### Citations

694 |
Markov processes: Characterization and convergence
- Ethier, Kurtz
- 1986
(Show Context)
Citation Context ...1 2 d∑ i,j=1 Σij ∂f ∂xi∂xj + d∑ i=1 bi(x) ∂f ∂xi . The following lemma says that the converse is also true. The proof simply verifies all the conditions in Echeverría’s theorem; see Theorem 4.9.17 of =-=[12]-=-. Lemma 1 If pi is a probability measure that satisfies (3.2), then it is a stationary distribution. Proof Clearly b is locally bounded. Because b is assumed to be Lipschitz, there exists some K > 0 s... |

383 |
Nonnegative Matrices in the Mathematical Sciences
- Berman, Plemmons
- 1994
(Show Context)
Citation Context ...ion operator conditioned on Z(0) having stationary distribution pi . (Harrison and Williams [15] consider an SRBM with reflection matrix R being an M-matrix, a class of matrices defined in Chap. 6 of =-=[1]-=-; but its proof can be straightforwardly carried over to a general reflection matrix.) For a Borel set A⊂Rd+, define νi (A)= Epi ∫ 1 0 1{Z(u)∈A} dYi(u), i = 1, . . . , d. Clearly, νi defines a finite ... |

103 |
Reflected Brownian motion on an orthant
- HARRISON, REIMAN
- 1981
(Show Context)
Citation Context ...Rd . For a signed measure pi on Rd , we use pi = pi+ − pi− to denote its Jordan decomposition and set |pi | = pi+ + pi−. 2 SRBMs Multidimensional SRBMs are first introduced in the pioneering paper of =-=[14]-=-; see [9, 18] for surveys on SRBMs. The state space for a d-dimensional SRBM Z = {Z(t), t ≥ 0} in Rd+. The data of the SRBM are a drift vector µ, a non-singular covariance matrix Σ , and a d × d “refl... |

89 |
Brownian models of open queueing networks with homogeneous customer populations, Stochastics 22
- Harrison, Williams
- 1987
(Show Context)
Citation Context ... stationary distribution. A necessary condition of the existence of the stationary distribution is R is non-singular and R−1µ< 0; (2.5) see, for example, [2] for a proof. It follows from the proof in =-=[15]-=- that the stationary distribution pi is unique, and each component of Epi (Y (1)) is finite and Epi (Y (1))= −R−1µ > 0, Epi is the expectation operator conditioned on Z(0) having stationary distributi... |

60 |
Brownian models of multiclass queueing networks: Current status and open problems, Queueing Systems 13
- Harrison, Nguyen
- 1993
(Show Context)
Citation Context ...ch.edu A.B. Dieker e-mail: ton.dieker@isye.gatech.edu 296 Queueing Syst (2011) 68:295–303 conventional heavy traffic when the number of servers at each station is fixed to be small; see, for example, =-=[13]-=-. A d-dimensional piecewise OU process is an unconstrained diffusion process that lives in Rd . These diffusion processes serve as diffusion approximations for many-server queues when the service time... |

53 |
The multiclass GI/PH/N queue in the Halfin-Whitt regime
- Puhalskii, Reiman
- 2000
(Show Context)
Citation Context ...ocess is an unconstrained diffusion process that lives in Rd . These diffusion processes serve as diffusion approximations for many-server queues when the service time distributions are of phase-type =-=[7, 17]-=-. The open problem was first stated as a conjecture in [3] for SRBMs in a twodimensional rectangle and in [4] for SRBMs in a d-dimensional orthant. The conjecture is a critical ingredient in determini... |

49 | Reflected Brownian motion in an orthant: numerical methods for steady-state analysis
- Dai, Harrison
- 1992
(Show Context)
Citation Context ...ximations for many-server queues when the service time distributions are of phase-type [7, 17]. The open problem was first stated as a conjecture in [3] for SRBMs in a twodimensional rectangle and in =-=[4]-=- for SRBMs in a d-dimensional orthant. The conjecture is a critical ingredient in determining stationary distributions of SRBMs, both numerically and analytically. For instance, numerical algorithms h... |

37 |
Semimartingale reflecting Brownian motions in the orthant
- Williams
- 1995
(Show Context)
Citation Context ... signed measure pi on Rd , we use pi = pi+ − pi− to denote its Jordan decomposition and set |pi | = pi+ + pi−. 2 SRBMs Multidimensional SRBMs are first introduced in the pioneering paper of [14]; see =-=[9, 18]-=- for surveys on SRBMs. The state space for a d-dimensional SRBM Z = {Z(t), t ≥ 0} in Rd+. The data of the SRBM are a drift vector µ, a non-singular covariance matrix Σ , and a d × d “reflection matrix... |

25 | Steady-state analysis of RBM in a rectangle: Numerical methods and a queueing application. The Annals of Applied Probability
- Dai, Harrison
- 1991
(Show Context)
Citation Context ... These diffusion processes serve as diffusion approximations for many-server queues when the service time distributions are of phase-type [7, 17]. The open problem was first stated as a conjecture in =-=[3]-=- for SRBMs in a twodimensional rectangle and in [4] for SRBMs in a d-dimensional orthant. The conjecture is a critical ingredient in determining stationary distributions of SRBMs, both numerically and... |

19 | Many-server diffusion limits for G/Ph/n +GI queues
- Dai, He, et al.
- 2010
(Show Context)
Citation Context ...ocess is an unconstrained diffusion process that lives in Rd . These diffusion processes serve as diffusion approximations for many-server queues when the service time distributions are of phase-type =-=[7, 17]-=-. The open problem was first stated as a conjecture in [3] for SRBMs in a twodimensional rectangle and in [4] for SRBMs in a d-dimensional orthant. The conjecture is a critical ingredient in determini... |

16 | Positive recurrence of reflecting Brownian motion in three dimensions
- Bramson, Dai, et al.
- 2010
(Show Context)
Citation Context ...on some filtered probability space (Ω, {Ft },P), such that (X,Y,Z) is adapted to {Ft }, X is an {Ft }-Brownian motion, and (X,Y,Z) satisfies (2.1)–(2.4) almost surely; see, for example, Appendix A of =-=[2]-=- for a complete definition. A d × d matrix R is said to be an S-matrix if there exists a d-vector w ≥ 0 such that Rw > 0, and R is said to be completely-S if each of its principal sub-matrices is an S... |

16 |
Characterization of the stationary distribution for a semimartingale reflecting Brownian motion in a convex polyhedron
- Dai, Kurtz
- 1994
(Show Context)
Citation Context ...ported on Fi , i = 1, . . . , d . If (pi,ν1, . . . ,νd) jointly satisfies BAR (2.6), then pi equals the stationary distribution of Z up to a multiplicative constant. Proposition 1 was first proved in =-=[6]-=- for SRBMs in polyhedron domains and more recently in [8] for a class of reflecting Brownian motions that do not necessarily have the semimartingale representation as in (2.1). We are now in a positio... |

13 | Positive recurrence of piecewise Ornstein-Uhlenbeck processes and common quadratic Lyapunov functions. Annals of Applied Probability
- Dieker, Gao
- 2012
(Show Context)
Citation Context ...3 The conditions, if any, should be mild so that they are satisfied for the piecewise OU process arising from the many-server diffusion approximations introduced in [7]. It has recently been shown in =-=[10]-=- that these piecewise OU processes have a unique stationary distribution. For these diffusions, the open problem thus reduces to showing that any pi satisfying (3.2) is proportional to the stationary ... |

11 | Reflected Brownian motion in a wedge: sum-of-exponential stationary densities
- Dieker, Moriarty
- 2009
(Show Context)
Citation Context ...cesses [5]. Moreover, in some known cases where a stationary distribution of an SRBM has been determined analytically, an essential and challenging step is to verify that the measure is unsigned; see =-=[11, 16]-=-. The validity of the conjecture would remove the need for this verification. The open problem could be discussed in a general setting such as general diffusion processes or even general Markov proces... |

7 |
Heavy traffic analysis of two coupled processors
- Knessl, Morrison
- 2003
(Show Context)
Citation Context ...cesses [5]. Moreover, in some known cases where a stationary distribution of an SRBM has been determined analytically, an essential and challenging step is to verify that the measure is unsigned; see =-=[11, 16]-=-. The validity of the conjecture would remove the need for this verification. The open problem could be discussed in a general setting such as general diffusion processes or even general Markov proces... |

2 |
S.: Computing stationary distributions for diffusion models of many-server queues
- Dai, He
- 2011
(Show Context)
Citation Context ...ary distribution of an SRBM. The convergence of these algorithms critically relies on the validity of the conjecture, and the open problem plays the same role in the context of piecewise OU processes =-=[5]-=-. Moreover, in some known cases where a stationary distribution of an SRBM has been determined analytically, an essential and challenging step is to verify that the measure is unsigned; see [11, 16]. ... |

2 |
T.G.: Characterization of the stationary distribution for a reflecting Brownian motion in a convex polyhedron
- Dai, Guettes, et al.
- 2010
(Show Context)
Citation Context ...ointly satisfies BAR (2.6), then pi equals the stationary distribution of Z up to a multiplicative constant. Proposition 1 was first proved in [6] for SRBMs in polyhedron domains and more recently in =-=[8]-=- for a class of reflecting Brownian motions that do not necessarily have the semimartingale representation as in (2.1). We are now in a position to formulate our first open problem. Open problem 1 Pro... |

2 |
Reflected Brownian motion
- Dieker
- 2010
(Show Context)
Citation Context ... signed measure pi on Rd , we use pi = pi+ − pi− to denote its Jordan decomposition and set |pi | = pi+ + pi−. 2 SRBMs Multidimensional SRBMs are first introduced in the pioneering paper of [14]; see =-=[9, 18]-=- for surveys on SRBMs. The state space for a d-dimensional SRBM Z = {Z(t), t ≥ 0} in Rd+. The data of the SRBM are a drift vector µ, a non-singular covariance matrix Σ , and a d × d “reflection matrix... |