the inner product of vectors μ and c. Then, using the notation,
1
γ (θ) = −θ , μ − θ , Σθ,
2
we have the following fact (see Proposition 6.1 of [16] for another derivation).
Proposition 2.1 Under conditions (i) and (ii), we have, for θ ∈ Rd ,
γ (θ)ϕ(θ ) = θ , Rϕ(θ ) ,
(2.3)
as long as ϕ(θ ) and ϕ(θ ) are finite, which holds at least for θ ≤ 0.
To facilitate this stationary equation in our arguments, we introduce the following
sets. Let
Γ = θ ∈ Rd ; γ (θ) > 0 ,
Γ = θ ∈ Rd ; γ (θ) = 0 ,
ΓA = θ ∈ Rd ; ∀j ∈ A, γj (θ ) < 0 ,
∂ΓA = θ ∈ Rd ; ∀j ∈ A, γj (θ ) = 0 , A ∈ 2J .
Note that Γ and ΓA are convex open sets, and Γ∅ = Rd . Furthermore, Γ is a bounded
set, that is, it is contained in some ball with a finite radius in Rd . For d = 2, ΓJ = ∅
under the stability condition, and we guess this is always the case.
3 Tail asymptotic behavior
Let Z be a random vector subject to the stationary distribution π . Let c be a vector
with unit length, which is called a direction vector. Define the decay rate α(c) in the
direction c ≥ 0 by
1
− log P Z, c > x ,
x→∞ x
α(c) = lim
as long as it is exists. Denote the corresponding limit sup and inf by α(c) and α(c),
respectively. If we can find a function f (x) such that, for some constant b > 0,
lim P Z, c > x /f (x) = b,
x→∞
then the marginal distribution of the π is said to have exact asymptotics f in the
direction c.
Our problem is to find this decay rate α(c) for all direction c ≥ 0 for the ddimensional SRBM. For this, we first consider the convergence domain of ϕ(θ ).
Denote it by D. Namely,
D = the interior of θ ∈ Rd ; ϕ(θ) < ∞ .
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255
If this domain is obtained, then we can expect that
α(c) = sup{x ≥ 0; xc ∈ D}.
(3.1)
Thus, it is important to get the domain D in terms of the modeling primitives. For
this, we consider the stationary equation (2.3). Let
γi (θ ) =
d
θ j Rj i ,
i = 1, 2, . . . , d,
j =1
and rewrite (2.3) as, for A ∈ 2J \ {∅}, where J = {1, 2, . . . , d},
γ (θ )ϕ(θ ) +
−γj (θ ) ϕj (θ ) =
γj (θ )ϕj (θ ).
j ∈A
(3.2)
j ∈J \A
We like to use this stationary equation to find D, but it is only valid for θ satisfying
ϕ(θ ) < ∞ and ϕ j (θ ) < ∞ for all j ∈ J . This is a circular argument.
To get out this situation, we will use an extended version of the stationary inequality in Lemma 6.1 of [15]. The following inequality can be proved in the same way as
the proof of Lemma 4.1 of [4].
Lemma 3.1 For each A, B ∈ 2J such that B ⊂ A, if θ ∈ Γ ∩ ΓB ∩ ∂ΓA\B and
ϕj (θ ) < ∞ for all j ∈ J \ A, we have
γ (θ )ϕ(θ ) +
−γj (θ ) ϕj (θ ) ≤
γj (θ )ϕj (θ ),
j ∈B
(3.3)
j ∈J \A
and therefore, ϕ(θ ) < ∞ and ϕ j (θ ) < ∞ for all j ∈ B.
One may wonder why we consider the smaller set B for the finiteness of ϕ j (θ ) for
j ∈ B. This is just because we have less constraints on θ for ΓB ∩ ∂ΓA\B than ΓA .
We need one more thing.
Conjecture 3.1 For each θ ∈ Rd , if ϕ(θ ) < ∞, then ϕj (θ ) < ∞ for all j ∈ J .
Intuitively, this conjecture must be true because ϕj is the moment generating function of the measure which is generated by a derivative of the stationary distribution ν.
However, we need a proof, which has been done for d = 2 in [4].
For each A ∈ 2J and C ⊂ Rd , define
◦
(j ) DA (C) =
θ ∈ Γ ∩ ΓB ∩ ∂ΓA\B ; ∀j ∈ J \ A, ∃η(j ) ∈ C, ∀k = j, θk ≤ ηk
,
B⊂A
(3.4)
where A◦ denotes the interior set A ⊂ Rd .
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If Conjecture 3.1 is true, then C ⊂ D implies DA (C) ⊂ D for any A ∈ 2J , where
Conjecture 3.1 is used for θ ∈ D to imply ϕj (θ ) < ∞ for all j ∈ J . Let
C0 = θ ∈ Rd ; θ < 0 .
Obviously, C0 ⊂ D. We inductively define Cn for n = 1, 2, . . . as
DA (Cn−1 ) ,
Cn = conv
(3.5)
A∈2J \{∅}
where conv(K) is the minimal convex set containing a set K. Since Cn is nondecreasing in n, its limits exists. Denote this limit by C∞ . By this construction, we can see
that C∞ is obtained as the minimal solution of the fixed point equation for C ⊂ Rd :
DA (C) .
(3.6)
C = conv
A∈2J \{∅}
Furthermore, C∞ ⊂ D if Conjecture 3.1 is true. This suggests:
Conjecture 3.2 D = {θ ∈ Rd ; ∃θ > θ , θ ∈ C∞ }, and the decay rate α(c) for the
marginal stationary distribution in the direction c is given by (3.1).
We have given the algorithm to get the solution C∞ of the fixed point equation
(3.6). However, this does not answer what geometrical expression it has. For d = 2,
this is answered using extreme points in [4]. For extending this idea, we introduce the
following fixed point equations for vectors in Rd . For each A ∈ 2J \ {J } and i ∈ A,
θ ((A,i),∞) = arg sup θi ≥ 0; θ ∈ Γ ∩ ΓA ,
θ∈Rd
∀j ∈ J \ A, ∃B ∈ 2J \ {J } , ∃ ∈ B, (B, ) = (A, i),
∀k = j, θk ≤ θk((B,),∞) .
(3.7)
We conjecture that these vectors determine the domain D in the following way.
Conjecture 3.3 (3.7) has the unique solution {θ ((A,i),∞) , A ∈ 2J \ {J }, i ∈ A}, and
D is obtained by (3.6) with
((A,j ),∞) ◦
DA (C∞ ) = θ ∈ Γ ∩ ΓA ; ∀j ∈ J \ A, ∀k = j, θk ≤ θk
, A ∈ 2J \ {J }.
(3.8)
This conjectured domain D is easier to get than that of Conjecture 3.2, but we still
need to solve fixed point equations. It is preferable to analytically get their solution.
For this, we consider the following extreme point (3.7).
θ ((A,i),c) = arg sup{θi ≥ 0; θ ∈ Γ ∩ ΓA },
θ∈Rd
A ∈ 2J \ {J }, i ∈ A.
(3.9)
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257
This corresponds with the convergence parameter of the moment generating function
of Markov additive kernel for the additive process in the ith coordinate direction with
the background process generated by removing the reflecting boundary on the kth
axis such that k ∈ A (see [14, 15] for this Markov additive process). We conjecture
Conjecture 3.4 For each i ∈ J , if
((J \{i},i),c)
θj
((J \{j },j ),c)
≤ θj
,
∀j = i,
then θ ((J \{i},i),∞) = θ ((J \{i},i,c) , and
((J \{j },j ),c) D = θ ∈ Rd ; θ < θ ∈ Γ ; ∀j ∈ J, θj ≤ θj
.
(3.10)
(3.11)
Of course, the condition (3.10) is not always satisfied. However, once it holds, we
can get the domain D in a simpler way.
4 Some special cases
We first consider the case for d = 2. In this case, J = {1, 2} and A = {1}, {2} for
A ∈ 2J \ {∅, J }. Conjecture 3.1 is proved in [4]. (3.7) can be written as
(({1},2),∞) ,
θ (({2},1),∞) = arg sup θ1 ≥ 0; θ ∈ Γ ∩ Γ{2} , θ2 ≤ θ2
θ∈Rd
(({2},1),∞) .
θ (({1},2),∞) = arg sup θ2 ≥ 0; θ ∈ Γ ∩ Γ{1} , θ1 ≤ θ1
θ∈Rd
This is equivalent to the fixed point equation (2.12) and (2.13) in [4], and therefore all
the conjectures are valid for d = 2. Moreover, exact asymptotics are obtained in [4].
We next consider the skew symmetric case. It is known that the stationary distribution π has a product form of exponential distributions if and only if
−1 T
2Σ = RΔ−1
R ΔΣ + ΔΣ ΔR R ,
(4.1)
where ΔA is the diagonal matrix with the diagonal entries of a square matrix A. See
[1, 3, 8] for details of this condition. In this case, let αi denote the exponent of the
exponential distribution for the ith coordinate, and let α = (α1 , . . . , αd )T . Then,
−1
α = −2Δ−1
Σ ΔR R μ.
(4.2)
For d = 2, this case has been studied in [1]. We here consider the general d ≥ 2.
We first note that the point α on the curve γ (α) = 0 since (4.1) and (4.2) imply
γ (α) = −α, μ −
1
α, RΔ−1
R ΔΣ α = 0.
2
We next consider the edge points of the first regions that are defined by
θ (i,e) = arg sup{θi ≥ 0; θ ∈ ΓJ −{i} },
θ∈Rd
i ∈ J.
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Queueing Syst (2011) 68:251–260
To find these points, we need to solve the following simultaneous equations:
θ , μ = −
1
θ , RΔ−1
R ΔΣ θ ,
2
[θ T R]j = 0,
j = i.
These equations imply
θ , μ = −
1
1 T −1
0, . . . , 0, θ T R i , 0, . . . , 0 Δ−1
R ΔΣ θ = − θ R i rii σii θi .
2
2
On the other hand, from (4.2),
θ , μ = −
1
1 T
−1
θ , RΔ−1
R ΔΣ α = − [θ R]i rii σii αi .
2
2
(i,e)
Hence, we have θi = αi . That is, θi
= αi .
Assume that R is an M-matrix, that is, R = (I − G)ΔR for a nonnegative matrix
G (see Sect. 7.2 of [3]). Then, we can see that ΓJ \{i} is a convex cone containing the
ith axis in its inside. Hence, if the point α is located in the interior
or at the boundary
of the convex cone that γj (θ ) ≥ 0 for all j ∈ J , that is, α ∈ dj =1 (Rd+ \ Γj ), then it
cannot be located under the hyperplane containing the θ (i,e) for all i, since the α must
be on the boundary of the convex region that
γ (θ ) ≥ 0. Hence, in this case, we have
θ (i,c) = θ (i,e) for all i ∈ J . Otherwise, α ∈ dj =1 Γj . Let i0 be such an index. Then,
θ (i0 ,c) = α, and θ (i,c) = θ (i,e) for i = i0 . Thus, we conclude that
C∞ = θ ∈ Rd ; γ (θ) ≥ 0, θ ≤ α ,
which verifies Conjecture 3.2 for this special model. Similar discussions can be found
in [13], in which the decay rates are conjectured for the generalized Jackson network.
5 Concluding remarks
We here give some remarks on how to get the decay rates once the convergence
domain Do is obtained. As shown in [4], we can prove that
lim sup
x→∞
1
log P (Z > xc) ≤ − sup c, θ ≥ 0; θ ∈ Do .
x
(5.1)
However, the lower bound generally requires the large deviations arguments (for example, see [11]). In [16], we have applied the analytic function method using a path
on the boundary γ (z) = 0 in the complex plane C2 . We expect the same method
can be applied for the d-dimensional case. If it works, then we can most likely get
the exact asymptotics as well, which has been obtained for the two dimensional case
(see [4]). Of course, there are many hard issues to be overcome for this, in particular
to get the decay rate in an arbitrary direction.
We end with a numerical example for the case of d = 3. This model is obtained from a Brownian fluid network. In the present formulation, it assumes that
Queueing Syst (2011) 68:251–260
259
Fig. 1 The regions separated by
γi (θ) = 0
μ = −(0.6, 0.7, 1)T , and
⎛
⎞
1 0 −0.2
0 ⎠.
R = ⎝ −1 1
0 0
1
Then, we have, θ (c,1) (0.435185, 0.087037, 0.087037)T , θ (c,2) (0.192308,
0.437843, 0.0384613)T and θ (c,3) (0.149999, 0.175001, 0.590037)T using Mathematica 7.0 (see Fig. 1). Thus, we have θ (i,c) = θ (c,i) for i = 1, 2, 3, and the decay rates the directions of the coordinate axes are approximately α1 = 0.435185,
α2 = 0.437843 and α1 = 0.590037 by our conjectures.
Acknowledgements This research is supported in part by the Japan Society for the Promotion of Science under grant No. 21510165.
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