Reflected Brownian motion

In probability theory, reflected Brownian motion (or regulated Brownian motion,^[1][2] both with the acronym RBM) is a Wiener process in a space with reflecting boundaries.^[3]

RBMs have been shown to describe queueing models experiencing heavy traffic^[2] as first proposed by Kingman^[4] and proven by Iglehart and Whitt.^[5][6]

Definition

A d–dimensional reflected Brownian motion Z is a stochastic process on uniquely defined by

• a d–dimensional drift vector μ
• a d×d non-singular covariance matrix Σ and
• a d×d reflection matrix R.^[7]

where X(t) is an unconstrained Brownian motion and^[8]

with Y(t) a d–dimensional vector where

• Y is continuous and non–decreasing with Y(0) = 0
• Y_j only increases at times for which Z_j = 0 for j = 1,2,...,d
• Z(t) ∈ , t ≥ 0.

The reflection matrix describes boundary behaviour. In the interior of the process behaves like a Wiener process, on the boundary "roughly speaking, Z is pushed in direction R^j whenever the boundary surface is hit, where R^j is the jth column of the matrix R."^[8]

Stability conditions

Stability conditions are known for RBMs in 1, 2, and 3 dimensions. "The problem of recurrence classification for SRBMs in four and higher dimensions remains open."^[8] In the special case where R is an M-matrix then necessary and sufficient conditions for stability are^[8]

1. R is a non-singular matrix and
2. R⁻¹μ < 0.

Marginal and stationary distribution

One dimension

The marginal distribution (transient distribution) of a one-dimensional Brownian motion starting at 0 restricted to positive values (a single reflecting barrier at 0) with drift μ and variance σ² is

for all t ≥ 0, (with Φ the cumulative distribution function of the normal distribution) which yields (for μ < 0) when taking t → ∞ an exponential distribution^[2]

For fixed t, the distribution of Z(t) coincides with the distribution of the running maximum M(t) of the Brownian motion,

But be aware that the distributions of the processes as a whole are very different. In particular, M(t) is increasing in t, which is not the case for Z(t).

The heat kernel for reflected Brownian motion at :

For the plane above

Multiple dimensions

The stationary distribution of a reflected Brownian motion in multiple dimensions is tractable analytically when there is a product form stationary distribution,^[9] which occurs when the process is stable and^[10]

where D = diag(Σ). In this case the probability density function is^[7]

where η_k = 2μ_kγ_k/Σ_kk and γ = R⁻¹μ. Closed-form expressions for situations where the product form condition does not hold can be computed numerically as described below in the simulation section.

Simulation

One dimension

In one dimension the simulated process is the absolute value of a Wiener process. The following MATLAB program creates a sample path.^[11]

%rbm.m
n=10^4; h=10^(-3); t=h.*(0:n); mu=-1;
X=zeros(1,n+1); M=X; B=X;
B(1)=3; X(1)=3;
for k=2:n+1
  Y=sqrt(h)*randn; U=rand(1);
  B(k)=B(k-1)+mu*h-Y;
  M=(Y + sqrt(Y^2-2*h*log(U)))/2;
  X(k)=max(M-Y,X(k-1)+h*mu-Y);
end
subplot(2,1,1)
plot(t,X,'k-');
subplot(2,1,2)
plot(t,X-B,'k-');

The error involved in discrete simulations has been quantified.^[12]

Multiple dimensions

QNET allows simulation of steady state RBMs.^[13][14][15]

Other boundary conditions

Feller described possible boundary condition for the process^[16][17][18]

• absorption^[16] or killed Brownian motion,^[19] a Dirichlet boundary condition
• instantaneous reflection,^[16] as described above a Neumann boundary condition
• elastic reflection, a Robin boundary condition
• delayed reflection^[16] (the time spent on the boundary is positive with probability one)
• partial reflection^[16] where the process is either immediately reflected or is absorbed
• sticky Brownian motion.^[20]

See also

• Skorokhod problem

References

Queueing theory
Single queueing nodes	D/M/1 queue M/D/1 queue M/D/c queue M/M/1 queue Burke's theorem M/M/c queue M/M/∞ queue M/G/1 queue Pollaczek–Khinchine formula Matrix analytic method M/G/k queue G/M/1 queue G/G/1 queue Kingman's formula Lindley equation Fork–join queue Bulk queue
Arrival processes	Poisson process Markovian arrival process Rational arrival process
Queueing networks	Jackson network Traffic equations Gordon–Newell theorem Mean value analysis Buzen's algorithm Kelly network G-network BCMP network
Service policies	FIFO LIFO Processor sharing Round-robin Shortest job first Shortest remaining time
Key concepts	Continuous-time Markov chain Kendall's notation Little's law Product-form solution Balance equation Quasireversibility Flow-equivalent server method Arrival theorem Decomposition method Beneš method
Limit theorems	Fluid limit Mean field theory Heavy traffic approximation Reflected Brownian motion
Extensions	Fluid queue Layered queueing network Polling system Adversarial queueing network Loss network Retrial queue
Information systems	Data buffer Erlang (unit) Erlang distribution Flow control (data) Message queue Network congestion Network scheduler Pipeline (software) Quality of service Scheduling (computing) Teletraffic engineering
Category