Attractiveness of Brownian queues in tandem
aa r X i v : . [ m a t h . P R ] M a r Attractiveness of Brownian Queues in Tandem
Eric A. Cator Sergio I. L´opez Leandro P. R. PimentelMarch 14, 2019
Abstract
Consider a sequence of n bi-infinite and stationary Brownian queues in tandem. Assumethat the arrival process entering in the first queue is a zero mean ergodic process. We provethat the departure process from the n -th queue converges in distribution to a Brownian mo-tion as n goes to infinity. In particular this implies that the Brownian motion is an attractiveinvariant measure for the Brownian queueing operator. Our proof exploits the relationshipbetween the Brownian queues in tandem and the last-passage Brownian percolation model,developing a coupling technique in the second setting. The result is also interpreted in therelated context of Brownian particles acting under one sided reflection. Keywords . Brownian queue, Tandem queues, Last-passage percolation, Exclusion pro-cess.
Subclass
Tandem queues systems (TQ) are classical models in queueing theory consolidated from manydecades of research and generalized to stochastic networks with diverse structures. A tandemqueue is a system of queues where there is an initial arrival process A and a sequence { S n } n ≥ of service processes, all independent. The system is defined recursively: the initial queue isfed from the arrival process A , and has departures determined by the service process S . For n ≥
2, the arrival process for the n -th queue is defined as the departure process of the ( n -1)-thqueue and the departures are determined by the service process S n . One fundamental result inqueueing theory is Burke’s theorem, which states that, given a Poisson process as arrival andan independent Poisson process as service (where the service intensity is strictly larger than thearrival one), the departure process is a Poisson process. This type of result, where there is aninvariant law of the process under the queueing operator, is known as an Output theorem in theliterature, and it allows to compute explicitly many features of tandem queues systems.It is natural to consider the convergence of the departure process law from the n -th queue,as n goes to infinity, when the initial arrival process is arbitrary. This was answered in [21]in the case when the service processes are Poisson: there is convergence to a Poisson process,under weak conditions on the initial arrival process. In [24] the result was generalized to thecase when the service processes are not Poisson but independent and identically distributed. Inthis work we study the same question when the service processes are Brownian motions.Let us start by introducing the Brownian Tandem Queues (TQ). We follow the notationintroduced in [22]. For real and continuous functions f ∈ C ( R ), set f ( x, y ) := f ( y ) − f ( x ).Let a = ( a ( x ) , x ∈ R ) denote some continuous arrival process and for µ > s (1) ( x ) := µx − B (1) ( x ), where B (1) = ( B (1) ( x ) , x ∈ R ) is a two-sided Brownian1otion independent of a . The queue length process is defined as q (1) ( x ) := sup z ≤ x n a ( z, x ) − s (1) ( z, x ) o . (1.1)In order for q (1) to be stable (positive recurrent), we impose that the service process s (1) has adrift larger than that of the arrival process. We do this by requiringlim x →−∞ a ( x ) x = 0 and lim x →∞ a ( x ) x = 0 . The departure process is defined by d (1) ( x, y ) := a ( x, y ) − q (1) ( x, y ) , (1.2)with the convention d (1) (0) = 0, and hence we put d (1) ( x ) := d (1) (0 , x ).The tandem queue model, in words, consists of a line of queues, where each queue uses asinput (arrival) process the output (departure) process of the queue that is just in front of it in theline. In this context we have an initial arrival process a and service processes { s ( n ) } n ∈ N where s ( n ) ( x ) = µx − B ( n ) ( x ) and (cid:8) B ( n ) : n ∈ N (cid:9) is a collection of independent (two-sided) Brownianmotions. One can define inductively the queue length and the departure process of the n -thBrownian queue. Assume that the departure process ( d ( n ) ( x ) : x ∈ R ) is already defined. Thenwe can define the queue length process of the n + 1-th Brownian queue as q ( n +1) ( x ) := sup z ≤ x n d ( n ) ( z, x ) − µ ( x − z ) + B ( n +1) ( z, x ) o , and the departure process from the n -th Brownian queue d ( n +1) ( x, y ) := d ( n ) ( x, y ) − q ( n +1) ( x, y ) , with the similar convention d ( n +1) (0) = 0 and d ( n +1) ( x ) := d ( n +1) (0 , x ).A measure on the space of continuous arrival functions with zero drift is called invariant forthe queueing operator (in equilibrium), if the departure process has the same law as the arrivalprocess. For the Brownian queue operator, the measure induced by an independent standardBrownian motion B is an invariant (ergodic) measure [22]. Our result is the uniqueness of sucha measure, by proving attractiveness: Theorem 1.1
Start the process of queues in tandem with a zero mean ergodic arrival process.Then lim n →∞ d ( n ) dist. = B . (1.3)In our way to prove Theorem 1.1 we will only use that B is an invariant ergodic measure for thequeue system. Uniqueness will follow from our method.Essential for our proof is the connection of the Brownian TQ model to two related Brownianmodels, namely the Brownian Last Passage Percolation (LPP) and the Totally AsymmetricBrownian Exclusion Process (TABEP). We will introduce these models in Section 3, and pointout the relationships between the three models.All of these models have been previously studied, and the connection between them has beenknown for a while. Hambly, Martin and O’Connell [11] defined the LPP Brownian model andderived concentration results for the associated Brownian growth model. The related Brownianparticle system model has been studied in [5, 6]: particles are driven by Brownian motions andeach particle is reflected (only) on its left closest particle. While models of Brownian motions2nteracting by exclusion on the real line have been an active research topic [13, 14, 23], Ferrari,Spohn and Weiss successfully constructed a strong version of a two-sided system with an infiniteamount of particles in a stationary regime [6], governed by an asymmetric Skorokhod’s typereflection, easily related to the LPP model. They accomplished it by some technique resemblingLoynes’ stability theorem for G/G/
In this article, we first revisit the connection between these three models: the LPP Brownianmodel, the TABEP process and the TQ Brownian system. The relationship between the LPPmodel and the TABEP process is mentioned in [6] while the relation between the LPP modeland the TQ system is described in [22]. This is completely analogous to the known relationshipbetween the standard Markovian Tandem Queues, LPP on Z with exponential weights and theTASEP (Totally Asymmetric Exclusion process). For the sake of completeness, these modelsare presented in Section 2.Relying on these relations, we prove a result concerning the uniqueness of the invariantmeasure for the Brownian queueing operator, by proving attractiveness to that measure. Inwords, if we start with some zero mean ergodic process as initial arrival process and let itpass through the Brownian queues in tandem then the departure process from the n -th queueconverges in distribution to a Brownian motion as n goes to infinity. This is precisely stated inTheorem 1.1.For this purpose, we only use that the invariant measure under the queueing operator isknown [12]: it is the random measure associated to the Brownian motion. The method of proofis a coupling technique developed in the LPP Brownian setting: starting with two differentinitial arrival processes (called mass profiles in the LPP Brownian model) we use the sameservice processes (the random environment in the LPP context) to define the coupled evolution.Then we can prove that the difference between the associated departure processes (mass profiles)at each stage of the tandem is converging to zero on compact sets. This is our main result:Theorem 3.2, which implies the desired conclusion in the queueing context, Theorem 1.1. Wepoint out that this result can also be translated to an attractiveness result for a semi-infiniteTABEP system, see Theorem 3.1.A key step in the method involves local comparison techniques which allow us to bound thedifference between mass profiles in terms of the so-called exit points in the LPP literature. Thisimplies that it is only necessary to control the exit points for a given system (done in Lemma 4.5)and then to control the difference between the exit points defined for each of the coupled systems.These exit points are naturally defined in the LPP context but we give an interpretation in thequeueing setting in the following. First, consider an arbitrary initial arrival process and a singlenode Brownian queue. The exit point associated to time x is the last time Z ( x,
1) before time x when the Brownian queue was empty. Given node n of a tandem Brownian queue system andsome time x , define I n − ( x, n ) as the last time the n -th queue was empty before time x , then I n − ( x, n ) to be the last time the ( n -1)-th queue was empty before time I n − ( x, n ), and so on,until we find the exit point Z ( x, n ) = I ( x, n ). Hence, the exit time can be found from thisiterative process of marking the beginning of the current excursion of the queue in each stage ofthe tandem system. This property is described in more detail in Subsection 4.1.Our method of proof differs substantially from the methods developed for discrete valued3ueueing systems: In [21] a coupling between the departure times in every step of the tandemqueue of each user is accomplished, while in [24] the waiting times of each user in every node ofthe tandem queue system are considered for the coupling.A rather simplified version of this result was presented in [15] where, using a path couplingof the departures processes, a non-stationary and one-sided (in time) system is studied withsome particular initial conditions. Those techniques are non applicable to the current bi-infinitestationary setting. In Section 2, we first review the classical discrete models. Then we define the Totally Asym-metric Brownian Exclusion Process (Subsection 3.1) and the Last Passage Percolation System(Subsection 3.2). In each of these Subsections, our result is stated in the corresponding context(Theorems 3.1 and 3.2) and the explicit relations between the models are shown. In Subsection3.2 the coupled dynamics are defined. In Section 3, we first present the definition of exit pointsand the results concerning its control (Subsection 4.1) and then proceed to show the comparisonresults and the proof of Theorem 3.2 (Subsections 4.2 and 4.3).
In this Section we review some fundamental relationship between the classical Markovian Tan-dem queue model (TQ) with the exponential last-passage percolation model (LPP) and theTotally Asymmetric Simple Exclusion Process (TASEP).Assume that we have K Markovian queues in tandem working under a FIFO discipline.At time zero, the first queue starts working with N users in the line while all the otherqueues are empty. Define a collection of rate one independent exponential random variables { X ( n, k ) } n =1 ,...,N,k =1 ,...,K where X ( n, k ) represents the service time of the n -th user at server k .Define D ( n, k ) as the time where the n -th user exits the k -th server. Note that server k onlystarts to serve user n after user n − k and the service from server k − n has been finished. Then we have the following recurrence structure: D ( n, k ) = X ( n, k ) + max( D ( n, k − , D ( n − , k )) , (2.1)with boundary conditions D (0 ,
0) = 0 and D ( n, k ) = 0 if n < k <
0. We will show how thisstructure is related with the aforementioned models.Consider a collection of i.i.d. random variables { W x : x ∈ ( Z + ) } (also called weights),distributed according to an exponential distribution function of parameter one. In last-passagesite percolation (LPP) models, each number W x is interpreted as the percolation (passage) timethrough vertex x = ( x (1) , x (2)). For a lattice vertex x = ( n, k ) in ( Z + ) , denote Γ( x ) theset of all up-right oriented paths γ = ( x , x . . . , x k ) from to x , i.e. x = , x k = x and x j +1 − x j ∈ { e , e } , for j = 0 , . . . , k −
1, where e = (1 ,
0) and e = (0 , γ is defined as W ( γ ) := k X j =0 W x i . The last-passage time between and x is defined as L ( x ) ≡ L ( n, k ) := max γ ∈ Γ( x ) W ( γ ) .
4y the up-right path structure and the dynamic programming principle, we have the followingBellman equation: L ( n, k ) = W ( n, k ) + max( L ( n, k − , L ( n − , k )) . (2.2)This equation is the same as (2.1) with the same boundary conditions, so the last passagepercolation function is an equivalent way to describe the departure times from a tandem queuesystem.Let us define the related interacting particle system. Let Ω be the space of binary sequences η : Z → { , } . The elements η in Ω will be configurations of particles. We will say that aconfiguration η such that η ( x ) = 1 has a particle at position x . If η ( x ) = 0 we say that position x is empty or that we have a hole in that position. The dynamics are defined by the infinitesimalgenerator L [ f ]( η ) = X x ∈ Z η ( x )(1 − η ( x − f ( η x,x − ) − f ( η )) , where η x,x − is defined as the configuration that is identical to η except for the positions x and x −
1, where the original values are exchanged. The interpretation is the following: fromeach possible site x we have a constant rate of jump of the particles (if there is no particle atsite x , nothing happens). Once the clock at position x rings, the particle in that place tries tojump to the site x − x − { η t : t ≥ } be a TASEP process with initial configuration η . Assumethe initial configuration η is such that η ( x ) = 1 [0 , ∞ ) ( x ) for every x ∈ Z , which means that allthe particles are at the right of the origin in consecutive positions. Label each particle with itsinitial position, and define x l ( t ) to be the position of the l -th particle at time t (so x l (0) = l forevery l ∈ N ). Define q l ( t ) := x l ( t ) − x l +1 ( t ) − , (2.3)that is, the number of users in server l at time is equal to the number of holes between par-ticles l and l + 1 at time t . Note that (2.3) translates exactly the movement of particles inthe exclusion process to the tandem queues dynamics: every time that the particle l moves tothe left, one user is entering into the l th queue. Moreover, if the particle l which is movingis not the first one, the number of users in the ( l -1)-th queue diminish by one, so the user isleaving that queue. The exclusion property for particles translates into the restriction of hav-ing a non-negative number of users in each queue. Consider now that holes are labeled in thestarting configuration η : the hole at position l < − l . The model is sym-metric in particles and holes: one can think of holes traveling to the right which satisfy theexclusion property between them. Therefore, using (2.3), we have another interpretation of thedeparture time D ( n, k ): it is exactly the time when particle k is exchanging position with hole n .The previous presented relationships are known and studied, see [18]. In the last two decadesgreat progress has been made for the LPP models and this has given insight to an importantquestion originally posed in queueing theory: the asymptotic distribution of the departure timeof the n -th user in line from the m -th queue (its order in the line of queues), when the wholesystem starts empty, by making m and n grow to infinity while keeping fixed the ratio betweenthem [9, 25]. On the other hand, strong results from queueing theory concerning the existenceand attractiveness of invariant measures under the queueing operator [19, 24] have been used to5hed light on difficult questions concerning LPP models, as for example the existence of semi-infinite geodesics and Busemann functions for the lattice model in Z with general distributedweights, see [7, 8]. Theorem 1.1 states our convergence result for the Brownian TQ. In this Section and the nextone we will restate basically the same result in the context of two different Brownian models.
Consider a semi-infinite system of Brownian interacting particles defined for all real times x .Take some stationary, ergodic and continuous process { X (0) ( x ) : x ∈ R } and define X (0) ( x ) asthe position of the leftmost particle at time x . We introduce a collection { B ( n ) : n ≥ } ofindependent two-sided standard Brownian motions. Then, for n ≥
1, define X ( n ) ( x ) = sup y ≤ x ( X ( n − ( y ) + B ( n ) ( x ) − B ( n ) ( y )) , x ∈ R . (3.1)The system { X ( n ) ( x ) : x ∈ R } n ≥ will be called the Totally Asymmetric Brownian ExclusionProcess (TABEP) with leftmost particle X (0) . By definition, the order of the particles is pre-served: X (0) ( x ) ≤ X (1) ( x ) ≤ ... for every real time x (choosing y = x in the argument of thesupremum in (3.1) shows that X ( n − ( x ) ≤ X ( n ) ( x )). Note that (3.1) implies that the TABEPis Markovian in n : conditionally on the information of the process X ( n ) , the process X ( n +1) isindependent from the collection { X ( k ) } k =1 ,...,n − . These two properties can be combined to givean informal interpretation: the n -th particle is obtained by reflecting an independent Brownianmotion to its left-side neighbor (the ( n -1)-th particle) and this is the only possible interactionbetween particles (note that a particle does not notice the particles to the right of it).A sufficient condition to have a well-defined system is that for some positive constant µ , X (0) satisfies lim inf x →−∞ X (0) ( x ) x ≥ µ and that lim sup x →∞ X (0) ( x ) x ≤ µ . (3.2)Note that the whole system is time stationary: one can prove inductively that the distributionof X ( n ) ( x ) does not depend on x , for every n ≥ µ Poisson process on [0 , ∞ ) and the left-most particle is given by X (0) ( x ) = B (0) ( x ) + µx, where { B (0) ( x ) : x ∈ R } is a Brownian motion. Using Burke’s theorem for Brownian motion[22], they constructed a stationary bi-infinite system of ordered particles ... ≤ X ( − ( x ) ≤ X (0) ( x ) ≤ X (1) ( x ) ≤ ... ∀ x ≥ x , the set of positions is distributed as a rate µ Poisson process on the line.Now we show the relation with the tandem Brownian queues. Consider a TABEP system { X ( n ) } n ≥ , defined by (3.1). Define the arrival process a ( x ) := µx − X (0) ( x ) and the service6rocesses s ( n ) ( x ) := µx − B ( n ) ( x ) for each n ≥ a has zero drift and s ( n ) ( x ) haspositive drift µ ). Then the associated first queue length process is given by q (1) ( x ) = sup y ≤ x ( X (0) ( y ) − X (0) ( x ) + B (1) ( x ) − B (1) ( y )) ∀ x ∈ R , the first departure process is d (1) ( x ) = q (1) (0) + X (0) (0) + µx − sup y ≤ x ( X (0) ( y ) + B (1) ( x ) − B (1) ( y )) ∀ x ∈ R , and, by (3.1), we conclude that d (1) ( x ) = X (0) (0) + q (1) (0) + µx − X (1) ( x ) (we are using theconvention d (1) (0) = 0).Analogous formulae hold for any n ≥
1, by induction: Suppose now that for a fixed natural k we have d ( k ) ( x ) = X (0) (0) + k X i =1 q ( i ) (0) + µx − X ( k ) ( x ) , ∀ x ∈ R . Then q ( k +1) ( x ) = sup y ≤ x ( d ( k ) ( y, x ) − s ( n ) ( y, x )) = sup y ≤ x ( X ( k ) ( y ) − X ( k ) ( x ) + B ( k ) ( x ) − B ( k ) ( y )) , ∀ x ∈ R . Since d ( k +1) (0) = d ( k ) (0) = 0, this implies that d ( k +1) ( x ) = d ( k ) ( x ) − q ( k +1) ( x ) + q ( k +1) (0) (3.3)= X (0) (0) + k +1 X i =1 q ( i ) (0) + sup y ≤ x ( X ( k ) ( y ) + B ( k ) ( x ) − B ( k ) ( y )) ∀ x ∈ R , (3.4)where we also used the induction hyphotesis. By (3.1) it follows that d ( k +1) ( x ) = X (0) (0) + k +1 X i =1 q ( i ) (0) + µx − X ( k +1) ( x ) , ∀ x ∈ R . An important remark is that, by using (3.1), we get that q ( n ) ( x ) = X ( n ) ( x ) − X ( n − ( x ) , so the distance between the particles n − n is equal to the n -th queue length process attime x . Thus (1.3) is equivalent to Theorem 3.1 below. Theorem 3.1
Start a two sided TABEP with an ergodic process as the leftmost particle whichsatisfies (3.2) for some positive constant µ . Then the limit of the (centered) n -th particle con-verges to a two-sided Brownian Motion with drift µ , that is lim n →∞ (cid:16) X ( n ) ( x ) − X ( n ) (0) (cid:17) dist. = B ( x ) + µx . (3.5) In this section we define the elements of the theory of last-passage percolation systems [3] withBrownian passage times, as developed in [11], and show its relationship with tandem Brownianqueues. Let ω := (cid:8) B ( n ) : n ∈ Z (cid:9) be a collection of i.i.d. two-sided Brownian motions. Definethe order “ < ” in R × Z as the coordinate-wise order. For x = ( x, k ) < y = ( y, l ) ∈ R × Z denote7( x , y ) the set of all real increasing sequences γ = ( x = z ≤ z ≤ · · · ≤ z l − k +1 = y ). Thepassage time of γ is defined as L ( γ ) := l − k X i =0 B ( k + i ) ( z i , z i +1 ) . The last-passage time between x and y is given by L ( x , y ) := sup γ ∈ Γ( x , y ) L ( γ ) . (3.6)The passage time of a path γ can be seen as a continuous real valued process X = ( X ( z ) : z ∈ Γ) where Γ = { z = ( z , ..., z l − k ) : x ≤ z ≤ · · · ≤ z l − k ≤ y } ⊆ R l − k . Since Γ is compact, by continuity, we have that the maximum is attained at some location.In [16] is proven that, for x and y fixed, the maximum is attained at a unique location withprobability one. However, it is not true that this uniqueness holds simultaneously for all points x , y ∈ R × N . To see an example, for x > Z ( x ) = { z ∈ [0 , x ] : B (0) (0 , z ) + B (1) ( z, x ) = L ( , ( x, } , where = (0 , W x := B (0) ( x ) − B (1) ( x ) and note that z ∈ Z ( x ) is equivalent to W z = sup u ∈ [0 ,x ] W u . Thus, by Levy’s theorem, we have that { x ≥ Z ( x ) > } dist. = { x ≥ √ l x is strictly increasing } , where l x is the local time of a standard Brownian motion.We will call the geodesic (or the maximizer) between x and y to be the path γ ( x , y ) suchthat L ( γ ( x , y )) = L ( x , y ) . To introduce the last-passage percolation system we consider an initial profile ν = ( ν ( x ) , x ∈ R ) such that ν (0) = 0 and lim inf y →−∞ ν ( y ) y > , (3.7)and define the (discrete time) evolution of ν as the Markov process ( M ( n ) ν , n ≥ M (0) ν = ν , L ν ( x, n ) := sup z ≤ x { ν ( z ) + L (( z, , ( x, n )) } and M ( n ) ν ( x ) := L ν ( x, n ) − L ν (0 , n ) . (3.8)The Markov property follows from the following fact: for all n ≥ k ∈ { , . . . , n − } L ν ( x, n ) − L ν (0 , k ) = sup z ≤ x n M ( k ) ν ( z ) + L (( z, k + 1) , ( x, n )) o , (3.9)which is an application of the dynamic programming principle. This is a graphical constructionof the process where the space-time random environment is given by the collection of Brownianmotions ω = (cid:8) B ( n ) : n ∈ Z (cid:9) . The variational formula expresses the profile at time n as afunction of the profile at time k < n plus some strip of the space-time environment which is8ndependent of the profile at time k . We note that this construction allows us to run the last-passage percolation system, started with two arbitrary initial profiles ν and ν , simultaneouslywith the same environment ω (basic coupling). Formally speaking, we define the joint process( M ( n ) ν , M ( n ) ν ) n ≥ by setting( x, n ) (cid:26) L ν ( x, n ) := sup z ≤ x { ν ( z ) + L (( z, , ( x, n )) } ,L ν ( x, n ) := sup z ≤ x { ν ( z ) + L (( z, , ( x, n )) } , (3.10)and putting M ( n ) ν i ( x ) := L ν i ( x, n ) − L ν i (0 , n ) for x real and i = 1 ,
2. Notice that L (( z, , ( x, n ))is a function that only depends on ω .The analogy with the queue system is as follows. Assume that ν ( x ) has drift µ and take a ( x ) = µx − ν ( x ) and s ( n ) ( x ) := µx − B ( n ) ( x ) . (3.11)Then q (1) ( x ) := sup z ≤ x n a ( z, x ) − s (1) ( z, x ) o = L ν ( x, − ν ( x )and d (1) ( x ) := a ( x ) + q (1) (0) − q (1) ( x ) = µx − M (1) ν ( x ) . From this, using definitions (1.1), (1.2), (3.8) and induction, one can check the analogous relationfor all n ≥ q ( n ) ( x ) = sup z ≤ x n d ( n − ( z, x ) − s ( n ) ( z, x ) o = L ν ( x, n ) − L ν ( x, n − d ( n ) ( x ) = a ( x ) + q ( n ) (0) − q ( n ) ( x ) = µx − M ( n ) ν ( x ) . Thus, (1.3) and (3.5) are consequences of (3.13) below. Define B µ ( x ) = µx + B ( x ) , where B is a standard Brownian motion. Using the invariance of the Brownian measure underthe queueing operator, it is immediate that B µ is invariant: M ( n ) µ ≡ M ( n ) B µ dist. := B µ , for all n ≥ . The main contribution of this article is the next theorem, from which (1.3) (and (3.5)) willfollow.
Theorem 3.2
Let µ ∈ (0 , ∞ ) and assume that, almost surely, lim inf x →−∞ ν ( x ) x ≥ µ and lim sup x →∞ ν ( x ) x ≤ µ . (3.12) Consider the basic coupling ( M ( n ) ν , M ( n ) µ ) n ≥ constructed by running the last-passage percolationsystem, started with ν and B µ , simultaneously with the same environment ω = (cid:8) B ( n ) : n ∈ Z (cid:9) .Then, for all compact K ⊆ R and ǫ > , lim n →∞ P (cid:18) sup x ∈ K | M ( n ) ν ( n, µ − n + x ) − M ( n ) µ ( n, µ − n + x ) | > ǫ (cid:19) = 0 . (3.13)It should be clear that an ergodic initial profile satisfies (3.12) almost surely (note that inthat case, we have translation invariance of the law of M ( n ) ν and M ( n ) µ , so that we can get ridof the translation by µ − n ). We note that (3.13) implies local convergence for initial profilesbeyond the ergodic condition: one could take a deterministic profile satisfying (3.12).9 Proofs
First proven in [1, 10], using that L ( , ( n, n )) has the same law as the largest eigenvalue of a n × n GUE random matrix, the shape theorem below is presented by Hambly et al. [11] as aconsequence of concentration results for the Brownian directed percolation paths:lim n →∞ n L (( , ( xn, tn )) a.s. = 2 √ xt. (4.1)Note that, by Brownian scaling, { L (( , ( rn, n )) : r ∈ [0 , x ] } dist. = (cid:8) √ xL (( , ( sn, n )) : s ∈ [0 , (cid:9) . (4.2) Remark 4.1
By Lemma in [11], there exist constants c , c ≥ such that P (cid:16)(cid:12)(cid:12)(cid:12) L ( , ( n, n )) n − (cid:12)(cid:12)(cid:12) ≥ y (cid:17) ≤ c exp {− c n ( y − ǫ n ) } , (4.3) for all n ≥ , and y > ǫ n , where ǫ n := 2 − E L ( , ( n, n )) n + 1 n / . Since ǫ n → , we can choose n large such that ǫ n < − δ and take y = 2 − δ . This implies thatthere exist constants c , c > such that for all δ > there exists N > such that P (cid:16)(cid:12)(cid:12)(cid:12) L ( , ( n, n )) n − (cid:12)(cid:12)(cid:12) ≥ δ (cid:17) ≤ c exp {− c nδ } , for all n ≥ N . We notice that a better upper bound could be produced by using the couplingmethod [2] to prove that E | L ( , ( n, n )) − n | = O ( n / ) , which would imply that ǫ n = O ( n − / ) . For the Brownian last-passage percolation model we haveall the ingredients necessary for the coupling method: we know explicitly the invariant regimeand the shape function. From now on we will treat ν as a fixed deterministic profile satisfying (3.12). Define the exitpoint from ( x, n ) as Z ν ( x, n ) = sup { z ≤ x : L ν ( x, n ) = ν ( z ) + L (( z, , ( x, n )) } . (4.4)We note that it is well defined. First, since we have the same asymptotic hypothesis (3.7) on theprofile ν , one can use similar arguments as in Proposition 4 . L ν ( x, n ) is well defined. By Brownian continuity, the map z → L (( z, , ( x, n )) is continuous,just as the profile ν (by hypothesis). Then the set { z ∈ C : L ν ( x, n ) = ν ( z ) + L (( z, , ( x, n )) } isnon empty for any compact set C . To prove that the supremum over z ≤ y can be restricted tosome compact set one can mimic the proof of Lemma 4 . exit point comes from the next geometrical interpretation in last-passage per-colation: Z ν ( x, n ) is the time before x when the path which maximizes the quantity ν ( z ) + L (( z, , ( x, n )) leaves the initial profile ν (that can be visualized on the line { ( x,
0) : x ∈ R } ) topercolate to the point ( x, n ). 10he exit point (4.4) can also be described in terms of the tandem queueing system. First, letus examine the interpretation for Z ν ( x, z ∗ be in { z ≤ x : L ν ( x,
1) = ν ( z ) + L (( z, , ( x, } .Then ν ( z ∗ ) + L (( z ∗ , , ( x, ≥ ν ( z ) + L (( z, , ( x, ∀ z ≤ x, and, by (3.11), this implies that a ( z ) − s (1) ( z ) ≥ a ( z ∗ ) − s (1) ( z ∗ ) ∀ z ≤ x. In other words, a ( z ∗ , x ) − s (1) ( z ∗ , x ) ≥ a ( z, x ) − s (1) ( z, x ) ∀ z ≤ x, so q (1) ( x ) = a ( z ∗ , x ) − s (1) ( z ∗ , x ) (by the definition (1.1)). This implies that q (1) ( z ∗ ) = 0, so thevalue Z ν ( x,
1) is the last time when the queue-length process q (1) was empty before time x . For n arbitrary, using the expression (3.9), one can check that the value Z ν ( x, n ) can be obtainedinductively: let I n − ( x, n ) be the last time when q ( n ) was empty before time x , then I n − ( x, n )is the last time when q ( n − was empty before time I n − ( x, n ), and so on, till we find the exitpoint Z ν ( x, n ) = I ( x, n ).In the next result we show that, in probability, the exit point is asymptotically sublinear. Lemma 4.2
Let µ ∈ (0 , ∞ ) and assume (3.12) . Then, for all C ∈ R and ǫ > , lim n →∞ P (cid:0) n − | Z ν ( µ − n + C, n ) | > ǫ (cid:1) = 0 . Proof:
By Brownian scaling (4.2), one can restrict the attention to µ = 1. For fixed δ >
0, take B δ and construct L δ and L ν simultaneously using the basic coupling (3.10). Since L (( z, , ( n + C, n )) ≤ L δ ( n + C, n ) − B δ ( z )and L ((1 , , ( n + C, n )) = L ((1 , , ( n + C, n )) + ν (0) ≤ L ν ( n + C, n )(recall that ν (0) = 0), we have that { Z ν ( n + C, n ) ≥ u } = {∃ z ∈ [ u, n + C ] : ν ( z ) + L (( z, , ( n + C, n )) = L ν ( n + C, n ) } is contained in the event {∃ z ∈ [ u, n + C ] : B δ ( z ) − ν ( z ) ≤ L δ ( n + C, n ) − L ((1 , , ( n + C, n )) } . By (3.12) there exists K > ν ( z ) ≤ (1 + δ/ z for all z > K . Hence, if u > K then { Z ν ( n + C, n ) ≥ u } is contained in the event {∃ z ∈ [ u, n + C ] : B − δ ( z ) ≤ L δ ( n + C, n ) − L ((1 , , ( n + C, n )) } . (4.5)Now we recenter the Brownian motion with drift at position u by writing B − δ ( z ) := B − δ ( u ) + ¯ B − δ ( z ) , where ¯ B − δ ( z ) := B − δ ( z ) − B − δ ( u ) for z ≥ u . Notice that { ¯ B − δ ( z ) : z ≥ u } has the samedistribution as the process { B − δ ( z ) : z ≥ } and it is independent of B − δ ( u ). Let A ( u ) := B − δ ( u ) + min z ≥ u ¯ B − δ ( z ) . B − δ has a positive drift, and its distribution is given byminus an exponential random variable of parameter 2 − δ (its value will not play an importantrole when n grows to infinity, since δ is fixed). Thus, by (4.5), { Z ν ( n + C, n ) ≥ u } ⊆ { A ( u ) ≤ L δ ( n + C, n ) − L ((1 , , ( n + C, n )) } ∀ u ≤ n + C . (4.6)The strategy is to show that if u = ǫn we can choose δ > ǫ >
0, to be defined later, we have that the event on the r.h.s.of (4.6) has probability bounded by P (cid:16) L ((1 , , ( n + C, n )) − n ≤ − ǫ n (cid:17) + P (cid:16) A ( u ) ≤ L δ ( n + C, n ) − n + ǫ n (cid:17) . By the shape theorem, lim n →∞ P (cid:16) L ((1 , , ( n + C, n )) − n ≤ − ǫ n (cid:17) = 0 . On the other hand, P (cid:16) A ( u ) ≤ L δ ( n + C, n ) − n + ǫ n (cid:17) ≤ P (cid:16) − δu − ǫ n ≤ L δ ( n + C, n ) − n (cid:17) + P (cid:16) A ( u ) ≤ − δu − ǫ n (cid:17) . (4.7)We now use the result in Section 4 of [22], where it is shown (in our notation) that L λ (0 , n ) − L λ (0 ,
0) (this is the vertical increment) is distributed as the sum of n independent exponentialrandom variables, each with expectation 1 /λ . We already know that x L λ ( x, n ) − L λ (0 , n )(the horizontal increment) is distributed as Brownian motion with drift λ . This shows us howto recenter L δ ( n + C, n ): P (cid:16) − δu − ǫ n ≤ L δ ( n + C, n ) − n (cid:17) = P (cid:18) ∆ − ǫ n ≤ L δ ( n + C, n ) − (cid:18) (1 + δ ) + 11 + δ (cid:19) n (cid:19) , where ∆ := 2 n − (cid:18) (1 + δ ) + 11 + δ (cid:19) n + δ u = − δ (1 + δ ) n + δ u> (cid:18) δ un − δ (cid:19) n . If u = ǫn and we pick δ := 4 − ǫ , we get the next lower bound for ∆:∆ > (cid:18) δ ǫ − δ (cid:19) n = ǫ n . Thus, for ǫ := ǫ , P (cid:16) − δu − ǫ n ≤ L δ ( n + C, n ) − n (cid:17) ≤ P (cid:18) − ǫ n ≤ L δ ( n + C, n ) − (cid:18) (1 + δ ) + 11 + δ (cid:19) n (cid:19) .
12e have already seen that L δ (0 , n ) − n/ (1 + δ ) has expectation 0 and variance of order n , andalso that L δ ( n + C, n ) − L δ (0 , n ) − (1 + δ ) n has expectation C (1 + δ ) and variance of order n , so we conclude thatlim n →∞ P (cid:18) − ǫ n ≤ L δ ( n + C, n ) − (cid:18) (1 + δ ) + 11 + δ (cid:19) n (cid:19) = 0 , and hence lim n →∞ P (cid:16) − δu − ǫ n ≤ L δ ( n + C, n ) − n (cid:17) = 0 . To bound the second summand in (4.7), take u = ǫn and writelim n →∞ P (cid:16) A ( u ) ≤ − δu − ǫ n (cid:17) = lim n →∞ P (cid:16) B ( ǫn ) n + min z ≥ ǫn ¯ B − δ ( z ) n ≤ − ǫ (cid:17) = 0 . By (4.5), this concludes the proof oflim n →∞ P ( Z ν ( n + C, n ) > ǫn ) = 0 . To get the analog result for { Z ν ( n + C, n ) < − ǫn } one just needs to adapt the same argument. (cid:4) . In the next lemmas we will always construct L ν and L ν simultaneously with the basic coupling(3.10). Lemma 4.3 If x < y and Z ν ( y, n ) ≤ Z ν ( x, n ) then L ν ( y, n ) − L ν ( x, n ) ≤ L ν ( y, n ) − L ν ( x, n ) . Proof:
Recall the definition of the geodesic γ ( x , y ) between two points x < y in R × Z in Subsection3.2. Denote by γ zn ( x ) to the geodesic between ( z,
1) and ( x, n ). Notice that L (( z, , ( x, n )) = L (( z, , ( y, m )) + L (( y, m ) , ( x, n )) , for any ( y, m ) ∈ γ zn ( x ).Assume that Z ν ( y, n ) ≤ Z ν ( x, n ), denote z ≡ Z ν ( y, ) and z ≡ Z ν ( x, n ). Let c be acrossing point between the two geodesics γ z n ( y ) and γ z n ( x ). Such a crossing point always existsbecause x ≤ y and z ≤ z (by assumption). We remark that, by superaddivity of L , L ν ( y, n ) ≥ ν ( z ) + L (( z , , ( y, n )) ≥ ν ( z ) + L (( z , , c ) + L ( c , ( y, n )) . We use this, and that (since c ∈ γ z n ( x )) ν ( z ) + L (( z , , c ) − L ν ( x, n ) = − L ( c , ( x, n )) , in the following inequality: M ( n ) ν ( x, y ) = L ν ( y, n ) − L ν ( x, n ) ≥ ν ( z ) + L (( z , , c ) + L ( c , ( y, n )) − L ν ( x, n )= L ( c , ( y, n )) − L ( c , ( x, n )) .
13y superaddivity, − L ( c , ( x, n )) ≥ L ν ( c ) − L ν ( x, n ) , and hence (since c ∈ γ z ( y, n )) M ( n ) ν ( x, y ) ≥ L ( c , ( y, n )) − L ( c , ( x, n )) ≥ L ( c , ( y, n )) + L ν ( c ) − L ν ( x, n )= L ν ( y, n ) − L ν ( x, n )= ∆ M ( n ) ν ( x, y ) . (cid:4) . Lemma 4.4
Assume that ν ( y ) − ν ( x ) ≤ ν ( y ) − ν ( x ) for all x < y . Then L ν ( y, n ) − L ν ( x, n ) ≤ L ν ( y, n ) − L ν ( x, n ) , ∀ x < y . Proof:
Denote z := Z ν ( y, n ) and z := Z ν ( x, n ) . If z ≤ z then it follows from Lemma 4.3 (we do not need to use the assumption). If z > z then L ν ( y, n ) − L ν ( x, n ) − (cid:0) L ν ( y, n ) − L ν ( x, n ) (cid:1) = L ν ( y, n ) − (cid:0) ν ( z ) + L (( z , , ( x, n )) (cid:1) − (cid:16)(cid:0) ν ( z ) + L (( z , , ( y, n )) (cid:1) − L ν ( x, n ) (cid:17) = L ν ( y, n ) − (cid:0) ν ( z ) + L (( z , , ( y, n )) (cid:1) − (cid:16)(cid:0) ν ( z ) + L (( z , , ( x, n )) (cid:1) − L ν ( x, n ) (cid:17) = L ν ( y, n ) − (cid:0) ν ( z ) + L (( z , , ( y, n )) (cid:1) + (cid:16) L ν ( x, n ) − (cid:0) ν ( z ) + L (( z , , ( x, n )) (cid:1)(cid:17) = L ν ( y, n ) − (cid:0) ν ( z ) + L (( z , , ( y, n )) (cid:1) + (cid:16) L ν ( x, n ) − (cid:0) ν ( z ) + L (( z , , ( x, n )) (cid:1)(cid:17) + (cid:0) ν ( z ) − ν ( z ) (cid:1) − (cid:0) ν ( z ) − ν ( z ) (cid:1) . By super-additivity, L ν ( y, n ) − (cid:0) ν ( z ) + L z ( y, n ) (cid:1) ≥ L ν ( x, n ) − (cid:0) ν ( z ) + L z ( x, n ) (cid:1) ≥ , while, by assumption, ν ( z ) − ν ( z ) ≥ ν ( z ) − ν ( z ) , since z > z . (cid:4) .14 .3 Proof of Theorem 3.2 Without lost of generality we will assume that µ = 1 (again by Brownian scaling (4.2)), andthat K = [0 , C ] with C >
0. We take as an initial profile a Brownian motion with drift 1, B ( x ) := x + B ( x ) , and also B µ ± := µ ± x + B ( x ) , with µ ± := 1 ± δ and δ >
0. Thus, B µ − ( y ) − B µ − ( x ) ≤ B ( y ) − B ( x ) ≤ B µ + ( y ) − B µ + ( x ) . Lemma 4.5
Let µ ∈ (0 , ∞ ) and assume (3.12) . Then, for all C > , lim n →∞ P (cid:16) Z µ − ( n + C, n ) ≤ Z ν ( n, n ) and Z ν ( n + C, n ) ≤ Z µ + ( n, n ) (cid:17) = 1 . Proof:
Let us first prove that lim n →∞ P (cid:0) Z µ − ( n + C, n ) ≤ Z ν ( n, n ) (cid:1) = 1 . For any ǫ > P (cid:0) Z µ − ( n + C, n ) > Z ν ( n, n ) (cid:1) ≤ P (cid:0) Z µ − ( n + C, n ) > − ǫn (cid:1) + P ( Z ν ( n, n ) < − ǫn ) . Thus, by Lemma 4.2, it is enough to show that (for fixed δ, C >
0) we can choose ǫ > n →∞ P (cid:0) Z µ − ( n + C, n ) ≤ − ǫn (cid:1) = 1 . (4.8)By shift invariance of Brownian Motion ( B ( x + C ) − B ( C ) dist. = B ( x )), P (cid:0) Z µ − ( n + C, n ) > − ǫn (cid:1) = P (cid:0) Z µ − ( n, n ) > − ǫn − C (cid:1) ≤ P (cid:0) Z µ − ( n, n ) > − ǫn (cid:1) , for n ≥ C/ǫ . Since n = µ − − n + (cid:0) − µ − − (cid:1) n and (cid:0) − µ − − (cid:1) < − δ , (recall δ ∈ (0 , / P (cid:0) Z µ − ( n, n ) > − ǫn (cid:1) ≤ P (cid:0) Z µ − ( µ − − n, n ) > ( δ − ǫ ) n (cid:1) . Hence, if ǫ < δ/
2, Lemma 4.2 implies (4.8). The proof oflim n →∞ P (cid:0) Z ν ( n + C, n ) ≤ Z µ + ( n, n ) (cid:1) = 1is analogous. (cid:4) .15f Z µ − ( n + C, n ) ≤ Z ν ( n, n ) and Z ν ( n + C, n ) ≤ Z µ + ( n, n ) then Z µ − ( n + x, n ) ≤ Z ν ( n, n ) and Z ν ( n + x, n ) ≤ Z µ + ( n, n ) for all x ∈ [0 , C ]. We use that Z ν ( y, n ) is a non-decreasing function of y (for fixed n ). By Lemma 4.3, M ( n ) µ − ( n, n + x ) ≤ M ( n ) ν ( n, n + x ) ≤ M ( n ) µ + ( n, n + x ) , for all x ∈ [0 , C ], and by Lemma 4.4, M ( n ) µ − ( n, n + x ) ≤ M ( n )1 ( n, n + x ) ≤ M ( n ) µ + ( n, n + x ) , for all x ∈ [0 , C ]. Therefore, | M ( n ) ν ( n, n + x ) − M ( n )1 ( n, n + x ) | ≤ M ( n ) µ + ( n, n + x ) − M ( n ) µ − ( n, n + x ) ≤ M ( n ) µ + ( n, n + C ) − M ( n ) µ − ( n, n + C ) , for all x ∈ [0 , C ]. We use that M ( n ) µ + ( n, n + x ) − M ( n ) µ − ( n, n + x ) is a non-decreasing function of x (Lemma 4.4). Hence, if Z µ − ( n + C, n ) ≤ Z ν ( n, n ) and Z ν ( n + C, n ) ≤ Z µ + ( n, n ) thensup x ∈ [0 ,C ] | M ( n ) ν ( n, n + x ) − M ( n )1 ( n, n + x ) | ≤ M ( n ) µ + ( n, n + C ) − M ( n ) µ − ( n, n + C ) . (4.9)Since M ( n ) µ + ( n, n + C ) − M ( n ) µ − ( n, n + C ) ≥ E (cid:16) M ( n ) µ + ( n, n + C ) − M ( n ) µ − ( n, n + C ) (cid:17) = ( µ + − µ − ) C = 2 δC , we have that P (cid:16) M ( n ) µ + ( n, n + C ) − M ( n ) µ − ( n, n + C ) > ǫ (cid:17) ≤ Cǫ δ .
Together with Lemma 4.5 and (4.9), this implies thatlim sup n →∞ P sup x ∈ [0 ,C ] | M ( n ) ν ( n, n + x ) − M ( n )1 ( n, n + x ) | > η ! ≤ Cǫ δ , under (3.12). Since δ > n →∞ P sup x ∈ [0 ,C ] | M ( n ) ν ( n, n + x ) − M ( n )1 ( n, n + x ) | > ǫ ! = 0under hypothesis (3.12), and Theorem 3.2 is proven. (cid:4) Conclusion
We proved that under mild conditions an initial flow passing through an infinite system of Brow-nian tandem queues converges in distribution to a Brownian Motion. The strong relationshipbetween the queueing system and the Last Passage Brownian Percolation model is fundamen-tal for the proof; since it allows to construct a coupling between different initial configurationsusing the concept of exit point in the LPP setting. This is a convenient way to manipulatethe busy periods associated to the tandem queues. One wonders if this relation, or the one16ith the TABEP system, could be useful to compute non asymptotic formulae for the queueingsystem. An example of this kind of result, whose interpretation in the Brownian tandem queuessetting has not yet been studied, is presented in [5], where an explicit determinantal formula isobtained for the joint distribution of particles in a periodic finite system of particles interactingby one-sided reflection.
Acknowledgements
The authors would like to thank an anonymous referee for her helpfulcomments that greatly improved the presentation and clarity of this work.
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