Reflecting Brownian motion in two dimensions: Exact asymptotics for the stationary distribution

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1 Reflecting Brownian motion in two dimensions: Exact asymptotics for the stationary distribution Jim Dai Joint work with Masakiyo Miyazawa July 8, INFORMS APS conference at Stockholm Jim Dai (Georgia Tech) Exact asymptotics July 8, / 31

2 Semimartingale reflecting Brownian motion (SRBM) -matrix if Rv > for some v. It is is an!-matrix. he n-dimensional non-negative orthant. Z(t) = X (t) + RY (t) for all t, X is a (µ, Σ) Brownian motion, Z(t) R n + for all t, n n covariance matrix, and an n- 3) proved the following: there exists a five conditions, and it is unique in Y ( ) is continuous and nondecreasing with Y () =, Y i ( ) only increases when Z i ( ) =, i = 1,..., n. Z 2 R 1 R 2 n) Z 1 Jim Dai (Georgia Tech) Exact asymptotics July 8, / 31

3 Semimartingale reflecting Brownian motion (SRBM) -matrix if Rv > for some v. It is is an!-matrix. he n-dimensional non-negative orthant. Z(t) = X (t) + RY (t) for all t, X is a (µ, Σ) Brownian motion, Z(t) R n + for all t, n n covariance matrix, and an n- 3) proved the following: there exists a five conditions, and it is unique in Y ( ) is continuous and nondecreasing with Y () =, Y i ( ) only increases when Z i ( ) =, i = 1,..., n. n) Z 2 R 1 R 2 Z 1 We focus on n = 2. R is assumed to be completely-s ( ) R = (R 1, R 2 r11 r ) = 12. r 21 r 22 Z exists and is unique in distribution; (Taylor-Williams 92) Jim Dai (Georgia Tech) Exact asymptotics July 8, / 31

4 A queueing network A Simple Queueing Network mean service time = 1 mean service time = 1 Poisson arrivals at average rate < deterministic service times service time variance a 2 Jim Dai (Georgia Tech) Exact asymptotics July 8, / 31

5 Its Approximating SRBM Its Approximating SRBM Z 2 Z 1 Z 1 (t) = X 1 (t) + Y 2 (t) Z 1 (t) = X 1 (t) + Y 1 (t), Z 2 (t) = X 2 (t) + Y 1 (t) Y 2 (t) Z 2 (t) = X 2 (t) Y 1 (t) + Y 2 (t) drift of X is µ = (ρ 1, ), drift of X is = ( -1, ) covariance of X is Σ = covariance of X is = a 2 ( ) ρ a 2 R. J. Williams (95), Semimartingale reflecting Brownian motions in the orthant, in Stochastic Networks, eds. F. P. Kelly and R. J. Williams, the IMA Volumes in Mathematics and its Applications, Vol. 71 (Springer, New York) R. J. Williams (96), On the approximation of queueing networks in heavy traffic, in Stochastic Networks: Theory and Applications, eds. F. P. Kelly, S. Zachary and I. Ziedens, Royal Statistical Society (Oxford Univ. Press, Oxford) Jim Dai (Georgia Tech) 9 Exact asymptotics July 8, / 31

6 Stationary distribution of an SRBM Proposition (Hobson-Rogers 94, Harrison-Hasenbein 9) Assume that r 11 >, r 22 >, r 11 r 22 r 12 r 21 >, (1) r 22 µ 1 r 12 µ 2 <, and r 11 µ 2 r 21 µ 1 <. (2) Then SRBM Z has a unique stationary distribution π. Condition (1): R is a P-matrix. Condition (2): R 1 µ <. Basic adjoint relationship: for each f Cb 2(R2 +) R 2 + ( 1 2, Σ f + µ, f ) π(dx) + [ ] 1 ν i (A) = E π 1 {Z(u) A} dy 1 (u) 2 i=1 R 2 + R i, f ν i (dx) =, defines the ith boundary measure. Jim Dai (Georgia Tech) Exact asymptotics July 8, / 31

7 Tail asymptotics of the stationary distribution Let Z( ) = (Z 1 ( ), Z 2 ( )) be the random vector that has the stationary distribution π. For any c R 2 +, set c, Z( ) = c 1 Z 1 ( ) + c 2 Z 2 ( ). We are interested in finding a function f c (x) that satisfies for some constant b >. P{ c, Z( ) > x} lim = b x f c (x) The function bf c (x) is said to be the exact asymptotic of P{ c, Z( ) > x}. Jim Dai (Georgia Tech) Exact asymptotics July 8, / 31

8 An overview Main Result For an SRBM satisfying (1) and (2), the exact asymptotic is given by f c (x) = x κc e αcx. The decay rate α c and the constant κ c can be computed explicitly from the primitive date (µ, Σ, R). The constant κ c must take one of the values 3/2, 1/2, or 1. Let p c (x) be the density of c, Z( ). In most cases, we have p c (x) lim = b for some b >. x f c (x) Jim Dai (Georgia Tech) Exact asymptotics July 8, / 31

9 A diversion: Laplace transform Let f : R + R + be a continuous and integrable function. Define f (z) = e zx f (x)dx, Rz < α f, where α f = sup{θ : f (θ) < }. f is analytic in Rz < α f. When f is a Gamma(k, α) density, namely, then α f = α and f (z) = f (x) = c Γ(k) x k 1 e αx, c g(z) for Rz < α. (α z) k Complex function g is an analytic extension of f. Jim Dai (Georgia Tech) Exact asymptotics July 8, / 31

10 A diversion: complex inversion Proposition (When k is a positive integer) Suppose that the analytic extension g of f satisfies that g(z) c (α z) k is analytic for Rz < α 1 for some α 1 > α. Then, under some mild conditions on g, f (x) c Γ(k) x k 1 e α x as x. (3) When α is the kth order pole of g(z), then (3) holds. Jim Dai (Georgia Tech) Exact asymptotics July 8, / 31

11 Convergence domain Define [ ϕ(θ 1, θ 2 ) = E e θ,z( ) ]. Then the transform of c, Z( ) has the following expression ψ c (z) = E ( e z c,z( ) ) = ϕ(zc). Define the convergence domain of ϕ D = interior of {θ R 2 : ϕ(θ) < }. When θ D, BAR gives the key relationship γ(θ)ϕ(θ) = γ 1 (θ)ϕ 1 (θ 2 ) + γ 2 (θ)ϕ 2 (θ 1 ). (4) Jim Dai (Georgia Tech) Exact asymptotics July 8, / 31

12 Geometric objects: Γ, Γ 1 and Γ 2 γ(θ) = θ, µ 1 θ, Σθ, 2 γ 1 (θ) = r 11 θ 1 + r 21 θ 2 = R 1, θ, γ 2 (θ) = r 12 θ 1 + r 22 θ 2 = R 2, θ, θ 2 γ 1 (θ) = Γ 2 θ (2,r) γ(θ) = (µ 1, µ 2 ) θ (1,r) γ 2 (θ) = θ 1 Γ 1 Jim Dai (Georgia Tech) Exact asymptotics July 8, / 31

13 Theorem 1: Convergence domain characterization Jim Dai (Georgia Tech) Exact asymptotics July 8, / 31

14 Convergence domain: Category I θ 2 θ (2,Γ) = γ 1 (θ) = θ (1,max) γ 2 (θ) = θ (1,Γ) = θ (1,r) θ 1 θ (2,Γ) the highest point in Γ 2 ; θ (1,Γ) the right-most point in Γ 1 Category I: θ (2,Γ) 1 < θ (1,Γ) 1 and θ (1,Γ) 2 < θ (2,Γ) 2, Jim Dai (Georgia Tech) Exact asymptotics July 8, / 31

15 Convergence domain: Category II θ 2 γ 2 (θ) = θ (1,r) γ 1 (θ) = θ (1,Γ) =θ (1,max) θ (2,Γ) = θ (2,r) θ 1 Category II: θ (2,Γ) θ (1,Γ), Category III: θ (1,Γ) θ (2,Γ). Jim Dai (Georgia Tech) Exact asymptotics July 8, / 31

16 Key steps in proving Theorem 1 ϕ(θ) < implies ϕ 1 (θ 2 ) < and ϕ 2 (θ 1 ) <. θ Γ 1 and ϕ 1 (θ 2 ) < imply ϕ(θ) <. θ Γ, ϕ 1 (θ 2 ) < and ϕ 2 (θ 1 ) < imply ϕ(θ) <. θ2 γ1(θ) = τ2 θ (2,c) =θ (2,r) 2nd Γ (τ1, τ2) 4th 3rd γ(θ) = θ (1,max) γ2(θ) = θ (1,c) =θ (1,r) Γ2 Γ1 1st point τ1 θ1 Jim Dai (Georgia Tech) Exact asymptotics July 8, / 31

17 An illustration for proving Theorems 2 and 3 Key relationship (4) gives γ(zc)ψ c (z) = γ 1 (zc)ϕ 1 (c 2 z) + γ 2 (zc)ϕ 2 (c 1 z) for Rz < α c. Letting γ(zc) = zζ c (z), we have ψ c (z) = γ 1(c)ϕ 1 (c 2 z) + γ 2 (c)ϕ 2 (c 1 z) ζ c (z) for Rz < α c. θ2 γ1(θ) = θ (2,Γ) =θ (2,r) τ2 η (2) c αcc η (1) θ (1,max) γ2(θ) = θ (1,Γ) =θ (1,r) θ1 τ1 ϕ 2 (z) is analytic on Rz < τ 1. ϕ 1 (z) is analytic on Rz < τ 2. α c c Γ. α c < min(τ 1 /c 1, τ 2 /c 2 ). ψ c (z) has a single pole at z = α c. f c (x) = e αcx. Jim Dai (Georgia Tech) Exact asymptotics July 8, / 31

18 Singularity properties of ϕ 2 (z) at z = τ 1 In Category I when τ 1 < θ (1,max) 1, single pole In Category I when τ 1 = θ (1,max) 1 When θ (1,max) θ (1,r), 1/2 analytic or 1/2 pole When θ (1,max) = θ (1,r), 1/2 pole In Category II when τ 1 < θ (1,max) 1, When θ (1,max) θ (1,r), single pole When θ (1,max) = θ (1,r), double pole In Category II when τ 1 = θ (1,max) 1 When θ (1,max) θ (1,r), 1/2 pole When θ (1,max) = θ (1,r), 1 pole Jim Dai (Georgia Tech) Exact asymptotics July 8, / 31

19 Theorem 2: exact asymptotics for Category I Jim Dai (Georgia Tech) Exact asymptotics July 8, / 31

20 Category I: β 1 < β < β 2 θ2 γ1(θ) = θ (2,Γ) =θ (2,r) τ2 η (2) c αcc η (1) τ 2 θ 2 γ 1(θ) = η (2) c α cc γ 2(θ) = θ (1,max) γ2(θ) = θ (1,Γ) =θ (1,r) θ (1,max) =θ (1,r) = η (1) τ1 θ1 τ 1 θ 1 ψ c (z) = γ 1(c)ϕ 1 (c 2 z) + γ 2 (c)ϕ 2 (c 1 z) ζ c (z) for Rz < α c. ϕ 2 (z) is analytic for Rz < τ 1 ϕ 1 (z) is analytic for Rz < τ 2 f c (x) = e αcx Jim Dai (Georgia Tech) Exact asymptotics July 8, / 31

21 Category I: β < β 1, τ 1 < θ (1,max) 1 τ 2 θ 2 γ 1 (θ) = θ (2,Γ) =θ (2,r) η (2) η (1) θ (1,max) γ 2 (θ) = θ (1,Γ) =θ (1,r) c α c c single pole at τ 1 for ϕ 2 (z) α c c Γ f c (x) = e αcx. τ 1 θ 1 ψ c (z) = γ 1(c)ϕ 1 (c 2 z) + γ 2 (c)ϕ 2 (c 1 z) ζ c (z) for Rz < α c. Jim Dai (Georgia Tech) Exact asymptotics July 8, / 31

22 Category I: β < β 1, τ 1 = θ (1,max) 1 θ 2 γ 1(θ) = τ 2 θ (2,Γ) = θ (2,r) η (2) γ 2(θ) = θ (1,r) θ (1,Γ) = θ (1,max) = η (1) τ 2 θ 2 γ 1(θ) = η (2) γ 2(θ) = θ (1,max) =θ (1,r) = η (1) α cc c α cc c τ 1 θ 1 τ 1 θ 1 1/2 pole at τ 1 for ϕ 2 (z) 1/2 pole at τ 1 for ϕ 2 (z) f c (x) = { x 3/2 e αcx if η (1) = θ (1,max) θ (1,r), x 1/2 e αcx if η (1) = θ (1,max) = θ (1,r). Jim Dai (Georgia Tech) Exact asymptotics July 8, / 31

23 Category I: β = β 1 < β 2, τ 1 < θ (1,max) 1 τ 2 θ 2 γ 1 (θ) = θ (2,Γ) =θ (2,r) η (2) c η (1) = α c c θ (1,max) γ 2 (θ) = θ (1,Γ) =θ (1,r) τ 1 θ 1 ψ c (z) = γ 1(c)ϕ 1 (c 2 z) + γ 2 (c)ϕ 2 (c 1 z) ζ c (z) for Rz < α c. f c (x) = xe αcx Jim Dai (Georgia Tech) Exact asymptotics July 8, / 31

24 Category I: β = β 1 < β 2, τ 1 = θ (1,max) 1 θ 2 γ 1(θ) = τ 2 θ (2,Γ) = θ (2,r) η (2) γ 2(θ) = θ (1,r) α cc θ (1,Γ) = θ (1,max) = η (1) τ 2 θ 2 γ 1(θ) = η (2) γ 2(θ) = α cc θ (1,max) =θ (1,r) = η (1) τ 1 θ 1 τ 1 θ 1 1/2 pole at τ 1 for ϕ 2 (z) 1/2 pole at τ 1 for ϕ 2 (z) f c (x) = { x 1/2 e αcx if η (1) = θ (1,max) θ (1,r), e αcx if η (1) = θ (1,max) = θ (1,r). Jim Dai (Georgia Tech) Exact asymptotics July 8, / 31

25 Category I: β = β 1 = β 2 f 2 (τ 1 ) θ 2 γ 1 (θ) = c θ (2,r) τ 2 τ θ (1,max) θ (1,r) γ 2 (θ) = τ 1 f 1 (τ 2 ) θ 1 ψ c (z) = γ 1(c)ϕ 1 (c 2 z) + γ 2 (c)ϕ 2 (c 1 z) for Rz < α c. ζ c (z) { xe αcx if η (1) = η (2) = τ Γ, f c (x) = e αcx if η (1) = η (2) = τ interior of Γ. Jim Dai (Georgia Tech) Exact asymptotics July 8, / 31

26 Theorem 3: exact asymptotics for Category II Jim Dai (Georgia Tech) Exact asymptotics July 8, / 31

27 Category II: β < β 1 : τ 1 < θ (1,max) 1 θ2 γ2(θ) = θ2 θ (1,r) γ1(θ) = θ (1,Γ) =θ (1,max) γ1(θ) = θ (1,max) γ2(θ) = τ2 θ (2,Γ) =θ (2,r) τ = η (1) = η (2) c αcc θ (2,Γ) = θ (2,r) τ2 τ = η (1) = η (2) = θ (1,r) c αcc τ1 θ1 τ1 θ1 single pole at τ 1 for ϕ 2 (z) f c (x) = { e αcx xe αcx double pole at τ 1 for ϕ 2 (z) if τ θ (1,r) if τ = θ (1,r) Jim Dai (Georgia Tech) Exact asymptotics July 8, / 31

28 Category II: β < β 1 : τ 1 = θ (1,max) 1 γ1(θ) = θ2 γ2(θ) = γ1(θ) = θ2 θ (1,r) γ2(θ) = θ (2,Γ) =θ (2,r) τ2 θ (1,Γ) = θ (1,max) = τ = η (1) = η (2) c αcc θ (2,Γ) =θ (2,r) τ2 θ (1,Γ) = θ (1,max) = θ (1,r) = τ = η (1) = η (2) c αcc τ1 θ1 τ1 θ1 1/2 pole at τ 1 for ϕ 2 (z) 1 pole at τ 1 for ϕ 2 (z) f c (x) = { x 1/2 e αcx, if τ θ (1,r), e αcx, if τ = θ (1,r). Jim Dai (Georgia Tech) Exact asymptotics July 8, / 31

29 Category II: β = β 1, τ 1 < θ (1,max) 1 θ2 γ2(θ) = θ2 θ (1,r) γ1(θ) = θ (1,Γ) =θ (1,max) γ1(θ) = θ (1,max) γ2(θ) = τ2 θ (2,Γ) =θ (2,r) τ = η (1) = η (2) αcc θ (2,Γ) = θ (2,r) τ2 αcc τ = η (1) = η (2) = θ (1,r) τ1 θ1 τ1 θ1 single pole at τ 1 for ϕ 2 (z) double pole at τ 1 for ϕ 2 (z) f c (x) = xe αcx. Jim Dai (Georgia Tech) Exact asymptotics July 8, / 31

30 Category II: β = β 1, τ 1 = θ (1,max) 1 γ1(θ) = θ2 γ2(θ) = γ1(θ) = θ2 θ (1,r) γ2(θ) = θ (2,Γ) =θ (2,r) τ2 αcc θ (1,Γ) = θ (1,max) = τ = η (1) = η (2) θ (2,Γ) =θ (2,r) τ2 αcc θ (1,Γ) = θ (1,max) = θ (1,r) = τ = η (1) = η (2) τ1 θ1 τ1 θ1 1/2 pole at τ 1 for ϕ 2 (z) 1 pole at τ 1 for ϕ 2 (z) f c (x) = xe αcx. Jim Dai (Georgia Tech) Exact asymptotics July 8, / 31

31 Category II: β 1 < β π/2 θ2 γ1(θ) = θ2 γ2(θ) = γ1(θ) = θ (2,Γ) = θ (2,r) τ2 αcc c θ (1,max) γ2(θ) = τ = η (1) = η (2) = θ (1,r) θ (2,Γ) =θ (2,r) τ2 αcc θ (1,r) c θ (1,Γ) = θ (1,max) = τ = η (1) = η (2) τ1 θ1 τ1 θ1 f c (x) = e αcx. Jim Dai (Georgia Tech) Exact asymptotics July 8, / 31

32 References J. G. Dai and M. Miyazawa (211), Semimartinagle Reflecting Brownian motion in two dimensions: Exact asymptotics for the stationary distribution, Stochastic Systems, to appear. M. Miyazawa and M. Kobayashi, M. (211). Conjectures on tail asymptotics of the stationary distribution for a multidimensional SRBM. Queueing Systems, to appear. Jim Dai (Georgia Tech) Exact asymptotics July 8, / 31