Stein s Method for Steady-State Approximations: Error Bounds and Engineering Solutions

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1 Stein s Method for Steady-State Approximations: Error Bounds and Engineering Solutions Jim Dai Joint work with Anton Braverman and Jiekun Feng Cornell University Workshop on Congestion Games Institute for Mathematical Sciences, Singapore December 16, 2015

2 M/M/n queue Erlang-C model exp(µ) exp(µ) Poisson(λ) arrival process Buffer exp(µ). exp(µ) Background Error Bounds Stein s Method High Order Other Systems 2/35

3 Markov chain and its transitions λ λ λ λ n 1 n n µ 2µ nµ nµ X = {X(t), t 0} is a CTMC on Z + = {0, 1,..., }. Generator ( ) G X f(i) = λ f(i + 1) f(i) ( ) + min(i, n)µ f(i 1) f(i) for i Z + Assume R λ/µ < n. Random variable X( ) has the stationary distribution. Background Error Bounds Stein s Method High Order Other Systems 3/35

4 M/M/1 queue: R =.95 X( ) is geometric: P{X( ) = i} = (1 R)R i, i Z + Probability Geometric Exponential Number of customers Continuous random variable Y ( ) exp(.05) Background Error Bounds Stein s Method High Order Other Systems 4/35

5 M/M/ queue: R = 500 X( ) is Poisson(500). Probability Poisson Normal Number of customers Continuous random variable Y ( ) N(500, 500) Background Error Bounds Stein s Method High Order Other Systems 5/35

6 M/M/n: distribution of X( )? Engineering solution: identify a continuous random variable Y ( ) Error bound: bound distance between X( ) and Y ( ) Stein s method: is able to achieve both Known results: M/M/n + M (Braverman, Dai & Feng 15) M/P h/n + M (Braverman, & Dai 15) A discrete time queue arising from hospital patient flow (Dai & Shi 15) Many systems having mean field limits (Ying 15)... (your papers) Background Error Bounds Stein s Method High Order Other Systems 6/35

7 Outline Error Bounds: Erlang-C model (Braverman-D-Feng 2015) Stein s Method: proof framework and solution technique Engineering Solution: High order approximations (Braverman & Dai 2015) Background Error Bounds Stein s Method High Order Other Systems 7/35

8 Erlang-C Model M/M/n Queue Steady-state number of customers in system X( ). Define X( ) = X( ) R R. X( ) lives on grid { x = δ(i R), i Z+ }, δ = 1/ R. Theorem 1 (Braverman-D-Feng 2015) For all n 1, λ > 0, µ > 0 with 1 R < n, where sup Eh( X( )) Eh(Y ( )) 157, h Lip(1) R Lip(1) = {h : R R, h(x) h(y) x y }. Background Error Bounds Stein s Method High Order Other Systems 8/35

9 The continuous random variable Y ( ) Y ( ) has density ( 1 x ) κ exp b(y)dy, (1) µ 0 where and b(x) = { µx, x ζ, µζ, x ζ ζ = R n R < 0. (2) Background Error Bounds Stein s Method High Order Other Systems 9/35

10 Discussions Corollary For all n 1, λ > 0 and µ > 0 with 1 λ/µ < n, EX( ) R REY ( ) 157. Not a limit theorem For µ = 1. n = 5 n = 500 λ EX( ) Error λ EX( ) Error Universal Background Error Bounds Stein s Method High Order Other Systems 10/35

11 Universal approximation Y ( ) depends on system parameters λ, n and µ: Y ( ) f(x) { N(0, 1) if x < ζ, Exponential( ζ ) if x ζ. X( ) < n behaves like a normal, and X( ) n behaves like an exponential. Gurvich, Huang, Mandelbaum (2014), Mathematics of Operations Research Glynn, Ward (2003), Queueing Systems Background Error Bounds Stein s Method High Order Other Systems 11/35

12 Approximating Diffusion Process A diffusion process Y = {Y (t) R, t 0} satisfies the stochastic differential equation t t Y (t) = Y (0) + b(y (s))ds + σ(y (s))db(s). 0 0 Y ( ) in (1) is the stationary distribution of diffusion process with drift b(x) in (2) and diffusion coefficient σ 2 (x) 2µ for now. Identifying Y ( ) is equivalent to identifying the generator of a diffusion process Gf(x) = 1 2 σ2 (x)f (x) + b(x)f (x) for f C 2 (R). Background Error Bounds Stein s Method High Order Other Systems 12/35

13 A Proof Outline for Theorem 1 Background Error Bounds Stein s Method High Order Other Systems 13/35

14 Stein Framework for Proofs Poisson (Stein) equation and gradient bounds Basic Adjoint Relationship (BAR) and the generator coupling State space collapse (SSC) Moment bounds Stein ( 72, 86) Louis Chen (75 ) Andrew Barbour (88 ) Chen, Goldstein, and Shao (2011) Gurvich (2014), Diffusion models and steady-state approximations for exponentially ergodic Markovian queues, Annals of Applied Probability. Background Error Bounds Stein s Method High Order Other Systems 14/35

15 Poisson Equation Generator of diffusion process Y = {Y (t), t 0} G Y f(x) = µf (x) + b(x)f (x) Given an h Lip(1), solve f = f h from the Poisson equation G Y f(x) = h(x) E[h(Y ( ))], x R. On each sample path of X( ) Key identity G Y f( X( )) = h( X( )) E[h(Y ( ))]. E[h( X( ))] E[h(Y ( ))] = E[G Y f( X( ))] Background Error Bounds Stein s Method High Order Other Systems 15/35

16 Basic Adjoint Relationship Suppose X = {X(t), t 0} is a CTMC on S = {1, 2, 3} with generator G = Unique π = (π(1), π(2), π(3)) that satisfies f(1) f(1) πg f(2) = 0 for each f = f(2) R 3. f(3) f(3) Alternatively, E [ Gf(X( )) ] = 0 for each f R 3. (3) When the state space is infinite, Glynn and Zeevi (2008, Kurtz Festschrift ) provides conditions on f for (3) to hold. Background Error Bounds Stein s Method High Order Other Systems 16/35

17 Generator Coupling From the Stein equation, E [ h( X( )) ] E [ h(y ( ))] =E [ G Y f h ( X( )) ] =E [ G Y f h ( X( )) ] E [ G Xf h ( X( )) ] [ =E G Y f h ( X( )) G Xf h ( X( )) ]. X( ) lives on grid { x = δ(i R), i Z+ }, δ = 1/ R. The generator of birth-death process X is ( ( ) ) ( ( ) ) G Xf h (x) = λ f h x + δ fh (x) + µ(i n) f h x δ fh (x). Background Error Bounds Stein s Method High Order Other Systems 17/35

18 Taylor Expansion To bound [ E G Y f h ( X( )) G Xf h ( X( )) ] one bounds G Y f h (x) G Xf h (x) for x = δ(i R) with i Z +. Conduct Taylor expansion ( ) G Xf h (x) = f h(x)δ λ µ(i n) f h (x)δ 2( ) λ + µ(i n) + higher order term ( ) = f h(x)δ λ µ(i n) f h (x)δ 2( 2λ) 1 2 f h (x)δ 2( ) λ µ(i n) + higher order term Background Error Bounds Stein s Method High Order Other Systems 18/35

19 Approximation One can check that δ 2 λ = µ and ( ) ( ) δ λ µ(i n) = µ (x + ζ) + ζ = b(x). From the CTMC generator we extract G Xf h (x) = f h(x)b(x) + µf h (x) Typical error term 1 δf h (x)b(x) + higher order terms 2 = G Y f h (x) 1 δf h (x)b(x) + higher order terms. 2 δe f h ( X( ))b( X( )), b(x) µ x Background Error Bounds Stein s Method High Order Other Systems 19/35

20 Gradient Bounds For all λ, n, and µ satisfying n 1 and 0 < λ < nµ, f h (x) 18 µ (1 + 1/ ζ ), x ζ, 1 µ ζ, x ζ, f h (x) { 1 µ ( / ζ ), x < ζ, 2/µ, x > ζ. Recall ζ = R n R. Background Error Bounds Stein s Method High Order Other Systems 20/35

21 Moment Bounds For all λ, n, and µ satisfying n 1 and 0 < R < n, Recall [ E ( X( )) 2 1( X( ) ] ζ) δ2 [ ] E X( )1( X( ) ζ) [ ] E X( )1( X( ) ζ) 2 ζ 3 + δ 3, δ2 3 + δ 3, [ ] E X( )1( X( ) ζ) 1 ζ + δ2 4 ζ + δ 2, P( X( ) ζ) (2 + δ) ζ. δ 2 = 1/R. Background Error Bounds Stein s Method High Order Other Systems 21/35

22 Moment Version of Theorem 1 Theorem 1 (b) Recall that ζ = (R n)/ R. There exists a constant C = C(m) such that for all n 1, λ > 0 and µ > 0 with 1 R < n, sup E( X( )) m E(Y ( )) m ( x R ζ m 1 ) C(m) R. For n = 500, µ = 1. λ E( X( )) 2 Error E( X( )) 10 Error Not universal Background Error Bounds Stein s Method High Order Other Systems 22/35

23 Kolmogorov Metric Version of Theorem 1 Theorem 1 (c) For all n 1, λ > 0 and µ > 0 with 1 R < n, sup P { } { } X( ) 190 x P Y ( ) x. R x R Background Error Bounds Stein s Method High Order Other Systems 23/35

24 An M/M/250 System Consider x = (i x( ))/ R, where i = 0, 1, 2, 3,... n = 250, mu = 1, lambda = Exact Const. Diffusion Coeff Figure: P ( X( ) = x), P(x 0.5 Y ( ) x + 0.5) Background Error Bounds Stein s Method High Order Other Systems 24/35

25 An M/M/5 System With only 5 servers, diffusion approximation not as good n = 5, mu = 1, lambda = 4 Exact Const. Diffusion Coeff x Figure: P ( X( ) = x), P(x 0.5 Y ( ) x + 0.5) Background Error Bounds Stein s Method High Order Other Systems 25/35

26 High Order Approximation M/M/5 n = 5, mu = 1, lambda = Exact Const. Diffusion Coeff. High Order Approximation x Figure: P(x 0.5 Y H ( ) x + 0.5) Background Error Bounds Stein s Method High Order Other Systems 26/35

27 High Order Approximation M/M/25 n = 25, mu = 1, lambda = Exact Const. Diffusion Coeff. High Order Approximation x Background Error Bounds Stein s Method High Order Other Systems 27/35

28 High order results n = 5 n = 500 λ EX( ) Error λ EX( ) Error Background Error Bounds Stein s Method High Order Other Systems 28/35

29 High order results Higher Moments n = 500, µ = 1 λ E( X( )) 2 Error E( X( )) 10 Error Background Error Bounds Stein s Method High Order Other Systems 29/35

30 Fixed ζ = 1/2 n λ E( X( )) 2 Error High Order Approx. Error λ = R increases by a factor of 10 Error decreases by a factor of 10 High order approx. error decreases by a factor of 10. Background Error Bounds Stein s Method High Order Other Systems 30/35

31 Deriving High order approximation Recall the Taylor expansion G Xf h (x) = f h(x)b(x) + µf h (x) 1 δb(x)f h (x) 2 + higher order terms Recall the Taylor expansion ( ) G Xf h (x) = f h(x)δ λ µ(i n) f h (x)δ 2( ) λ + µ(i n) f h (x)δ 3( ) λ µ(i n) + fourth order term ( ) = b(x)f (x) + µ δb(x)/2 f (x) δ3 f h (x)b(x) + fourth order term Use entire second order term and bound δe f h ( X( ))b( X( )), b(x) µ x Background Error Bounds Stein s Method High Order Other Systems 31/35

32 High Order Approximation Y H ( ) corresponds to diffusion process with generator G YH f(x) = 1 2 σ2 (x)f (x) + b(x)f (x), f C 2 (R), σ 2 (x) = µ + ( µ δb(x) ) 1(x R) µ, x R. Previously, used σ 2 (x) = 2µ. Theorem 2 (High Order Approximation, Braverman-D 2015) C W2 > 0 (explicit) such that for all n 1, 1 R < n, sup h W 2 Eh( X( )) 1 Eh(Y H ( )) C W2 R, W 2 = {h : R R, h(x) h(y) x y, h (x) h (y) x y }. Background Error Bounds Stein s Method High Order Other Systems 32/35

33 Other Works M/P h/n + M systems with phase-type service time distribution and customer abandonment Halfin-Whitt or QED regime (Braverman-D 2015) Erlang-A system (Braverman-D-Feng 2015) A discrete time queue arising from hospital patient flow management (Dai & Shi 15 ) Mean field analysis (Ying 2015) Background Error Bounds Stein s Method High Order Other Systems 33/35

34 References Braverman, Dai, Stein s method for steady-state diffusion approximations of M/P h/n + M systems, Braverman, Dai, Feng, Stein s method for steady-state diffusion approximations: an introduction, working paper. Gurvich 2014, Diffusion models and steady-state approximations for exponentially ergodic Markovian queues, Annals of Applied Probability, Vol. 24, Gurvich, Huang, Mandelbaum, 2014, Excursion-Based Universal Approximations for the Erlang-A Queue in Steady-State, Mathematics of Operations Research, Vol. 39(2), Background Error Bounds Stein s Method High Order Other Systems 34/35

35 References Ying, On the Rate of Convergence of Mean-Field Models: Stein s Method Meets the Perturbation Theory, J. G. Dai and Pengyi Shi (2015), A Two-Time-Scale Approach to Time-Varying Queues in Hospital Inpatient Flow Management, Submitted to Operations Research. Background Error Bounds Stein s Method High Order Other Systems 35/35