ON THE SUPREMUM DISTRIBUTION OF INTEGRATED STATIONARY GAUS- SIAN PROCESSES WITH NEGATIVE LINEAR DRIFT

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1 Advances in Applied Probabiliy March 1999) ON THE SUPREMUM DISTRIBUTION OF INTEGRATED STATIONARY GAUS- SIAN PROCESSES WITH NEGATIVE LINEAR DRIFT JINWOO CHOE, Purdue Universiy NESS B. SHROFF, Purdue Universiy Absrac In his paper we sudy he supremum disribuion of a class of Gaussian processes having saionary incremens and negaive drif using key resuls from Ereme Value Theory. We focus on deriving an asympoic upper bound o he ail of he supremum disribuion of such processes. Our bound is valid for boh discree- and coninuous-ime processes. We discuss he imporance of he bound, is applicabiliy o queueing problems, and show numerical eamples o illusrae is performance. KEYWORDS: SUPREMUM DISTRIBUTION; GAUSSIAN PROCESS; STATIONARY INCREMENT WITH LINEAR DRIFT; QUEUE LENGTH DISTRIBUTION; EXTREME VALUE THEORY; ASYMPTOTIC UPPER BOUND; TAIL PROBABILITY AMS 1991 SUBJECT CLASSIFICATION: PRIMARY 6G15 SECONDARY 6G7;6K25 1. Inroducion Consider a coninuous-ime sochasic process {X : } or a discree-ime sochasic process {X n : n = 1, 2,...} described by he following equaions. 1) Coninuous-ime process : X = ξ s ds κ [, )), Posal address: School of Elecrical and Compuer Engineering, Purdue Universiy, Wes Lafayee, IN , U.S.A. Posal address: School of Elecrical and Compuer Engineering, Purdue Universiy, Wes Lafayee, IN , U.S.A. Tel: , Fa: , shroff@ecn.purdue.edu. Please address correspondence o his auhor. 1

2 2) Discree-ime process : X n = J. Choe and N. B. Shroff n ξ m κn n {, 1, 2,...}). Here ξ is a cenered zero-mean) saionary Gaussian process and κ is a posiive consan ha deermines he drif of X. Since ξ is a cenered saionary Gaussian process, he sochasic process X is a Gaussian process wih saionary incremens and negaive linear drif. In his paper we are ineresed in sudying he supremum disribuion of his process X. Specifically, we will derive an asympoic upper bound o he ail of he supremum disribuion of X under he following condiions on C ξ, he auocovariance funcion of he cenered saionary Gaussian process ξ. m=1 C1) Coninuous-ime: C ξ τ) := E{ξ ξ +τ } is absoluely inegrable and C ξτ)dτ >. Discree-ime: C ξ l) := E{ξ n ξ n+l } is absoluely summable and l= C ξl) >. C2) Coninuous-ime: τc ξ τ) is absoluely inegrable. Discree-ime: lc ξ l) is absoluely summable. C3) Coninuous-ime: τc ξ τ) > and τc ξτ)dτ + C ξ τ)dτ > for all, ). Discree-ime: l=1 lc ξl) > and m l=1 lc ξl) + l=m+1 mc ξl) > for all m = 1, 2,.... For noaional simpliciy, we define w Θ := sup θ Θ w θ we will no specify he inde range Θ when i includes he enire domain of w θ ). The sudy of he ail disribuion P{ X > }) is moivaed by is applicabiliy o queueing sysems and high-speed elecommunicaion neworks [6, 7, 8]. In paricular, when κ and ξ are appropriaely defined, one can show ha he seady sae queue lengh disribuion of a queueing sysem is equal o he supremum disribuion of X [12, 14]. Therefore, similar problems have been sudied in he queueing cone. For eample, using Large Deviaion echniques i has been shown for very general classes of saionary processes ξ ha he limi η := lim 1 log P{ X > }) eiss and is finie [12], ha is, 3) log P{ X > }) η, where f g means lim f) g) has been shown for saionary ergodic Gaussian processes ξ n [1]: = 1. Also, in he discree-ime case he following sronger resul 4) P{ X > }) Ce η as, In his paper, we say a posiive-valued funcion f asympoically bounds a posiive-valued funcion g from above, if lim sup g)/f) 1 or from below, if lim inf g)/f) 1). 2

3 On he supremum disribuion of Inegraed Saionary Gaussian processes wih linear drif ha is, he ail of he supremum disribuion of X is asympoically eponenial. However, he asympoic consan C is in general difficul o obain and approimaions have been suggesed o evaluae i. An imporan resul of his paper is he derivaion of an asympoic upper bound, of an eponenial form as in 4), for a fairly large class of Gaussian processes ξ given by C1) C3). This resul can be saed in he form of he following heorem. Theorem 1 Under condiions C1) C3), lim sup e 2κ S P{ X > }) e 2κ2 D S 2. In oher words, P{ X > }) is asympoically bounded from above by e 2κ S + κd S ). Here, S and D are posiive consans ha will be defined laer in Secion 3. Noe ha from large deviaion sudies, η in 3) has already been shown o be 2κ/S under condiion C1) [12]. Hence, his heorem considerably srenghens he large deviaion resul in 3) under condiions C1) C3). Furher, his heorem also provides he upper bound e 2κ2 D/S 2 o he asympoic consan C which is a useful parameer for nework dimensioning. In he coninuous-ime case, 4) has been shown in a more limied seing e.g., when ξ is an Ornsein-Uhlenbeck process [19], or when X is a Brownian Moion process wih negaive drif [16]). In his paper, for he coninuous-ime case, our asympoic upper bound will also be used o show ha here eiss a consan η such ha c 1 e η P{ X > }) c 2 e η for some consans c 1, c 2, and all large enough. The paper is organized as follows. In Secion 2, we firs inroduce fundamenal resuls from he Ereme Value Theory for Gaussian processes; in Secion 3, we derive an asympoic upper bound o P{ X > }). To avoid redundancy, we derive he bound only for he coninuous-ime case and refer o [8] for he derivaions in discree-ime; in Secion 4, we discuss he imporance of he bound in analyzing he behavior of a queueing sysem; finally, in Secion 5 we briefly illusrae he performance of he bound hrough numerical eamples. 2. Resuls from Ereme Value Theory Our sudy of he supremum disribuion of X is based on he Ereme Value Theory lieraure. The following wo heorems from [2]) play key roles in our sudy. 3

4 J. Choe and N. B. Shroff Theorem 2 Borell s Inequaliy) Le {ζ : T } be a cenered Gaussian process wih sample pah bounded a.s.; ha is ζ < a.s. Then E{ ζ } is finie and for all > E{ ζ }, P{ ζ > }) 2e E{ ζ }) 2 2 σ 2, where σ 2 := sup T E{ζ 2 }. Theorem 3 Slepian s Inequaliy) Le {ζ : T } and {υ : T } be wo cenered Gaussian processes on an inde se T wih sample pah bounded a.s. If E{ζ 2} = E{υ2 } and E{ζ s ζ ) 2 } E{υ s υ ) 2 } for all s, T, hen for all P{ ζ > }) P{ υ > }). In addiion o Theorems 2 and 3, we inroduce anoher imporan resul from [2, Corollary 4.15], which provides us a way o bound E{ ζ } and will be used ogeher wih Theorem 2 o derive a bound for he ail probabiliy P{ ζ > }). Theorem 4 Le {ζ : T } be a cenered Gaussian process and define a pseudo-meric d on T as d 1, 2 ) := E{ζ 1 ζ 2 ) 2 }. Also, le Nɛ) be he minimum number of closed d-balls of radius ɛ needed o cover T, hen here eiss a universal consan K such ha E{ ζ } K log Nɛ)dɛ. 3. Asympoic Upper Bound for P{ X > }) In his secion, we derive an asympoic upper bound o he ail probabiliy P{ X > }) for he saionary Gaussian processes ξ ha saisfy C1) C3). This secion consiss of wo pars. We firs obain several preliminary resuls in Secion 3.1, and hen from hese resuls we derive our main resuls in Secion 3.2. Since he proofs for he discree-ime case are essenially similar o hose for he coninuous-ime case, we provide derivaions only for he coninuous-ime case. The deailed proofs for he discree-ime case can be found in [8] Preliminaries We assume ha ξ is a cenered saionary Gaussian process wih a coninuous auocovariance funcion C ξ τ). Also, we assume ξ o be a separable and measurable Gaussian Noe ha d is no a meric, since d 1, 2 ) = does no necessarily imply 1 = 2. 4

5 On he supremum disribuion of Inegraed Saionary Gaussian processes wih linear drif process in order for X o be a well defined sochasic process. From 1), he mean and he auocovariance funcion of X can be obained as 5) E{X } = κ, and C X 1, 2 ) := E{X 1 + κ 1 )X 2 + κ 2 )} = 2 1 C ξ τ 2 τ 1 )dτ 1 dτ 2. We now define a few parameers which will be used eensively hroughou he paper. 6) S := C ξ τ)dτ, D := 2 τc ξ τ)dτ, and D := 2 τ C ξ τ) dτ. In he following proposiion, we show several imporan properies of he variance and he auocovariance funcion of X, which will laer be used in deriving our bounds. Proposiion 1 a) Var{X} b) C X 1, 2 ) = 1 2 is a coninuous and differeniable funcion for >. Furher, d d ) Var{X } Var{X } lim = 2 2 τc ξ τ)dτ =. Var{X1 } + Var{X 2 } Var{X 1 2 } ). c) Le α 1, hen under condiion C1), In paricular, lim Var{X } = S. d) Under condiions C1) and C2), C X α, ) C X, α) lim = lim = S. for >, and Var{X 1 } Var{X 2 } 1 2 D 1 2 for all 1, 2 >, and 1 2 lim S Var{X ) } = D. e) Under condiions C1) C3), Var{X} < S and here eiss a o > such ha Var{X } Var{X s } = sup <s s for all o. Noe ha, from he coninuiy of he auocovariance funcion, he process ξ can always be replaced wih is separable and measurable version [11]. 5

6 J. Choe and N. B. Shroff Proof of Proposiion 1. a) From 5), we have 7) Var{X } = 1 = 2 1 τ C ξ τ 2 τ 1 )dτ 1 dτ 2 ) C ξ τ)dτ by seing τ = τ 2 τ 1 ). Differeniaing boh sides of 7), we ge ) d Var{X } = 2 8) d 2 τc ξ τ)dτ. Also, noe ha 1 τ )C ξτ) C ξ τ) C ξ ) for τ [, ]. Therefore, lim Var{X } lim C ξ )dτ =. b) c) Wihou loss of generaliy W.L.O.G.), assume 2 > 1. Then 2C X 1, 2 ) = = = C ξ τ 2 τ 1 )dτ 1 dτ 2 + C ξ τ 2 τ 1 )dτ 1 dτ C ξ τ 2 τ 1 )dτ 1 dτ 2 + C ξ τ 2 τ 1 )dτ 1 dτ C ξ τ 2 τ 1 )dτ 1 dτ 2 = Var{X 2 } + Var{X 1 } Var{X 2 1 }. C ξ τ 2 τ 1 )dτ 1 dτ 2 C ξ τ 2 τ 1 )dτ 1 dτ 2 1 C ξ τ 2 τ 1 )dτ 1 dτ 2 C ξ τ 2 τ 1 )dτ 1 dτ 2 From he symmery of he auocovariance funcion, i suffices o show ha lim C X α,) = S. Le h τ) be defined as C ξ τ) h τ) = 1 + τ ) Cξ τ) if τ [, ), 1 τ α 1) ) C ξ τ) if τ [, α 1)], if τ α 1), α], oherwise. Then, again, by changing he variables of inegraion τ = τ 2 τ 1 ), we obain C X α, ) = 1 α C ξ τ 1 τ 2 )dτ 1 dτ 2 = h τ)dτ. However, from he definiion of h, we know ha lim h τ) = C ξ τ) and h τ) C ξ τ). Therefore, from condiion C1) and he Dominaed Convergence Theorem, i follows ha C X α, ) lim = 6 C ξ τ)dτ = S.

7 On he supremum disribuion of Inegraed Saionary Gaussian processes wih linear drif d) W.L.O.G. assume 2 > 1 >. From 7), we have Var{X 2 } 2 Var{X 1 } 1 = 2 2 = ) τ2 ) C ξ τ)dτ 1 τc ξ τ)dτ ) ) 1 τ1 C ξ τ)dτ 1 2 τ) ) C ξ τ)dτ. Since 12 τ) 2 1 τ for τ [ 1, 2 ], i follows ha Var{X 2 } Var{X 1 } ) ) Now, le h τ) be defined for τ by τc ξ τ) h τ) = C ξ τ) 2 τ C ξ τ) dτ τ) τ C ξ τ) dτ 2 1 ) D 1 2. if τ [, ), if τ [, ). Then, from 7) and from he definiion of S and h τ), we ge S Var{X ) } = 2 C ξ τ)dτ = 2 h τ)dτ. 1 τ ) ) C ξ τ)dτ ) C ξ τ) dτ On he oher hand, from he definiion of h τ), we know ha h τ) τc ξ τ) as and h τ) τ C ξ τ). Therefore, from condiion C2) and he Dominaed Convergence Theorem, lim S Var{X ) } = 2 τc ξ τ)dτ = D. e) From 7) and he definiion of S, S Var{X } = 2 C ξ τ)dτ 1 τ ) )C ξτ)dτ = 2 ) τc ξ τ)dτ + C ξ τ)dτ > for all > from condiion C3)). Therefore, 9) Var{X } < S for all >. Now, from he Dominaed Convergence Theorem and condiions C2) and C3), i follows ha lim τc ξ τ)dτ = 7 τc ξ τ)dτ >.

8 J. Choe and N. B. Shroff The above equaion wih 8) implies ha here eiss a 1 > such ha d d Var{X} ) > for all 1 ; ha is, Var{X } is an increasing funcion for 1. Le a := sup,1] Var{X }. From 9), he coninuiy of Var{X} and he fac ha lim Var{X } =, i hen follows ha a < S. Therefore, since Var{X} S as, here eiss a o > 1 such ha Var{Xo } o > a. Le o, hen for s 1, Var{X s } s a < Var{X o } o from he definiion of o ) Var{X } Also, since Var{X} is increasing on [ 1, ), o, Var{X} Var{X = sup s} <s s. because Var{X} Var{X s} s Var{X} is increasing on [ 1, )). for s 1, ). Therefore, for all In his paper, we will sudy he supremum disribuion of X hrough he Gaussian process {Y ) : } defined for each > by Y ) X + κ) := = ξ sds. + κ + κ The following relaion beween X and Y ) key role in sudying he ail probabiliy P{ X > }). direcly comes from he definiion of Y ) and plays a 1) For any and any >, X > if and only if Y ) >. I also immediaely follows ha Y ) C ) Y 1, 2 ) can be obained in erms of C X as 11) C ) Y is a cenered Gaussian process and is auocovariance funcion 1, 2 ) := E{Y ) 1 Y ) 2 } = C X 1, 2 ) + κ 1 ) + κ 2 ). Now, le σ, 2 be he variance of Y ). I can hen be epressed in erms of Var{X } as 12) σ 2, = Var{X } + κ) 2. Hence, from Proposiion 1c), we have lim σ, 2 =. Since σ2, is a coninuous funcion of from Proposiion 1a)), here is a finie value = ˆ a which σ 2, aains is maimum σ 2 noe ha σ 2 denoes he supremum of σ2, over he ime inde ). In he ne proposiion Proposiion 2), we show an imporan propery of ˆ. Before we proceed, for noaional simpliciy, we define a funcion g) for as if =, g) := Var{X } S if >. 8

9 On he supremum disribuion of Inegraed Saionary Gaussian processes wih linear drif Noe ha from Proposiion 1a), g) is a coninuous funcion of [, ), and σ, 2 can be wrien in erms of S and g) as 13) Proposiion 2 Under condiion C1), σ 2, = S + κ) 2 g). ˆ κ as. Furher, under condiions C1) and C2), he following sronger resul holds. ˆ κ lim ɛ = for all ɛ >. Proof of Proposiion 2. From Proposiion 1c), i follows ha lim g) = 1. Le G := sup g) G is finie and no less han 1). Since σ 2, aains is maimum a = ˆ, i follows ha 14) Sg κ ) 4κ By solving 14) for ˆ, we have 2 G g κ ) 1 2 G G g κ ) = σ 2, κ σ2,ˆ = Sˆ gˆ ) + κˆ ) 2 Sˆ G + κˆ ) 2. ) ) g κ ) 1 κ ˆ 2 G g κ ) G G g κ ) g κ ) 1 ) ) Since g κ) 1 as, his implies ha ˆ consequenly gˆ ) 1) as. Now, since S +κ) 2 he following relaion should hold. 15) 2 gˆ ) g κ ) 1 2 aains is maimum S 4κ a = κ, we know from 14) ha g κ ) gˆ ) and gˆ ) gˆ ) g κ ) ) ) g κ ) 1 κ ˆ 2 gˆ ) g κ ) Since boh gˆ ) and g κ ) approach 1 as, i follows from 15) ha gˆ ) gˆ ) g κ ) κ. g κ ) 1 ) ) κ. κˆ lim = 1. Thus, we have proved he firs par of he proposiion. We ne prove he second par of he proposiion for which he auocovariance funcion C ξ saisfies boh condiions C1) and C2). From Proposiion 1d), noe ha 16) g 1 ) g 2 ) = 1 Var{X 2 } S Var{X 1 } 2 D 2 1. S

10 J. Choe and N. B. Shroff Since boh g κ ) and ˆ κ approach 1 as increases, we know ha g κ ), ˆ κ [ 1 2, 2] for all sufficienly large. Therefore, for sufficienly large, ˆ 2 gˆ κ κg κ ) ) g κ ) + gˆ ) gˆ ) g ) κ ) from 15)) 4 D ˆ κ G + D ˆ κ ) from 16) and he definiion of G) κ Sˆ κ Sˆ κ D ˆ κ = 4 G + D ˆ κ ) κ Sˆ Sˆ 4 D S + 2G D ˆ κ 17). S Now assume ha lim ˆ κ ɛ ɛ 2 = for some ɛ > from he fac ha ˆ κ holds wih any ɛ > 1). Then, from 17), ˆ κ 4 D 2G + D ˆ κ S ɛ, as. S ɛ 2 as, his Therefore, lim ˆ κ ɛ 2 =. Thus, by inducion we have ˆ κ lim ɛ =, for all ɛ >. The following proposiion is a direc resul of Proposiions 1c) and 2, and describes he limi of σ 2 as. Proposiion 3 Under condiion C1), lim σ2 = S 4κ. Proof of Proposiion 3. From 12), we have σ 2 = Var{Xˆ } + κˆ ) 2 = 1 κ Var{Xˆ } κˆ ˆ κˆ ). 2 Since Var{Xˆ } S Proposiion 1c)), ˆ lim σ 2 = S 4κ. κˆ 1 Proposiion 2), as, i follows ha 1

11 On he supremum disribuion of Inegraed Saionary Gaussian processes wih linear drif 3.2. Main Resul In his secion alhough we provide proofs only for he coninuous-ime case, all he resuls are also valid for he discree-ime case wih he process {Y ) n he parameers S and D redefined as : n =, 1,...} and 18) Y ) n := Xn + κn), S := + κn l= C ξ l), and D := 2 lc ξ l). l=1 Now as menioned in Secion 1, i has been shown for many classes of saionary processes ξ, ha 3) holds for some η [12]. In paricular, for he case when ξ is a saionary Gaussian process ha saisfies C1), i has been shown ha η = 2κ S, ha is, 19) 1 lim log P{ X > }) = 2κ S. Le us ne consider a simple lower bound o he ail probabiliy P{ X > }) epressed in erms of he maimum variance σ 2. From 1), i follows ha However, noe ha Y ) ˆ 2) P{ X > }) = P{ Y ) > }) P{Y ) ˆ > }). is a cenered Gaussian random variable wih variance σ 2. Therefore, ) Ψ P{ X > }), σ where Ψ) := 1 2π e y2 2 dy is he ail of he sandard Gaussian disribuion. I is imporan o ) noe ha Ψ is he probabiliy ha Y ) is greaer han a = ˆ, which is ha value σ of for which he variance of Y ) aains is maimum σ 2. In he Ereme Value Theory for Gaussian processes, i has been frequenly emphasized ha he maimum variance of a cenered Gaussian process wih nonconsan variance, is a very imporan facor in sudying he supremum disribuion of he Gaussian process as can be seen in Borell s inequaliy) [2, 3, 17, 2]. Also, i has been found ha if {ζ : T } is a cenered Gaussian process wih nonconsan variance which aains is maimum variance a = ˆ, P{ ζ > }) he ail of he supremum disribuion of ζ can ofen be closely approimaed by he ail probabiliy P{ζˆ > }). Therefore, i would no be surprising if he lower bound, given by 2), accuraely approimaes he ail probabiliy P{ X > }). In fac, he lower bound has been used o approimae he ail probabiliy in [6, 7] and found o be quie accurae over a wide range of. Addiionally, i has also been shown in [6] ha 21) ) 1 lim log Ψ = 2κ σ S. 11

12 J. Choe and N. B. Shroff Therefore, from 19) and 21), he lower bound is asympoically similar o he ail probabiliy in he logarihmic sense; ha is, log Ψ ) log P{ X > }) as. σ Qualiaively, he above observaions on he lower bound suggess ha he ail probabiliy P{ X > }) is concenraed on or around he maimum variance inde ˆ. However, similariy in he ) logarihmic sense does no imply ha Ψ = P{Xˆ > }) P{ X > }) as. In σ fac, his relaion does no hold in general [8]. Therefore, a naural quesion o ask is wheher and how) we can choose some neighborhood F around ˆ for each such ha P{ Y ) F > }) P{ Y ) > }) as. The following heorem gives us an answer o his quesion, and will be used o obain an asympoic upper bound o P{ X > }). Theorem 5 Under condiion C1), for any α > 1, P{ X [ ακ lim, α κ ] > }) P{ Y ) [ ακ = lim, α κ ] > }) P{ X > }) P{ Y ) > = 1. }) Proof of Theorem 5. second equaliy, i suffices o show ha The firs equaliy direcly follows from 1). Now, in order o show he P{ Y ) [ ακ lim, α > }) κ ]c P{ Y ) > = for all α > 1, }) where A c denoes he complemenary se of A. Le α > 1. Since g) 1 as, here eiss a o such ha g) α+1 2 α for all o. Now, le G := sup g), hen here eiss an o > ακ o such ha S o G + κ o ) 2 S α 2κα + 1) for all o. Since SG +κ) 2 is an increasing funcion of on [, κ ], his fac in conjuncion wih 13) implies ha 22) σ 2, SG + κ) 2 S og + κ o ) 2 S α 2κα + 1) for all o and o. Furher, i can be easily verified ha 23) S + κ) 2 Sα κα + 1) 2 for [ ακ, α κ ]c. From he definiion of o and 23), we have 24) σ, 2 = Sg) + κ) 2 S α 2κα + 1) for o and o, ακ ) α κ, ). 12

13 On he supremum disribuion of Inegraed Saionary Gaussian processes wih linear drif Hence, from 22) and 24), i follows ha 25) σ 2 [ ακ, α S α κ ]c 2κα + 1) for all o. We now define a pseudo-meric d ) on [, ) as d ) 1, 2 ) := E{Y ) 2 Y ) 1 ) 2 }. Also, le B ) ɛ ) := {s [, ) : d ), s) ɛ} be a d ) -ball of radius of ɛ cenered a, and le N ) ɛ) be he minimum number of d ) -balls of radius of ɛ needed o cover [, ). Since Var{Y ) } SG +κ) SG ) 2 4κ and since Y =, B ) SG ɛ ) cover [, ) when ɛ 4κ. Therefore, for all >, SG 26) N ) ɛ) = 1 if ɛ 4κ. SG Now, assume ha ɛ < 4κ and 2 > 1. Then, d ) 1, 2 ) = = { X2 E +κ2) +κ 2 { X2 E +κ2) E +κ 2 ) } X1 +κ 2 1) +κ 1 { X2 X1 +κ2 1)) +κ 2 ) 2 } + X1 +κ 1) X1 +κ +κ 2 + 1) X1 +κ +κ 2 1) E +κ 1 ) 2 } { ) } κ2 1) X 1 +κ1) 2 +κ 2)+κ 1) 27) = +κ Var{X2 2 X 1 )} + κ2 1) +κ 2)+κ 1) Var{X1 }. However, since he saionariy of ξ implies ha Var{X 2 X 1 )} = Var{X 2 1 }, Var{X 2 X 1 )} and Var{X 1 } are bounded by SG 2 1 ) and SG 1, respecively. Therefore, from 27) d ) 1, 2 ) SG2 1 ) + κ 2 1 ) SG1 + κ 2 + κ 1 ) + κ 2 ) SG κ ) SG κ 2 + κ 1 ) + κ 2 ) SG + 1 ) SG 2 2SG from he fac ha +κ 2 1 and +κ) 1 2 κ ). This implies ha if 2 1 2SG ɛ2, hen d ) 1, 2 ) ɛ. Consequenly, 28) [ 2SG ɛ2, + 2SG ɛ2 ] B ) ɛ ). Also, i can easily be shown ha Var{Y ) } ɛ 2 for SG ɛ 2 κ. Since Y ) 2 =, his implies ha 29) [ SG ɛ 2 κ 2, ) B ) ɛ ). 13

14 J. Choe and N. B. Shroff Therefore, from 28) and 29), d ) -balls of radius of ɛ cenered a i i =, 1,..., SG ɛ 2 κ 2 / ɛ2 SG ) covers [, ), where w is he smalles ineger ha is larger han or equal o w and Hence, for ɛ < inequaliy: 3) if i =, i = i SG ɛ2 2SG ɛ2 oherwise. SG 4κ, he minimum number of d) -balls o cover [, ) is bounded by he following N ) ɛ) From 26) and 3), Nɛ) defined by SG ɛ 2 κ 2 / ɛ2 + 1 S2 G 2 SG κ 2 ɛ S 2 G 2 SG κ Nɛ) := 2 ɛ + 2 if ɛ < 4 4κ, 1 oherwise, bounds N ) ɛ) for all, ɛ >. Now, le M := K log 1 2 Nɛ)dɛ, where K is he universal consan in Theorem 4 i can easily be shown ha he inegral is finie). Then, from Theorem 4, E{ Y ) } M for all >. Hence, by applying Theorem 2 o Y ) P{ Y ) [ ακ, α κ ]c > }) 2e on [ ακ, α κ ]c, we ge E { Y ) [ }) 2, α ακ κ ]c 2 σ 2 [ ακ, α κ ]c 2e κ E{ Y ) }) 2 α+1) S α from 25) and he fac ha Y ) [ 2e κ M) 2 α+1) S α ακ, α for sufficienly large. κ ]c Y ) ) Therefore, i direcly follows ha 31) lim sup 1 log P{ Y ) [ ακ, α > }) lim κ M) 2 α + 1) κ ]c S κα + 1) = α S α. Since κα+1) S α < 2κ S for all α > 1, 19) and 31) imply ha P{ Y ) [ ακ lim, α > }) κ ]c P{ Y ) > =. }) 14

15 On he supremum disribuion of Inegraed Saionary Gaussian processes wih linear drif We will now use Theorem 5 and a well known propery of Brownian Moion process o obain an asympoic upper bound o P{ X > }). Le {B : } be he sandard Brownian Moion Wiener) process, and le {V : } be defined as V := ab b. This process is ofen called Brownian Moion process wih drif and has been sudied eensively. In paricular, he supremum disribuion of V has been found in a simple closed form see, for eample, [16, page 199]) as 32) P{ V > }) = P { here eiss a such ha B > b a + }) = e 2b a a 2. We are now ready o prove Theorem 1, which provides a simple single-eponenial based asympoic upper bound o P{ X > }), when ξ saisfies condiions C1) C3). As will soon be eviden, his bound is obained by comparing P{ X > }) and he ail of he supremum disribuion of a Brownian Moion process wih drif, hrough Slepian s inequaliy. For he reader s convenience we resae Theorem 1. Theorem 1 Under condiions C1) C3), lim sup e 2κ S P{ X > }) e 2κ2 D S 2. In oher words, P{ X > }) is asympoically bounded from above by e 2κ S + κd S ). Proof of Theorem 1. for each > by Using his definiion, C ) Z Le V = SB κ and define a cenered Gaussian process {Z ) : } Z ) g)v + κ) g)sb := =. + κ + κ ) he auocovariance funcion of Z can easily be obained as 33) C ) Z 1, 2 ) = S min{ 1, 2 } g 1 )g 2 ). + κ 1 ) + κ 2 ) From 13) and 33), we can verify ha he variance of Z ) and >. is equal o ha of Y ) for any An ineresing fac is ha even hough V canno be epressed in he form of 1), Proposiion 2, Proposiion 3, and Theorem 5 hold wih X, κ, and S replaced by V, b, and a 2, respecively. From he simple auocovariance funcion C V 1, 2 ) = a 2 min{ 1, 2 } of V, hese resuls can be obained in almos he same way as or usually easier han) in he case of X. 15

16 Now, le α > 1 and consider Y ) eiss a o > such ha for all o, and Z ) on he inerval [ ακ, α κ J. Choe and N. B. Shroff ]. From Proposiion 1e), here 34) Var{X s } s Var{X } for all s <. Hence if we assume ha 2 > 1 o, hen C X 1, 2 ) 1 = 1 This implies ha 35) Var{X 1 } + Var{X 2 } Var{X }) 1 = 1 Var{X1 } + Var{X 2 } Var{X1 } + Var{X ) 2 } from 34)) Var{X 1 }Var{X 2 } 1 2 since Var{X} ). S min{ 1, 2 } g 1 )g 2 ) = 1 Var{X 1 }Var{X 2 } 1 2 C X 1, 2 ) if 2 > 1 o. Therefore, from 11), 33), and 35), and from he fac ha Var{Y ) any ακ o ha E{Y ) 1 Hence, from Theorem 3, from Proposiion 1b)) Var{X2 } Var{X )) 2 1 } from he definiion of g)) Y ) 2 ) 2 } E{Z ) 1 Z ) 2 ) 2 } for all 1, 2 [ ακ, α κ ]. } = Var{Z ) }, i follows for 36) P{ Y ) [ ακ, α κ ] > }) P{ Z ) [ ακ, α κ ] > }) for all ακ o. We now obain an upper bound o P{ Z ) [ ακ, α κ ] > }) for ακ o. 37) P{ Z ) [ ακ, α κ ] > }) = P{Z ) > for some [ ακ, α κ ]}) = P{ Sg)B > + κ for some [ ακ, α κ ]}) from he definiion of V and Z ) ) P{ Sg α κ )B > + κ for some }) since g) is increasing on [ ακ, α κ ]) = e 2κ Sg α κ ) from 32)). 16

17 On he supremum disribuion of Inegraed Saionary Gaussian processes wih linear drif Hence, from 36) and 37), 38) P{ Y ) [ ακ, α κ ] > }) e 2κ Sg α κ ) for ακ o. Furher, from Proposiion 1d) and he fac ha g) 1 as, we have 39) e 2κ S e 2κ Sg α κ ) = e 2κ Sg α κ ) 1 1 S ακ = e 2κ 2 S 2 αg α α κ ) κ S α 1 κ { }) Var X ακ }) from he definiion of g)) { Var X α κ e 2κ2 D αs 2 as. Therefore, from Theorem 5 and from 1), 38) and 39), i follows ha lim sup Since α > 1 is arbirary, he heorem follows. e 2κ S P{ X > }) e 2κ2 D αs 2. An ineresing observaion is ha he asympoic upper bound given in Theorem 1 can also be achieved by a simple epression given in erms of he maimum variance σ 2. Proposiion 4 Under condiions C1) and C2), e 2 σ 2 e 2κ S + κd S ) as. Proof of Proposiion 4. From 12) and he definiion of ˆ, we have Therefore, 4) Since ˆ κ, σ 2 = Var{Xˆ } + κˆ ) 2. ) 2κ S 4κ ˆ 2 σ 2 = ˆ S Var{Xˆ } ˆ Sκ2 ˆ κ ) 2 ˆ. 2S Var{Xˆ } ˆ Var{Xˆ } ˆ S, Proposiions 1 and 2, and from 4) we ge Hence, lim e 2κ S e 2 σ 2 = e 2κ2 D S 2. S Var{Xˆ } ˆ ) ˆ D, and κ ˆ ) 2 ˆ 2κ lim S 2 σ 2 = D 2κ2 S 2. as from Proposiion 4 and Theorem 1 ell us ha when he process ξ saisfies condiions C1) C3), he ail of he supremum disribuion is asympoically bounded by e 2 σ 2. Noe ha he class 17

18 J. Choe and N. B. Shroff Infinie Buffer λ Fluid inpu rae a ime Q Amoun of fluid a ime Server µ Service rae a ime Figure 1. A fluid queueing sysem wih an infinie buffer and a server. λ is he insananeous amoun of fluid work) fed ino he sysem a ime, µ is he maimum rae a which fluid can be served a ime, and Q is he amoun of fluid in he queue a ime. of saionary Gaussian processes ha saisfy condiions C1) C3) is fairly large. For eample, any auocovariance funcion ha vanishes faser han τ ɛ l ɛ ) for some ɛ > 2, saisfy condiions C1) and C2) of course, ecep for hose wih S = ). Also, condiion C3) which is somewha more resricive, is saisfied by any nonnegaive auocovariance funcion. Hence, he fac ha an asympoic upper bound o P{ X > }) can be obained merely from σ 2, again indicaes he imporance of he maimum variance in sudying he supremum disribuion of Gaussian processes. In he ne secion, we will discuss he applicaions and imporance of he asympoic upper bound for he sudy of queueing sysems. 4. Applicaion o Queueing Sysems Consider a queueing sysem shown in Figure 1. Le Λ be an increasing funcion defined in such a way ha Λ Λ s is he amoun of fluid ha arrives ino he sysem during he ime inerval s, ]. Similarly, we define M o be an increasing funcion such ha M M s is he maimum amoun of fluid ha can be served during he ime inerval s, ]. Then assuming ha he queue is empy a =, Q he amoun of fluid in he sysem workload) a ime can be epressed as where N := Λ M see for eample [12, 14]). Q = sup N N s ), s If we assume ha Λ and M are independen sochasic processes wih saionary incremens, hen { P{Q > }) = P sup s }) N N s ) > { }) = P sup s N N s ) > 18

19 On he supremum disribuion of Inegraed Saionary Gaussian processes wih linear drif { }) 41) P sup N N s ) > s as. Hence, P{Q > }) := lim P{Q > }) = P { sup s N N s ) > }). In oher words, he seady sae limiing) queue lengh disribuion coincides wih he disribuion of sup s N N s ). The ail of he seady sae queue lengh disribuion is an imporan measure of nework congesion and very useful in he design and conrol of communicaion neworks. Now le λ be defined as he insananeous rae of fluid inpu and µ as he maimum rae a which fluid can be served a ime. Then, N N s can be given by 42) Coninuous-ime : N N s = s ν udu, Discree-ime : N N s = m=s+1 ν m, and where ν := λ µ is he ne inpu rae ino he queue noe ha ν can ake on boh posiive and negaive values). Hence, from 41) and 42), i follows under he saionariy of ν or under he saionariy and independence of λ and µ ) ha Coninuous-ime : { }) P{Q > }) = P sup ν sds >, and Discree-ime : P{Q > }) = P { n sup n m= ν m > }). Fluid queueing models have frequenly been employed for he analysis of mulipleers in emerging high-speed communicaions such as Asynchronous Transfer Mode ATM) neworks [1, 13]. In hese applicaions, he saionary process λ models he aggregae raffic inpu o a mulipleer, and µ is ofen fied o a consan µ o represen he link capaciy of he mulipleer which is usually no ime-varying. Since commercial ATM mulipleers and swiches are already equipped wih very high-capaciy links, many raffic sources can be served a a mulipleer. Therefore, he ne inpu raffic he aggregae raffic inpu minus he link capaciy of he mulipleer, which corresponds o ν ) can usually be accuraely characerized by a saionary Gaussian process [6, 7]. Furher, i has been found ha some imporan ypes of individual raffic sources hemselves can be modeled as a saionary Gaussian process [15]. Once he ne inpu raffic is characerized by a saionary Gaussian process, as we will discuss ne, our asympoic analysis of P{ X > }) can be direcly applied o sudy P{Q > }), he ail of he queue lengh disribuion, in such neworks. Assuming ha ν is a saionary Gaussian process, i is easy o see ha he seady sae queue lengh disribuion is equal o he supremum disribuion of X given by 1) or 2)) wih ξ and κ defined as 43) Coninuous-ime : ξ = ν E{ν }, and κ = E{ν } or Discree-ime : ξ n = ν 1 n E{ν }, and κ = E{ν }. 19

20 J. Choe and N. B. Shroff Therefore, when he ne raffic inpu can be effecively characerized by a saionary Gaussian process ha saisfies condiions C1) C3), Theorem 1 provides us an asympoic upper bound o P{Q > }), he ail of he queue lengh disribuion. Here i should be noed ha while µ = µ for high-speed ATM neworks, i may no be rue for oher neworks; however, all our resuls are also valid for general ime-varying µ as long as he ne inpu rae can be effecively modeled as a Gaussian process. Now le us briefly discuss he relevance of our work in he cone of he eising lieraure. Discree-Time Case: As menioned in Secion 1, in he discree-ime seing [1], i has been shown for saionary ergodic Gaussian ne inpu processes ν n ha 44) P{Q > }) = P{ X > }) Ce 2κ S as, where ξ n and κ are given by 43), and S defined by 18). From he above relaion, Ce 2κ S been suggesed as an approimaion o P{Q > }) for large. This approimaion is ofen called he asympoic approimaion. However, since he eac value of he asympoic consan C canno be obained in general, he following simpler approimaion obained by seing C = 1) has also been suggesed: P{Q > }) e 2κ S. This approimaion is he well known effecive bandwidh approimaion, which can be eended o fairly general classes of ne inpu processes ν [12, 13]. In recen papers, however, i has been argued ha he effecive bandwidh approimaion does no accoun for he advanage of mulipleing and could lead o significan underuilizaion of he nework [9, 18]. Therefore, here is renewed ineres in he accurae approimaions and bounds for he asympoic consan C. I is imporan o noe ha he decay rae of he asympoic upper bound given in Theorem 1 coincides wih he decay rae of he ail P{Q > }) which is equal o 2κ S. asympoic upper bound provides us an upper bound e 2κ2 D S 2 has Therefore, he o he asympoic consan C when ν n is a saionary Gaussian process ha saisfies condiions C1) C3). As previously menioned, a fairly large class of saionary Gaussian processes saisfy hese condiions. Hence, he upper bound o he asympoic consan is epeced o help us o beer eploi he advanage of mulipleing when designing hese neworks. Coninuous-Time Case: In conras o he discree-ime case, 44) has been shown o be valid in he coninuous-ime case only for a very limied class of saionary Gaussian processes ν. Therefore, 2

21 On he supremum disribuion of Inegraed Saionary Gaussian processes wih linear drif obaining an asympoic resul for he ail probabiliy, which is similar o 44), is very imporan. In he following par of his secion, we show how our asympoic upper bound can be used o obain an asympoic resul for P{Q > }) which is nearly comparable o 44). Using he resuls for he discree-ime case, we can show ha here eiss an asympoic lower bound o he ail probabiliy P{ X > }) of he form Ce 2κ S, ha is, lim inf e 2κ S P{ X > }) >. Now, consider he coninuous-ime process X epressed by 1). Given a >, an asympoic lower bound o he ail probabiliy P{ X > }) can be found by looking a he sampled sochasic process {Ẋn = X n : n =, 1, 2,...}. Noe ha Ẋn can be epressed as Ẋ n = = n m=1 n m=1 m m 1) ξ m κn, ξ s ds κn where ξ m := m m 1) ξ sds and κ := κ. ξn is a saionary Gaussian process from is definiion) and C ξl) is auocovariance funcion can be obained in erms of C ξ τ) as from which one can verify ha C ξl) = Ṡ := Hence, from 44) here eiss a c 1 > such ha τ )C ξ τ + l )dτ, C ξl) = C ξ τ)dτ = S. P{ Ẋ > }) c 1e 2 κ Ṡ = c 1 e 2κ S. Therefore, we ge lim inf 2κ e S P{ X > }) lim inf 2κ e S P{ Ẋ > }) since X Ẋ = X {,,2,...}) 45) = c 1 > from 4)). Now, by combining Theorem 1 and 45), i follows ha for saionary Gaussian processes ξ ha saisfy condiions C1) C3), c 1 lim inf 2κ e S P{ X > }) lim sup 21 e 2κ S P{ X > }) e 2κ2 D S 2,

22 J. Choe and N. B. Shroff κ=8 κ=16 Asympoic Upper Bound Tail Probabiliy Figure 2. The ail probabiliy P{ X > }) esimaed hrough simulaion and is asympoic upper bound e 2κ S + κd S ) for a coninuous-ime process X epressed by 1). In his eample, he auocovariance funcion of ξ is given as C ξ τ) = 8 e τ + 4 e τ 2 and κ is se o wo differen values, 8 and 16. where D is defined by 6). Therefore, if we le c 2 := e 2κ2 D S 2, hen he above equaion implies ha for a fluid queue whose ne inpu rae ν = ξ κ) is a saionary Gaussian process ha saisfies condiions C1) C3), for any ɛ > 1, 46) c 1 ɛ 2κ e S P{Q > }) = P{ X > }) ɛc2 e 2κ S for all sufficienly large. Even hough he above relaion is no as srong as 44), i ells us ha P{Q > }) is asympoically enclosed wihin an eponenial envelope when condiions C1) C3) are saisfied by he ne inpu rae ν. 5. Numerical Eamples In his secion we provide wo numerical eamples o illusrae he performance of he asympoic upper bound P{ X > }) e 2κ S + κd S ). Our analyical resuls are compared wih simulaion resuls using he Imporance Sampling echnique described in [5], which has been developed o esimae he queue lengh disribuion efficienly. Therefore, o esimae P{ X > }), we use he fac ha he supremum disribuion of X is equal o he queue lengh disribuion if ξ and κ are relaed o ν by 43). Also, in order o show he accuracy of he simulaion esimaes, 99% confidence inervals are compued by he mehod of bach mean [4], and displayed as verical segmens around he esimaes of he ail probabiliy. 22

23 On he supremum disribuion of Inegraed Saionary Gaussian processes wih linear drif κ=8 Asympoic Upper Bound Tail Probabiliy κ=16 Figure 3. The ail probabiliy P{ X > }) esimaed hrough simulaion and is asympoic upper bound e 2κ S + κd S ) for a discree-ime process X n epressed by 2). In his eample, he auocovariance funcion of ξ n is given as C ξ l) = 25.9 l l and κ is se o wo differen values, 8 and 16. In he firs eample, we consider a coninuous-ime process X given by 1) where ξ is a saionary Gaussian process wih auocovariance funcion C ξ τ) = 8 e τ +4 e 2 τ. Since he queueing) simulaion wih a Gaussian ne inpu rae canno be performed in coninuous-ime, we show he ail probabiliy P{ Ẋ > }) insead of P{ X > }) where Ẋn is he sampled sequence of X inroduced in he previous secion. More precisely, we se o.5 o obain Ẋn = X n from X. In Figure 2, we compare he ail probabiliies P{ Ẋ > }) esimaed via simulaion, and he asympoic upper bounds given in Theorem 1 for κ = 8 and κ = 16. Remember ha he decay raes of he eac ail probabiliy and he asympoic upper bound are equal o 2κ S. Therefore, as one can see in he figure, he simulaion and analyical curves are parallel o each oher for large. Also noe ha he he asympoic upper bound is fairly close o he ail probabiliy for sufficienly large. Alhough he ail probabiliy P{ X > }) canno be direcly esimaed hrough simulaion, i is bounded by P{ Ẋ > }) from below. Hence, for his case, we can conclude ha he envelope given by 46) is fairly narrow. In he second eample, we consider a discree-ime process X n given by 2) where ξ n is a saionary Gaussian process wih is auocovariance funcion C ξ l) = 25.9 l l. In Figure 3, we show he ail probabiliy and he asympoic upper bound again for κ = 8 and κ = 16. As in he previous eample, he eac ail probabiliy curve esimaed by simulaion is parallel o he asympoic upper bound for large values of. Also, from he figure, we can deduce ha 23

24 J. Choe and N. B. Shroff he asympoe of he ail probabiliy as described by 44), here is an eponenial asympoe of he ail probabiliy in he discree-ime case) will be quie close o he bound. This suggess ha e 2κ2 D S 2 is a igh upper bound o he asympoic consan C in 44) which can be used as a dimensioning parameer for nework design and conrol. Eensive eperimenaion wih a wide variey of differen processes ξ n has indicaed ha he upper bound o he asympoic consan is usually quie igh [8]. I should be noed, however, ha he asympoic consan C is bu one imporan parameer in nework design and conrol. Using a single eponenial approimaion of he form in 44) may no be enough o accuraely predic P{Q > }) over a large range of [6, 7, 9]. We are currenly developing analyical echniques o address his problem. References [1] Addie, R. G. and Zukerman, M. 1994). An Approimaion for Performance Evaluaion of Saionary Single Server Queues. IEEE Transacions on Communicaions 42, [2] Adler, R. J. 199). An Inroducion o Coninuiy, Erema, and Relaed Topics for General Gaussian Processes. Insiue of Mahemaical Saisics, Hayward, CA. [3] Berman, S. M. and Kono, N. 1989). The Maimum of a Gaussian Process wih Nonconsan Variance: A Sharp Bound for The Disribuion Tail. The Annals of Probabiliy 17, [4] Braley, P., Fo, B. L. and Schrage, L. E. 1987). A Guide o Simulaion second ed. Springer-Verlag, New York. [5] Chang, C.-S., Heidelberger, P., Juneja, S. and Shahabuddin, P. 1994). Effecive bandwidh and fas simulaion of ATM inree neworks. Performance Evaluaion 2, [6] Choe, J. and Shroff, N. B. 1997). A Cenral Limi Theorem Based Approach o Analyze Queue Behavior in ATM Neworks. In Proceedings of he 15h Inernaional Teleraffic Congress. [7] Choe, J. and Shroff, N. B. 1997). A New Mehod o Deermine he Queue Lengh Disribuion a an ATM Mulipleer. In Proceedings of IEEE INFOCOM. pp

25 On he supremum disribuion of Inegraed Saionary Gaussian processes wih linear drif [8] Choe, J. and Shroff, N. B. 1998). A Cenral Limi Theorem Based Approach for Analyzing Queue Behavior in High-Speed Neworks. Technical repor. Purdue Universiy, Wes Lafayee, IN. submied o IEEE/ACM Transacions on Neworking. [9] Choudhury, G. L., Lucanoni, D. M. and Whi, W. 1996). Squeezing he Mos Ou of ATM. IEEE Transacions on Communicaions 44, [1] Elwalid, A. I. 1991). Markov Modulaed Rae Processes for Modeling, Analysis and Conrol of Communicaion Neworks. PhD hesis. Graduae School of Ars and Sciences, Columbia Universiy. [11] Gihman, I. I. and Skorohod, A. V. 1974). The Theory of Sochasic Processes. I. Springer- Verlag, New York. [12] Glynn, P. W. and Whi, W. 1994). Logarihmic asympoics for seady-sae ail probabiliies in a single-server queue. Journal of Applied Probabiliy [13] Kesidis, G., Walrand, J., and Chang, C.-S. 1993). Effecive Bandwidh for Muliclass Markov Fluid and oher ATM Sources. IEEE/ACM Transacions on Neworking 1, [14] Loynes, R. M. 1962). The Sabiliy of a Queue wih Non-independen Iner-arrival and Service Times. Proc. Cambridge Philos. Soc. 58, [15] Maglaris, B., Anasassiou, D., Sen, P., Karlsson, G. and Robbins, J. D. 1988). Performance Models of Saisical Mulipleing in Packe Video Communicaion. IEEE Transacions on Communicaions 36, [16] Ross, S. M. 1983). Sochasic Processes. John Wiley & Son, New York. [17] Samorodnisky, G. 1991). Probabiliy ails of Gaussian Erema. Sochasic Processes and heir Applicaions 38, [18] Shroff, N. and Schwarz, M. 1996). Improved Loss Calculaions a an ATM Mulipleer. In Proceedings of IEEE INFOCOM. vol. 2. pp [19] Simonian, A. 1991). Saionary Analysis of a Fluid Queue wih Inpu Rae Varying as an Ornsein-Uhlenbeck Process. SIAM Journal on Applied Mahemaics 51, [2] Talagrand, M. 1988). Small ails for he supremum of a Gaussian process. Ann. Ins. Henri Poincare 24,