Convergence of a queueing system in heavy traffic with general patience-time distributions

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1 Available olie at Stochastic Processes ad their Applicatios 121 (211) Covergece of a queueig system i heavy traffic with geeral patiece-time distributios Chihoo Lee a, Aada Weerasighe b, a Departmet of Statistics, Colorado State Uiversity, Fort Collis, CO 8523, USA b Departmet of Mathematics, Iowa State Uiversity, Ames, IA 511, USA Received 8 October 21; received i revised form 3 April 211; accepted 7 July 211 Available olie 21 July 211 Abstract We aalyze a sequece of sigle-server queueig systems with impatiet customers i heavy traffic. Our state process is the offered waitig time, ad the customer arrival process has a state depedet itesity. Service times ad customer patiet-times are idepedet; i.i.d. with geeral distributios subject to mild costraits. We establish the heavy traffic approximatio for the scaled offered waitig time process ad obtai a diffusio process as the heavy traffic limit. The drift coefficiet of this limitig diffusio is iflueced by the sequece of patiece-time distributios i a o-liear fashio. We also establish a asymptotic relatioship betwee the scaled versio of offered waitig time ad queue-legth. As a cosequece, we obtai the heavy traffic limit of the scaled queue-legth. We itroduce a ifiite-horizo discouted cost fuctioal whose ruig cost depeds o the offered waitig time ad server idle time processes. Uder mild assumptios, we show that the expected value of this cost fuctioal for the -th system coverges to that of the limitig diffusio process as teds to ifiity. c 211 Elsevier B.V. All rights reserved. MSC: primary 6K25; 68M2; 9B22; secodary 9B18 Keywords: Stochastic cotrol; Cotrolled queueig systems; Heavy traffic theory; Diffusio approximatios; Customer abadomet; Customer impatiece; Reegig 1. Itroductio I this article, we study a heavy traffic approximatio result for a sequece of sigleserver queueig systems with impatiet customers. Customers are served uder the First-Come Correspodig author. address: aada@iastate.edu (A. Weerasighe) /$ - see frot matter c 211 Elsevier B.V. All rights reserved. doi:1.116/j.spa

2 258 C. Lee, A. Weerasighe / Stochastic Processes ad their Applicatios 121 (211) First-Serve (FCFS) service disciplie. I the -th system, where = 1, 2, 3,..., the arrival process has a dyamic itesity which depeds o the offered waitig time ad this itesity is of order O() for large, ad the service-times are i.i.d. with a geeral distributio where the mea service-time is of order O(1/) for large. The customers abado the system if the service is ot iitiated withi their patiece-time. I the -th system, customers act idepedetly, ad their patiece-times are i.i.d. distributed ad this distributio may deped o. I may real world examples, such as telephoe call ceters or iteret traffic, customers may ot observe the actual queue-legth but ofte approximate waitig time is available to them. I our model, offered waitig time (or the workload process) is the basic state process ad the arrival itesity of the customers is depedet o it. To motivate this work, cosider a processig facility where each customer or job arrives with a deadlie. Upo the arrival of each customer, a system maager lears about the customer deadlie as well as the required service time. Hece, the iformatio o offered waitig time is available to the maager ad accordigly, the maager ca ifluece the arrival itesity by meas of admissio cotrol. I practice, customer abadomet is a well documeted sigificat feature of the queueig systems. I the queueig models, Palm [23] iitiated the importace of icorporatig this feature. I the telephoe call ceter settig with may-server systems, such models are cosidered i [13,19,1,11,36,29,22,25]. For sigle server settig, Ward ad co-authors addressed several performace evaluatio issues of such systems i [3,31,26]. For geeral queueig systems i heavy traffic (with or without customer abadomet), there are umerous articles that address the issue of system optimizatio ad [3,5,14,15] is a partial list of such articles. The results established i this article are closely related to the works of [26,3,31], but they differ i three mai aspects: first, i the -th system, the itesity of our arrival process is ocostat ad may deped o the curret value of the offered waitig time. Loosely speakig, system maager may exercise adjustmets of order O( ) to the admissio rate of the -th system without disturbig the delicate balace i heavy traffic coditios. But such adjustmets have a ifluece o the drift coefficiet of the limitig diffusio process as described i our Theorem 4.1. I cotrolled queueig systems, such adjustmets are kow as thi cotrol ad we refer to [1,15] for such problems. Secod, our assumptios o patiece-time distributios are quite geeral. I Markovia abadomet regimes [3] ad also i [31] (for may-server queues i Halfi Whitt heavy traffic regime see [4,11,1,21,13,22,25]) where the same patiece-time distributio is used i the modelig, oly the behavior of patiece-time distributio i a eighborhood of origi effects the dyamics of the limitig diffusio. But, i a iterestig article [26], Reed ad Ward cosider the patiece time distributio of the -th system to have a hazard rate itesity depedet o (see [25] for a may-server Halfi Whitt heavy traffic case). They provide statistical data i support of their choice. The dyamics of their limitig diffusio process depeds o the etire patiece-time distributio fuctio. Our results icorporate both of these scearios i the same geeral framework as illustrated i the examples of Sectio 3, ad our assumptios ca be satisfied by may other classes of patiece-time distributios. The heavy traffic limit for the diffusio-scaled waitig time process is established i Theorem 4.1 ad it describes the effect of patiece-time distributios o the limitig diffusio. Oe key igrediet i our proof of Theorem 4.1 is the martigale fuctioal cetral limit theorem, ad this approach helps us to accommodate these geeral assumptios. This is i cotrast with the proofs i [26]. The diffusio-scaled offered waitig time process tured out to be the reflected process uder a geeralized Skorokhod map itroduced i Sectio 4.3. The martigale cetral limit theorem helps

3 C. Lee, A. Weerasighe / Stochastic Processes ad their Applicatios 121 (211) us to establish the weak covergece of the iput process related to this geeralized Skorokhod map, where the output is the above described reflected process. The, the cotiuity properties of the geeralized Skorokhod map yield the weak covergece of the diffusio-scaled offered waitig time process ad also idetify the diffusio limit. Third, we use martigale momet iequalities to obtai momet bouds for the iput process. The agai we employ the martigale cetral limit theorem ad Theorem 4.1 to establish the covergece of the expected value of a ifiite-horizo discouted cost fuctioal of the -th system to that of the limitig diffusio process as teds to ifiity. Such covergece results for the expected value of the cost fuctioals are importat i derivig asymptotically optimal strategies for the system optimizatio problems i heavy traffic regimes. We refer to [32,15] ad [5,4,22] i may-server Halfi Whitt heavy traffic regime) for such results related to cotrolled queueig systems. We ited to use the results obtaied here to address such a cotrolled system optimizatio problem i a future article. This article is orgaized as follows. I Sectio 2 we itroduce the basic model ad the key martigale relevat to the arrival process. Such a martigale formulatio is used i [35] for the heavy traffic aalysis of queue-legth processes, whe the arrival ad service rates are depedet o queue-legth. I Sectio 3, we speed up the arrival rates to be of order O() ad to balace this ad to obtai heavy traffic coditios, we make the average service time i the -th system to be 1. We carefully lay out our assumptios o arrival itesities, service times ad patiece-time distributios. Sectio 4 addresses the weak covergece of scaled offered waitig time processes i heavy traffic. We establish the fluid limit first ad the use it to obtai the diffusio limit for the scaled offered waitig time process. Mai result i this sectio is Theorem 4.1, ad we use martigale fuctioal cetral limit theorem to obtai this weak covergece result. I Sectio 5, we establish the asymptotic relatioship betwee the scaled queue legth ad scaled offered waitig time processes. Here we follow the proof of a similar result i [26], but supplemet it with ecessary additioal estimates to accommodate our geeral assumptios. We prove the covergece of a ifiite horizo discouted cost fuctioal of the -th system to that of the limitig diffusio uder heavy traffic i Sectio 6. I this cost fuctioal, the ruig cost fuctio depeds o offered waitig time, ad there is also a cost related to server idle time. Sice the ruig cost fuctio is ubouded ad is of polyomial growth, we eed a few additioal assumptios there. To reach our coclusio, we establish ecessary momet estimates ad combie them with the weak covergece result i Theorem 4.1. For cotrolled queueig etworks, such covergece results are obtaied i [32,15] ad i the case of may-server systems, we refer to [5]. I the Appedix we provide a detailed costructio of the arrival process with arrival itesity depedet o the offered waitig time. The followig otatio is used. The set of positive itegers is deoted by N, the set of real umbers by R ad oegative real umbers by R +. Let R d be the d-dimesioal Euclidea space. For a, b R, let a b =. mi{a, b} ad a + = max{a, }, a = mi{a, }. We use [a] to deote the iteger part of a R. If (M(t)) t is a martigale the we deote the associated quadratic variatio of M o the iterval [, T ] by [M](T ). The covergece i distributio of radom variables (with values i some Polish space) Φ to Φ will be deoted as Φ Φ. Whe sup s t f (s) f (s) as, for all t, we say that f f uiformly o compact sets. For a real valued fuctio f defied o some metric space X ad T R +, defie f T = sup x [,T ] f (x). Fially, let D[, ) deote the class of right cotiuous fuctios havig left limit defied from [, ) to R, equipped with the usual Skorokhod topology.

4 251 C. Lee, A. Weerasighe / Stochastic Processes ad their Applicatios 121 (211) Basic model First we describe the queueig model with FCFS service disciplie ad customer abadomet o a probability space (Ω, F, P). Let A(t) be the umber of customers arrived at the statio by time t. The radom variable t j represets the arrival time of the j-th customer, ad we assume E(t j ) <. Service time of the j-th customer is represeted by the radom variable v j. We assume that the customers are impatiet ad the j-th customer will leave the system after waitig a radom time d j if the service does ot begi by the. The sequeces (v j ) ad (d j ) are assumed to be i.i.d. ad idepedet of each other, E(v 1 ) = 1 ad var(v 1 ) = σs 2 <. We let F be the cumulative distributio fuctio of d 1. The amout of time a icomig customer at time t has to wait for service depeds upo the service times of the o-abadoig customers, who are already waitig i the queue. Similar to [26], we defie the offered waitig time process A(t) V (t) v j 1 [V (t j )<d j ] 1 [V (s)>] (s)ds. (2.1) The process {V (t) : t } is o-egative, has sample paths which are right cotiuous with left limits (RCLL), ad also at each arrival epoch t j, it has a upward jump of size v j. O the time iterval [t j, t j+1 ), V (t) is cotiuous, o-icreasig ad satisfies V (t) = max{, V (t j ) (t t j )}. Fig. 1 shows a typical sample path of the process {V (t)} t. The quatity V (t) ca be iterpreted as the time eeded to empty the system from time t owards if there are o arrivals after time t, ad hece it is also kow as the workload at time t. We ote that oce V (t ) is kow the V (t) is well defied o the ext iterval [t, t +1 ) (see below (2.11) for more details). Next, we defie the σ -fields ( F ). Let F σ (t 1 ), ad for 1 let F σ ((t 1, v 1, d 1 ),..., (t, v, d ), t +1 ) F. (2.2) Notice that V (t ) is F 1 -measurable ad the abadomet time d of the -th customer is idepedet of F 1. Hece, P[V (t ) d F 1 ] = F(V (t )) (2.3) holds almost surely, where F is the distributio fuctio of d. We itroduce two martigales (M v ()) ad (M d ()) with respect to the filtratio ( F ) 1 itroduced i (2.2). We let M v () M d () (v j 1)1 [V (t j )<d j ] (2.4) 1[V (t j ) d j ] E[1 [V (t j ) d j ] F j 1 ] (2.5) for all N. Clearly, M d () is a F -martigale (see also [26]). Here we show that M v () also is a F -martigale. Sice V (t +1 ) ad d +1 are measurable with respect to σ ( F, d +1 ) ad v +1 is idepedet of σ ( F, d +1 ), it follows that E (v +1 1)1 [V (t+1 )<d +1 ] σ ( F, d +1 ) = 1 [V (t+1 )<d +1 ]E(v +1 1) =.

5 C. Lee, A. Weerasighe / Stochastic Processes ad their Applicatios 121 (211) Fig. 1. A typical sample path of V (t). Now coditioig both sides of (2.4) with respect to F, we ca see that M v () is a F - martigale as well. Usig (2.3) i (2.5), we also see that for all N M d () = 1[V (t j ) d j ] F(V (t j )). (2.6) Usig (2.1) ad (2.3) (2.6) ad after simple algebraic maipulatios, we obtai the followig system equatio: V (t) + F(V (s ))da(s) = (A(t) t) + M v (A(t)) M d (A(t)) + I (t), (2.7) for all t, where I (t) 1 [V (s)=] (s)ds, ad I (t) represets the idle time at the statio durig time iterval [, t]. Next, we describe a filtratio (G t ) t which represets the iformatio gathered over time by the system maager. We begi with a discrete filtratio ( F ) give by F {, Ω} ad (2.8) F = σ ((t 1, v 1, d 1 ),..., (t, v, d )) for 1. (2.9) It is easy to verify that for each t, A(t) is a stoppig time with respect to the filtratio ( F ), where A( ) is the arrival process with arrival times (t j ) ad F F for all ad the filtratio ( F ) is give i (2.2). Next, we itroduce the filtratio (G t ) t by G t F A(t) for all t. (2.1) Let λ( ) be a give Borel measurable fuctio defied o [, ) which satisfies the coditio < ϵ < λ(x) < C for all x. Here ϵ ad C are positive costats. I our aalysis, we assume that A(t) λ(v (s))ds : t (2.11)

6 2512 C. Lee, A. Weerasighe / Stochastic Processes ad their Applicatios 121 (211) is a martigale with respect to the filtratio (G t ) t. We itroduce yet aother filtratio (G t ) t where G t σ (A(s), V (s) : s t). (2.12) Notice that, oce the value of V (t ) is kow, the process V (t) ca be obtaied o the iterval [t, t +1 ) as explaied earlier ad hece for all t t < t +1, the quatity t λ(v (s))ds is also kow by the time t. Moreover, whe t t < t +1, V (t) is a fuctioal of the radom variables (t 1, v 1, d 1 ),..., (t, v, d ). Cosequetly, G t G t for all t ad the process A(t) t λ(v (s))ds is a martigale with respect to (G t) t as well. I our proofs, we commoly use this martigale property with respect to (G t ) t, while the martigale property with respect to (G t ) t filtratio will be used oly i the proof of Lemma We idicate the costructio of such a arrival process A( ) ad several of its properties i the Appedix. We ote that sice A( ) is a poit process with (G t )-itesity λ(v (t)), we ca use the radom chage of time method (see Theorem T16 ad Lemma L17 i Sectio 6 of Chapter 2, [7]) to obtai the coveiet represetatio A(t) = Y λ(v (s))ds, (2.13) where Y ( ) is a uit-rate Poisso process. This represetatio helps us i several estimates. 3. Heavy traffic regime We cosider a sequece of queueig systems idexed by N. I our aalysis, basic state process of the -th system will be the offered waitig time process V ( ). The arrival rate λ (V ( )) of the -th system is state depedet ad the j-th customer arrival occurs at time t j. The cumulative umber of customer arrivals i [, t] i the system is give by A (t). Whe becomes large, arrival rate of the -th system becomes large ad thus to obtai heavy traffic coditios, we eed to make the service time of the -th system small as described below. For the j-th arrival i the -th system, service time is v j v j /, ad the abadomet time is deoted by d j. As described i [26], the basic equatio of the offered waitig time process {V (t) : t } is give by V (t) = 1 A (t) v j 1 [V (t j )<d j ] 1 [V (s)>](s)ds, (3.1) where A ( ) is the arrival process. We itroduce the filtratio {Gt : t } of the -th system by Gt σ (A (s), V (s) : s t). We also itroduce the filtratio (G t ) as similar to (2.1) ad this filtratio represets the iformatio available to the system maager over time. Next, we defie the discrete time filtratio ( F i ) i by F σ (t 1 ) ad F i σ ((t 1, v 1, d 1 ),..., (t i, v i, d i ), t i+1 ) (3.2) for i 1. Next, we defie the associated cotiuous time filtratio (F t ) t by Ft F [t] σ ((t 1, v 1, d 1 ),..., (t [t], v [t], d [t] ), t [t]+1). (3.3)

7 C. Lee, A. Weerasighe / Stochastic Processes ad their Applicatios 121 (211) Now we describe our basic assumptios: Assumptio 3.1. (i) The sequeces (v j ) j 1 ad (d j ) j 1 are idepedet, o-egative, i.i.d. radom variables with v j v j for all j 1, E(v j) = 1 ad E(v j 1) 2 = σs 2 >. Furthermore, for each j 1, the radom variables v j ad d j are idepedet of F j 1. (ii) The arrival process A ( ) of the -th system has a associated itesity process λ (V ( )); that is, A (t) λ (V (s))ds : t (3.4) is a (G t )-martigale. Sice this process is adapted to (G t ) ad G t G t, it is a (G t )- martigale as well. Assumptio 3.2. (i) The fuctio λ ( ) is Borel measurable o [, ) ad there exist two positive costats ϵ, C > (idepedet of ad x) such that < ϵ < λ (x) < C for all x ad 1. (ii) For each K >, lim sup x [,K ] λ (x) 1 =. (iii) There exist small δ > ad M > such that sup 1 sup x [,δ ] (λ (x) 1) + M <. (iv) There exists a o-egative, locally Lipschitz cotiuous fuctio u( ) defied o [, ) such that lim sup x 1 λ u(x) =, x [,K ] for each K >. Assumptio 3.3. Let F ( ) be the right cotiuous abadomet distributio fuctio of the i.i.d. sequece (d j ) j 1. Assume that F () = ad there exists a o-egative, locally Lipschitz cotiuous fuctio H( ) such that lim sup x F H(x) =, x [,K ] for each K >. As a cosequece, we have H() = ad lim F (x/ ) = for each x. Remark 3.4. We provide cocrete examples that satisfy the above set of assumptios. 1. A example of arrival rate fuctio λ ( ): Let u( ) be o-egative, locally Lipschitz cotiuous ad λ (x) = 1 u( x) + θ (x), where θ ( ) is a bouded fuctio such that lim θ K = for each K >. 2. Examples of abadomet distributio fuctios (F ): (a) Let F F for all, ad F be differetiable with a bouded derivative o [, δ] for some δ >. Hece, let H(x) = F ()x i Assumptio 3.3.

8 2514 C. Lee, A. Weerasighe / Stochastic Processes ad their Applicatios 121 (211) (b) We may take F (x) = 1 exp( x h( u)du) for x, where h is a o-egative cotiuous fuctio as i (14) of [26]. I this case, H(x) = x h(u)du ad it satisfies Assumptio 3.3 sice h is cotiuous. Ideed, for ay geeral sequece (F ), if F ( x ) coverges to a o-egative fuctio h(x) uiformly o compact sets, the (F ) satisfies Assumptio 3.3 with the limitig fuctio H(x) = x h(u)du. (c) Here we provide a simple example to illustrate that there ca be may limitig fuctios H( ) other tha the oes described i (a) ad (b) above. Let H( ) be ay o-egative, o-decreasig, locally Lipschitz cotiuous fuctio which satisfies H() = ad H(+ ) = +. We let F (x) = 1 mi{h( x), } for all x. The, for each 1, F () =, F (+ ) = 1 ad F is a cotiuous, o-decreasig probability distributio fuctio. It is evidet that the sequece of distributio fuctios F satisfies Assumptio 3.3 with limitig fuctio H( ). Remark 3.5. To describe a specific example of a heavy traffic regime usig the same arrival process, we ca cosider the system ( A( ), V ( )) satisfyig (2.1), (2.7) (2.11). The we ca scale these processes as described ext. First, we itroduce the filtratio (Gt ) by G t G t for each 1, where G t = σ (A(s), V (s) : s t). Now let A (t) A(t) ad V (t) V (t) for all t. The usig (2.11) ad by a chage of variable i itegratio, it easily follows that {A (t) t λ (V (s))ds : t } is a (Gt )-martigale. Throughout, oe ca cosider the arrival itesity λ ( ) as a cotrol process related to the -th system. I a future article, we ited to address a optimal cotrol problem associated with this heavy traffic regime, which miimizes a prescribed cost fuctioal. We refer to [2,3,15] for related thi cotrol problems ad also refer to Chapter VII of [7]. It will be helpful to defie fluid-scaled ad diffusio-scaled quatities to carry out our aalysis. We let Ā (t) A (t) ad A (t) 1 A (t) λ (V (s))ds for all t. We also itroduce the diffusio-scaled offered waitig time process (3.5) V (t) V (t) for all t. (3.6) Sice V ( ) ad V ( ) are RCLL processes (ad hece with coutably may discotiuities), whe itegrated with respect to Lebesgue measure, it follows that f (V (s ))ds = f V (s ) t ds = f V (s) t ds = f (V (s)) ds for all t, where f is ay bouded Borel-measurable fuctio. Hece, throughout this article, we use t λ (V (s))ds or t λ V (s) ds appropriately, whe itegrated with respect to Lebesgue measure.

9 C. Lee, A. Weerasighe / Stochastic Processes ad their Applicatios 121 (211) We also defie the diffusio-scaled martigales with respect to the filtratio (Ft ) (see (3.3)), give by M v (t) 1 [t] (v j 1)1 [V (t j )<d j ], M d (t) 1 [t] 1 [V (t j ) dj ] E(1 [V (t j ) dj ] F j 1 ). Usig (3.1) ad (3.4) ad the state equatio described i (2.7), ad after simple algebraic maipulatios, we obtai V (t) + 1 F (V (s ))da (s) = 1 A (t) λ (V (s))ds + 1 M v (Ā (t)) M d (Ā (t)) where I (t) = t 1 [V (s)=]ds for all t. 4. Weak covergece 4.1. Fluid limits + (3.7) [λ (V (s)) 1]ds + I (t), (3.8) Throughout we use T defied by f T = sup t [,T ] f (s) for ay f i D[, ). Our aim here is first to establish the fluid limit lim V T = i probability for each T >. We ited to employ several properties of the Skorokhod map Γ (see, for example, [2,8,33,16]) i the discussio below. The Skorokhod map Γ : D[, ) D[, ) is explicitly defied by Γ ( f )(t) = f (t) + sup ( f (s)) + for all t. (4.1) s [,t] Give a fuctio f i D[, ), the pair (Γ ( f ), sup s [, ] ( f (s)) + ) is called the Skorokhod decompositio of f ad this decompositio is uique. I (3.8), we let X (t) 1 (A (t) t) + 1 M v (Ā (t)) M d (Ā (t)) 1 F (V (s ))da (s). (4.2) Thus, by (3.8), (4.1) ad (4.2), we observe that (V, I ) is the Skorokhod decompositio of the process X ad thus V (t) = Γ (X )(t), for all t. (4.3) Theorem 4.1 (Fluid limit). For each T >, V T as. (4.4)

10 2516 C. Lee, A. Weerasighe / Stochastic Processes ad their Applicatios 121 (211) Proof. First we show that lim 1 A T = a.s., (4.5) for each T >. For the -th system, we cosider the martigale A ( ) described i (3.5). Usig a radom time chage theorem for poit processes (use Theorem T16 i page 41 of [7] with F t σ (A (s), V (s) : s t) ad Lemma L17 therei ad the fact that λ (x) > ϵ > to guaratee λ (V (s))ds = + a.s.), there is a uit-rate Poisso process Y ( ) such that A (t) = Y ( t λ (V (s))ds) for all t. Here Y (t) (Y (t) t)/ for all 1. Thus, usig part (i) of Assumptio 3.2, we have A T Y C T ad we ca estimate P[ 1 A T > ϵ] for ϵ > arbitrary. Sice Y also is a martigale, usig Doob s iequality we have [ ] 1 P A T > ϵ P Y C T > ϵ E[φ( Y (C T ) )] φ(ϵ, ) where φ( ) is a o-egative, covex, strictly icreasig fuctio o R +. Let θ > 1/2 be fixed. The there is a real umber x θ > so that e x < (1 + x) + θ x 2 for < x < x θ. We pick ay α > so that < α < x θ ad let φ(x) e αx for all x >. The by a elemetary computatio, we obtai E[φ( Y (C T ) )] φ(ϵ ) e θα2 C T e ϵ. (See also Theorem 5.18, page 114 of Che ad Yao [9].) Cosequetly, [ ] 1 P A T > ϵ e θα2 C T e ϵ, where θ > 1/2, α > ad C > are costats idepedet of. Now we ca apply Borel Catelli lemma to coclude the a.s. limit i (4.5). Hece, there is (ω) N such that A (T ) for all (ω). This together with Assumptio 3.2(i) implies that A (T ) A (T ) + C T + C T K 1 for all (ω), for some costat K 1 > which is idepedet of. Next, usig (3.1) V T 1 A (T ) v j 1 K 1 v j for all (ω). But lim 1 K1 v j exists a.s. by SLLN ad hece V T K 2 T for all 1 (ω) ad for some costat K 2 > which is idepedet of. This, together with Assumptio 3.2(ii), implies that T λ (V (s)) 1 ds sup λ (x) 1 T as. x [,K 2 T ] T Hece lim λ (V (s)) 1 ds = a.s. Sice sup Ā (t) t 1 sup A (t) + t [,T ] t [,T ] T λ (V (s)) 1 ds,

11 C. Lee, A. Weerasighe / Stochastic Processes ad their Applicatios 121 (211) usig the above fact with (4.5), we obtai lim sup t [,T ] Ā (t) t = a.s. (4.6) Next, we cosider the martigale term 1 ( M v(t) M d (t)). Notice that E [ M v ](T ) 1 [T ] 2 E(v j 1) 2 σ s 2T as. Similarly, it follows that E [ M d](t ) 4T as. We cosider the vector valued martigale M (t) = ( M v(t)/, M d(t)/ ) ad defie M (t) sup s [,t] M (s) for all t. Usig Doob s iequality for the submartigale M (t) 2 oce more, we obtai E[sup t [,T ] M (t) 2 ] CT/ where C > is a geeric costat idepedet of. We coclude that lim E sup t [,T ] M (t) 2 =. Cosequetly, (M (T ))2 as. This, together with (4.6) ad the radom chage of time theorem (cf. Sectio 14, [6]), implies that M (Ā (T )) (4.7) as. Hece, usig (4.6) ad (4.7), we have sup Ā (t) t + M (Ā (T )) i probability, (4.8) t [,T ] as. Let T (t) X (t) + 1 F (V (s ))da (s) = 1 (A (t) t) + 1 M v (Ā (t)) M d (Ā (t)), (4.9) where X is described i (4.2). With (4.8) i had ad usig (4.2), we observe that lim T T = i probability, (4.1) for each T >. By (4.2), we have T (t) X (t) for all t ad T (t) X (t) is a oegative, o-decreasig process i D[, ). Therefore, we ca use the compariso theorem for the Skorokhod map Γ (Propositios 3.4 ad 3.5 of [8]) to coclude that V (t) = Γ (X )(t) Γ (T )(t) for all t. (4.11) Sice Γ (T ) T coclude 2 T T by the Lipschitz cotiuity of Γ, usig (4.1) ad (4.11) we ca lim V T = i probability. (4.12) This completes the proof. Remark 4.2. I Theorem 6.5 of Sectio 6, we are able to show that lim E V m T = for some m > 2, uder a additioal hypothesis give i (6.6).

12 2518 C. Lee, A. Weerasighe / Stochastic Processes ad their Applicatios 121 (211) Diffusio limits Here we ited to establish the weak covergece of the process V ( ) defied i (3.6) to a (reflected) diffusio process. We eed to obtai several techical results to achieve this objective. Our first propositio is a improvemet of (4.4). Usig (3.5) (3.8), we ca describe the state equatio for V ( ) by V (t) + 1 F V (s ) da (s) = A (t) + M v (Ā (t)) M d (Ā (t)) + V (s) λ 1 ds + I (t), (4.13) where I (t) = t 1 [V (s)=] (s)ds. Notice that ( V, I ) is ideed the Skorokhod decompositio (see the lie below (4.1)) of the process X ( ) where X is described i (4.2). The, Γ ( X )(t) = V (t) for all t where Γ is give i (4.1). We use this fact i the followig propositio. Propositio 4.3. We have for each T >, lim K lim sup P V T > K =. (4.14) Proof. We itroduce X (t) X (t) ad Z (t) X (t) + 1 V (s ) F da (s) + λ V (s) 1 ds for all t, where X is defied i (4.2) ad x = mi{x, }. Notice that {Z (t) X (t) : t } is a o-egative, o-decreasig process ad thus by a compariso argumet as i (4.11), we obtai V (t) Γ (Z )(t) for all t. Cosequetly, usig the Lipschitz cotiuity of Γ, we get But V T 2 Z T for all T. Z (t) = A (t) + M v (Ā (t)) M d (Ā (t)) + ad hece we have V T C 1 A T + sup M v (Ā (t)) + sup t [,T ] + T (λ (V (s)) 1) + ds t [,T ] (λ (V (s)) 1) + ds for all t, M d (Ā (t)), (4.15) where C 1 > is a geeric costat idepedet of T. To estimate P[ V T > K ] for K >, we estimate the probability correspodig to each term i the right had side of (4.15). Throughout, we cosider K > to be a geeric costat. First, we estimate P[ A T > K ]. Usig the same

13 C. Lee, A. Weerasighe / Stochastic Processes ad their Applicatios 121 (211) techique used i the proof of (4.5), we obtai P A T > K E Y (C T ) 2 K 2 CT K 2, where C > is the costat as i Assumptio 3.2 (i) ad C > is a geeric costat idepedet of K. Here Y (t) (Y (t) t)/ for all t ad Y ( ) is a uit-rate Poisso process. Hece lim K lim sup P A T > K =. (4.16) Next we cosider P[sup t [,T ] M v(ā (t)) > K ], ad here we ited to use (4.6). We have P sup M v (Ā (t)) > K t [,T ] P P sup M v (Ā (t)) > K, Ā (T ) 2T t [,T ] + P[Ā (T ) > 2T ] Notice that the quadratic variatio process [ M v ] satisfies sup M v (t) > K + P[Ā (T ) > 2T ]. t [,2T ] [ M v ](t) 1 [t] (v j 1) 2 ad hece E([ M v ](2T )) 2T σ s 2, where σs 2 E(v j 1) 2 > is a fiite costat. Thus, from Doob s maximal iequality for submartigales (cf. [18]) we obtai P sup t [,2T ] M v(t) > K CT/K 2, where C > is a costat idepedet of T ad. Hece lim K lim sup P sup t [,2T ] M v(t) > K = ad by (4.6), lim P[Ā (T ) > 2T ] =. Thus we have lim K lim sup P sup M v (Ā (t)) > K t [,T ] =. (4.17) The proof of lim K lim sup P sup t [,T ] M d(ā (t)) > K = is very similar to that of (4.17). For the last term i the right had side of (4.15), we ited to use (4.4). Recall δ > ad M > are as i Assumptio 3.2(iii). The we have [ T ] P (λ (V (s)) 1) + ds > K [ T ] P (λ (V (s)) 1) + ds > K, V T < δ + P [ V T δ ] P [MT > K, V T δ ] + P [ V T δ ]. Notice that lim K P[MT > K, V T δ ] = ad by (4.12) we obtai lim P [ V T δ ] =. Cosequetly, [ T ] lim lim sup P (λ (V (s)) 1) + ds > K =. (4.18) K Now, (4.15) (4.18) imply (4.14) ad this completes the proof.

14 252 C. Lee, A. Weerasighe / Stochastic Processes ad their Applicatios 121 (211) Next, we itroduce R (i) i 1 [V (t j ) d j ], (4.19) which represets the umber of customers who abadoed the system amog the first i customers. We also defie its fluid-scaled term R (t) 1 R ([t]) = 1 [t] 1 [V (t j ) d j ] (4.2) for all t. We ited to show R ( ). I the case of costat itesity, this is ideed proved i the Lemma 5.5 of [26]. But, our proof maily uses the previous propositio ad martigale property of A. Lemma 4.4. For each T >, lim E[ R (T )] =. (4.21) Proof. Cosider the martigale {A (t) : t } ad the stoppig times {t[t ] : 1}. Let M > be a costat ad τ t[t ] M. The A (τ ) A (t[t ] ) = [T ]. Sice τ is a bouded stoppig time, E[A (τ )] =. Thus E A (τ ) τ λ (V (s))ds ad usig Assumptio 3.2, we have ϵ E[τ ] E[A (τ )] [T ], which implies E[τ ] T/ϵ. By lettig M +, we have E[t [T ] ] C 1T, (4.22) where C 1 > is a geeric costat. Next, we estimate P[max 1 j [T ] V (t j ) K ]. Let ϵ > be arbitrary. We pick a large costat C 2 such that < C 1T C 2 T < 4 ϵ. The we have [ ] P max V (t j ) K 1 j [T ] [ P max 1 j [T ] P V C2 T > K + ϵ 4, ] V (t j ) K, t [T ] < C 2T + P[t[T ] C 2T ] where the secod iequality follows from Chebyshev s iequality ad (4.22). Also, lim K lim sup P V C2 T K = by (4.14). Hece, there exists a K > such that for all K > K, lim sup P[ V C2 T K ] < ϵ/4 ad as a cosequece we have [ ] lim sup P max V (t j ) K < ϵ for all K > K. (4.23) 1 j [T ] 2

15 C. Lee, A. Weerasighe / Stochastic Processes ad their Applicatios 121 (211) To estimate E[ R (T )], we pick K > K ad cosider P[V (t j ) > d j ], where j = 1, 2,..., [T ]. The it follows that [ P[V (t j ) > d j ] P V (t j ) > d j > K ] + P [d j K ] [ P max 1 j [T ] V (t j ) > K ] + F K. By Assumptio 3.3, lim F ( K ) = ad cosequetly, there is a 1 such that sup P[V (t j ) > d j ] < ϵ 1 j [T ] for all. Hece by (4.2), E[ R (T )] 1 [T ]ϵ T ϵ ad we coclude that lim E[ R (T )] =. This completes the proof. Our ext step is to show that the term 1 t F V (s ) da (s) i the state equatio (4.13) ca be well approximated by t H( V (s))ds, where H( ) is give i Assumptio 3.3. Lemma 4.5. We have for each T >, 1 t V (s ) sup F da (s) t [,T ] H(V (s))ds i probability as. (4.24) Proof. We recall Ā (t) = 1 A (t) ad it satisfies (4.6). Hece we ca write 1 t = F V (s ) da (s) H(V (s))ds V (s ) t F (dā (s) ds) + F V (s ) H(V (s)) ds. (4.25) To obtai (4.24), we estimate the right had side of (4.25) usig (4.6) ad Assumptio 3.3. First we ote that { M A(t) Ā (t) t λ (V (s))ds : t } is a martigale ad its quadratic variatio is give by [ M A](T ) = 1 Ā(T ). By radom time chage theorem of poit processes (see (2.13) ad the proof of (4.5)), Ā (T ) = 1 T Y λ (V (s))ds 1 Y (C T ), where Y is a uit-rate Poisso process ad C > is as i Assumptio 3.2(i). Thus, [ M A ](T ) 1 2 Y (C T ) ad dā (t) dt = d M A (t) + (λ (V (t)) 1)dt ad the first term o the right side of (4.25) is equal to t V (s ) F d M A (s) + t V (s ) F (λ (V (s)) 1)ds. (4.26)

16 2522 C. Lee, A. Weerasighe / Stochastic Processes ad their Applicatios 121 (211) V We cosider a arbitrary δ >. Sice F (x) 1 for all x ad F (t ) is a predictable process, the itegral t F V (s ) d M A (s) defies a martigale ad its quadratic variatio process is give by t F2 V (s ) d[ M A ](s) (see [24]). Hece usig Doob s maximal iequality, we have P sup t [,T ] δ 2 E T F 2 V (s ) F d M A (s) > δ V (s ) d[ M A ](s) δ 2 E F ( V T )[ M A ](T ) 1 δ 2 E [F ( V T )Y (C T )] 1 δ 2 E[F 2 ( V T )]E[Y 2 (C T )] C 1T δ 2 1/2 E[F 2 ( V T )] 1/2. (4.27) I the above estimatio, for the last two iequalities, we have used Cauchy Schwarz iequality ad the fact that E[Y 2(C T )] C1 22 T 2 for some geeric costat C 1 > idepedet of ad T. Next, we will show E[F 2( V T )] approaches as. By Assumptio 3.3, there exist ad M 1 > such that x sup F < M 1 for all. x [,K ] We cosider > ad the [ ] E F 2 ( V T ) = E F 2 ( V T )1 [ V T K ] [ ] + E F 2 ( V T )1 [ V T > K ] M2 1 + P[ V T > K ]. Now, lettig ad the K ad usig (4.14), we obtai lim E[F 2( V T )] =. Cosequetly, by (4.27), we coclude that lim P t V (s ) F d M A (t) > δ =. (4.28) sup t [,T ] Similar to the previous estimatio, we obtai [ T ] P F (V (s )) λ (V (s)) 1 ds > δ [ T P F (V (s )) λ (V (s)) 1 ds > δ, ] V T K

17 C. Lee, A. Weerasighe / Stochastic Processes ad their Applicatios 121 (211) P[ V T > K ] T ] P [M 1 λ (V (s)) 1 ds > δ + P[ V T > K ]. (4.29) T I the derivatio of (4.6), we have obtaied lim λ (V (s)) 1 ds = a.s. This together with (4.14) implies that the right had side of (4.29) teds to zero as. This yields [ T ] lim P F (V (s )) λ (V (s)) 1 ds > δ =. (4.3) Cosequetly, usig (4.26), (4.28) ad (4.3), we obtai for each T > t lim P V (s ) sup F (dā (s) ds) > δ =. (4.31) t [,T ] Fially, we ited to establish t lim P F V (s ) sup H(V (s)) ds > δ =. (4.32) t [,T ] Pick ϵ > so that < ϵ < T δ. By Assumptio 3.3, we take ay K > ad the there is a 1 N such that sup x [,K ] F ( x ) H(x) < ϵ for all 1. We cosider > 1 ad estimate T F V (s ) P H(V (s)) ds > δ T F V (s ) P H(V (s)) ds > δ, V T K + P[ V T > K ] P[ϵT > δ, V T K ] + P[ V T > K ]. Sice ϵt < δ, the first term of the above is for all > 1. Also, lim K lim P[ V T > K ] =. Hece (4.32) follows. Therefore, (4.31) ad (4.32) yield (4.24). This completes the proof. Our ext lemma shows that the term t (1 λ (V (s)/ ))ds ca be well approximated by t u( V (s))ds, where the fuctio u( ) is as give i Assumptio 3.2. Lemma 4.6. We have for each T >, T V (s) 1 λ u(v (s)) ds i probability as, (4.33) ad cosequetly, t V (s) sup 1 λ u(v (s)) ds t [,T ] i probability as. (4.34)

18 2524 C. Lee, A. Weerasighe / Stochastic Processes ad their Applicatios 121 (211) Proof. Fix T >. Let δ > ad pick ϵ > small so that ϵt < δ. Let K > be arbitrary. By Assumptio 3.2(iv), there is (K ) so that sup x [,K ] (1 λ (x/ )) u(x) < ϵ wheever. Thus, o the set [ V T < K ], we have T V (s) 1 λ u(v (s)) ds < ϵt < δ, for all. Followig a estimatio similar to that of Lemma 4.5, we ca have T V (s) lim sup P 1 λ u(v (s)) ds > δ lim sup P[ V T > K ]. Hece, usig (4.14), desired coclusio (4.33) follows. The followig result is a immediate cosequece of Lemmas 4.5 ad 4.6. Therefore, we omit the proof. Lemma 4.7. For all t, let ϵ (t) 1 V (s ) F da (s) + H(V (s))ds V (s) 1 λ u(v (s)) ds. (4.35) The for each T >, ϵ T i probability as. To discuss the weak covergece of the process {V (t) : t }, we rewrite the state equatio (4.13) i the followig form: V (t) = ξ (t) ϵ (t) u(v (s))ds H(V (s))ds + I (t), (4.36) where ξ (t) A (t) + M v (Ā (t)) M d (Ā (t)), (4.37) ad I (t), ϵ (t) are give i (3.8) ad (4.35), respectively Geeralized Skorokhod map ad weak covergece Followig Sectio 4 of [26], we itroduce the geeralized Skorokhod map. Defiitio 4.8. Let p : [, ) [, ) be a locally Lipschitz cotiuous fuctio. The for a give x i D[, ) with x(), there exists a uique pair of fuctios (z, l) such that z, l are also i D[, ) ad (i) z(t) = x(t) t p(z(u))du + l(t), z(t), for all t, (ii) l() =, l( ) is o-decreasig, ad z(t)dl(t) =.

19 C. Lee, A. Weerasighe / Stochastic Processes ad their Applicatios 121 (211) We use the otatio i [26] ad itroduce two fuctios φ p : D[, ) D[, ) ad ψ p : D[, ) D[, ) give by (φ p, ψ p )(x) = (z, l). (4.38) This describes the geeralized Skorokhod decompositio of the fuctio x i D[, ). The map φ p : D[, ) D[, ) is kow as the geeralized Skorokhod map. Sice (4.36) describes precisely this decompositio, it is easy to observe that (φ p, ψ p )(ξ ϵ ) = (V, I ), (4.39) where p(x) = u(x) + H(x) for all x, ad i this case p( ) is a locally Lipschitz cotiuous fuctio. I [26], the fuctio p( ) is of the form p(x) = x h(u)du where h( ) is a o-egative cotiuous fuctio. But their discussio o existece ad uiqueess of the pair (z, l) for a give x i D[, ) as well as o the cotiuity properties of (φ p, ψ p ) i D[, ) edowed with the Skorokhod J 1 -topology holds for a o-egative, locally Lipschitz cotiuous fuctio p( ) with a few mior chages i their proofs. We state these results i the followig propositio ad idicate the ecessary chages required i the proofs give i [26]. Propositio 4.9 (Lemma 4.1 ad Propositio 4.1 of Reed ad Ward [26]). Let p : [, ) [, ) be a o-egative, locally Lipschitz cotiuous fuctio. The the followig results hold. (i) For each x i D[, ) with x(), there exists a uique pair of fuctios (z, l) satisfyig the Defiitio 4.8. (ii) The fuctios φ p ad ψ p defied i (4.38) are cotiuous o D[, ), whe it is edowed with the Skorokhod s J 1 -topology. Proof. Proofs of the above statemets essetially follow from those of Lemma 4.1 ad Propositio 4.1 of [26] with the chages described below. Give x i D[, ), Picard s iteratio scheme was used i Lemma 4.1 of [26] to obtai a uique solutio to w(t) = x(t) p(γ (w)(s))ds, for t, (4.4) where Γ is the Skorokhod map defied i (4.1). Give x i D[, T ], they itroduce the iterative scheme by w (t) o [, T ] ad w (t) = x(t) p(γ (w 1 )(s))ds, for all t T ad 1. (4.41) I this situatio, we eed to establish the boud sup 1 sup t [,T ] w (t) M <, where M > is a costat depedig o x. Sice x(t) w (t) = t p(γ (w 1)(s))ds is a o-egative, o-decreasig fuctio, by a compariso result for the Skorokhod map (cf. [2]), we have Γ (w )(t) Γ (x)(t) for all t T. Next, itroduce p (y) max z [,y] p(z), the we have p(γ (w )(t)) p (Γ (w )(t)) p (Γ (x)(t)) for all t T, ad hece by (4.41), this implies x(t) w (t) t p (Γ (x)(s))ds. By (4.41), x(t) w (t) for all t T sice p( ) is o-egative. Therefore, x(t) t p (Γ (x)(s))ds w (t) x(t) for all t T ad 1. Hece, the required boud sup 1 sup t [,T ] w (t) M <

20 2526 C. Lee, A. Weerasighe / Stochastic Processes ad their Applicatios 121 (211) holds for some M >. Now oe ca follow the proof of covergece of the sequece (w ) to a fuctio w i D[, T ] as i [26]. Moreover, sup t [,T ] w(t) M < holds. The uiqueess of the solutio w to (4.4) for a give x i D[, ) with x() is quite straightforward. Assume w 1 ad w 2 are two solutios to (4.4). The followig the same proof above, we have sup t [,T ] w i (t) M for i = 1, 2, ad cosequetly sup t [,T ] Γ (w i )(t) 2M where M > is a costat. Let K M be the Lipschitz costat of p( ) o the iterval [, 2M]. t The usig (4.4) for w 1 ad w 2, we obtai w 1 w 2 t 2K M w 1 w 2 s ds, where t deotes the sup orm o [, t]. Hece, by Growall s iequality, it follows that w 1 w 2 t = for all t T ad thus the uiqueess of w i (4.4) follows. I [26], the map M p is defied from D[, ) to D[, ) so that M p (x) = w, where w is the uique solutio to (4.4). The the cotiuity of M p o D[, ), whe this space is edowed with the Skorokhod s J 1 -topology, essetially follows from the same proofs i parts (ii) ad (iii) of Lemma 4.1 i [26]. Next, otice that give x i D[, ) with x(), the pair (z, l) defied i (4.38) satisfies z = Γ (w) Γ (M p (x)) ad l = w Γ (w) = M p (x) Γ (M p (x)). Hece, φ p (x) = Γ M p (x) ad ψ p (x) = M p (x) Γ M p (x) for each x i D[, ) with x(). Sice the Skorokhod map Γ is Lipschitz cotiuous o D[, ), the proof of part (ii) of the propositio is straightforward. To obtai the weak covergece of (V ( )) 1 ad to idetify the limit, we ited to show that ξ ( ) σ W ( ) as i D[, ), where W ( ) is a stadard Browia motio. The Lemma 4.7 together with the cotiuous mappig theorem implies that ξ ( ) ϵ ( ) σ W ( ) as i D[, ). Sice both fuctios φ p ad ψ p are cotiuous o D[, ), whe this space is edowed with the Skorokhod s J 1 -topology, we ca establish the followig theorem for the weak covergece of the process (V ( )) 1. Theorem 4.1 (Diffusio Limit). The process (V, I ) coverges weakly to (Z, L) as i D 2 [, ), where (Z, L) is the uique strog solutio to the reflected stochastic differetial equatio Z(t) = σ W (t) u(z(s))ds H(Z(s))ds + L(t), (4.42) for all t. Here, W ( ) is a stadard Browia motio ad σ > is a costat which satisfies σ 2 = 1 + σs 2. The fuctios u( ) ad H( ) are described i the Assumptios 3.2 ad 3.3. The process Z( ) is o-egative ad has cotiuous sample paths. Here, L( ) is the local-time process of Z at the origi. The process L( ) is uique, cotiuous, o-decreasig process such that L() = ad Z(s)dL(s) =, for all t, (4.43) ad that Z(t) for t. Proof. Recall that the process ϵ ( ) i (4.39) coverges to uiformly o compact sets i probability as show i Lemma 4.7. We ited to show ξ ( ) σ W ( ) i D[, ) i Propositio 4.12 ad we assume this fact i this proof. Here W is a stadard oe-dimesioal Browia motio. Hece, by the cotiuous mappig theorem, we ca coclude ξ ϵ weakly

21 C. Lee, A. Weerasighe / Stochastic Processes ad their Applicatios 121 (211) coverges to σ W. Therefore, by the cotiuity properties of the mappig (φ p, ψ p ) i (4.39) (see Propositio 4.1 i [26]), we have (φ p, ψ p )(ξ ϵ ) (φ p, ψ p )(σ W ) as. Sice the reflected stochastic differetial equatio i (4.42) ad (4.43) has a uique pathwise solutio, (φ p, ψ p )(σ W ) (Z, L) ad the proof of the Theorem 4.1 is complete. We begi with a techical lemma that will be used i the proof of Propositio Lemma Let H ( ) be the process defied by H (t) = 1 t [t]+1 ([t] + 1) λ (V (s))ds (4.44) for all t. Itroduce the vector-valued process {M (t) = ( H (t), M v(t), M d (t)) : t }, where the processes M v ad Md are defied i (3.7). The the followig results hold: (i) (M (t), Ft ) is a mea zero martigale, where the filtratio (F t ) is defied i (3.3). (ii) For each t, the quadratic variatio processes have the followig limits i probability: (a) lim [ H, H ](t) = t, (b) lim [ M v, M v](t) = σ s 2t, (c) lim [ M d, M d ](t) =, (d) lim [ H, M v](t) = lim [ H, M d](t) = lim [ M v, M d ](t) =. I part (b), σs 2 is give by σ s 2 = E(v 1 1) 2. Proof. We already kow M v ad M d are (F t )-martigales from the discussio after (2.4) ad (2.5). To prove part (i), it remais to show that H is also a (Ft )-martigale. Sice H ( ) has piecewise costat paths with possible jumps at the times k, we cosider H (i) = 1 t i+1 (i + 1) λ (V (s))ds (4.45) for i =, 1, 2,.... Notice that H (t) = H ([t]) for all t ad H is adapted to the filtratio ( F i ) i defied i (3.2). We show that (H (i), F i ) is a martigale ad from this, it follows that ( H (t), Ft ) also is a martigale. Followig the discussio i (A.1) ad (A.2), we ited to itroduce two filtratios (Gt ) t ad (G t ) t. Let F {, Ω} ad F j = σ ((t1, v 1, d 1 ),..., (t j, v j, d j )) for j 1 as i (2.9). The, it is easy to check that A (t) is a ( F j ) j -stoppig time for each t. Now we itroduce the two filtratios (G t ) t ad (G t ) t by G t σ (A (s), V (s) : s t), G t F A(t) for all t. (4.46) For each i, the jump time ti of the process A ( ) is clearly a (G t )-stoppig time ad E[t i ] is also fiite as i (4.22). Thus the filtratio (G t i ) i 1 is well defied. Sice A (t) is a (G t )-martigale as i (3.4), we have E[A (ti+2 ) G t i+1] = A (ti+1 ) for each i =, 1, 2,.... (4.47)

22 2528 C. Lee, A. Weerasighe / Stochastic Processes ad their Applicatios 121 (211) Next, we observe that A (ti+1 ) = H (i) ad F i G t i+1 for each i =, 1, 2,.... By coditioig both sides of (4.47) with respect to F i, we obtai that (H (i), F i ) is a martigale. This completes the proof of part (i). For part (ii), first otice that H ca be writte as H (t) = 1 [t] t j+1 1 λ (V (s))ds (4.48) j= t j for all t, where t. Recall that usig (2.13), we ca write A (t) = Y ( t λ (V (s))ds) for all t, where Y is a uit-rate Poisso process. Let (e j ) j 1 be the sequece of jump times of Y ad defie the sequece (η j ) j 1 by η1 e 1 ad η j e j e j 1 for all j 2. The (η j ) is a i.i.d. sequece of expoetial radom variables with parameter 1. With the above represetatio, t j λ (V (s))ds = e j ad hece H ca be writte as Therefore, H (t) = 1 [t] (1 η j ). (4.49) j= [ H, H ](t) = 1 [t] (1 η j )2. (4.5) j= Let (η j ) be a geeric i.i.d. sequece of expoetial radom variables with parameter 1. The for each ϵ >, P [ H, H ](t) t 1 [t] < ϵ = P (1 η j ) 2 t < ϵ j= 1 ad by strog law of large umbers, lim [t] j= (1 η j) 2 = t a.s. Cosequetly, for each t, lim [ H, H ](t) = t i probability. Next, we cosider the quadratic variatio process [ M v, M v ](t). Usig (3.7), we obtai [ M v, M v ](t) = 1 [t] (v j 1) 2 1 [V (t j )<d j ]. Let S (t) = 1 [t] (v j 1) 2. The S (t) [ M v, M v ](t) = 1 [t] (v j 1) 2 1 [V (t j ) d j ]. Sice v j is idepedet of σ ( F j 1 {d j }) ad 1 [V (t j ) d j ] is measurable with respect to this σ -algebra, we have E (v j 1) 2 1 [V (t j ) dj ] σ ( F j 1 {d j }) = σs 2 1 [V (t j ) d j ].

23 C. Lee, A. Weerasighe / Stochastic Processes ad their Applicatios 121 (211) Takig the expected value i both sides, we have E[(v j 1) 2 1 [V (t j ) d j ] ] = σs 2E[1 [V (t j ) d j ] ]. Cosequetly, E S (t) [ M v, M v ](t) = σ 2 s E [t] 1 [V (t j ) d j ] = σs 2 E[ R (t)], where R (t) is give i (4.2). By Lemma 4.4, we have lim E[ R (t)] = ad thus lim E S (t) [ M v, M v](t) =. O the other had, (v j) is a i.i.d. sequece with E(v j 1) 2 = σs 2. Therefore, by strog law of large umbers, lim S (t) = σs 2 t a.s. Usig these two facts, we ca coclude lim [ M v, M v](t) = σ s 2 t i probability for each t >. Usig (3.7), we have E([ M d, M d ](t)) = 1 [t] E 2 1 [V (t j ) d j ] E(1 [V (t j ) d j ] F j 1 ). Sice E(1 [V (t j ) d j ] F j 1 ) 1, we obtai 2 E 1 [V (t j ) d j ] E(1 [V (t j ) d j ] F j 1 ) 2E[1[V (t j ) d j ] ]. Therefore, E([ M d, M d](t)) 2E[ R (t)], where R (t) is give i (4.2). Usig (4.21), we have lim E([ M d, M d](t)) = ad thus lim [ M d, M d ](t) = i probability for each t >. Similar to the above computatios, we have [ M v, M d ](t) = 1 [t] (v j 1)1 [V (t j )<d j ] E 1 [V (t j ) d j ] F j 1. But V (t j ) ad d j are measurable i σ ( F j 1 {d j }) ad v j is idepedet of σ ( F j 1 {d j }). Also, E v j 1 E(v j 1) 2 = σ s. Hece we ca easily obtai E [ M v, M d ](t) σ se[ R (t)] as by (4.21). Thus, lim [ M v, M d ](t) = i probability for each t >. From (4.48) ad (3.7), we obtai [ H, M v ](t) = 1 [t] (v j 1)1 [V (t 1 j )<d j ] j+1 t j λ (V (s))ds. (4.51) Let U (t) = [ H, M v](t). We claim that (U (t), Ft ) is a martigale. Clearly, {U (t)} is adapted to (Ft ). Usig the otatio i (4.49), we ca write t j+1 (v j 1)1 [V (t 1 j )<d j ] λ (V (s))ds = (v j 1)(1 η t j )1 [V (t j )<d j ]. j This term is itegrable sice E(v j 1) 2 = σ 2 s < ad E(1 η j )2 = 1. This term is also equal to (v j 1)1 [V (t j )<d j ] (A (t j+1 ) A (t j )). Usig the fact that v j, V (t j ) ad d j

24 253 C. Lee, A. Weerasighe / Stochastic Processes ad their Applicatios 121 (211) are G t -measurable ad by (4.47), we see that j j+1 E (v j 1)1 [V(tj )<dj ] 1 λ (V (s))ds G t j t j But F j 1 G t ad therefore by coditioig o F j 1, we have {U (t)} is a (Ft )-martigale. j Cosequetly, E([U, U ](t)) 1 =. [t] E (v j 1) 2 (A (t j+1 ) A (t j ))2. Sice (A (t), G t ) is a martigale (recall (3.4)) ad the quadratic variatio process is give by [A, A ](t) = 1 A (t), we have E (A (t j+1 ) A (t j ))2 G t = 1 j. Also, (v j 1) is G t -measurable, ad hece j E (v j 1) 2 (A (t j+1 ) A (t j ))2 G t = 1 j (v j 1) 2. Cosequetly, E[(v j 1) 2 (A (t j+1 ) A (t j ))2 ] = σs 2 / ad we deduce that E([U, U ](t)) σ 2 s [t] 2 as. Therefore, U (t) = [ H, M v](t) i probability. The proof of lim [ H, M d ](t) = i probability is similar to that of the previous result ad therefore we omit it. This completes the proof of part (ii) of lemma. Propositio Let ξ be defied by (4.37). The the process ξ ( ) coverges weakly to σ W ( ) i D[, ) as, where W ( ) is a stadard Browia motio ad σ > is a costat give by σ 2 = 1 + σ 2 s. Here, σ 2 s = E(v 1 1) 2 is a costat as i Assumptio 3.1. Proof. We cosider the vector-valued process {(A (t), M v (Ā (t)), M d (Ā (t))) : t }, where Ā (t) = 1 A (t) for all t. We ited to show that this process coverges weakly to (W 1, σ s W 2, ) i D 3 [, ), where W 1 ad W 2 are idepedet stadard Browia motios. Cosider process H defied i (4.44). The H (Ā (t)) = A (t k+1 ) if t k t < t k+1. (4.52) Notice that the vector-valued process M (t) = ( H (t), M v (t), M d (t)) for t is a (F t )- martigale by part (i) of Lemma Our approach here is to use the martigale fuctioal cetral limit theorem (cf. Theorem 1.4, Chapter 7 i [12] or Theorem 2.1 i [34]) to establish the weak covergece of M to (W 1, σ s W 2, ) ad the to apply radom time chage theorem (cf. Sectio 14 of [6]) to coclude M (Ā (t)) also coverges to (W 1, σ s W 2, ). Fially, we establish that for each T >, sup t [,T ] A (t) H(Ā (t)) coverges to zero i probability. The as a cosequece of this, (A ( ), M v (Ā ( )), M d (Ā ( ))) coverges weakly to (W 1, σ s W 2, ) i D 3 [, ).

25 C. Lee, A. Weerasighe / Stochastic Processes ad their Applicatios 121 (211) To implemet the sketch of the proof give above, we cosider the vector-valued martigale (M (t), Ft ) ad apply the martigale fuctioal cetral limit theorem, Theorem 1.4 of Chapter 7 i [12]. We ited to verify the assumptio i the quoted Theorem 1.4, part (a). First, we show that for each T >, lim E sup M (t) M (t ) t [,T ] sup H (t) H (t ) t [,T ] =. Usig the represetatio (4.49) for H, we ca write 1 E = E max 1 1 j T η j [ ] 1 1/2 E max 1 1 j T η j 2, where (η j ) is a i.i.d. sequece of expoetially distributed radom variables with parameter 1. If (η j ) is a geeric sequece of i.i.d. expoetially distributed radom variables with parameter 1, the 1 E(max 1 j T 1 η j 2 ) = 1 E(max 1 j T 1 η j 2 ) ad sice E(1 η j ) 2 = 1, by (A.5) (see the Appedix), we have lim 1 E(max 1 j T 1 η j 2 ) =. Hece lim E[sup t [,T ] H (t) H (t ) ] =. Similarly, 1 E P sup M v (t) M v (t ) E t [,T ] sup t [,T ] [ 1 E max v j 1 1 j T max v j j T ] 1/2. Sice (v j ) is i.i.d. ad E(v j 1) 2 = σs 2 1 <, agai by (A.5), lim E(max 1 j T v j 1 2 ) =. For 1, let q = (q i j (t) : t ) 1 i, j 3 be the symmetric 3 3 matrix-valued process such that q i j represets the (i, j)-th quadratic-covariatio process of the martigale M = ( H, M v, M d ) (for example, q12 (t) [ H, M v ](t) for t ). I part (ii) of Lemma 4.11, we have established that lim q i j (t) = c i j t i probability for 1 i, j 3 ad the costat matrix C = (c i j ) 3 3 is described by the diagoal matrix C = diag(1, σs 2, ) (see Remark 1.5 i page 34 of [12]). Hece, the assumptios of the martigale fuctioal cetral limit theorem, Theorem 1.4, part (a) i pages of [12] are satisfied. Thus, we ca coclude that M coverges weakly to (W 1, σ s W 2, ) as, where W 1 ad W 2 are idepedet stadard Browia motios. By (4.6), sup t [,T ] Ā (t) t as for each T >, ad hece by the radom time chage theorem (Sectio 14, [6]), M Ā also coverges weakly to (W 1, σ s W 2, ) i D 3 [, ) as. Now, to establish the weak covergece of the process (A (t), M v(ā (t)), M d(ā (t))) it remais to estimate sup t [,T ] H (Ā (t)) A (t) for each T >. Let ϵ >. Notice that H (Ā (t)) A (t) > ϵ