On the Rate of Convergence of the Power-of-Two-Choices to its Mean-Field Limit
11 On the Rate of Convergence of thePower-of-Two-Choices to its Mean-Field Limit
Lei Ying, ECEE, Arizona State University
Abstract
This paper studies the rate of convergence of the power-of-two-choices, a celebrated randomizedload balancing algorithm for many-server queueing systems, to its mean field limit. The convergenceto the mean-field limit has been proved in the literature, but the rate of convergence remained tobe an open problem. This paper establishes that the sequence of stationary distributions, indexed by M , the number of servers in the system, converges in mean-square to its mean-field limit with rate O (cid:16) ( log M ) ( log log M ) M (cid:17) . I. I ntroduction
The power-of-two-choices is a celebrated randomized load balancing algorithm for the super-market model , under which each incoming job is routed to the shorter of two randomly sampledservers from M servers. It has been shown in [ ], [ ] that in the infinite server regime (also calledthe mean-field limit), the power-of-two-choices reduces the mean queue length (at each server)from Θ (cid:16) − ρ (cid:17) to Θ (cid:16) log − ρ (cid:17) , where ρ is the load of the system. Besides the convergence ofthe stationary distributions to its mean-field limit, and the process-level convergence over afinite time interval has been studied in [ ], the convergence analysis with general service-timedistributions can be found in [ ], and the analysis with heterogeneous servers can be found in[ ]. This seminal work has also inspired new randomized load balancing algorithms in recentyears (see, e.g., [ ]–[ ]).The convergence proofs in [ ], [ ] are based on the interchange of the limits, which includethe following steps: ( ) proving the convergence of the continuous-time Markov chain (CTMC)to the solution of a dynamical system (the mean-field model) over a finite time interval [ t ] as M → ∞ , i.e., lim M → ∞ sup ≤ s ≤ t d ( x ( M ) ( s ) , x ( s )) = x ( M ) is the CTMC, x is the solution of the dynamical system, and d ( · , · ) is some measureof distance; ( ) proving that the mean-field model converges to a unique equilibrium pointstarting from any initial condition, i.e., lim t → ∞ x ( t ) = x ∗ ,where x ∗ is the equilibrium point; and ( ) establishing the convergence of the stationarydistributions to its mean-field limit via the interchange of the limits, i.e.,lim M → ∞ lim t → ∞ x ( M ) ( t ) = lim t → ∞ lim M → ∞ x ( M ) ( t ) = x ∗ .This approach based on the interchange of the limits and process-level convergence is an indirect method for proving the convergence of stationary distributions, so does not answer a r X i v : . [ c s . N I] M a y the rate of convergence (i.e., the approximation error of using the mean-field limit for thefinite-size system). Over last few years, a new approach has emerged for quantifying therate of convergence of queueing systems to its mean-field or diffusion limit based on Stein’smethod. Gurvich [ ] used the method for steady-state approximations for exponentiallyergodic Markovian queues. In [ ] a modular framework has been developed by Bravermanand Dai for steady-state diffusion approximations and has been used for quantifying the rateof convergence to diffusion models for M / Ph / n + M queuing systems. An approach similarto Stein’s method has also been used by Stolyar in [ ] to show the tightness of diffusion-scaled stationary distributions. An introduction to Stein’s method for steady-state diffusionapproximations can be found in [ ]. Based on Stein’s method and the perturbation theory,Ying recently established the rate of convergence of a class of stochastic systems that can berepresented by finite-dimensional population processes [ ].This paper exploits the method in [ ]. However, the result in [ ] only applies to continuous-time Markov chains (CTMCs) with finite state space so that the CTMC can be representedas a finite-dimensional population process. The main contributions of this paper are two-fold. First, it establishes the rate of convergence of the power-of-two-choices under the super-market model, where the state space of the CTMC is countable instead of finite, and thecorresponding population process is an infinite dimensional process. From the best of ourknowledge, the rate of convergence of the power-of-two-choices to its mean-field limit has notbeen established. In fact, little is known for the rate of convergence of infinite-dimensionalqueueing systems (e.g., [ ], [ ]) to their mean-field limits. Another contribution of this paperis that it demonstrates that the approach based on Stein’s method and the perturbation theory[ ] can be extended to infinite-dimensional systems. While general conditions such as thosein [ ] appear difficult to establish, the approach itself can be applied for analyzing otherinfinite-dimensional systems. In particular, this paper considers a so-called imperfect mean-field model [ ], which is a finite-dimensional, truncated version of the dynamical systemcorresponding to the infinite-dimensional CTMCs (the perfect mean-field model). Specifically,this paper constructs a truncated dynamical system with dimensional n = Θ ( log M ) . Theequilibrium point of the truncated system is consistent with the first n -dimension of the mean-field limit. Then we prove the approximation error from the mean-field model is O (cid:18) n ( log n ) M (cid:19) and the error from the truncation is O ( ρ n n log n ) . After establishing the two facts above, therate of convergence in mean square is obtained by choosing n = Θ ( log M ) .II. T he power - of - two - choices and its mean - field limit This paper studies the super market model in [ ], [ ] with M identical servers, each main-taining a separate queue. Assume jobs arrive at the system following a Poisson process withrate λ M and the processing time of each job is exponentially distributed with mean processingtime µ =
1. Let Q m ( t ) denote the queue size of server m at time t . For each incoming job, therouter (or called a scheduler) routes the job using the power-of-two-choices algorithm, whichrandomly samples two servers and dispatches the job to the server with a smaller queue size.In this setting, Q ( t ) is a CTMC and has a unique stationary distribution when λ < ].Let s ( M ) k ( t ) denote the fraction of servers with queue size at least k at time instant t , where thesuperscript M denotes the system size. s ( M ) ( t ) is also a CTMC with the following transition rates: Q s , s (cid:48) = M ( s k − s k + ) , if s (cid:48) = s − k M λ M (cid:0) s k − − s k (cid:1) , if s (cid:48) = s + k M ∑ ∞ k = − λ M (cid:0) s k − − s k (cid:1) − M ( s k − s k + ) , if s (cid:48) = s
0, otherwise. , ( )where k is a n × k th element is 1 and the others are 0. Note that thefirst term is for the event that a departure occurs at a queue with size k so s k decreases by1/ M . The second term is for the event that an arrival occurs and it is routed to a queue withsize k −
1. It has been shown in [ ] that s ( M ) k ( ∞ ) converges weakly to s ∗ k , where s ∗ k = λ k − ( )is the equilibrium point of the following mean-field model:˙ s k = λ ( s k − − s k ) − ( s k − s k + ) ∀ k ≥ s = tein ’ s method for the rate of convergence Note that the system above is an infinite dimensional system. Analysis of perturbed infinite-dimensional nonlinear systems is a challenging problem. So instead of analyzing the infinitedimensional system, we consider an n -dimensional truncated system defined as follows:˙˜ s k = (cid:26) λ ( ˜ s k − − ˜ s k ) − ( ˜ s k − ˜ s k + ) , n − ≥ k ≥ λ ( ˜ s n − − ˜ s n ) − ( ˜ s n − s ∗ n + ) , k = n . , ( )where ˜ s ( t ) = t ≥
0. It is easy to verify that s ∗ defined in ( ) remains to be the uniqueequilibirium point of this truncated system. Furthermore, ˜ s k ( t ) ≤ t given the initialcondition of the system satisfies 1 = ˜ s ( ) ≥ ˜ s ( ) ≥ ˜ s ( ) ≥ · · · ≥ ˜ s n ( ) as shown in Lemma . Lemma . Consider the dynamical system defined in ( ). Given that the initial condition satisfies = ˜ s ( ) ≥ ˜ s ( ) ≥ ˜ s ( ) ≥ · · · ≥ ˜ s n ( ) , we have ˜ s k ( t ) ≤ for any t ≥ and ≤ k ≤ n . Proof:
Note that max ≤ k ≤ n ˜ s k ( ) ≤
1. Furthermore, if˜ s j ( t ) = = max ≤ k ≤ n ˜ s k ( t ) and sup 0 ≤ τ ≤ t max ≤ k ≤ n ˜ s k ( τ ) ≤ s j ( t ) ≤
0. Therefore, we conclude that for any t ≥ ≤ k ≤ n ˜ s k ( t ) ≤ x k = ˜ s k − s ∗ k for 1 ≤ k ≤ n , so˙ x k = f k ( x ) ( ): = (cid:40) λ ( (cid:0) x k − + s ∗ k − (cid:1) − (cid:0) x k + s ∗ k (cid:1) ) − ( x k + s ∗ k − x k + − s ∗ k + ) , 1 ≤ k ≤ n − λ ( (cid:0) x n − + s ∗ n − (cid:1) − ( x n + s ∗ n ) ) − ( x n + s ∗ n − s ∗ n + ) , k = n = − λ (cid:0) x + s ∗ x (cid:1) − ( x − x ) , k = λ ( (cid:0) x k − + s ∗ k − x k − (cid:1) − (cid:0) x k + s ∗ k x k (cid:1) ) − ( x k − x k + ) , 2 ≤ k ≤ n − λ ( (cid:0) x n − + s ∗ n − x n − (cid:1) − (cid:0) x n + s ∗ n x n (cid:1) ) − x n , k = n ( ) The unique equilibrium point for the system is x = Lemma . Under the dynamical system defined in ( ), − s ∗ k ≤ x k ( t ) ≤ − s ∗ k for ≤ k ≤ n and allt ≥ Proof:
According to Lemma , 0 ≤ x k ( t ) + s ∗ k = s k ( t ) ≤ t ≥
0, so the lemmaholds.For ease of notation, we always use x to denote an infinite-dimensional vector in this sectionand define f k ( x ) = k > n . Let g ( x ) be the solution to the Poisson equation (cid:79) g ( x ) · ˙ x = (cid:79) g ( x ) · f ( x ) = n ∑ k = x k . ( )Then, g ( x ) = − (cid:90) ∞ n ∑ k = ( x k ( t , x )) dt when the integral is finite [ ], [ ], where x k ( t , x ) is the solution of the dynamical system ( )with x as the initial condition. Note that the solution only depends on the first n componentsof the initial condition x . The integral is finite since the system is exponentially stable as shownin Lemma in Section IV and ∑ nk = | x k ( ) | ≤ n according to Lemma .Now let G M denote the generator of the M th CTMC. Define Q x , y = Q x + s ∗ , y + s ∗ and q x , y = M Q x , y .Then, G M g ( x ) = ∑ y : y (cid:54) = x Q x , y ( x ) ( g ( y ) − g ( x ))= M ∑ y : y (cid:54) = x q x , y ( x ) ( g ( y ) − g ( x )) .It has been proved in [ ], [ ] that x ( M ) has a stationary distribution. We use E M [ · ] throughoutto denote the stationary expectation of the system with M servers. Based on the basic adjointrelation (BAR) [ ], E M [ G M g ( x )] = E M (cid:34) M ∑ y : y (cid:54) = x q x , y ( x ) ( g ( y ) − g ( x )) (cid:35) =
0. ( )Then by taking expectation of the Poisson equation ( ) and adding ( ) to the equation, weobtain E M (cid:34) n ∑ k = x k (cid:35) = E M (cid:34) (cid:79) g ( x ) · f ( x ) − M ∑ y : y (cid:54) = x q x , y ( x ) ( g ( y ) − g ( x )) (cid:35) .From the transition rates of the CTMC under the power-of-two-choices ( ), we obtain ∑ y : y (cid:54) = x q x , y M ( y k − x k ) = λ (cid:16)(cid:16) x k − + s ∗ k − x k − (cid:17) − (cid:16) x k + s ∗ k x k (cid:17)(cid:17) − ( x k − x k + ) . The definition of f k ( x ) ( ) further implies f k ( x ) = (cid:26) ∑ y : y (cid:54) = x q x , y M ( y k − x k ) , if 1 ≤ k < n ∑ y : y (cid:54) = x q x , y M ( y n − x n ) − x n + , if k = n . ,and (cid:79) g ( x ) · f ( x ) = n ∑ k = ∂ g ∂ x k (cid:32) ∑ y : y (cid:54) = x q x , y M ( y k − x k ) (cid:33) − ∂ g ∂ x n ( x ) x n + = ∑ y : y (cid:54) = x q x , y M n ∑ k = ∂ g ∂ x k ( y k − x k ) − ∂ g ∂ x n ( x ) x n + = ∑ y : y (cid:54) = x q x , y M (cid:79) g ( x ) · ( y − x ) − ∂ g ∂ x n ( x ) x n + ,where the last equality holds because ∂ g ∂ x k = k > n . Therefore, we have E M (cid:34) n ∑ k = x k (cid:35) ( ) = E M (cid:34) − ∂ g ∂ x n ( x ) x n + − M ∑ y : y (cid:54) = x q x , y ( x ) ( g ( y ) − g ( x ) − (cid:79) g ( x ) · ( y − x )) (cid:35) . ( )Since the mean-field model ( ) is a truncated system, we have the additional term − ∂ g ∂ x n ( x ) x n + compared to the similar equation in [ ]. In the remaining of this paper, we will derive anupper bound on the mean-square error E M (cid:2) ∑ ∞ k = x k (cid:3) by first deriving an upper bound on E M (cid:2) ∑ nk = x k (cid:3) . The result will be proved based on a sequence of convergence results of thedynamical system ( ) to be proved in Section IV. Lemma . Under the power-of-two-choices and given n log n = o ( M ) , we haveE M (cid:34) n ∑ k = x k (cid:35) = O ( λ n n log n ) + O (cid:18) n ( log n ) M (cid:19) . ( ) Proof:
The proof is based on several convergence properties of the dynamical system ( ).The proofs of these convergence properties can be found in Section IV.According to Lemma and Corollary in Section IV, both ∑ nk = | x k ( t , x ) | and ∑ nk = | (cid:79) x k ( t , x ) | decay exponentially as t increases. Furthermore, | y − x | = M for any x and y such that Q x , y (cid:54) = ). Therefore (cid:90) ∞ x k ( t , x ) (cid:79) x k ( t , x ) · ( y − x ) dt is finite, and by exchanging the order of integration and differentiation, we obtain (cid:79) g ( x ) · ( y − x ) = (cid:90) ∞ n ∑ k = x k ( t , x ) (cid:79) x k ( t , x ) · ( y − x ) dt . ( ) Hence, we have − ( g ( y ) − g ( x ) − (cid:79) g ( x ) · ( y − x ))= (cid:90) ∞ n ∑ k = (cid:16) ( x k ( t , y )) − ( x k ( t , x )) − x k ( t , x ) (cid:79) x k ( t , x ) · ( y − x ) (cid:17) dt . ( )We now define e k ( t ) = x k ( t , y ) − x k ( t , x ) − (cid:79) x k ( t , x ) · ( y − x ) ,i.e., x k ( t , y ) = e k ( t ) + x k ( t , x ) + (cid:79) x k ( t , x ) · ( y − x ) ,so ( x k ( t , y )) − ( x k ( t , x )) − x k ( t , x ) (cid:79) x k ( t , x ) · ( y − x )= ( e k ( t ) + x k ( t , x ) + (cid:79) x k ( t , x ) · ( y − x )) − ( x k ( t , x )) − x k ( t , x ) (cid:79) x k ( t , x ) · ( y − x )= e k ( t ) + ( (cid:79) x k ( t , x ) · ( y − x )) + e k ( t ) (cid:79) x k ( t , x ) · ( y − x ) + e k ( t ) x k ( t , x )= e k ( t ) ( e k ( t ) + (cid:79) x k ( t , x ) · ( y − x ) + x k ( t , x )) + ( (cid:79) x k ( t , x ) · ( y − x )) .According to Lemmas and in Section IV, | e k ( t ) | ≤ n ∑ k = | e k ( t ) | = O (cid:18) n log nM (cid:19) .Under the power-of-two-choices algorithm, ∑ ∞ k = | x k − y k | = M for any x and y such that Q x , y (cid:54) =
0. Furthermore, | (cid:79) x k ( t , x ) | is bounded by n according to Corollary and | x k ( t , x ) | is boundedby | x ( ) | ≤ n according to Lemma . Therefore, given n log n = o ( M ) , e.g., n = Θ ( log M ) , wecan choose a sufficiently large ˜ M such that for any M ≥ ˜ M , | e k ( t ) + (cid:79) x k ( t , x ) · ( y − x ) + ( x k ( t , x )) | ≤ + | x k ( t , x ) | ≤ + | x ( t , x ) | ≤ n ,where the last inequality follows from Lemma , which implies that | g ( y ) − g ( x ) − (cid:79) g ( x ) · ( y − x ) |≤ n (cid:90) ∞ ∑ k | e k ( t ) | dt + (cid:90) ∞ ∑ k ( (cid:79) x k ( t , x ) · ( y − x )) dt . ( )In Theorem , we will establish that (cid:90) ∞ n ∑ k = | e k ( t ) | dt = O (cid:18) ( n log n ) M (cid:19) .Under the power-of-two-choices, for any x and y such that Q x , y (cid:54) =
0, there exists j such that | x j − y j | = M and | x h − x h | = h (cid:54) = j . Therefore, (cid:90) ∞ n ∑ k = ( (cid:79) x k ( t , x ) · ( y − x )) dt = M (cid:90) ∞ n ∑ k = (cid:32) ∂∂ x j x k ( t , x ) (cid:33) dt ≤ M (cid:90) ∞ (cid:32) n ∑ k = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂∂ x j x k ( t , x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:33) dt From Corollary , we have (cid:90) ∞ (cid:32) n ∑ k = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂∂ x j x k ( t , x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:33) dt = O ( n log n ) .Therefore, we can conclude that | g ( y ) − g ( x ) − (cid:79) g ( x ) · ( y − x ) | = O (cid:18) n ( log n ) M (cid:19) . ( )Furthermore, from Corollary , we have (cid:12)(cid:12)(cid:12)(cid:12) ∂ g ∂ x n ( x ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − (cid:90) ∞ n ∑ k = x k ( t , x ) ∂∂ x n x k ( t , x ) dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:90) ∞ n ∑ k = (cid:12)(cid:12)(cid:12)(cid:12) ∂∂ x n x k ( t , x ) (cid:12)(cid:12)(cid:12)(cid:12) dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = O ( n log n ) .It has been shown in [ ] that E M [ s n ] ≤ λ n for any n ≥
0, so when n is sufficiently large, E M [ | x n + | ] ≤ E M [ | s n + | ] + s ∗ n + ≤ λ n + + λ n + − ≤ λ n .Therefore, we conclude E M (cid:34) n ∑ k = ( x k ) (cid:35) = O ( λ n n log n ) + O (cid:18) n ( log n ) M (cid:19) .From the lemma above, we can conclude the following theorem. Theorem . Under the power-of-two-choices with λ < E M (cid:34) ∞ ∑ k = (cid:12)(cid:12)(cid:12) s ( M ) k − s ∗ k (cid:12)(cid:12)(cid:12) (cid:35) = E M (cid:34) ∞ ∑ k = x k (cid:35) = O (cid:18) ( log M ) ( log log M ) M (cid:19) . Proof:
By choosing n = M log λ in equation ( ), we obtain E M (cid:34) n ∑ k = x k (cid:35) = O (cid:18) ( log M ) ( log log M ) M (cid:19) .Since E M [ s k ] ≤ λ k and 0 ≤ s k ≤ k , E M (cid:34) ∞ ∑ k = n + x k (cid:35) ≤ E M (cid:34) ∞ ∑ k = n + s k + ( s ∗ k ) (cid:35) ≤ E M (cid:34) ∞ ∑ k = n + s k + ( s ∗ k ) (cid:35) ≤ λ n − λ ,where the last inequality holds when n is sufficiently large. The theorem holds. IV. C onvergence P roperties of the D ynamical S ystem ( )In this section, we analyze the dynamical system defined by ( ), and present the conver-gence properties used in the previous section. Since the system is an n -dimensional truncatedsystem of the original infinite-dimensional dynamical system, in this section, all vectors are n -dimensional instead of infinite-dimensional. Lemma . Under the dynamical system defined in ( ), we have for any t ≥ | x ( t ) | ≤ | x ( ) | . Proof:
The proof follows the proof of Theorem . in [ ]. Define the Lyapunov function tobe V ( t ) = n ∑ k = | x k ( t ) | .Note that d | x k ( t ) | dt is well-defined for x k ( t ) (cid:54) =
0. When x k ( t ) =
0, we consider the upper right-hand derivative as in [ ], i.e., d | x k ( t ) | dt = lim τ → t + | x k ( τ ) | − | x k ( t ) | τ − t = (cid:26) ˙ x k ( t ) , if ˙ x k ( t ) ≥ − ˙ x k ( t ) , if ˙ x k ( t ) < k such that n > k >
1. If x k ( t ) >
0, or x k ( t ) = x k ≥
0, then d | x k ( t ) | dt = dx k ( t ) dt = λ (cid:16)(cid:16) x k − + s ∗ k − x k − (cid:17) − (cid:16) x k + s ∗ k | x k | (cid:17)(cid:17) − ( | x k | − x k + )= λ (cid:16) x k − + s ∗ k − x k − (cid:17) − λ (cid:16) x k + s ∗ k | x k | (cid:17) − | x k | + x k + .Define ˙ V ( t ) = ∑ nk = W k ( t ) such that W k ( t ) includes all the terms involving x k ( t ) . Then when x k ( t ) >
0, or x k ( t ) = x k ≥
0, we have W k ( t ) ≤ λ (cid:16) x k + s ∗ k | x k | (cid:17) − λ (cid:16) x k + s ∗ k | x k | (cid:17) − | x k | + | x k | = d | x k + ( t ) | dt and the last term comes from d | x k − ( t ) | dt .If x k ( t ) <
0, or x k ( t ) = x k <
0, then d | x k ( t ) | dt = − dx k ( t ) dt = − λ (cid:16) x k − + s ∗ k − x k − (cid:17) + λ (cid:16) x k − s ∗ k | x k | (cid:17) − | x k | − x k + ,so W k ( t ) ≤ λ (cid:16) − x k + s ∗ k | x k | (cid:17) + λ (cid:16) x k − s ∗ k | x k | (cid:17) − | x k | + | x k | = d | x k − ( t ) | dt , and the first term comes from d | x k + ( t ) | dt and the factthat x k ( t ) ≥ − s ∗ k when x k ( t ) <
0, so (cid:0) − x k + s ∗ k | x k | (cid:1) > W k ( t ) ≤ k = k = n . From the discussion above,we have ˙ V ( t ) = n ∑ k = W k ≤ t ≥ Define k λ = log (cid:32) log (cid:16) −√ λ √ λ (cid:17) log λ + (cid:33) log 2 ,where implies that λ (cid:16) s ∗ k λ + (cid:17) = λ (cid:16) λ k λ − + (cid:17) ≤ √ λ .Furthermore, define a sequence { w k } such that w = w = w k = (cid:32) + k λ k ∑ j = ( max {
1, 4 λ } ) j − (cid:33) , 2 ≤ k ≤ k λ w k = (cid:18) + k − k λ n (cid:19) w k λ , k λ < k ≤ n ,and δ = − √ λ n . Lemma . Under the dynamical system defined in ( ), we have for any t ≥ n ∑ k = w k | x k ( t ) | ≤ (cid:32) n ∑ k = w k | x k ( ) | (cid:33) e − δ t . Proof:
The proof follows the idea of Theorem . in [ ]. Define V ( t ) = ∑ nk = w k | x k ( t ) | and˙ V ( t ) = ∑ nk = W k ( t ) such that W k ( t ) includes all the terms involving x k ( t ) . The lemma is provedby showing that W k ( t ) ≤ − δ w k | x k ( t ) | . ( )We consider the case where x k ( t ) >
0, or x k ( t ) = x k ≥
0. According to the proof ofLemma , we have W k ( t ) ≤ w k + λ (cid:16) x k + s ∗ k | x k | (cid:17) − w k λ (cid:16) x k + s ∗ k | x k | (cid:17) − w k | x k | + w k − | x k | .So ( ) holds if w k + λ (cid:16) x k + s ∗ k | x k | (cid:17) − w k λ (cid:16) x k + s ∗ k | x k | (cid:17) − w k | x k | + w k − | x k | ≤ − δ w k | x k | ,in other words, if w k + − w k ≤ ( − δ ) w k − w k − λ ( | x k | + s ∗ k ) . ( )We now prove ( ) by considering the following three cases. When 1 ≤ k ≤ k λ −
1, we have w k + − w k = k λ ( max {
1, 4 λ } ) k ( − δ ) w k − w k − λ ( | x k | + s ∗ k ) ≥ w k − w k − − δ w k λ = k λ ( max { λ } ) k − − δ w k λ .So inequality ( ) holds if3 λ ≤ max {
1, 4 λ } − δ w k k λ ( max {
1, 4 λ } ) k ,which can be established by proving δ w k k λ ( max {
1, 4 λ } ) k ≤ λ .Since δ = −√ λ n and w k ≤ k ≤ k λ −
1, the inequality above holds when n is sufficientlylarge.When n ≥ k ≥ k λ +
1, according to the definition of k λ , λ ( | x k | + s ∗ k ) ≤ √ λ . Therefore, wehave w k + − w k = w k λ n ( − δ ) w k − w k − λ ( | x k | + s ∗ k ) ≥ w k − w k − − δ w k √ λ = w k λ n − δ w k √ λ .So inequality ( ) holds if √ λ w k λ ≤ w k λ − δ w k n ,in other words, if w k ≤ ( − √ λ ) w k λ δ n = w k λ .Since w k ≤ w k λ >
1, the inequality above holds.When k = k λ , according to the definition of k λ , λ ( | x k | + s ∗ k ) ≤ √ λ . Therefore, we have w k + − w k = w k λ n ( − δ ) w k − w k − λ ( | x k | + s ∗ k ) ≥ w k − w k − − δ w k √ λ = k λ ( max { λ } ) k λ − − δ w k √ λ .So inequality ( ) holds if √ λ w k λ ≤ nk λ ( max {
1, 4 λ } ) k λ − − δ w k n ,in other words, if w k ≤ − √ λ (cid:32) nk λ ( max {
1, 4 λ } ) k λ − − √ λ w k λ (cid:33) . Since w k ≤ n is sufficiently large.From the discussion above, we conclude that˙ V ( t ) ≤ − n ∑ k = δ w k | x k ( t ) | = − δ V ( t ) ,so the lemma holds.For the ease of notation, in the following analysis, we use x ( t , 0 ) = x ( t , x ) and x ( t , (cid:101) ) = x ( t , y ) ,where (cid:101) = M , which is also to be consistent with notation used in [ ] such that x ( t , (cid:101) ) isthe perturbed version of x ( t , 0 ) . We consider the finite Taylor series for x ( t , (cid:101) ) : = x ( t , y ) interms of (cid:101) : x ( t , (cid:101) ) = x ( ) ( t ) + (cid:101) x ( ) ( t ) + e ( t ) , ( )and x ( (cid:101) ) = x + (cid:101) z , ( )where x ( ) ( t ) = x ( t , 0 ) and x ( ) ( t ) = d x d (cid:101) ( t , (cid:101) ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:101) = ,and z = M ( y − x ) (again, all three vectors are n -dimensional). Substituting ( ) into thedynamical system equation, we get˙ x ( t , (cid:101) ) = ˙ x ( ) ( t ) + (cid:101) ˙ x ( ) ( t ) + ˙ e ( t ) = f ( x ( t , (cid:101) )) ( ) = h ( ) ( x ( ) ( t )) + h ( ) ( x ( ≤ ) ( t )) (cid:101) + R e ( t , (cid:101) ) , ( )where x ( ≤ ) = (cid:16) x ( ) , x ( ) (cid:17) . The zero-order term h ( ) is given by˙ x ( ) ( t ) = h ( ) (cid:16) x ( ) ( t ) (cid:17) = f (cid:16) x ( ) ( t ) (cid:17) with x ( ) ( ) = x ,which is the nominal system without the perturbation on the initial condition. The first-orderterm is given by h ( ) (cid:16) x ( ≤ ) ( t ) (cid:17) = dd (cid:101) f ( x ( t , (cid:101) )) (cid:12)(cid:12)(cid:12)(cid:12) (cid:101) = = ∂ f ∂ x ( x ( t , (cid:101) )) d x d (cid:101) ( t , (cid:101) ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:101) = = ∂ f ∂ x ( x ( ) ( t )) x ( ) ( t ) .Therefore, we have ˙ x ( ) ( t ) = ∂ f ∂ x ( x ( ) ( t )) x ( ) ( t ) with x ( ) ( ) = z , ( )which implies that˙ x ( ) k = g k ( x ( ) ) = − λ (cid:16) x ( ) + s ∗ k (cid:17) x ( ) − x ( ) + x ( ) , k = λ (cid:16) x ( ) k − + s ∗ k − (cid:17) x ( ) k − − λ (cid:16) x ( ) k + s ∗ k (cid:17) x ( ) k − x ( ) k + x ( ) k + , 2 ≤ k ≤ n − λ (cid:16) x ( ) n − + s ∗ n − (cid:17) x ( ) n − − λ (cid:16) x ( ) n + s ∗ n (cid:17) x ( ) n − x ( ) n , k = n . ( ) Lemma . Under the dynamical system defined by ( ), (cid:12)(cid:12)(cid:12) x ( ) ( t ) (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) x ( ) ( ) (cid:12)(cid:12)(cid:12) . Proof:
First recall that x ( ) k ( t ) + s ∗ k ≥ t ≥ k according to Lemma . Define V ( t ) = n ∑ k = (cid:12)(cid:12)(cid:12) x ( ) ( t ) (cid:12)(cid:12)(cid:12) .Following the proof of Lemma , we obtain that d | x ( ) k ( t ) | dt ≤ − λ (cid:16) x ( ) + s ∗ k (cid:17) (cid:12)(cid:12)(cid:12) x ( ) (cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12) x ( ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) x ( ) (cid:12)(cid:12)(cid:12) , k = λ (cid:16) x ( ) k − + s ∗ k − (cid:17) (cid:12)(cid:12)(cid:12) x ( ) k − (cid:12)(cid:12)(cid:12) − λ (cid:16) x ( ) k + s ∗ k (cid:17) (cid:12)(cid:12)(cid:12) x ( ) k (cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12) x ( ) k (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) x ( ) k + (cid:12)(cid:12)(cid:12) , 2 ≤ k ≤ n − λ (cid:16) x ( ) n − + s ∗ n − (cid:17) (cid:12)(cid:12)(cid:12) x ( ) n − (cid:12)(cid:12)(cid:12) − λ (cid:16) x ( ) n + s ∗ n (cid:17) (cid:12)(cid:12)(cid:12) x ( ) n (cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12) x ( ) n (cid:12)(cid:12)(cid:12) , k = n . ( )Therefore ˙ V ( t ) ≤ − λ (cid:16) x ( ) n + s ∗ n (cid:17) (cid:12)(cid:12)(cid:12) x ( ) n (cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12) x ( ) (cid:12)(cid:12)(cid:12) ≤ k = log (cid:18) log 8log 1 λ + (cid:19) log 2 , a sequence of ˜ w k such that˜ w = w k = (cid:32) + k k ∑ j = ( max {
1, 5 λ } ) j − (cid:33) , 2 ≤ k ≤ ˜ k ˜ w k = (cid:18) + k − ˜ kn (cid:19) ˜ w ˜ k , ˜ k < k ≤ n .Furthermore, we define ˜ δ = − λ n and ˜ t = δ log ( n ) ≥ δ log ( | x ( ) | ) . Lemma . There exists ˜ n independent of M such that for any n ≥ ˜ n and t ≥ ˜ t , we have n ∑ k = ˜ w k (cid:12)(cid:12)(cid:12) x ( ) k ( t ) (cid:12)(cid:12)(cid:12) ≤ n ∑ k = ˜ w k (cid:12)(cid:12)(cid:12) x ( ) k ( ˜ t ) (cid:12)(cid:12)(cid:12) e − ˜ δ ( t − ˜ t ) . Proof:
Since ˜ k = log (cid:18) log 8log 1 λ + (cid:19) log 2 , so s ∗ k ≤ s ∗ ˜ k = λ ˜ k − ≤ for any k ≥ ˜ k . Now according toLemma and the fact that 1 ≤ w k ≤
4, we have n ∑ k = | x ( ) k ( t ) | ≤ n ∑ k = w k | x ( ) k ( t ) | ≤ (cid:32) n ∑ k = w k | x k ( ) | (cid:33) e − δ t ≤ (cid:32) n ∑ k = | x k ( ) | (cid:33) e − δ t .Therefore, when t ≥ ˜ t ≥ δ log ( | x ( ) | ) , | x ( ) ( t ) | ≤
18 .We further define the following Lyapunov function V ( x ( ) ) = n ∑ k = ˜ w k (cid:12)(cid:12)(cid:12) x ( ) k (cid:12)(cid:12)(cid:12) .Following the proof of Lemma or the proof of Theorem . in [ ], we obtain that˙ V ( x ( ) ) ≤ n ∑ k = − (cid:16) λ ˜ w k (cid:16) x ( ) k + s ∗ k (cid:17) + ˜ w k − λ ˜ w k + (cid:16) x ( ) k + s ∗ k (cid:17) − ˜ w k − (cid:17) (cid:12)(cid:12)(cid:12) x ( ) k (cid:12)(cid:12)(cid:12) .So the lemma holds by proving − (cid:16) λ ˜ w k (cid:16) x ( ) k + s ∗ k (cid:17) + ˜ w k − λ ˜ w k + (cid:16) x ( ) k + s ∗ k (cid:17) − ˜ w k − (cid:17) ≤ − ˜ δ ˜ w k ,i.e., by proving ˜ w k + − ˜ w k ≤ ˜ w k − ˜ w k − − ˜ δ ˜ w k λ (cid:16) x ( ) k + s ∗ k (cid:17) . ( )We now prove ( ) by considering the following three cases.When 1 ≤ k ≤ ˜ k −
1, we have ˜ w k + − ˜ w k = k ( max {
1, 5 λ } ) k ˜ w k − ˜ w k − − ˜ δ ˜ w k λ (cid:16) x ( ) k + s ∗ k (cid:17) ≥ k ( max { λ } ) k − − ˜ δ ˜ w k λ .So inequality ( ) holds if4 λ ≤ max {
1, 5 λ } − ˜ δ ˜ w k ˜ k ( max {
1, 5 λ } ) k ,which can be established by proving˜ δ ˜ w k ˜ k ( max {
1, 5 λ } ) k ≤ λ .Since ˜ δ = − λ n and ˜ w k ≤ k ≤ ˜ k −
1, the inequality above holds when n is sufficientlylarge. When n ≥ k ≥ ˜ k +
1, according to the definition of ˜ k , s ∗ k ≤ . Furthermore, given t ≥ ˜ t , | x ( ) k | ≤ for any k . Therefore, we have˜ w k + − ˜ w k = ˜ w ˜ k n ( − δ ) ˜ w k − ˜ w k − λ ( | x k | + s ∗ k ) ≥ ˜ w k − ˜ w k − − δ ˜ w k λ = ˜ w ˜ k n − δ ˜ w k λ .So inequality ( ) holds if 12 λ ˜ w ˜ k ≤ ˜ w ˜ k − ˜ δ ˜ w k n ,in other words, if ˜ w k ≤ (cid:16) − λ (cid:17) ˜ w ˜ k ˜ δ n = ( − λ ) ˜ w ˜ k − λ ≤ w ˜ k .Since ˜ w k ≤ w ˜ k > ) holds.When k = ˜ k , according to the definition of ˜ k and ˜ t , we have 2 λ ( | x k | + s ∗ k ) ≤ λ for t ≥ ˜ t .Therefore, we have ˜ w ˜ k + − ˜ w ˜ k = ˜ w ˜ k n ( − δ ) ˜ w ˜ k − ˜ w ˜ k − λ ( | x ˜ k | + s ∗ ˜ k ) ≥ ˜ w ˜ k − ˜ w ˜ k − − ˜ δ ˜ w ˜ k λ = k ( max { λ } ) ˜ k − − ˜ δ ˜ w ˜ k λ .So inequality ( ) holds if λ w ˜ k ≤ n ˜ k ( max {
1, 5 λ } ) ˜ k − − ˜ δ ˜ w k n ,in other words, if (cid:18) λ + − λ (cid:19) ˜ w ˜ k ≤ n ˜ k ( max {
1, 5 λ } ) ˜ k − .Since ˜ w ˜ k ≤ n is sufficiently large.From the analysis above, we conclude when t ≥ ˜ t ,˙ V ( t ) ≤ − ˜ δ V ( t ) and V ( t ) ≤ V ( ˜ t ) e − ˜ δ ( t − ˜ t ) . Corollary . (cid:12)(cid:12)(cid:12) x ( ) ( t ) (cid:12)(cid:12)(cid:12) ≤ (cid:40)
1, 0 ≤ t ≤ ˜ t min (cid:110)
1, 4 e − ˜ δ ( t − ˜ t ) (cid:111) , t ≥ ˜ t . Proof:
Note that (cid:12)(cid:12)(cid:12) x ( ) ( ) (cid:12)(cid:12)(cid:12) = | z | ∈ {
1, 0 } under the power-of-two-choices. The case for t ≤ ˜ t holds according to Lemma . Since 1 ≤ ˜ w k ≤ t ≥ ˜ t . Corollary . n ∑ k = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ y j x k ( t , y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:40)
1, 0 ≤ t ≤ ˜ t min (cid:110)
1, 4 e − ˜ δ ( t − ˜ t ) (cid:111) , t ≥ ˜ t . Proof:
Notice that ∂ y j x k ( t , y ) = lim (cid:101) → x k ( t , y + (cid:101) j ) − x k ( t , y ) (cid:101) ,where j is an n × ( j ) j = ( j ) k = k (cid:54) = j . Therefore, ∂ z j x k ( t , y ) = x ( ) k ( t ) with x ( ) ( ) = j and x ( ) ( t ) = x ( t , y ) . The corollary follows from the corollary above.We next study e ( t ) = x ( t , (cid:101) ) − x ( ) ( t ) − (cid:101) x ( ) ( t ) . According to its definition and ( ), we have˙ e ( t ) = f ( x ( t , (cid:101) )) − f (cid:16) x ( ) ( t ) (cid:17) − (cid:101) ∂ f ∂ x ( x ( ) ( t )) x ( ) ( t ) e ( ) = Lemma . Assume n log n = o ( M ) . For any ≤ t ≤ ˜ t , | e ( t ) | ≤ tM = O (cid:18) n log nM (cid:19) . Proof:
We first have for 1 < k < n ,˙ e k ( t )= f k (cid:16) x ( ) ( t ) + (cid:101) x ( ) ( t ) + e ( t ) (cid:17) − f k (cid:16) x ( ) ( t ) (cid:17) − (cid:101) n ∑ j = ∂ f k ∂ x j ( x ( ) ( t )) x ( ) j ( t )= λ (cid:18)(cid:16) x ( ) k − ( t ) + (cid:101) x ( ) k − ( t ) + e k − ( t ) (cid:17) + s ∗ k − (cid:16) x ( ) k − ( t ) + (cid:101) x ( ) k − ( t ) + e k − ( t ) (cid:17)(cid:19) − λ (cid:18)(cid:16) x ( ) k ( t ) + (cid:101) x ( ) k ( t ) + e k ( t ) (cid:17) + s ∗ k (cid:16) x ( ) k ( t ) + (cid:101) x ( ) k ( t ) + e k ( t ) (cid:17)(cid:19) − (cid:16) x ( ) k ( t ) + (cid:101) x ( ) k ( t ) + e k ( t ) − x ( ) k + ( t ) − (cid:101) x ( ) k + ( t ) − e k + ( t ) (cid:17) − λ (cid:18)(cid:16) x ( ) k − ( t ) (cid:17) + s ∗ k − x ( ) k − ( t ) (cid:19) + λ (cid:18)(cid:16) x ( ) k ( t ) (cid:17) + s ∗ k x ( ) k ( t ) (cid:19) + (cid:16) x ( ) k ( t ) − x ( ) k + ( t ) (cid:17) − (cid:101)λ (cid:16) x ( ) k − + s ∗ k − (cid:17) x ( ) k − + (cid:101)λ (cid:16) x ( ) k + s ∗ k (cid:17) x ( ) k + (cid:101) x ( ) k − (cid:101) x ( ) k + = λ (cid:16) e k − + (cid:16) x ( ) k − + s ∗ k − + (cid:101) x ( ) k − (cid:17) e k − − e k − (cid:16) x ( ) k + s ∗ k + (cid:101) x ( ) k (cid:17) e k (cid:17) − ( e k − e k + ) + λ(cid:101) (cid:18)(cid:16) x ( ) k − (cid:17) − (cid:16) x ( ) k (cid:17) (cid:19) = λ (cid:16) x ( ) k − + s ∗ k − (cid:17) e k − − λ (cid:16) x ( ) k + s ∗ k (cid:17) e k − ( e k − e k + )+ λ (cid:16) e k − + (cid:101) x ( ) k − e k − − e k − (cid:101) x ( ) k e k (cid:17) + λ(cid:101) (cid:18)(cid:16) x ( ) k − (cid:17) − (cid:16) x ( ) k (cid:17) (cid:19) = g k ( e ) + λ (cid:16) e k − + (cid:101) x ( ) k − e k − − e k − (cid:101) x ( ) k e k (cid:17) + λ(cid:101) (cid:18)(cid:16) x ( ) k − (cid:17) − (cid:16) x ( ) k (cid:17) (cid:19) ,where the last equality holds according to the definition of g k ( · ) in ( ). The same equationholds for k = k = n .Define V ( t ) = | e ( t ) | . Now following the proof of Lemma , we can obtain˙ V ( t ) ≤ n ∑ k = λ (cid:16) e k + (cid:101) (cid:12)(cid:12)(cid:12) x ( ) k (cid:12)(cid:12)(cid:12) | e k | (cid:17) + λ(cid:101) (cid:16) x ( ) k (cid:17) .Assume | e ( t ) | ≤ tM for t ≤ ˜ t , we have that for sufficiently large M ,˙ V ( t ) ≤ λ ˜ tM + λ M ≤ M ,where the last inequality holds because n log n = o ( M ) and ˜ t = Θ ( n log n ) . The inequalityabove implies that | e ( t ) | ≤ tM and the lemma holds. Lemma . For any t ≥ ˜ t , we have | e ( t ) | ≤ | e ( ˜ t ) | exp (cid:0) − δ (cid:48) ( t − ˜ t ) (cid:1) + λ(cid:101) exp (cid:0) − ˜ δ ( t + ˜ t ) (cid:1) δ (cid:0) − exp (cid:0) − ˜ δ ( t − ˜ t ) (cid:1)(cid:1) = O (cid:18) n log nM (cid:19) . Proof:
We now consider t ≥ ˜ t . Define Lyapunov function V ( e ( t )) = n ∑ k = ˜ w k | e k ( t ) | for ˜ w k defined previously. We first have˙ e k ( t )= f k ( x ( t , (cid:101) )) − f k (cid:16) x ( ) ( t ) (cid:17) − (cid:101) ∂ f k ∂ x ( x ( ) ( t )) x ( ) ( t )= f k (cid:16) x ( ) ( t ) + (cid:101) x ( ) ( t ) + e ( t ) (cid:17) − f k (cid:16) x ( ) ( t ) (cid:17) − (cid:101) ∂ f k ∂ x ( x ( ) ( t )) x ( ) ( t )= λ (cid:16) (cid:16) x ( ) k − + s ∗ k − + e k − + (cid:101) x ( ) k − (cid:17) e k − − (cid:16) x ( ) k + s ∗ k + e k + (cid:101) x ( ) k (cid:17) e k (cid:17) − ( e k − e k + ) + (cid:18) λ(cid:101) (cid:16) x ( ) k − (cid:17) − λ(cid:101) (cid:16) x ( ) k (cid:17) (cid:19) .Define ˙ V ( t ) = n ∑ k = W k ( t ) + W ( t ) ,where W k ( t ) includes all the terms involving e k ( t ) and W ( t ) includes all the remaining terms.Note that | e k ( t ) | = O ( n log n / M ) according to the previous lemma and (cid:101) | x ( ) k | = O ( M ) ,both of which can be made arbitrarily small by choosing sufficiently large M . Therefore,following the analysis of Lemma , we have n ∑ k = W k ( t ) ≤ − ˜ δ V ( t ) ,which implies that ˙ V ( e ( t )) ≤ − ˜ δ V ( e ( t )) + λ(cid:101) n ∑ k = (cid:16) x ( ) k (cid:17) .Define A ( t ) = n ∑ k = (cid:16) x ( ) k (cid:17) .By the comparison lemma in [ ], we have V ( t ) ≤ φ ( t − ˜ t , 0 ) V ( ˜ t ) + λ(cid:101) (cid:90) t − ˜ t φ ( t − ˜ t , τ ) A ( τ + ˜ t ) d τ ( )where the transition function φ ( t , τ ) is φ ( t , 0 ) = exp (cid:0) − δ (cid:48) t (cid:1) .According to Corollary , we have A ( t ) = n ∑ k = (cid:16) x ( ) k (cid:17) ≤ (cid:32) n ∑ k = (cid:12)(cid:12)(cid:12) x ( ) k (cid:12)(cid:12)(cid:12)(cid:33) ≤ e − δ t . Substituting the bounds on φ ( t , τ ) and A ( τ ) , we obtain V ( t ) ≤ V ( ˜ t ) exp (cid:0) − ˜ δ ( t − ˜ t ) (cid:1) + λ(cid:101) (cid:90) t − ˜ t exp (cid:0) − ˜ δ ( t − ˜ t − τ ) − δ ( τ + ˜ t ) (cid:1) d τ = V ( ˜ t ) exp (cid:0) − ˜ δ ( t − ˜ t ) (cid:1) + λ(cid:101) exp (cid:0) − ˜ δ ( t + ˜ t ) (cid:1) (cid:90) t − ˜ t exp (cid:0) − ˜ δτ (cid:1) d τ = V ( ˜ t ) exp (cid:0) − ˜ δ ( t − ˜ t ) (cid:1) + λ(cid:101) exp (cid:0) − ˜ δ ( t + ˜ t ) (cid:1) δ (cid:0) − exp (cid:0) − ˜ δ ( t − ˜ t ) (cid:1)(cid:1) .The lemma holds due to the definition of V ( t ) and the fact that 0 ≤ ˜ w k ≤ k . Theorem . For sufficiently large M , we have (cid:90) ∞ | e ( t ) | dt = O (cid:32) ( n log n ) M (cid:33) . Proof:
Combining the previous two lemmas, we obtain (cid:90) ∞ | e ( t ) | dt = (cid:90) ˜ t | e ( t ) | dt + (cid:90) ∞ ˜ t | e ( t ) | dt ≤ t M + | e ( ˜ t ) | ˜ δ + λ ˜ δ (cid:101) ≤ t M + t ˜ δ M + λ ˜ δ (cid:101) .Since ˜ t = Θ ( n log n ) and ˜ δ = Θ ( n ) , the theorem holds.V. C onclusions This paper proved that the stationary distribution of the power-of-two-choices convergesin mean-square to its mean-field limit with rate O (cid:16) ( log M ) ( log log M ) M (cid:17) . The proof was basedon Stein’s method and the perturbation theory. The proof extended the result in [ ] toinfinite-dimensional systems. Besides quantifying the rate of convergence for the power-of-two-choices, the approach based on truncated mean-field models has the potential to be appliedto understand the rate of convergence other infinite-dimensional CTMCs to their mean-fieldlimits. A cknowledgement The author thanks Prof. Jim Dai, Anton Braverman, and Dheeraj Narasimha for very helpfuldiscussions. This work was supported in part by the NSF under Grant ECCS- . R eferences [ ] M. Mitzenmacher, “The power of two choices in randomized load balancing,” Ph.D. dissertation, University of Californiaat Berkeley, .[ ] N. D. Vvedenskaya, R. L. Dobrushin, and F. I. Karpelevich, “Queueing system with selection of the shortest of twoqueues: An asymptotic approach,” Problemy Peredachi Informatsii , vol. , no. , pp. – , .[ ] M. J. Luczak and J. Norris, “Strong approximation for the supermarket model,” Ann. Appl. Probab. , vol. , no. , pp. – , .[ ] M. Bramson, Y. Lu, and B. Prabhakar, “Asymptotic independence of queues under randomized load balancing,” QueueingSystems , vol. , no. , pp. – , .[ ] A. Mukhopadhyay and R. R. Mazumdar, “Analysis of load balancing in large heterogeneous processor sharing systems,” arXiv preprint arXiv: . , .[ ] J. N. Tsitsiklis and K. Xu, “On the power of (even a little) resource pooling,” Stochastic Systems , vol. , no. , pp. – , .[ ] Y. Lu, Q. Xie, G. Kliot, A. Geller, J. R. Larus, and A. Greenberg, “Join-Idle-Queue: A novel load balancing algorithm fordynamically scalable web services,” Performance Evaluation , vol. , no. , pp. – , .[ ] L. Ying, R. Srikant, and X. Kang, “The power of slightly more than one sample in randomized load balancing,” in Proc.IEEE Int. Conf. Computer Communications (INFOCOM) , Hong Kong, .[ ] Q. Xie, X. Dong, Y. Lu, and R. Srikant, “Power of d choices for large-scale bin packing: A loss model,” in Proc. Ann.ACM SIGMETRICS Conf. , .[ ] A. Mukhopadhyay, A. Karthik, R. R. Mazumdar, and F. Guillemin, “Mean field and propagation of chaos in multi-classheterogeneous loss models,” Perform. Eval. , vol. , pp. – , .[ ] I. Gurvich et al. , “Diffusion models and steady-state approximations for exponentially ergodic markovian queues,” Adv.in Appl. Probab. , vol. , no. , pp. – , .[ ] A. Braverman and J. Dai, “Stein’s method for steady-state diffusion approximations of m / ph / n + m systems,” arXivpreprint arXiv: . , .[ ] A. L. Stolyar et al. , “Tightness of stationary distributions of a flexible-server system in the halfin-whitt asymptotic regime,” Stochastic Systems , vol. , no. , pp. – , .[ ] A. Braverman, J. Dai, and J. Feng, “Stein’s method for steady-state diffusion approximations: an introduction throughthe Erlang-A and Erlang-C models,” arXiv preprint arXiv: . , .[ ] L. Ying, “On the approximation error of mean-field models,” in Proc. Ann. ACM SIGMETRICS Conf. , Antibes Juan-les-Pins, France, .[ ] A. D. Barbour, “Stein’s method and Poisson process convergence,” J. Appl. Probab. , pp. – , .[ ] F. Gotze, “On the rate of convergence in the multivariate clt,” Ann. Probab. , pp. – , .[ ] P. W. Glynn and A. Zeevi, “Bounding stationary expectations of Markov processes,” in Markov processes and related topics:a Festschrift for Thomas G. Kurtz . Institute of Mathematical Statistics, , pp. – .[ ] H. K. Khalil, Nonlinear systems . Prentice Hall,2001