Multiclass Hammersley-Aldous-Diaconis process and multiclass-customer queues

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www.imstat.org/aihp 2009, Vol. 45, No. 1, 250–265

DOI: 10.1214/08-AIHP168

Multiclass Hammersley–Aldous–Diaconis process and multiclass-customer queues

Pablo A. Ferrari

^a

and James B. Martin

^b

aInstituto de Matemática e Estatística, Universidade de São Paulo, Caixa Postal 66281, 05311-970 São Paulo, Brazil. E-mail: pablo@ime.usp.br bDepartment of Statistics, 1 South Parks Road, Oxford OX1 3TG, United Kingdom. E-mail: martin@stats.ox.ac.uk

Received 27 July 2007; revised 29 January 2008; accepted 1 February 2008

Abstract. In the Hammersley–Aldous–Diaconis process, infinitely many particles sit inRand at most one particle is allowed at each position. A particle atx, whose nearest neighbor to the right is aty, jumps at ratey−xto a position uniformly distributed in the interval(x, y). The basic coupling between trajectories with different initial configuration induces a process with different classes of particles. We show that the invariant measures for the two-class process can be obtained as follows. First, a stationary M/M/1 queue is constructed as a function of two homogeneous Poisson processes, the arrivals with rateλand the (attempted) services with rateρ > λ. Then put first class particles at the instants of departures (effective services) and second class particles at the instants of unused services. The procedure is generalized for then-class case by usingn−1 queues in tandem withn−1 priority types of customers. Amulti-lineprocess is introduced; it consists of a coupling (different from Liggett’s basic coupling), having as invariant measure the product of Poisson processes. The definition of the multi-line process involves thedual pointsof the space–time Poisson process used in the graphical construction of the reversed process. The coupled process is a transformation of the multi-line process and its invariant measure is the transformation described above of the product measure.

Résumé. Dans un processus de Hammersley nous considérons une infinité de particules sur la droite réelle; et il ne peut pas y avoir plus d’une particule sur chaque position. Une particule située enxet ayant pour plus proche voisine (sur sa droite) une particule située eny, saute avec un tauxy−xà une position aléatoire choisie uniformément dans l’interval(x, y). Le couplage basique entre des trajectoires ayant des configurations initiales différentes induit un processus avec des particules de classes différentes. Nous donnons une construction explicite de la mesure invariante pour le processus ayantnclasses de particules. Pour démontrer que la mesure est invariante nous introduisons un autre processus appelé “multi-ligne”. La mesure invariante pour ce processus est un produit de plusieurs processus de Poisson. La définition du processus multi-ligne met en jeu les “points duaux” (de l’espace-temps), qui apparaient naturellement dans la construction graphique du processus renversé par rapport au temps.

MSC:60K35; 60K25; 90B22

Keywords:Multi-class Hammersley–Aldous–Diaconis process; Multiclass queuing system; Invariant measures

1. The Hammersley–Aldous–Diaconis process

The state spaceX is an appropriate subset of locally finite subsets of R. Elements ofX are called configurations and elements of a configurationη∈X are calledparticles. TheHammersley–Aldous–Diaconis(HAD)process[1,15]

can be informally described by saying that a particle sitting atr∈Rwaits an exponentially distributed random time with rate equal to the distance to the nearest particle to its right, located atr> r (say) to jump to a site uniformly distributed in the interval[r, r]. Alternatively, bells ring at space–time points at rate 1 and, when a bell rings at(r, t ), the nearest particle to the left ofrat timet−jumps tor.

A Harris graphical construction of the process is the following: Let ω∈Ω be a homogeneous rate-1 Poisson process in the space–time space R×R⁺ (or later inR²), whereΩ is the set of locally finite subsets of R². We

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Fig. 1. Harris construction. Stars represent the space–time Poisson points inω. Balls represent particle positions at times 0 andt.

shall usepointsto refer to space–time elements of ω. If η is the particle configuration at timet−and(r, t )∈ω, then at time t the configuration jumps to η\ {u} ∪ {r}, where u=u(η, r) is the nearest particle in η to the left of r. See Figure 1. This construction is well defined in a finite region as the points can be well ordered by time [1]. The construction inR was performed by Aldous and Diaconis [1] and then by Seppäläinen [21] in the state spaceX = {η: lim_s→∞|η∪ [0, s]|²/s= ∞}that we adopt. Here| · |counts the number of elements of a finite set.

Homogeneous Poisson processes inRgive mass 1 toX.

For fixed initial configurationηand pointsω, the process(ηt, t≥0)is a deterministic function ofηandωdenoted η_t=Φ(t, η, ω),t≥0. In this case we say that the process isgoverned byωwith initial configurationη. It satisfies

η_t=Φ(t−s, η_s, τ_sω), (1)

for all 0≤s < t, whereτ is the time translation operator defined byτ_sω= {(x, t−s), (x, t )∈ω}.

Coupled and multiclass process

A joint construction ofHADprocesses((η_tⁱ, t≥0), i=1, . . . , n)with initial configurationsη¹, . . . , ηⁿ governed by the same pointsωis calledcoupled process. It is defined by

ηⁱ_t=Φ t, ηⁱ, ω

. (2)

We slightly abuse notation, writingη=(η¹, . . . , ηⁿ)andη_t =Φ(t, η, ω)∈Xⁿ. In the coupled process when a point is at(x, t ), the closest particle to the left ofxin each marginal jumps toxsimultaneously.

Consider initial ordered particle configurationsη¹⊂ · · · ⊂ηⁿand run the coupled process. The order is maintained at later times. DefineXⁿ^↑= {(η¹, . . . , ηⁿ)∈X: η¹⊂ · · · ⊂ηⁿ}. DefineR:Xⁿ^↑→Xⁿby

(Rη)^k=η^k\η^k−¹. (3)

The process ξ_t =Rη_t is called themulticlass process. Particles in ξ_tⁱ are called i-class particles. The multiclass process is just a convention to describe a coupled process with ordered initial configurations. The mapRis invertible;

its inverse is given by(R⁻¹ξ )^k=ξ¹∪ · · · ∪ξ^k. The processξ_t governed byωwith initial configurationξ is defined by

ξt=Υ (t, ξ, ω)=RΦ

t, R⁻¹ξ, ω

. (4)

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Invariant measures

The Poisson process with densityλis an invariant measure for theHADprocess for allλ >0 [1]. Our main result is to construct invariant measures for the multiclass process. The resulting measure coincides with the law of the departure process of a stationary multiclass-customer queue system.

LetAandS be particle configurations inX. Think ofRas time and construct a continuous time random walk (Z_r, r∈R)jumping one unit up at times in Aand one unit down at times inS. We fixZ₀=0 but since we are interested in the increments, the position at a given time is not important. The increments satisfy

Z_r−Z_s=A∩ [r, s)−S∩ [r, s). (5)

Assume thatZ_r → ∓∞asr→ ±∞; this impliesZ_r visits each site a finite number of times. LetU (A, S)be the times inSthatZ_r attains a new record down andD(A, S)its complement:

U (A, S):=

r∈S: Z_r< inf

r<rZ_r

,

(6) D(A, S):=S\U (A, S).

We interpretAas arrival times of customers to a one-server queue andSas the times at which service occurs. At these service times there are two possibilities: either there is a customer in the system, producing adeparture, or there is no customer, in this case there is anunused service. We collect the departure times inDand the unused service times inU. In particular, suppose thatAandSare stationary Poisson processes, independent from each other, with ratesλ < λ, respectively. ThenZ_r satisfies the conditions above. LetQ_r=(Z_r−infr≤rZ_r); then the processQ_r is stationary Markov, and satisfies

Qr−Qr−=1{r∈A} −1{r∈D: Qr−>0}. (7)

The transition rates of this Markov process areλfrom stateito statei+1 for alli≥0, andλ from stateito state i−1 for alli≥1. HenceQis the queue-length process of a stationaryM/M/1 queue, with arrival and service rates λandλ,respectively. The pointsD(A, S)are the departure times from this queue (the times when the queue-length decreases by 1). Figure 2 illustrates the original Poisson processesAandS, the random walkZand the queueQ.

The operatorDhas the following properties:

(i) D(α¹, α²)⊂α²;

(ii) Ifα˜¹⊂α¹,thenD(α˜¹, α²)⊂D(α¹, α²);

(iii) Ifα¹ andα² are independent one-dimensional Poisson processes of densitiesρ¹ andρ² withρ¹< ρ², then D(α¹, α²)is also a one-dimensional Poisson process with densityρ¹.

Fig. 2. Arrivals and services. Each arrival is linked to its departure time, running the queue according to the FIFO (first-in-first-out) schedule.

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Part (iii) is Burke’s theorem for anM/M/1 queue [3].

Let ν^λ be the law of a Poisson process of rate λ. For ρ¹<· · ·< ρⁿ define ν:=ν^ρ¹ × · · · ×ν^ρⁿ. Let α= (α¹, . . . , αⁿ)∈Xⁿbe a multi-line configuration with lawν. Define a sequence of operatorsD⁽ⁿ⁾:Xⁿ→Xas follows:

LetD⁽¹⁾(α¹)=α¹, and then recursively forn≥2, let D⁽ⁿ⁾

α¹, α², . . . , αⁿ

=D D⁽ⁿ⁻¹⁾

α¹, . . . , αⁿ⁻¹ , αⁿ

. (8)

The configurationD⁽ⁿ⁾(α¹, α², . . . , αⁿ)represents the departure process from a system of(n−1)queues in tandem.

The arrival process to the first queue isα¹. The service process of thekth queue isα^k⁺¹, fork=1, . . . , n−1. Finally, fork=2, . . . , n−1, the arrival process to thekth queue is given by the departure process of the(k−1)st queue. This is known as a system of·/M/1 queues in tandem.

Note D⁽²⁾(α¹, α²)=D(α¹, α²). By applying (i)–(iii) above repeatedly, we obtain that D⁽ⁿ⁾(α¹, . . . , αⁿ)⊂ D⁽ⁿ⁻¹⁾(α², . . . , αⁿ)⊂ · · · ⊂αⁿ and if α¹, . . . , αⁿ are independent one-dimensional Poisson processes of densities ρ¹<· · ·< ρⁿ, thenD⁽ⁿ⁾(α¹, . . . , αⁿ)also is a one-dimensional Poisson process with densityρ¹.

Define the configurationη=(η¹, . . . , ηⁿ)by η^k=D⁽ⁿ⁻^k⁺¹⁾

α^k, α^k⁺¹, . . . , αⁿ

. (9)

By constructionη^k⊂η^k+¹for allk=1, . . . , n−1 and for eachk,η^k has marginal distributionν^ρ^k.

Define the mapC:Xⁿ→Xⁿ^↑byCα=η. Define the measureπonXⁿ^↑as the law ofη(that is,π=Cν). Define the multiclass measureμas the law ofξ=Rη, that is,μ=RCν. CallM=RC. See Fig. 3.

Theorem 1. Ifαhas lawνwithρ¹<· · ·< ρⁿ,then the lawMνofξ =Mαis invariant for the multiclass processξ_t defined in(4).Equivalently,the lawCνofCαis invariant for the coupled processη_t defined in(2).

To prove Theorem 1 we introduce later a new processα_t=(α_t¹, . . . , α_tⁿ)∈Xⁿcalled amulti-line process. We show that the product measureνis invariant forα_t and that ifαis the initial configuration for the multi-line processα_t then (Cα_t, t≥0)is the coupled process with initial configurationCα. As a consequence,Cν is invariant for the coupled process andMνis invariant for the multiclass process. Those results are proven in Propositions 4 and 5.

The right density of a configurationη∈X is the limit asrgoes to infinity of the number ofη-particles in[0, r] divided byr, if this limit exists. Analogously, we define the left density and call simply thedensityofηthe result when the left and right density are the same. The density of an ergodic measureνinXis the unique valueλsuch that the set of configurations with densityλhasνprobability one.

Fig. 3. Construction of a coupled configurationηand a multiclass configurationξfrom a multi-line configurationα.ξ^kparticles are filled differ- ently according to its class. In the middle picture, arrivals have been linked with their departure times in each server, according to the FIFO schedule and the customer class.

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Uniqueness

The next result says that the measureCνis the unique invariant measure for the coupled process in the set of ergodic measures onXⁿwith marginal densitiesρ. Its domain of attraction includes the ergodic measures with densitiesρ. Let ρ¹<· · ·< ρⁿ,ρ=(ρ¹, . . . , ρⁿ)andρ=(ρ¹, ρ²−ρ¹, . . . , ρⁿ−ρⁿ⁻¹). LetMⁿ(ρ)be the set of ergodic measures onXⁿsuch that thekth marginal has densityρ^kfork=1, . . . , n.

Theorem 2. Assume the conditions of Theorem1.Then,

(1) The measureMν is the unique invariant measure for the multiclass process inMⁿ(ρ).The multiclass process starting with a measure inMⁿ(ρ)converges weakly toMνast→ ∞.

(2) The measureCνis the unique invariant measure for the coupled process inMⁿ(ρ).The coupled process starting with a measure inMⁿ(ρ)converges weakly toCνast→ ∞.

For allλ >0 and almost all realizationsωof the points, there is a unique stationary realization of theHADprocess at densityλgoverned byω. This is shown by the next theorem; notice that the notationη_t in this theorem is used for theHADprocess onX, that is, withn=1.

Theorem 3. For eachλ >0there exists an essentially unique functionH_λmapping elementsωofΩ toHADtrajec- tories(η_t, t∈R)such that:

(i) The induced law of(ηt, t∈R)=Hλ(ω)is stationary in time.

(ii) The marginal law ofηt for eachtis space-ergodic with particle densityλ.

(iii) With probability1,(ηt, t∈R)is aHADevolution governed byω.

(Here “essentially unique” means that ifH_λ is another function satisfying the three conditions,thenHλ(ω)=H_λ(ω) with probability1.)Then in fact the marginal law ofηt for eachtisν^λ.

To stress the dependence onλcallη_t^λthe process constructed in Theorem 3. The construction impliesη^λ_t ⊂η_t^λ if λ < λ. The union ofη^λ_t inλis a countable dense set ofR. It consists on the space coordinates of the points inωwith time coordinate less thant.

Theorems 2 and 3 are proven in Section 4, based on Proposition 10 which considers the coupled process(η_t¹, η²_t) starting with two independent configurations with the same densityλ. If one of the configurations is a Poisson process and the other comes from an ergodic distribution, then, with probability one, the densityc(t )of positions where the two configurations differ at timet is deterministic and converges to zero as t grows. The proof of the proposition follows an argument of Ekhaus and Gray [7] as implemented by Mountford and Prabhakar [19].

Dual points

The stationary realization(η^λ_t, t∈R)of theHADprocess governed byωand densityλof Theorem 3 induces a point configuration onR×Rconsisting on the space–time positions of the particles just before jumps; see Figure 4. Cator and Groeneboom [4] call themdual pointsand prove that they have the same Poisson law asωfor allλ >0. The proof is based on two facts: (a) the reverseHADprocess with respect toν^λis also aHADprocess, with jumps to the left instead of to the right and (b) a trajectory of the process uniquely determines the points governing it.

Multi-line process

The dual points of a stationary realization of theHAD process and a new density produce a new stationary realization of theHAD process. Givenn densities and repeating the proceduren−1 times, we constructnstationary realizations of theHADprocess at the given densities coupled in such a way that thekth realization is governed by the dual points of the(k+1)th realization. The resulting process, called themulti-line process, is constructed as follows:

Fix densitiesρ¹, . . . , ρⁿ and a realization of the pointsω. Callα_·ⁿ=(αⁿ_t, t∈R)the stationary realization of the

HADprocess at densityρⁿgoverned byω(which is well defined by Theorem 3). Fork=n−1, . . . ,1 setω^k=dual

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Fig. 4. The dual points are represented by circles.

points generated byρ^kandω^k⁺¹andα_·^k⁻¹=the stationary realization of theHADprocess at densityρ^k⁻¹governed by ω^k. Define(α_t, t ∈R) as themulti-line (stationary)process given by α_t =(α_t¹, . . . , α_tⁿ). The generator of the multi-line process is given later in (16). The multi-line process can be constructed for times in[0,∞)for any initial distribution; we do so in Section 2. In this case then−1 first marginals do not follow theHADdynamics unless the initial distribution is a product of Poisson processes. We prove two results in Section 2:

Proposition 4. Letρ=(ρ¹, . . . , ρ^k)be positive densities.The product of Poisson processesν^ρ¹ × · · · ×ν^ρ^k is the unique invariant measure for the multi-line process with marginal densitiesρ.

Proposition 5. Take a multi-line configurationαsuch that the multiclass configurationMαis well defined.Letα_t be the multi-line process with initial configurationαgoverned byω.Then(Mα_t, t≥0)is the multiclass process with initial distributionMαand(Cα_t, t≥0)is the coupled process with initial distributionCα,both governed byω.

The proof of Theorem 1 is a consequence of Proposition 4 (without the uniqueness part) and Proposition 5.

Prähofer and Spohn [20] and Patrik Ferrari [13] introduce amultilayer dynamics based on the dual points and related to our multi-line process by a rotation of 45^◦(roughly speaking). See for example the pictures in [13], partic- ularly Figure 2. In the context of these multilayer dynamics, all the densitiesρ_k introduced would be the same (and equal to 1, say).

Multiclass Burke’s theorem

The invariant measure for the multiclassHADis also a fixed point for a multiclass·/M/1 queue.

To make this more precise we need to introduce more notation in the construction ofηas a functionCofα. As in [11] we label multiclass configurationsξ^k,ias functions of the multi-line configurationα. Letξ^1,1=α¹and define

ξ^k,i:=D

ξ^k⁻^1,i, α^k\

ξ^k,1∪ · · · ∪ξ^k,i⁻¹ ,

(10) ξ^k,k:=U

α^k⁻¹, α^k

fork=1, . . . , nandi=1, . . . , k. Interpret these configurations as the service times of customers ofkclasses in the kth system; at times inξ^k⁻^1,icustomers of classiarrive to thekth system and are served at times inξ^k,i. The service schedule respects the classes: first-class customers are served first, second-class customers are served when there are no first-class customers waiting, etc. End of service occurs at times inα^k. The unused service times of thekth system are inξ^k,k. One can think that thekth system has infinitely many customers of classkwaiting, so that they are served at the service times unused by the lower class clients. With this interpretationξ^1,1=α¹are the service times of a system with infinitely many customers in queue at all times. Assuming FIFO schedule (first-in-first-out) for customers of the

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same class we can link each time inα^kwith the corresponding service time inα^k⁺¹. This is the meaning of the links in Figure 3. In each linek, the particles linked to some particle in the first line will be first-class particles; particles linked to particles in the second line but not to particles in the first line are second-class particles and so on.

Let M:Xⁿ → Xⁿ be the map between multi-line configuration α and the multiclass configuration ξ = (ξ^n,1, . . . , ξ^n,n)using (6) and (10):

Mα:=

ξ^n,1, . . . , ξ^n,n

, (11)

whereXⁿ⊂Xⁿ is the subspace ofX whereMis well defined. This coincides with the mapM=RC defined just after (9).

Thenth multiclass stationary queuing system(Qⁿ_r, r∈R)in{0,1, . . .}^{^1,...,n⁻¹^}is defined as a deterministic function of the arrivals(ξⁿ⁻^1,1, . . . , ξⁿ⁻^1,n⁻¹)and the servicesαⁿby

Qⁿ_r(j )−Qⁿ_r₋(j )=1

r∈ξⁿ⁻^1,j −1

r∈αⁿ: Qⁿ_r₋(i)=0 fori < j;Qⁿ_r₋(j ) >0 , (12) whereQⁿ_r(j )is the number of customers of classj in thenth system at timer. The existence of a process satisfying (12) can be proved by induction onnby observing that at timesrinξ^n,nthe queue is empty:Qⁿ_r(j )=0 for allj < n.

The processQⁿ_r is not Markov unless the arrival process is a product of homogeneous Poisson and the service process is also Poisson. The departure times of classj in systemnare given [equivalently to (10)] by

ξ^n,j=

r∈αⁿ: Qⁿ_r₋(i)=0 fori < jandQⁿ_r₋(j ) >0 (13) forj=1, . . . , n−1. As beforeξ^n,nis just the set of unused service times in thenth system.

Burke’s theorem [3] says that the departures of a stationaryM/M/1 queue have the same Poisson law as the arrivals. It applies to our case for n=2: Q²_r(1) is an M/M/1 queue with arrivals ξ^1,1, Poisson of rate ρ¹ and departures ξ^2,1. The extension to the multiclass system says that the departures of the (n−1)classes of the nth system(ξ^n,1, . . . , ξ^n,n−¹)have the same law as the arrivals(ξⁿ⁻^1,1, . . . , ξⁿ⁻^1,n−¹)to the same system. This is one of the consequences of the multiclass Burke’s theorem as follows:

Theorem 6 (Multiclass Burke). Fixk≥1and forn≥kletα∈Xⁿhave lawνwithρ¹<· · ·< ρⁿ.Then the law of (ξ^n,1, . . . , ξ^n,k)(=firstkcoordinates ofMα)is independent ofn.

Proof. Letη⁽ⁿ⁾_t be the coupled process with initial invariant distributionCν⁽ⁿ⁾, whereν⁽ⁿ⁾is the product measure in Xⁿwith marginalsν^ρ^k,ρ¹<· · ·< ρ^k. Since the evolution of the firstkcoordinates is Markovian, the marginal law of the firstkcoordinates ofη_t⁽ⁿ⁾underCν⁽ⁿ⁾is invariant for the firstkcoordinates of the processη⁽ⁿ⁾_t . By uniqueness of the invariant measure of Theorem 2, the law of these marginals must coincide withCν^(k), the invariant measure forη^(k)_t , the process withklines and the same densities. Since(ξ^n,1, . . . , ξ^n,k)=firstkcoordinates ofRη⁽ⁿ⁾, its law coincides with the law ofRη^(k)=Mα^(k), whereα^(k)is the multi-line configuration with lawν^(k). The process in a cycle

Similar results can be proven for the multiclass process in a cycleRN. The state spaceXN is the set of finite configurations contained in[0, N]. The pointsωare restricted toRN×R⁺. When the configuration at timet−isη_t₋=η and there is a point at(x, t ), if there are noηparticles to the left ofx, then the rightmost particle ofηjumps tox. The coupled process is defined as in (2). Take two finite particle configurationsANandSNinRNwith|SN|>|AN|. Ex- tend these configurations periodically to two infinite configurationsAandS. Construct(Z_t, t∈R), a periodic process satisfying (5). The resulting periodic configurations of departuresD(A, S)and unused servicesU (A, S)are periodic and induce configurationsD_N andU_N inRN. We construct a (unique) queueQ_r,r∈RN such that it satisfies (7), which has valueQ_r=0 forr∈U_N; actuallyQ_r so constructed is the minimal process satisfying (7).

Given a multi-line configurationαinX_Nⁿ we construct a multiclass configurationξ=MNα∈X_Nⁿ, whereMN is defined as the mapM substitutingD andU byD_N andU_N. The analogues to Theorems 1, 2 and 6 hold for this process. Letting=(¹, . . . , ⁿ), the setXⁿ(ρ)of Theorem 2 must be substituted byX_Nⁿ(), the set of configurations with exactly^kparticles in thekth cycle. The proof of the analogue to Theorem 2 is easy in this case.

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Regeneration properties of the invariant measure

In the two-class invariant measure, the second-class particles are regeneration events. More precisely, letμonX² be a translation invariant measure with marginal densitiesρ¹, ρ². The Palm measure “conditioned to have a second class particle at the origin” is the measureμˆ defined byμfˆ =(1/|I|ρ²)

dμ(ξ )

r∈ξ²∩If (θrξ ), for any measurable bounded setI⊂R, where|I|is the Lebesgue measure ofIandθris translation byr. The multiclass invariant measure Mνof Theorem 1 conditioned to have a second-class particle at the origin satisfies that: (a) the configuration to the left of the origin is independent of the configuration to its right; (b) both the distribution of first-class particles to the right of the origin and the positions of the first plus second-class particles to the left of the origin are Poisson processes (or product measures in the discrete-space case). These properties were proven by [5] for the two-class invariant measure for the totally asymmetric simple exclusion process (TASEP); alternative probabilistic proofs can be found in [9]. Angel [2] shows that these properties follow easily from the two-class representationξ =Mα that he called “collapsing procedure”. In Section 5 we show that as in the discrete case [11], there areregeneration stringsfor the multiclass invariant measureMν.

Microscopic shocks

The process with two classes of particles is a crucial tool to define shock measures [8,10]. Taking a configuration (ξ¹, ξ²)distributed according to the invariant measureMνfor the two-class process conditioned to have aξ²particle at the origin and considering the configuration consisting of all particles (of any class) to the left of the origin and the first-class particles to the right of it, the distribution of the resulting configuration is invariant for the process as seen from an isolated second-class particle. We discuss this item in Section 6.

Multiclass HAD process inZand other discrete processes

The approach has been implemented for a discrete-space multiclassHADprocess and other discrete-space processes onZ. In fact, this research started with the extension of Angel’s [2] result to the multiclass process for the discrete

HAD andTASEP we performed in [11] and [12]. There are two main points in this paper: one is to show that the approach works also in continuous space (theHAD process is the most natural model to studyTASEP like questions inR); the other is to show some results that are only announced in [12]. In particular, we show the uniqueness of the multiclass invariant measure with given marginal densities. The use of dual points is more intuitive in the continuous

HADthan in the discrete processes. Indeed, the dual points in this case are identified by the trajectory of the process, while in the discrete cases it is necessary to appeal auxiliary spin–flip processes to identify the dual points.

The existence of an invariant measure for the two-class TASEP is first shown by Liggett [16,17] and then computed by Derrida, Janowsky and Lebowitz [5], see also [9,22]. The description of the two-class invariant measures by Angel [2] and Duchi and Schaeffer [6] was the starting point of our multiclass version [11,12].

2. The multi-line process

The multi-line process has configurationsαt=(α¹_t, . . . , αⁿ_t)inXⁿ. Given a multi-line configurationαand a position x∈Rletxⁿbe the position of the closest particle ofαⁿto the left ofxand inductively for linesk=n−1, . . . ,1 let x^kbe the position of the closestα^kparticle to the left ofx^k⁺¹. When the configuration at timetisαand there is anω point in(x, t ), theαⁿ-particle atxⁿ jumps tox at timet and simultaneously theα^k particle located atx^k jumps to x^k⁺¹fork=n−1, . . . ,1. The functionJ:Xⁿ×R→Xⁿdefined by

J (α, x)k

=α^k∪

x^k⁺¹ \

x^k (14)

maps the multi-line configurationαbefore the jumps produced byx to the configuration after the jumps; see upper part of Figure 5. The configurationα_tof the multi-line process at timetis a function of the initial configurationαand the pointsωcalledΨ:

αt:=Ψ (t, α, ω). (15)

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Fig. 5. Effect of a point atxfor the multi-line process and the reverse jump (due to a point atyin the reverse).

The generator of the process is given by Lf (α)=

Rdx f

J (α, x)

−f (α)

. (16)

This definition is equivalent to the one given in the introduction. For t such that (x, t ) is in ωdenote xⁿ⁺¹(t )= x andx^k(t )the position at time t− of the particle in the kth line jumping due to the point in (x, t ). Then ω^k= {(x^k(t ), t ): (x, t )∈ω}are the dual points ofω^k⁺¹which in turn govern the processα_t^k.

We define another multi-lineHADprocessα^∗_t inXⁿand show that it is the reverse ofα_t with respect to the product of Poisson processesν. Forα_t^∗the points govern the first line but produce jumps to the left: when a point appears aty, it calls the closestα¹particle to the right ofy, located aty¹. Simultaneously the closestα²particle to the right ofy¹, located aty²jumps toy¹, and so on. Callingy0=y, the positions(y0, . . . , yn)are defined as a function ofαandy.

The multi-line configuration obtained after the jumps produced by a point atyis calledJ^∗(α, y); itskth line is given by

J^∗(α, y)k

=α^k∪

y^k⁻¹ \

y^k . (17)

The operatorsJ andJ^∗are the inverse of each other in the following (equivalent) senses:

J^∗

J (α, x), x¹

=α; J

J^∗(α, y), yⁿ

=α (18)

(in Figure 5,x¹=yandyⁿ=x). The generator ofα_t^∗is given by L^∗f (α)=

Rdy f

J^∗(α, y)

−f (α)

. (19)

Proposition 7. For any choice of positive densitiesρ=(ρ¹, . . . , ρⁿ),the multi-line process with generatorL^∗is the reverse of the multi-line process with generatorLwith respect to the product measureνonXⁿwith densitiesρ.As a consequence,νis invariant for bothLandL^∗.

Proof. The proof is based on the observations: (a) the configurationsαandJ (α, r)and the configuration after a jump produced by a Poisson point atrhave the same probability weight; (b) the rate to jump fromαtoJ (α, r)is the same as the rate in the reverse process to jump fromα=J (α, r)toJ^∗(α, y)=α; (c) the rate of exiting any configuration αis the same for the direct and the reverse process. We formalize this with a generator computation.

It suffices to show that for bounded functionsf andgdepending on finite regions inRⁿ,ν(gLf )=ν(fL^∗g); that

is,

ν(dα)

Rg(α)dx f

J (α, x)

−f (α)

=

ν(dα)

Rf (α)dx g

J^∗(α, x)

−g(α) . Since the term subtracting in both terms is the same, it suffices to prove that

ν(dα)

Rg(α)f

J (α, x) dx=

ν(dα)

Rg(α)f

J^∗(α, x)

dx. (20)

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By conditioning on the positions of the jumping particles(x¹, . . . , xⁿ)the left term in (20) can be written as

Rdx

x¹<···<xⁿ<x

n

k=1

ρ^ke⁻^ρ^k^(x^k⁺¹⁻^x^k⁾

dx¹· · ·dxⁿ

ν_x(dα)g(α)f

J^∗(α, x)

, (21)

whereν_x is the measureνconditioned to have at each lineka particle at positionx^k and no particle in(x^k, x^k⁺¹).

Change variables: callα=J (α, x)so that, by (18)α=J^∗(α, y)andy^k=x^k⁺¹,k=0, . . . , nto get that (21) equals

Rdy

y<y¹<···<yⁿ

n

k=1

ρ^ke⁻^ρ^k^(y^k⁻^y^k⁻¹⁾

dy¹· · ·dyⁿ

ν^y dα

g J^∗

α, y f

α

, (22)

whereν^y is the measureν conditioned to have at each lineka particle at positiony^k and no particles in(y^k⁻¹, y^k).

Expression (22) is just the right-hand side of (20).

The set ofdual pointsof theHADprocess starting with a Poisson process of rateλis a Poisson process onR×R⁺. This is proven in Theorem 3.1 in [4] under the title “Burke’s theorem for Hammersley processes”. We state and prove this fact in our context.

Proposition 8. Ifη∈X has lawν^λandωis a rate-1homogeneous Poisson process inR×R⁺,then the dual points of the trajectory(η_t, t ≥0)form a rate-1 homogeneous Poisson process inR×R⁺.Furthermore for t >0, η_T is independent of the dual points contained inR× [0, T].

Proof. By Proposition 7 (withn=1), the reverse HAD process with respect to ν^λ is just aHAD process with the drift to the left. Assume we have constructed the process (η_t,0≤t≤T ) with initial lawν^λ and Poisson pointsω onR× [0, T]. The reverse processη_s^∗=η_T₋_s₋, 0≤s≤T, is governed by the dual points inR× [0, T]. Since the reverse process started in equilibriumν^ρ, the points governing the reverse process must be also Poisson inR× [0, T].

The dual points inR× [0, T]are the points governing the future of the reverse processη^∗_t :=ηT−t− which is in equilibrium. Hence it is a Poisson process independent of the initial configurationη₀^∗=ηT. In the proof of Theorem 3 we use Proposition 8 to construct a stationary trajectory(η_t, t∈R)of theHADprocess governed by points ω with marginal lawν^λ. The dual points of this trajectory are, asω, a homogeneous Poisson process inR².

3. Multi-line and multiclass processes

In this section we prove that the projection of the multi-line process on the multiclass space using the mapMis just the multiclass process. The analogous statement is true for the coupled process. Recall from (15) that given an initial multi-line configurationα and a homogeneous rate-1 Poisson processω inR×R² the process(α_t, t ≥0), where α_t=Ψ (t, α, ω)is a realization of the multi-line process. On the other hand, given an initial configurationξ and the sameω, the process(ξ_t, t≥0)withξ_t=Φ(t, ξ, ω)is the multiclass process.

Proposition 9. AssumeCαis well defined.Then almost surely:

C

Ψ (t, α, ω)

=Φ(t, Cα, ω), (23)

M

Ψ (t, α, ω)

=Υ (t, Mα, ω). (24)

In particular,the law ofCα_t is the same as the law ofη_t.

Proof. Expression (24) follows then from (23), the identityM=RC and (4). Fixx∈R; it suffices to show

H (Cα, x)=CJ (α, x), (25)

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Fig. 6. Effect of a point atxfor the multi-line process and the coupled process.α=J (α, x)andη=H (η, x). Takingη=Cαwe see thatη=Cα.

whereH (η, x)is the coupled Hammersley configuration obtained after a point at(x, t )(for somet) ifηt−=ηand J (α, x)is the resulting multi-line configuration after a point at(x, t )ifα_t₋=α: itskth coordinates satisfy

H (η, x)k

=η\

z^k ∪ {x}, J (α, x)k

=α^k\ x^k ∪

x^k⁺¹ ,

wherez^kis the position of the closestη^k particle to the left ofxandx^k is the position of the particle jumping on line kin the multi-line process; herexⁿ⁺¹=x.

Letη=Cα be the coupled configuration obtained from αand denoteα=J (α, x) andη=H (η, x). With this notation (25) is Cα=η; see Figure 6. We first prove (25) forn=2 lines and then use this case to proceed by induction.

The casen=2. The second line of both processes is governed by the point atx with the same Hammersley rule.

This implies they coincide in the second line:(Cα)²=(η)². There is anα²service time atxand anα¹particle atx². This particle is served in line 2 at timexunless there is anotherα¹particle to the left ofx²served at that time. In any case, anα¹particle is served atxin line 2. This means(Cα)¹has a particle atx. To see that(Cα)¹has no particle at z¹and that otherwise it is equal toηwe consider two cases depending on whetherx²serves anα¹particle or not.

(1) Theα² time atx² serves anα¹particle. Hence, there is anη¹ particle atx². Since there are noα²services betweenx²andx (by definition ofx²), there cannot beη¹particles either andη¹=η¹\ {x²} ∪ {x}. Let’s check that alsoCαis of this form. Letybe the position of theα¹particle served atx²in line 2. Clearlyy≤x¹. Theα¹particles to the left ofyare served in line 2 beforex²and hence its service time is not affected ifx¹andx²are translated to the right. This meansCαcoincides withηatx and to the left ofx. Theα¹particles to the right ofy are served afterx.

The displacement of the service time fromx²toxjust changes the service time of the particle aty, leaving the other service times unchanged. The displacement of theα¹particle fromx¹tox²does not change the order or arrival of the particles or their service times atα², as those times are afterx. This impliesCα=ηin this case.

(2) Theα² time atx² does not serve anα¹ particle. Theα¹ particle atx¹ is served at some time zbeforex². There are noα¹particles betweenx¹andx². The translation of theα¹particle fromx¹tox²and theα²particle from x² tox does not change the service times of theα¹particles to the left ofx¹, which are served beforex². By the FIFO schedule, particles that arrive afterx²are served after particles that arrive before. Hence, the translations do not change the service times of the particles that arrive afterx².

The induction step. From the definition ofC, we have η_t^k=D⁽ⁿ⁻^k⁺¹⁾

α^k_t, . . . , αⁿ_t

. (26)

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From the observation just before (9),η_t^k has distributionν^ρ^k. So we need to show that the RHS of (26) is a HAD

trajectory governed byω. Since(α^k, . . . , αⁿ)is itself a multi-line process (withn−k+1 lines) governed byω, it is enough to show that, for anyn,D⁽ⁿ⁾(α_t¹, . . . , α_tⁿ)is aHADtrajectory governed byω. But from the definitions ofD⁽ⁿ⁾ and of the multi-line process,

D⁽ⁿ⁾

α¹_t, . . . , αⁿ_t

=D⁽²⁾ D⁽ⁿ⁻¹⁾

α¹_t, . . . , α_tⁿ⁻¹ , αⁿ_t

.

This and the fact that(α¹_t, . . . , α_tⁿ⁻¹)is an(n−1)-line multi-line process governed byωⁿ⁻¹concludes the induction

step.

4. Uniqueness of the invariant measure

In this section we prove Theorems 2 and 3 using the following proposition. It says that the marginals at timet of the coupled Hammersley process with initial independent marginals will coincide ast→ ∞if the first initial marginal is a Poisson process and the second one is an ergodic distribution with the same density as the first.

Proposition 10. Letλ >0 andη,η˜∈X² be independent particle configurations with lawν^λ,the Poisson process andν˜^λ,an ergodic process inX with densityλ,respectively.Let(η_t,η˜_t)be the coupled process with initial configu- ration(η,η).˜ Then with probability one,bothηt\ ˜ηt andη˜t\ηt have a deterministic density denotedc(t )=c(t; ˜ν^λ) which is decreasing and converges to0ast→ ∞.

An immediate consequence of the proposition and the translation invariance of the law of the coupled process at each given time is that the probability of the event{Λ∩η_t=Λ∩ ˜η_t}converges to 1 ast→ ∞for any bounded intervalΛ⊂R.

Before proving the proposition we show how it implies Theorems 2 and 3.

Proof of Theorem 2. Assumeηandη˜ inXⁿ are independent with laws μandμ, respectively and let˜ (η_t,η˜_t)be a 2n-coordinates coupled process starting with configuration(η,η). Assume˜ μandμ˜ are invariant for the coupled process and that the marginal law of bothη^k andη˜^kis a Poisson process with rateρ^kfor allk. Then, for boundedf depending on a bounded intervalΛ:

|μf − ˜μf| =

μ(dη)f (η)−

˜

μ(dη)f (˜ η)˜

(27)

=

μ(dη)μ(d˜ η)˜ E

f (η_t)−f (η˜_t) (28)

≤ μ(dη)μ(d˜ η)˜ f_∞P(Λ∩η_t=Λ∩ ˜η_t) (29)

which tends to zero by Proposition 10 applied to each coordinate. This impliesμ= ˜μ.

Proof of Theorem 3. First construct a double infinite realization and the corresponding points. Fix particlesηwith law Poisson of rateλand Poisson pointsω⁻∪ω⁺, the subsets of points with negative and positive time coordinates, respectively. Run the processη_t=Φ(t, η, ω⁺)fort≥0 using the pointsω⁺. Run the reverse process backwardsη^∗_−t= Φ^∗(−t, η, ω⁻)starting from the same configuration using the points ω⁻. HereΦ^∗(−t, η, ω⁻)=Φ(t, η,TR(ω⁻)), where TR(ω)= {(x,−t ), (x, t )∈ω}are the points ofωreflected with respect to the line{t=0}. Let the dual points D⁻(ω⁻, η)be the positions of the particles of η^∗_t just before jumps. By Proposition 8, D⁻(ω⁻, η) is a Poisson process of points. Letω=D⁻(ω⁻, η)∪ω⁺be the configuration consisting on the dual points of the reverse process for negative times and the original points for positive times. The pointsωare Poisson and govern a stationary process having configuration η at time zero. This constructs simultaneously Poisson points ω and a stationary trajectory governed byω.