Asymptotic Coupling of an SPDE, with Applications to Many-Server Queues Mohammadreza Aghajani joint work with Kavita Ramanan Brown University March 2014 Mohammadreza Aghajanijoint work Asymptotic with Kavita Coupling Ramanan of(brown an SPDE, University) with Applications Marchto2014 Many-Server 1 / 16 Qu
Many-Server Queues λ N 1 2 3 μ μ μ N μ Where do they arise? Call Centers Health Care Data Centers Relevant Steady state performance: P ss {all N servers are busy} P ss {waiting time > t seconds} ohammadreza Aghajanijoint work Asymptotic with Kavita Coupling Ramanan of(brown an SPDE, University) with Applications Marchto2014 Many-Server 2 / 16 Qu
Asymptotic Analysis Exact analysis for finite N is typically infeasible. state variable appropriately centered/scaled steady state distribution Classic pre-limit result Process Level Convergence Convergence of Stationary Dist. Unique Invariant Distribution limit process (Markov) invariant distirbution Mohammadreza Aghajanijoint work Asymptotic with Kavita Coupling Ramanan of(brown an SPDE, University) with Applications Marchto2014 Many-Server 3 / 16 Qu
Exponential Service Distribution Exponential service time case is studied in [Halfin-Whitt 81] Markov Process # of customers in system at time t Classic pre-limit result Process Level Convergence Convergence of Stationary Dist. Unique Invariant Distribution Gaussian Exponential (1-dimensional diffusion) Mohammadreza Aghajanijoint work Asymptotic with Kavita Coupling Ramanan of(brown an SPDE, University) with Applications Marchto2014 Many-Server 4 / 16 Qu
General Service Distribution Statistical data shows that service times are generally distributed (Lognormal, Pareto, etc. see e.g. [Brown et al. 05]) Goal: To extend the result to general service distribution Challenges {X (N) = # of customers in system} is no longer a Markov Process need to keep track of residual times or ages of customers in service to make the process Markovian Dimension of any finite-dim. Markovian representation grows with N Results using X N obtained by [Puhalskii and Reed], [Reed], [Mandelbaum and Momcilovic], [Dai and He], etc. However, few results on stationary distribution (beyond phase-type distributions) A way out: Common State Space (infinite-dimensional) Mohammadreza Aghajanijoint work Asymptotic with Kavita Coupling Ramanan of(brown an SPDE, University) with Applications Marchto2014 Many-Server 5 / 16 Qu
A Measure-Valued Representation A measure-valued state descriptor [Kaspi-Ramanan 11]: (X (N), ν (N) ) ν (N) (N) Entry into Service K Departure Process D (N) (N) E age a j (N) t E (N) represents the cumulative external arrivals a (N) j represents age of the jth customer to enter service ν (N) keeps track of the ages of all the customers in service ν (N) t = j δ (N) a (t) j Mohammadreza Aghajanijoint work Asymptotic with Kavita Coupling Ramanan of(brown an SPDE, University) with Applications Marchto2014 Many-Server 6 / 16 Qu
A Central Limit Theorem An important functional: Z (N) t (r) = Take the state variable Y (N) t j in service 1 G(a j (t) + r), r (0, ). 1 G(a j (t)) = (X (N) t, Z (N) t ) R H 1 (0, ) Lemma (Aghajani and R 13, Corollary of Kaspi and R 13) Under Assumptions on G, if ŷ (N) 0 y 0 in R H 1 (0, ), then for every t 0, Ŷ (N) t Y t in R H 1 (0, ), for some continuous Markov process {Y t = (X t, Z t ); t 0}. To make Y have all the desired properties, the choice of space is subtle. ohammadreza Aghajanijoint work Asymptotic with Kavita Coupling Ramanan of(brown an SPDE, University) with Applications Marchto2014 Many-Server 7 / 16 Qu
Main focus: Limit Process When z 0 H 1 (0, ) C 1 [0, ), the limit process is the unique solution of the coupled Ito process/spde: dx t = dw t + db t βdt + Z t(0)dt, dz t (r) = [ Z t(r) Ḡ(r)Z t(0) ] dt dm t (r) + Ḡ(r)dZ t(0), with boundary condition Z t (0) = X t = X t 0. M is a martingale measure, correlated to the Brownian motion W. W and B are independent Brownian motions. Remarks: Unusual SPDE boundary conditions enter the drift term. The process lies in the infinite-dimensional space R H 1 (0, ). For general initial condition z 0, Y is a solution to a slightly more complicated SPDE. We are the interested in the invariant distribution of the Markov process Y. Mohammadreza Aghajanijoint work Asymptotic with Kavita Coupling Ramanan of(brown an SPDE, University) with Applications Marchto2014 Many-Server 8 / 16 Qu
Invariant Distribution: Existence Theorem (Aghajani and R 13) Under assumptions on G, the limit process has an invariant distribution. Main tool: Krylov-Bogoliubov [ 37] Theorem. Need to show the tightness for family of occupation measures µ(γ) = 1 T T 0 1 Γ (Y t ) dt Starting from a particular initial condition introduced in [Gamarnik and Goldberg 11], we use the bounds they obtained on X (N). ohammadreza Aghajanijoint work Asymptotic with Kavita Coupling Ramanan of(brown an SPDE, University) with Applications Marchto2014 Many-Server 9 / 16 Qu
Invariant Distribution: Uniqueness Key challenge: State Space Y =. R H 1 is infinite dimensional Traditional recurrence methods are not easily applicable. In some cases, traditional methods fail [Hairer et. al. 11]. We invoke the asymptotic coupling method (Hairer, Mattingly, Sheutzow, Bakhtin, et al.) Theorem (Hairer et al. 11, continuous-time version) Assume for a Markov kernel {P t } on a polish state space Y, there exists a measurable set A Y with the following two properties: (I) µ(a) > 0 for any invariant probability measure µ of P t. (II) For every y, ỹ A, there exists a measurable map Γ y,ỹ : A A M(P [0, ) δ y, P [0, ) δỹ), such that Γ y,ỹ (D) > 0. Then {P t } has at most one invariant probability measure. P [0, ) δ y M(Y [0, ) ): distribution of a Markov process with kernel P starting at y. M(µ 1, µ 2 ) = {γ M(Y [0, ) Y [0, ) ), γ i µ i, i = 1, 2.} D = {(x, y) Y [0, ) Y [0, ) ; lim t d(x t, y t) = 0} ohammadreza Aghajanijoint work Asymptotic with Kavita Coupling Ramanan of(brown an SPDE, University) with Applications Marchto 2014 Many-Server 10 / 16 Qu
Invariant Distribution: Uniqueness To utilize the the asymptotic coupling theorem, we have to: 1. Define the set A y 2. Construct the joint ~y 3. Prove the convergence with positive probability Then Γ y,ỹ = Law(Y y, Ỹ ỹ) is a legitimate asymptotic coupling. Mohammadreza Aghajanijoint work Asymptotic with Kavita Coupling Ramanan of(brown an SPDE, University) with Applications Marchto 2014 Many-Server 11 / 16 Qu
Invariant Distribution: Uniqueness Theorem (Aghajani and R 13) Under assumptions on G, the limit process has at most one invariant distribution. Proof idea. Let y = (x 0, z 0 ) and ỹ = ( x 0, ỹ 0 ). As defined before, dx t = dw t + db t βdt + Z t(0)dt, Now define dz t(r) = [ Z t(r) Ḡ(r)Z t(0) ] dt dm t(r) + Ḡ(r)dZt(0), d X t = dw t + d B t βdt + Z t(0)dt, d Z [ t(r) = Z t (r) Ḡ(r) Z t(0)] dt dm t(r) + Ḡ(r)d Z t(0), where B t = B t + t 0 ( Z s(0) λ X s ) ds. ohammadreza Aghajanijoint work Asymptotic with Kavita Coupling Ramanan of(brown an SPDE, University) with Applications Marchto 2014 Many-Server 12 / 16 Qu
Invariant Distribution: Uniqueness Define A = {(x, z) Y; x 0}. For every invariant dist. µ of P, µ(a) > 0. Lemma When y, ỹ A, we have Z (0) L 2 Asymptotic Convergence: Xt = x 0e λt X t 0. Z t(r) = z 0(t + r) + Ḡ(r) X t + t 0 Using Lemma (2), Zt 0 in H 1 (0, ). t 0 X s g(t + r s)ds Z s(0)ḡ(t + r s)ds. (1) Distribution of Ỹ : By Lemma 2, the drift Z (0) λ X satisfies the Novikov condition. By Girsanov s Theorem, the distribution of B is equivalent to a Brownian motion, and therefore, Ỹ P δỹ. ohammadreza Aghajanijoint work Asymptotic with Kavita Coupling Ramanan of(brown an SPDE, University) with Applications Marchto 2014 Many-Server 13 / 16 Qu
Summary and Conclusion Key Challenge: Choosing the right space for Z Space Markov Property SPDE Charac. Uniqueness of Stat. Dist. C[0, ) Yes No Unknown C 1 [0, ) Yes Yes Unknown L 2 (0, ) Unknown No Yes H 1 (0, ) Yes Yes Yes Summary and Conclusions: Introduced a more tractable SPDE framework for the study of diffusion limits of many-server queues Use of the asymptotic coupling method (as opposed to Lyapunov function methods) to establishing stability properties of queueing networks: more suitable for establishing uniqueness of stationary distributions of infinite-dimensional processes Strengthened the Gamarnik-Goldberg tightness result to convergence of the X-marginal Mohammadreza Aghajanijoint work Asymptotic with Kavita Coupling Ramanan of(brown an SPDE, University) with Applications Marchto 2014 Many-Server 14 / 16 Qu
Open Problem The main open problem: to characterize the unique invariant distribution of Y A possible approach: to use the generator of the process Y. Lemma For all F : R H 1 (0, ) C 1 [0, ) R with a representation F (x, z) = f(x, z (n) ) for some n 1, r 1,..., r n [0, ), z (n) = (z(r 1 ),..., z(r n)) and f C 1 (R n+1 ), LF (x, z) =f x(x, z (n) ){z (0) β} + n f j (x, z (n) ){z (r j ) Ḡ(r j)m(x, z)} j=1 n + f xx(x, z (n) ) + b j (x)f xj (x, z (n) ) + 1 n c i,j (x)f i,j (x, z (n) ). 2 j=1 i,j=1 where b i and c i,j s are piecewise constant functions with discontinuity only at x = 0, and { β if x 0 m(x, z) = z (0) if x > 0. ohammadreza Aghajanijoint work Asymptotic with Kavita Coupling Ramanan of(brown an SPDE, University) with Applications Marchto 2014 Many-Server 15 / 16 Qu
Selected References Halfin, S. and Whitt, W. (1981). Heavy-traffic limit theorems for queues with many servers, Oper. Res. 29, 567-588. Hairer, M., Mattingly, J. C. and Scheutzow, M. (2011). Asymptotic coupling and a general form of Harris theorem with applications to stochastic delay equations. Probab. Theory Related Fields 149, 223-259. Puhalskii, A.A. and Reiman, M.I (2000) The Multiclass GI/P H/N Queue in the Halfin-Whitt Regime. Adv. Appl. Probab. 32, no. 2, pp. 564 595. Gamarnik, D. and Momcilovic, P. (2008) Steady-state analysis of a multi-server queue in the Halfin-Whitt regime, Advances in Applied Probability, 40, 548577. Reed, J. (2009) The G/GI/N queue in the Halfin-Whitt regime, Ann. Appl. Probab. 19 No. 6, 2211-2269. Puhalskii, A.A. and Reed, J. (2010) On many servers queues in heavy traffic, Ann. Appl. Probab. 20 No.1, 129-195. Kang, W. and Ramanan, K. (2010) Fluid limits of many-server queues with reneging Ann. Appl. Probab. 20:2204-2260 Kaspi, H. and Ramanan, K. (2013) Law of large numbers limits for many-server queues, Ann. Appl. Probab. 21 No. 1, 33-114.. Kaspi, H. and Ramanan, K. (2013) SPDE limits of many-server queues, Ann. Appl. Probab. 23 No. 1, 145-229. Mohammadreza Aghajanijoint work Asymptotic with Kavita Coupling Ramanan of(brown an SPDE, University) with Applications Marchto 2014 Many-Server 16 / 16 Qu