Perturbation analysis of an M/M/1 queue in a diffusion random environment

PER TURBA TION ANAL YSIS O F AN M / M / 1 QUEUE IN A DIFFUSION RAN DO M ENVIRONMENT CHRISTINE FRICKER, F ABRICE GUILLEMIN, AND PHILIPPE R OBER T Abstract. W e study in this pap er an M / M / 1 queue whose serve r rate de- pends up on the state of an independen t Ornstein-Uhl en beck diffusion p ro cess ( X ( t )) so that its v alue at time t is µφ ( X ( t )), where φ ( x ) is some bounded function and µ > 0. W e first establish the differential syste m for the condi- tional probability densit y functions of the couple ( L ( t ) , X ( t )) in the stationary regime, where L ( t ) is the num b er of customers in the system at time t . By assuming that φ ( x ) is defined by φ ( x ) = 1 − ε (( x ∧ a/ε ) ∨ ( − b/ε )) for some posi tive real num bers a , b and ε , we sho w that the abov e differential system has a unique solution under some condition on a and b . W e then show t hat this solution is close, in some appropriate sense, to the solution to the differential system obtained when φ is replaced w i th Φ( x ) = 1 − εx for sufficiently small ε . W e finally p erform a p erturbation analysis of this latter solution for s mall ε . This all o ws us to chec k at the first order the v alidity of the so-called reduced service r ate approximat ion, stating that eve rything happens as if the server rate were constan t and equal to µ (1 − ε E ( X ( t ))). 1. Introduction W e conside r in this pap er an M / M / 1 queue with a server ra te v a rying in time. W e sp ecifically as s ume that the se rver rate at time t is equal to µφ ( X ( t )) for some function φ and some auxiliary pro c ess ( X ( t ) ). Throughout this pap e r, w e shall assume that the mo dulating pro cess ( X ( t )) is an Ornstein-Uhlenbeck pro cess with mean m > 0, drift co efficient α > 0, and diffusion co efficient σ > 0. This pro cess satisfies Itˆ o’s sto chastic eq uation (1) dX ( t ) = − α ( X ( t ) − m ) dt + σ dB ( t ) , where ( B ( t )) is a standa r d B rownian motion. The stationar y distribution of the pro cess ( X ( t )) is a no rmal distribution with mean m and v a riance σ 2 / (2 α ); the asso ciated proba bilit y density function is defined on the whole o f R a nd is given b y (2) n ( x ) def. = 1 σ r α π exp  − α ( x − m ) 2 σ 2  . Throughout this pap e r, we shall assume that the Ornstein-Uhlenbeck pro cess is stationary . If L ( t ) = j deno tes the num b er of custo mer s in the M / M / 1 queue and X ( t ) = x at time t , then the transitions of the pro ces s ( L ( t )) are g iven by j →  j + 1 with r ate λ, j − 1 with r ate µφ ( X ( t )) . Key wor ds and phr ases. M / M / 1 queue, Self-Adjoint Op erators, Perturbation Analysis, Po wer Series Expansion, Reduced Service R ate. 1 2 CHRISTINE FRICKER, F ABRICE GUILLEMIN, AND PHILIPP E ROBER T In the fo llowing, we shall as sume that the condition ρ def. = λ/µ < E ( φ ( X (0))) ≤ 1 is satisfied so that it is stra ightf orward to show the exis tence o f a s ta tionary proba- bilit y distribution for the Ma rko v pro cess ( X ( t ) , L ( t )); see Meyn and Tweedie [12] for ex a mple. The study of the above system is motiv ated b y the pr oblem of bandwidth sha ring in telecommunication netw o rks and the co e x istence on the same transmiss io n links of ela s tic traffic, which a da pts to the level of cong estion o f the netw ork b y achieving a fair sharing of the av ailable bandwidth, and unresp onsive tr affic, which c o nsumes bandwidth without taking care of other traffic. See for instance [11 ] for a dis cussion ab out bandwidth sharing in pack et netw orks. The choice of an Or nstein–Uhlenbeck pro cess as mo dulating pro c e ss is natural for several reasons: ma thematically this is a standard “t ypical” diffusion pr o cess with a n e q uilibrium distribution and secondly it can b e seen as a ce ntered approximation of the n um ber of jobs of an M / M / ∞ queue (the unresp onsive tr affic), see for example Borovko v [4] o r Iglehar t [6] o r Chapter 6 of Rob ert [17]. One o f the ob jectives of this paper is to inv es tig ate the s o-called Reduced Service Rate (RSR) pr o pe rty for which the system would be hav e as if the server ra te were equal to the mean v alue µ E ( φ ( X 0 )). Even though some res ults can b e esta blished for arbitrary pe r turbation functions φ ( x ), we shall pay specia l a tten tion in the following to the case when the function φ ( x ) has the form (3) φ ( x ) = 1 − ε (( x ∧ ( a/ε )) ∨ ( − b/ε )) for so me small 0 < ε < 1 and real num ber s 0 < a < 1 and b > 0, where w e us e the notation a ∨ b = max( a, b ) and a ∧ b = min( a, b ). The choice o f the bo unded per turbation function is discussed at the end of the pap er. As it will b e seen, one of the imp ortant tec hnical problems encountered in the per turbation a nalysis is the ex istence of a reaso nably smo oth density pro ba bilit y function for the couple ( X ( t ) , L ( t )) in the stationa r y reg ime. Conditions on ε for ensuring the existence a nd the uniqueness of a density pr obability function will b e established in the following via Hilb ertia n analysis. More precisely , let p j ( x ) denote the stationary probability densit y function that the pr o cess ( L ( t )) is in state j a nd the pro cess ( X ( t )) is in state x a nd let P denote the vector whose j th co mpo nen t is p j ( x ) /ρ j . In a fist step, we show that P is s olution to an equation of the type (4) Ω f + V ( φ ) f = 0 , where Ω is a selfa djoin t second order differential op era to r on D ′ ( R ) N , D ′ ( R ) denoting the set of dis tr ibutions in R . Unfortunately , the o per ator V ( φ ) is not selfadjoint so that Ka to’s per turbation theory for selfadjoint o pe r ators cannot b e applied. Nevertheless, w e prov e that the a bove eq uation has a unique non n ull s mo o th solution P ∈ C 2 ( R ) N for sufficiently small ε . In addition, we prov e that when replacing φ ( x ) with Φ( x ) = 1 − εx , we obtain a n eq ua tion of the t yp e (5) Ω f + ε V (Φ) f = 0 , which ha s a unique no n null smo o th solution for sufficiently small ε , V (Φ) b e ing independent of ε . Denoting this solution by g , w e prov e that P a nd g a re close to each other for some adeq uate norm when ε is small. W e then per form a power series expa nsion in ε of g and we determine the radius of convergence of this ser ies. By ex plicitly computing the tw o firs t terms of the series, this even tua lly enables us PER TURBA TION ANAL YSIS OF AN M / M / 1 QUEUE 3 to prov e the v alidity of the reduced serv ice ra te approximation a t the first order for the s y stem. The pr oblem cons ider ed in this pap er falls into the framework of queueing sys- tems w ith time v ar ying ser ver ra te, which hav e b e en s tudied in the q ueueing liter- ature in many different situations. In N ´ u˜ nez-Que ija and Boxma [15], the authors consider a queueing system wher e prio rity is given to some flows driven by Marko v Mo dulated Poisson Pro c esses (MMPP) with finite state spaces and the low pr iority flows s hare the r emaining se r ver capa city acc ording to the pro ces s or shar ing disci- pline. By assuming that arriv a ls are Poisson and service times are ex po nent ially distributed, the authors so lve the system via a ma trix analysis. Similar mo dels hav e been in vestigated in N´ u ˜ nez-Queija [13, 14] by still using the qua si-birth and death pro cess ass o ciated with the system a nd a matrix analys is. The integration of elastic and streaming flows has b een studied by Delcoigne et a l. [5], where sto chastic bo unds for the mean num b er o f active flows have b een established. Mo re r ecently , priority queueing sy s tems with fast dynamics, which can b e describ ed by means of quasi birth and death pro cesses , have b een studied via a p erturbatio n a nalysis of a Markov chain by Altman et al [1]. A proba bilistic a nalysis of these queues with v arying service rate has been prese nted in An tunes e t al. [2, 3]. Our p oint of view in this pa per is co mpletely different since a functional analysis appro ach is used to tackle the p erturbation analysis . The key difficulty for the cas e consider ed in the present pa per is that the asso ciated Ma r ko v chain has an infinite s tate s pace. This pap er is org anized as follows: In Section 2, we establish the basic s ystem of partial differential equations for the joint pr obability density functions of the pro- cess ( X ( t ) , L ( t )). W e re call in Section 3 so me basic results on the genera tors of the Ornstein-Uhlenbeck pro cess a nd the o ccupation pro cess in an M / M / 1 q ue ue. In Section 4 it is prov ed that this sy s tem ha s a unique solution with c o nv enient regular - it y prop erties in an adeq ua te Hilb ert s pace when ε is sufficiently small. In Section 5, we car ry o ut a p er turbation a na lysis for the p erturbatio n function Φ( x ) = 1 − εx , we s how that when replacing φ ( x ) with Φ( x ) = 1 − εx , the cor resp onding differ- ent ial system has also a smo oth s olution in the underlying Hilber t space when ε is sufficiently small. W e then prove that the solutions to the differential sy s tems for φ defined b y Equatio n (3) a nd Φ are c lo se to each other in some appr opriate sense. By expanding the so lution of the second differen tial system in p ow er ser ies of ε , we show that at the first order the so-ca lled Reduced Ser vice Rate prop erty for the orig inal system holds; the subsequent terms of the as so ciated expansio n are also expr e ssed. Some concluding remar ks are presented in Section 6. 2. Fundament al differential pr oblem 2.1. Notation and di fferen tial system . The goal of this section is to establis h the fundamental differential system for the conditiona l pro bability densit y functions p j ( x ), j ≥ 0, in the sta tio nary regime, where p j ( x ) is the probability that the pro cess ( L ( t )) is in state j knowing that the O rnstein-Uhlenbeck pro c e ss ( X ( t )) is in state x . As long as we do not have proved r egularity results, these functions have to b e considered in the sense of distributions , i.e., for all j ≥ 0, p j ( x ) ∈ D ′ ( R ), wher e D ′ ( R ) is the set of distributions in R . The distributions p j ( x ), j ≥ 0, ar e forma lly defined as follows: for every infinitely differentiable function with compact supp o rt 4 CHRISTINE FRICKER, F ABRICE GUILLEMIN, AND PHILIPP E ROBER T ϕ ( x ) (deno ted, for shor t, ϕ ∈ C ∞ 0 ( R )) Z R p j ( x ) n ( x ) ϕ ( x ) dx = − Z R ϕ ′ ( x ) P ( X (0 ) ≤ x, L (0) = j ) dx, where n ( x ) is the normal distribution given by Eq uation (2). Throughout this paper , we shall use the following notation. The functional space L 2 ( R , n ) =  f : R → R : Z R f ( x ) 2 n ( x ) dx < ∞  is a Hilb ert space equipp e d with the sca lar pro duct defined fo r f , g ∈ L 2 ( R , n ) by ( f , g ) 2 = Z R f ( x ) g ( x ) n ( x ) dx and the no rm of an element f ∈ L 2 ( R , n ) is k f k 2 = p ( f , g ) 2 . If H is a sepa r able Hilber t space equipp ed with the s calar pro duct ( ., . ) H and ass o ciated norm k . k H , we define the Hilb ert space L 2 ( R , n ; H ) =  ( f j ( x ) , j ≥ 0) ∈ L 2 ( R ; n ) N : Z R k f ( x ) k 2 H n ( x ) dx < ∞  equipp e d with the scala r pro duct ( f , g ) = Z R ( f ( x ) , g ( x )) H n ( x ) dx. Finally , let ℓ 2 ( ρ ) be the Hilb ert space comp osed of those sequenc e s ( c j , j ≥ 0 ) ta king v alues in R and such that P ∞ j =0 c 2 j ρ j < ∞ , and equipped with the scala r pro duct defined by: if c = ( c j ) and d = ( d j ) in ℓ 2 ( ρ ), ( c, d ) ρ = P ∞ j =0 c j d j ρ j ; the as so ciated norm is defined by: for c ∈ ℓ 2 ( ρ ), k c k ρ = p ( c, c ) ρ . Let e j denote the sequence with all entries eq ual to 0 except the j th one equal to 1. The family ( e j , j ≥ 0) is a ba sis for ℓ 2 ( ρ ). The s pace ℓ 2 1 ( ρ ) deno tes the s ubspace of ℓ 2 ( ρ ) spa nned by the vectors e j for j ≥ 1. In a first step, we determine the infinitesimal g enerator of the Ma rko v pro ces s ( X ( t ) , L ( t )) taking v alues in R × N . W e sp ecifically hav e the following res ult. Lemma 1. The pr o c ess ( X ( t ) , L ( t )) is a Markov pr o c ess in R × N with infi nitesimal gener ator G define d by (6) G f ( x, j ) = σ 2 2 ∂ 2 f ∂ x 2 ( x, j ) − α ( x − m ) ∂ f ∂ x ( x, j ) + λ ( f ( x, j + 1) − f ( x, j )) + µφ ( x ) 1 { j > 0 } ( f ( x, j − 1) − f ( x, j )) , for every function f ( x, j ) fr om R × N in R , twic e differ en tiable with r esp e ct t o the first variable. Pr o of. According to Equation (1), the infinitesimal genera to r of an Ornstein-Uh- lenbe ck pro c e ss applied to some twice differentiable function g on R is g iven by (7) H g = σ 2 2 ∂ 2 g ∂ x 2 ( x ) − α ( x − m ) ∂ g ∂ x ( x ) . The s econd part o f E q uation (6) cor resp onds to the infinitesimal generato r of the nu m be r of cus tomers in a clas sical M / M / 1 queue with a rriv al ra te λ and serv ice rate µφ ( x ), when the Or ns tein-Uhlen be ck pro cess is in sta te x .  PER TURBA TION ANAL YSIS OF AN M / M / 1 QUEUE 5 F or finite sequences f = ( f j ( x )) of infinitely differentiable functions with compact suppo rt, the equation for the inv a riant measure for the Ma rko v pro c ess ( L ( t ) , X ( t )) is given by P j ≥ 0 R R G f ( x, j ) p j ( x ) n ( x ) dx = 0, that is, X j ≥ 0 Z ∞ −∞  σ 2 2 d 2 f j dx 2 − α ( x − m ) d f j dx + µ 1 { j > 0 } φ ( x ) f j − 1 ( x ) − ( λ + µφ ( x ) 1 { j > 0 } ) f j ( x ) + λf j +1 ( x )  p j ( x ) n ( x ) dx = 0 . Via integration by parts, we obtain for every finite sequence f = ( f j ( x )) of infinitely differentiable functions with compa ct supp ort X j ≥ 0  Z ∞ −∞  σ 2 2 d 2 P j dx 2 − α ( x − m ) dP j dx  f j ( x ) n ( x ) dx + Z ∞ −∞  µ 1 { j > 0 } φ ( x ) f j − 1 ( x ) − ( λ + µφ ( x ) 1 { j > 0 } ) f j ( x ) + λf j +1 ( x )  p j ( x ) n ( x ) dx  = 0 and then X j ≥ 0 Z ∞ −∞  σ 2 2 d 2 P j dx 2 − α ( x − m ) dP j dx + µ 1 { j > 0 } P j − 1 ( x ) − ( λ + µφ ( x ) 1 { j > 0 } ) P j ( x ) + λφ ( x ) P j +1 ( x )  f j ( x ) n ( x ) dx = 0 . This implies the following res ult. Prop ositio n 1 . The family ( P j ( x ) def = p j ( x ) /ρ j , j ≥ 0 ) ∈ D ′ ( R ) N is solution in the sense of distributions t o the fol lowing infinite differ ent ial syst em: for j ≥ 0 , (8) σ 2 2 d 2 P j dx 2 − α ( x − m ) dP j dx + µ 1 { j > 0 } P j − 1 ( x ) − ( λ + µφ ( x ) 1 { j > 0 } ) P j ( x ) + λφ ( x ) P j +1 ( x ) = 0 . 2.2. Additional prop erties. F or the system consider e d in this pap er, w e hav e µφ ( x ) > µ (1 − a ) fo r all x ∈ R and 0 < a < 1. Classical sto chastic order ing arguments imply that the pro c e s s ( L ( t )) is sto chastically dominated for the s trong ordering s ense by the queuing pr o cess o f the M / M / 1 queue with input ra te λ and service rate µ (1 − a ). Hence, if the solution ( P j ( x ) , j ≥ 0) of the infinite differ e ntial system (8) is r elated to the conditio nal probability density functions ( p j ( x ) , j ≥ 0) of the couple ( X (0) , L (0)) a s P j ( x ) = p j ( x ) /ρ j for a ll j ≥ 0, then (9) ∀ x ∈ R , ∀ j ≥ 0 , P j ( x ) ≤ 1 (1 − a ) j . If a < 1 − √ ρ , it is ea sily chec ked that for all x ∈ R , the sequence ( P j ( x ) , j ≥ 0) is in the Hilb ert space ℓ 2 ( ρ ). In a ddition, for a ll j ≥ 0 (10) Z R p j ( x ) 2 n ( x ) dx ≤ P ( L (0) = j ) < ∞ , since p j ( x ) = P ( L (0) = j | X (0) = x ) ≤ 1. It follows that for all j ≥ 0, the function p j ( x ) should b e in the space L 2 ( R , n ). 6 CHRISTINE FRICKER, F ABRICE GUILLEMIN, AND PHILIPP E ROBER T Hence, if the solution ( P j ( x ) , j ≥ 0) of the infinite differential sys tem (8) is related to the conditional probability density functions ( p j ( x ) , j ≥ 0) of the co u- ple ( X (0) , L (0)) as sp ecified ab ov e, then P j ( x ) ∈ L 2 ( R , n ) for all j ≥ 0. F rom inequality (10), we a ls o deduce that if a < 1 − √ ρ , ∞ X j =0 Z R  p j ( x ) ρ j  2 n ( x ) dxρ j < ∞ since P ( L (0) = j ) ≤ ρ j / (1 − a ) j . If follows from the ab ov e remarks that to show the regula rity of the conditional probability density functions p j ( x ) for j ≥ 0 under the assumption a < 1 − √ ρ , we are led to prov e that the differential systems admits a unique regula r solution in the s pa ce L 2 ( R , n ; l 2 ( ρ )). In the next section, we review some pr op erties of the op era tors asso ciated with the genera tors of the Ornstein-Uhlenbeck pr o cess a nd the Ma rko v pro cess describing the num ber of customers in an M / M / 1 queue. 3. Some resul ts on the opera tors associa ted with th e genera tors of the Ornstein-Uhlenbeck process and the M / M / 1 queue It is w ell kno wn in the liter ature (see for instance [19]) tha t the op erato r H defined by Eq uation (7) is s elfadjoint in the Hilb ert space L 2 ( R , n ). The eigenv alues of this o pe r ator are the n umber s − αj , j ≥ 0, and the normalized eigenvector asso ciated with the eigenv alue − αj is the function h j given by (11) h j ( x ) = 1 p 2 j j ! √ π H j ( √ 2 α ( x − m ) /σ ) , where H j ( x ) is the j th Hermite p olynomial. The sequenc e ( h j , j ≥ 0) is an or- thonormal bas is of L 2 ( R , n ). The domain of the o per ator H is the set D ( H ) =  f ∈ H 2 ( R , n ) : x 2 f ∈ L 2 ( R , n )  , where H 2 ( R , n ) is the Sob olev space defined as follows: H 2 ( R , n ) = { f ∈ C 1 ( R ) : f , f ′ ∈ L 2 ( R , n ) and the weak der iv ative f ′′ ∈ L 2 ( R , n ) } . While the oper ator H is well known in the litera tur e, less information is av ailable on t The op era tor A a s so ciated with the Mar ko v pro cess de s cribing the num b er of customers in an M / M / 1 queue and defined in ℓ 2 ( ρ ) by the infinite matrix (12) A =       − λ λ 0 . . . µ − ( λ + µ ) λ 0 . . 0 µ − ( λ + µ ) λ 0 . 0 0 µ − ( λ + µ ) λ . . . . . . .       . has alrea dy been studied in the technical literature, notably in [7] (see a lso [8]). F rom these r eferences, we know that the o per ator A is s elfadjoint. Asso ciated with PER TURBA TION ANAL YSIS OF AN M / M / 1 QUEUE 7 the o per ator A is the o p e rator A 1 defined in ℓ 2 1 ( ρ ) by the infinite matrix g iven by (13) A 1 =       − ( λ + µ ) λ 0 . . . µ − ( λ + µ ) λ 0 . . 0 µ − ( λ + µ ) λ 0 . 0 0 µ − ( λ + µ ) λ . . . . . . .       . Note that the ab ove matr ix is the generator of the Marko v pr o cess describing the nu m be r of customers in an M / M / 1 queue and absorb ed at state 0 (see [7]). Finally , let A [ N ] 1 denote the truncated op er ator ass o ciated with the finite matr ix (14) A [ N ] 1 =        − ( λ + µ ) λ 0 . . . µ − ( λ + µ ) λ 0 . . 0 µ − ( λ + µ ) λ 0 . 0 0 0 . . . . . . . . . . . µ − µ        . The ab ov e matrix is the g enerator of the Mar kov pro cess des cribing the num ber of customers in the finite ca pa city M / M / 1 / N queue and absor bed at state 0. ( See [9] for a related mo del.) In the following, we recall the spe ctral prop erties o f the op erators A , A 1 and A [ N ] 1 ; see [7, 8, 9] for details Lemma 2. The op er ator A in ℓ 2 ( ρ ) is b ounde d and symmetric, and then selfad joint. In p articular, for al l f ∈ ℓ 2 ( ρ ) , (15) 0 ≤ ( − Af , f ) ρ ≤ µ (1 + √ ρ ) 2 k f k 2 ρ , which implies that the op er ator − A is monotonic (i.e., ( − Af , f ) ≥ 0 for al l f ∈ ℓ 2 ( ρ ) ). Ther e exists a un ique normalize d m e asur e dψ ( z ) , r eferr e d to as sp e ctr al me asur e, whose supp ort is the sp e ctrum σ ( A ) of op er ator A , and a family of sp ac es {H z ( ρ ) } , z ∈ σ ( A ) , such that • the Hilb ert sp ac e ℓ 2 ( ρ ) is e qual to the dir e ct su m of the sp ac es H z ( ρ ) , i.e., every f ∈ ℓ 2 ( ρ ) c an b e de c omp ose d into a family ( f z , z ∈ σ ( A )) , wher e f z ∈ H z ( ρ ) and R k f z k 2 ρ dψ ( z ) < ∞ . Mor e over, ( f , g ) ρ = Z ( f z , g z ) ρ dψ ( z ) . • The op er ator A is su ch ( Af ) z = z f z for z ∈ σ ( A ) , wher e ( Af ) z is t he pr oje ction of ( Af ) on the sp ac e H z ( ρ ) . The sp e ctra l me asur e dψ ( x ) is sp e cific al ly given by (16) Z h ( x ) dψ ( x ) = (1 − ρ ) h (0) − √ ρ π Z − µ (1 − √ ρ ) 2 − µ (1 + √ ρ ) 2 h ( x ) x s 1 −  x + λ + µ 2 √ λµ  2 dx, for any smo oth function h . The sp e ctrum of the op er ator A is σ ( A ) = [ − µ (1 + √ ρ ) 2 , − µ (1 − √ ρ ) 2 ] ∪ { 0 } . The op er ator A has a unique eigenvalue e qual to 0 , and the eigensp ac e H 0 ( ρ ) is sp anne d by the ve ctor e with al l c omp onents e qual to 1. F or z ∈ ( − ( √ λ + √ µ ) 2 , − ( √ λ − 8 CHRISTINE FRICKER, F ABRICE GUILLEMIN, AND PHILIPP E ROBER T √ µ ) 2 ) , the s p ac e H z ( ρ ) is the ve ctor sp ac e sp anne d by the ve ctor Q ( z ) , whose c om- p onents Q j ( z ) , j ≥ 0 ar e define d by the fol lowing r e cursion: (17)    Q 0 ( z ) = 1 , Q 1 ( z ) = ( z + λ ) /λ µQ j +1 ( z ) − ( z + λ + µ ) Q j ( z ) + µQ j − 1 ( z ) = 0 , j ≥ 1 . The ve ctors ( Q ( z )) for z ∈ ( − ( √ λ + √ µ ) 2 , − ( √ µ − √ λ ) 2 ) form an ortho gonal family with weight function dψ ( z ) : for al l j, k , Z R Q j ( x ) Q k ( x ) dψ ( x ) = 1 ρ j δ j,k , wher e δ j,k is the Kr one cker symb ol, e qual to 1 if j = k and 0 if j 6 = k . Note the polyno mials Q j ( x ) appea ring in the ab ov e res ult a re k nown a s p er- turb ed Chebyshev po lynomials in the literature on or thogonal p olynomials [18]. F or the op erator A 1 , we hav e the following result, where we us e Chebyshev p oly- nomials of the second kind ( U n ( x )) defined by the recurs ion (18)    U 0 ( x ) = 1 , Q 1 ( x ) = 2 x U j +1 ( x ) = 2 xU j ( x ) − U j − 1 ( x ) , j ≥ 1 . Lemma 3. The op er ator A 1 in the su bsp ac e span( e j , j ≥ 1) is b ounde d and sym- metric, and then selfadjoint. In p articular, for al l f ∈ span( e j , j ≥ 1) , (19) µ (1 − √ ρ ) 2 k f k 2 ρ ≤ ( − A 1 f , f ) ρ ≤ µ (1 + √ ρ ) 2 k f k 2 ρ . The asso ciate d normalize d sp e ctr al me asu r e dψ 1 ( z ) is given by (20) dψ 1 ( x ) = 2 π s 1 −  x + λ + µ 2 √ λµ  2 1 { x ∈ ( − µ (1+ √ ρ ) 2 , − µ (1 − √ ρ ) 2 ) } dx 2 √ λµ . The sp e ctrum of the op er ator A 1 is diffuse (ther e ar e no eigenvalues) and e qual to the interval σ ( A 1 ) = [ − µ (1 + √ ρ ) 2 , − µ (1 − √ ρ ) 2 ] . The H ilb ert sp ac e span( e j , j ≥ 1) is e qual to t he dir e ct sum of the sp ac es H (1) z ( ρ ) , for z ∈ ( − µ (1 + √ ρ ) 2 , − µ (1 − √ ρ ) 2 ) , wher e t he sp ac e H (1) z ( ρ ) is the ve ctor s p ac e sp anne d by t he ve ctor Q (1) ( z ) , whose c omp onents Q (1) j ( z ) , j ≥ 0 ar e define d by the fol lowing r e cursion: (21)      Q (1) 0 ( z ) = 1 , Q (1) 1 ( z ) = ( z + λ + µ ) /λ µQ (1) j +1 ( z ) − ( z + λ + µ ) Q (1) j ( z ) + µQ (1) j − 1 ( z ) = 0 , j ≥ 1 . The ve ctors ( Q (1) ( z )) for z ∈ ( − ( √ λ + √ µ ) 2 , − ( √ µ − √ λ ) 2 ) form an ortho go- nal family with weight function dψ 1 ( z ) . The p olynomials ( Q (1) j ( z )) ar e r elate d to Chebyshev p olynomials as fol lows: for j ≥ 0 , Q (1) j ( z ) = 1 ρ j / 2 U j  z + λ + µ 2 √ λµ  . Finally , for the op erator A [ N ] 1 , we hav e the following result. PER TURBA TION ANAL YSIS OF AN M / M / 1 QUEUE 9 Lemma 4. The op er ator A [ N ] 1 is symmetric (and then selfadjo int) in the ve ctor sp ac e span( e 1 , . . . , e N ) e quipp e d with the sc alar pr o duct induc e d by ( ., . ) ρ . The eigen- values of the op er ator A [ N ] 1 ar e t he solutions to the p olynomial e quation Q (1) N +1 ( − x ; 1) = Q (1) N ( − x ; 1) , wher e the p olynomials Q (1) j ( x ) ar e define d by t he r e cursion (21) . The eigenvalues ar e denote d by − x [ N ] j , j = 1 , . . . , N with x [ N ] 1 < x [ N ] 2 < . . . < x [ N ] N . The ve c- tors Q (1 ,N ) ( x j ) , j = 1 , . . . , N , form an ortho gonal b asis of span( e 1 , . . . , e N ) , wher e Q (1 ,N ) ( x j ) is the ve ctor with the k th c omp onent e qual to Q (1) k ( x j ) , k , j = 1 , . . . , N , wher e t he p olynomials ( Q (1) k ( z )) ar e define d by Equation (21) . The op era tor A na turally induce s in L 2 ( R , n ; l 2 ( ρ )) an opera tor that w e still denote b y A . The same prope r ty is v alid for the op erator s A 1 and A [ N ] 1 in the spaces L 2 ( R , n ; span( e j , j ≥ 1)) and L 2 ( R , n ; span( e j , j = 1 , . . . , N )), resp ectively . Similarly , the op erato r H induces in L 2 ( R , n ; l 2 ( ρ )) an op era tor that we still de- note b y H and which is de fined a s follows: for f ∈ L 2 ( R , n ; l 2 ( ρ )), H f is the element with the j th comp onent equal to H f j . This op era tor also induces in L 2 ( R , n ; span( e j , j ≥ 1 )) and L 2 ( R , n ; span( e j , j = 1 , . . . , N )) op era tors denoted by H 1 and H [ N ] 1 , res pectively . The op er a tors A , A 1 , A [ N ] 1 , H , H 1 and H [ N ] 1 are clearly selfadjoint in the spaces where they a re defined. With the ab ov e definitions, the fundamental differential system (8) reads (22) ( H + A ) f + V f = 0 , where the op erator V is defined b y: for f ∈ L 2 ( R , n ; l 2 ( ρ )), V f = ( φ ( x ) − 1) B f , where B is the op er ator ass o ciated with the infinite matrix B =       0 λ 0 . . . . 0 − µ λ 0 . . . 0 0 − µ λ 0 . . 0 0 0 − µ λ 0 . . . . . . . .       . In the nota tion of Equa tion (4), we hav e Ω = H + A a nd V ( ε ) = − ε (( x ∧ ( a/ε )) ∨ ( − b/ε )) B . The matrices B 1 =       − µ λ 0 . . . . 0 − µ λ 0 . . . 0 0 − µ λ 0 . . 0 0 0 − µ λ 0 . . . . . . . .       . and B [ N ] 1 =      − µ λ 0 . . . 0 − µ λ 0 . . 0 0 . . . . . . . . . . . . . . 0 − µ      define the op e r ators B 1 and B [ N ] 1 in the spaces L 2 ( R , n ; span( e j , j ≥ 1)) and L 2 ( R , n ; span( e j , j = 1 , . . . , N )), resp ectively . It is straig ht forwardly check ed that the o per ator B , B 1 and B [ N ] 1 are b ounded with a norm less than or equa l to 10 CHRISTINE FRICKER, F ABRICE GUILLEMIN, AND PHILIPP E ROBER T µ (1 + √ ρ ). With the op er ators B 1 and B [ N ] 1 are asso ciated the op er a tors V 1 and V [ N ] 1 induced by V in the spa ces L 2 ( R , n ; span( e j , j ≥ 1)) and L 2 ( R , n ; span( e j , j = 1 , . . . , N )), resp ectively . In the next section, we prov e that the differen tia l system (8) (or equiv alently Equation (22 )) has a unique solution in L 2 ( R , n ; ℓ 2 ( ρ )). 4. Existence and uniqueness of a solution When we re fer to the existence of a density probability density function satisfying the differe ntial system (8), we think of a vector ( P j ( x )) s uch tha t every function P j ( x ) is twice contin uo usly differentiable over R (i.e., P j ( x ) ∈ C 2 ( R ) for all j ≥ 0). But, this differen tial s ystem may hav e a solution, which is in L 2 ( R , n ; ℓ 2 ( ρ )) but with comp onents no t in C 2 ( R ). In the following, we prove that the differential system (8) ha s a unique s o lution in L 2 ( R , n ; ℓ 2 ( ρ )) and then we show a t the end of the s e ction that the comp onents of the solutio n are C 2 ( R ) functions. If f ∈ L 2 ( R , n ; ℓ 2 ( ρ )) is so lution to Equation (22), then (23) ( H 1 + A 1 ) f 1 + V 1 f 1 = − µe 1 ( f 0 ) , where f 1 is the pro jection of f on the spa ce L 2 ( R , n, span( e j , j ≥ 1)) a nd e 1 ( f 0 ) is the element of L 2 ( R , n, span( e j , j ≥ 1)) with the first c o mpo nent equal to f 0 ( x ) and all other comp onents equal to 0 . The op erator ( H 1 + A 1 ) is self-adjoint and invertible. It is straightforwardly chec ked that the norm k ( H 1 + A 1 ) − 1 k def = sup {   ( H 1 + A 1 ) − 1 f   : f ∈ L 2 ( R , n, span( e j , j ≥ 1)) , k f k = 1 } is such that k ( H 1 + A 1 ) − 1 k ≤ 1 µ (1 − √ ρ ) 2 , since by Lemma 3 and the monotonicity of the op era tor − H 1 , we hav e for all f in L 2 ( R , n, span( e j , j ≥ 1 )) ( − ( H 1 + A 1 ) f , f ) ≥ ( − H 1 f , f ) + ( − A 1 f , f ) ≥ − µ (1 − √ ρ ) 2 k f k 2 . In a ddition, if f ∈ L 2 ( R , n ; ℓ 2 ( ρ )) is a so lution to Equation (22), then the function P ∞ j =0 ρ j f j ( x ) ∈ L 2 ( n ) sinc e by Sch warz inequality Z ∞ −∞       ∞ X j =0 ρ j f j ( x )       2 n ( x ) dx ≤ 1 1 − ρ Z ∞ −∞ ∞ X j =0 | f j ( x ) | 2 ρ j n ( x ) dx = 1 1 − ρ k f k 2 < ∞ . By summing all the lines o f Equation (22), w e s ee that the function P ∞ j =0 ρ j f j ( x ) has to b e solution to the e quation H g = 0 in L 2 ( R , n ). By using the ab ove observ atio ns, w e prov e the existence and uniqueness of a non trivial solution in L 2 ( R , n ; ℓ 2 ( ρ )) to Equa tion (22) by showing the following r esults: (1) If ε satisfies some condition, Equation (23) has a solution for a ll f 0 ∈ L 2 ( n ); if ( f j ) is the so lution, we set (24) K f 0 = ∞ X j =1 ρ j f j ∈ L 2 ( R , n ) . PER TURBA TION ANAL YSIS OF AN M / M / 1 QUEUE 11 (2) If f is a non triv ial solution to Eq uation (22) , then f 0 is a no n tr ivial solution to e q uation (25) f 0 + K f 0 = 1 . (3) Equa tion (25) has a unique non triv ia l solutio n. T o pr ov e the last p oint, we intend to use the F redholm alter native, which r equires that the op erato r K is compact. Lemma 5. U nder the c ondition (26) a ∨ b < (1 − √ ρ ) 2 1 + √ ρ , wher e a ∨ b = max( a, b ) , Equation (23) has a unique solution in L 2 ( R , n ; ℓ 2 1 ( ρ )) . Pr o of. Equation (23) can b e rewritten as ( I + ( H 1 + A 1 ) − 1 V 1 ) f = − µ ( H 1 + A 1 ) − 1 e 1 ( f 0 ) . Since V 1 is bounded s o that for all f ∈ L 2 ( R , n ; ℓ 2 1 ( ρ )), | ( V 1 f , f ) | ≤ µ ( a ∨ b )(1 + √ ρ ) k f k 2 , we deduce tha t the o per ator ( H 1 + A 1 ) − 1 V 1 is b ounded with a no r m less than o r equal to ( a ∨ b )(1 + √ ρ ) / (1 − √ ρ ) 2 . Under Condition (26), the norm of op erator ( H 1 + A 1 ) − 1 V 1 is less than 1 and we then deduce that ( I + ( H 1 + A 1 ) − 1 V 1 ) is inv ertible [16] and Equation (23) ha s a unique solution in L 2 ( R , n ; ℓ 2 1 ( ρ )).  It is worth noting that under Condition (26), w e hav e a < (1 − √ ρ ). Mo reov er, the ab ove result ensures that the op erator K is well defined b y Equation (24). No w, we prove that under the s ame conditio n, the op er ator K is monotonic. Lemma 6. Under Condition (26) , the op er ator K is monotonic, which implies that ther e exists at most one non trivial solution to Equation (25) . Pr o of. W e fir st note that from Equation (2 4 ), we hav e ( K f 0 , f 0 ) 2 = ( − µ ( H 1 + A 1 + V 1 ) − 1 e 1 ( f 0 ) , e ( f 0 )) , where e ( f 0 ) is the vector with all entries equal to f 0 . The ab ov e equation can be rewritten a s ( K f 0 , f 0 ) 2 = ( − µ ( I + A − 1 1 ( H 1 + V 1 )) − 1 A − 1 1 e 1 ( f 0 ) , e ( f 0 )) . Since − µA − 1 1 e 1 ( f 0 ) = e ( f 0 ), we have (27) ( K f 0 , f 0 ) 2 = (( I + A − 1 1 ( H 1 + V 1 )) − 1 e ( f 0 ) , e ( f 0 )) and hence ( K f 0 , f 0 ) 2 ≥ 0 . Indeed, for a ll f ∈ L 2 ( R , n ; ℓ 2 1 ( ρ )) (( I + A − 1 1 H 1 + A − 1 1 V 1 ) f , f ) = (( I + A − 1 1 H 1 ) f , f ) + ( A − 1 1 V 1 f , f ) ≥  1 − ( a ∨ b )(1 + √ ρ ) (1 − √ ρ ) 2  k f k 2 , where we hav e used the fact that the op erator A − 1 1 H 1 is monotonic, k V 1 k ≤ µ ( a ∨ b )(1 + √ ρ ), a nd k A − 1 1 k ≤ 1 / ( µ (1 − √ ρ ) 2 ). The a b ove inequality implies that (( I + A − 1 1 H 1 + A − 1 1 V 1 ) − 1 f , f ) ≥  1 − ( a ∨ b )(1 + √ ρ ) (1 − √ ρ ) 2  k ( I + A − 1 1 H 1 + A − 1 1 V 1 ) − 1 f k 2 ≥ 0 , and Inequa lity (2 7) follows.  12 CHRISTINE FRICKER, F ABRICE GUILLEMIN, AND PHILIPP E ROBER T W e now turn to the co mpa ctness of the o per ator K . The ma jor difficulty co mes from the fact that the op era tor ( H 1 + A 1 ) − 1 is not compact, since we know that the spe c trum of this self-adjoint op er a tor is no t discrete. How ever, b y truncating the infinite matrix defined b y Equa tion (12), w e can int ro duce co mpact o p e r ators and subsequently pr ov e that the op era tor K is compact. Let us fix s ome N > 0. W e first pr ov e the following technical lemma. Lemma 7. F or N > 0 , the op er ator ( H [ N ] 1 + A [ N ] 1 ) − 1 in L 2 ( R , n ; span( e 1 , . . . , e N )) is c omp act. Pr o of. By using Lemma 4 and the orthono rmal basis ( h n ) is L 2 ( R , n ), we know that the family e j,k ( x ) = h k ( x ) Q (1 ,N ) ( x j ) fo r k ≥ 0 a nd j = 1 , . . . , N for ms an orthogo nal bas is of L 2 ( R , n ; span( e 1 , . . . , e N )). In pa r ticular, we have ( H [ N ] 1 + A [ N ] 1 ) − 1 e j,k ( x ) = − 1 k α + x [ N ] j e j,k ( x ) . The op erator ( H [ N ] 1 + A [ N ] 1 ) then app ear s as the norm limit as M → ∞ of the finite rank op erator s ( H [ N ,M ] 1 + A [ N ,M ] 1 ) − 1 defined in the vector spa ce span( e j,k ( x ) , j = 1 , . . . , N , k = 0 , . . . , M ) by ( H [ N ,M ] 1 + A [ N ,M ] 1 ) − 1 e j,k ( x ) = − 1 αk + x [ N ] j e j,k ( x ) and the result follows.  Lemma 8. U nder Condition (26) , the op er ator K is c omp act. Pr o of. Let us co nsider a b ounded sequence ( f i 0 ) in L 2 ( R , n ). (W ithout lo ss of generality , we assume that k f i 0 k 2 = 1.) Since the ope rator ( H [ N ] 1 + A [ N ] 1 ) − 1 V [ N ] 1 is bo unded with a nor m less than or equal to ( a ∨ b ) µ (1 + √ ρ ) /x [ N ] 1 , and the op erator ( H [ N ] 1 + A [ N ] 1 ) − 1 is compact, we deduce tha t the oper ator ( H [ N ] 1 + A [ N ] 1 + V [ N ] 1 ) − 1 is compa ct. Let f i = ( f i j ( x )) denote the vector − µ ( H 1 + A 1 + V 1 ) − 1 e 1 ( f i 0 ). Since the ope r ator ( H 1 + A 1 + V 1 ) − 1 is bo unded with a norm less than o r equa l to 1 / ( µ ((1 − √ ρ ) 2 − a ∨ bε (1 + √ ρ ))), the vector ( f i ) is such that (28) v u u t N X j =1 k f i j k 2 2 ρ j ≤ k f i k ≤ 1 ((1 − √ ρ ) 2 − a ∨ b (1 + √ ρ )) . W e hav e    f i 1 . . . f i N    = ( H [ N ] 1 + A [ N ] 1 + V [ N ] 1 ) − 1        − µf i 0 0 . . . 0 − λφf i N +1 + λf i N        In view of inequality (2 8), the sequence appea ring in the right hand side of the ab ov e equation is b ounded. Hence, since the op e r ator ( H [ N ] 1 + A [ N ] 1 + V [ N ] 1 ) − 1 is PER TURBA TION ANAL YSIS OF AN M / M / 1 QUEUE 13 compact, it is p ossible to extract a sub- s equence ( f i k ) s uch that ( H [ N ] 1 + A [ N ] 1 + V [ N ] 1 ) − 1        − µf i k 0 0 . . . 0 − λφf i k N +1 + λf i k N        →        f ∞ 1 f ∞ 2 . . . f ∞ N − 1 f ∞ N        as k → ∞ in L 2 ( R , n ; span( e 1 , . . . , e N )), where the vector a ppe a ring in the right hand side of the ab ov e eq ua tion is in L 2 ( R , n ; span( e 1 , . . . , e N )). In pa rticular, we have f i k N → f ∞ N in L 2 ( R , n ) as k → ∞ . This implies that    f i k N +1 f i k N +2 . . .    = ( H [ N ] 1 + A [ N ] 1 + V [ N ] 1 ) − 1    − µf i k N 0 . . .    → ( H [ N ] 1 + A [ N ] 1 + V [ N ] 1 ) − 1    − µf ∞ N 0 . . .    as k → ∞ in L 2 ( R , n, ℓ 2 1 ( ρ )), since the op era tor ( H [ N ] 1 + A [ N ] 1 + V [ N ] 1 ) − 1 is b ounded. W e set    f ∞ N +1 f ∞ N +2 . . .    = ( H [ N ] 1 + A [ N ] 1 + V [ N ] 1 ) − 1    − µf ∞ N 0 . . .    The vector with the j th comp onent eq ual to f ∞ j is in L 2 ( R , n, ℓ 2 ( ρ )) and w e hav e K f i k 0 → P ∞ j =1 ρ j f ∞ j in L 2 ( R , n ) as k → ∞ . W e then deduce that from every bo unded sequence ( f i 0 ) in L 2 ( R , n ), we can extract a s ub- sequence ( f i k 0 ) such that K f i k 0 is conv er ging in L 2 ( R , n ). The op erator K is hence compa ct.  By using the a b ove lemmas and the F redholm alternative, we can sta te the following res ult. Prop ositio n 2. U n der Condition (26) , Equation (25) has a un ique solut ion. This establishes that Equation (22) has a un ique non trivial solution in L 2 ( R , n, ℓ 2 ( ρ )) if Condition (26) is satisfie d. The ab ov e result has b een established for the p erturbatio n function φ ( x ) = 1 − ε (( x ∧ ( a/ε )) ∨ ( − ( b/ε ))) and we hav e exploited the fact that the function | 1 − φ ( x ) | is bo unded by a ∨ b . In fact, it is p ossible to prov e a similar r e sult w he n φ is replaced with Φ( x ) = 1 − εx . Let us define the op erator W in L 2 ( R , n ; ℓ 2 ( ρ )) by: if f = ( f j ( x ) , j ≥ 0) W f = − εxB f . Note that the domain of W is g iven by D ( W ) =    ( f j ( x ) , j ≥ 0) ∈ L 2 ( R , n ) N : Z R ∞ X j =0 x 2 f j ( x ) 2 n ( x ) dx < ∞    . In the notation of Equation (5), we have V (Φ) = − xB . W e can then state the following r esult, whose pr o of is given in Appendix A . 14 CHRISTINE FRICKER, F ABRICE GUILLEMIN, AND PHILIPP E ROBER T Prop ositio n 3. Under the c ondition (29) 2 ε (1 + √ ρ ) (1 − √ ρ ) 2  m + σ √ α  < 1 , the e quation (30) ( H + A + W ) f = 0 has a unique s olut ion in L 2 ( R , n, ℓ 2 ( ρ )) . T o prov e the e x istence and the uniqueness of the solutio ns to Equatio ns (22) and (30), we hav e only supp osed that the comp onents of the solutions are in D ( H ), in particula r the compo nent s ar e in H 2 ( R , n ). B ut, by ex amining the differen tial systems satisfied by the different compone nts, notably by taking into account the contin uity of the functions φ ( x ) and Φ( x ), these co mpo nent s ar e clear ly in C 2 ( R ). T o conclude this section, let us ment ion that the solution to the differen tial system (8) have the following probabilistic interpretation: for all j ≥ 0, in the stationary reg ime P j ( x ) = P ( L ( t ) = j | X ( t ) = x ) /ρ j . In particular , we hav e for all j ≥ 0 (31) 0 ≤ P j ( x ) ≤ 1 and for all x ∈ R (32) ∞ X j =0 P j ( x ) ρ j = 1 . 5. Per turba tion anal ysis The go a l of this section is to prov e the following Reduced Ser v ice Rate approxi- mation. Theorem 1. F or su fficiently smal l ε , t he first or der exp ansion of the gener ating function of the st ationary distribution of ( L ( t )) is given by E  u L ( t )  = 1 − ρ 1 − ρu − ρ (1 − u ) (1 − ρu ) 2 mε + o ( ε ) . Therefore, E ( u L ( t ) ) ∼ E ( u L ε ), where L ε has the stationary distribution of the nu m be r o f customers in an M / M / 1 queue when the server rate is 1 − εm . This shows a princ iple of reduced service rate approximation, i.e., everything happens as if the server rate were fixed eq ual to 1 − mε . T o s how the ab ove result, we pro ceed as follows: (1) W e compare the solutions to Equations (22) and (30) when Conditions (26) and (29) are s a tisfied. In pa rticular, we compute an upper b ound for the norm o f their difference. (2) W e develop the solution g to Equation (30) in p ow er series ex pansion of ε . In pa rticular, we explic itly compute the tw o first ter ms. (3) W e finally prove Theor em 1. Throughout this section, we denote by P and g the solutio ns to Equa tions (22) and (30) in L 2 ( R , n ; ℓ 2 ( ρ )), re s pec tiv ely . PER TURBA TION ANAL YSIS OF AN M / M / 1 QUEUE 15 5.1. Compariso n of the sol utions P and g . The so lutions to E quations (22) and (30) when they exist are close to each o ther when ε is small. W e sp ecifically hav e the following res ult. Prop ositio n 4. Assume that Conditions (26 ) and (29) ar e satisfie d. These solu- tions P and g ar e such that (33) k P − g k ≤ D ( ε ) , wher e t he function D ( ε ) is given by (34) D ( ε ) = M  1 + r ρ 1 − ρ + λ r ρ 1 − ρ M  ∆( ε ) with M = 1 µ (1 − √ ρ ) 2  1 + (1 + √ ρ ) (1 − √ ρ ) 2  m + σ √ α  , ∆( ε ) 2 = ( µ 2 + 3 λµ ) Z R  ( εx − a ) 2 1 { x ≥ a/ε } + ( εx + b ) 2 1 { x ≤− b/ε }  n ( x ) dx. The function D ( ε ) is O  ε 5 / 2 exp( − α ( a ∧ b ) 2 / (2 σ 2 ε 2 ))  when ε → 0 . Pr o of. The vector P is s uc h that ( H + W + A ) P + ( V − W ) P = 0 and hence, if P 1 denotes the pro jection of P on the s pace spa n( e j , j ≥ 1), ( H 1 + A 1 + W 1 ) P 1 = ( W 1 − V 1 ) P 1 − µe 1 ( P 0 ) and then (35) P 1 = ( H 1 + A 1 + W 1 ) − 1 ( W 1 − V 1 ) P 1 − µ ( H 1 + A 1 + W 1 ) − 1 e 1 ( P 0 ) . It follows that ∞ X j =1 P j ( x ) ρ j ≡ ( P 1 , e 1 ) ρ = (( H 1 + A 1 + W 1 ) − 1 ( W 1 − V 1 ) P 1 , e 1 ) ρ − µ (( H 1 + A 1 + W 1 ) − 1 e 1 ( P 0 ) , e 1 ) ρ . W e hav e − µ (( H 1 + A 1 + W 1 ) − 1 e 1 ( P 0 ) , e 1 ) ρ = K ′ g 0 , where the op era to r K ′ is defined as the op erator K (defined by Equa tion (24)) but by replacing V with W . In addition, s ince P satisfies ( P , e ) ρ = 1, we co me up with the co nclusion that P 0 verifies 1 − P 0 = (( H 1 + A 1 + V 1 ) − 1 ( W 1 − V 1 ) P 1 , e 1 ) ρ + K ′ P 0 . Since g 0 + K ′ g 0 = 1, we obtain ( P 0 − g 0 ) + K ′ ( P 0 − g 0 ) = (( H 1 + A 1 + W 1 ) − 1 ( W 1 − V 1 ) P 1 , e 1 ) ρ . Since the op erato r K ′ is monoto nic, the ab ov e e quation implies that k P 0 − g 0 k 2 ≤ r ρ 1 − ρ k ( H 1 + A 1 + W 1 ) − 1 kk ( W 1 − V 1 ) P 1 k . 16 CHRISTINE FRICKER, F ABRICE GUILLEMIN, AND PHILIPP E ROBER T The norm k ( W 1 − V 1 ) P 1 k is g iven by k ( W 1 − V 1 ) P 1 k 2 = ∞ X j =1 Z R  ( εx − a ) 2 1 { x ≥ a/ε } + ( εx + b ) 2 1 { x ≤− b/ε }  ( µP j ( x ) − λP j +1 ( x )) 2 ρ j n ( x ) dx and by using Equa tion (31), we o btain k ( W 1 − V 1 ) P 1 k 2 ≤ ( µ 2 + 3 λµ ) Z R  ( εx − a ) 2 1 { x ≥ a/ε } + ( εx + b ) 2 1 { x ≤− b/ε }  n ( x ) dx Simple co mputations show that Z R ( εx − a ) 2 1 { x ≥ a/ε } n ( x ) dx ∼ 5 ε 5 σ 5 8 α 5 / 2 a 3 √ π e − αa 2 ε 2 σ 2 , Z R ( εx + b ) 2 1 { x ≤− b/ε } n ( x ) dx ∼ 5 ε 5 σ 5 8 α 5 / 2 b 3 √ π e − αb 2 ε 2 σ 2 , when ε → 0. The ter m k ( W 1 − V 1 ) P 1 k is hence O (exp ( ε 5 / 2 exp( − α ( a ∧ b ) 2 / (2 σ 2 ε 2 ))) when ε → 0. In addition, sinc e by using Equatio n (61) in Appendix A k ( H 1 + A 1 + W 1 ) − 1 k ≤ 1 µ (1 − √ ρ ) 2  1 + (1 + √ ρ ) (1 − √ ρ ) 2  m + σ √ α  , we deduce tha t k f 0 − g 0 k 2 is dominated by a term, which is O (exp( ε 5 / 2 exp( − α ( a ∧ b ) 2 / (2 σ 2 ε 2 ))) when ε → 0. Finally , by using the fact that ( H 1 + A 1 + W 1 ) g = − µe 1 ( g 0 ), we deduce from Equation (35) that P 1 − g 1 = ( H 1 + A 1 + W 1 ) − 1 ( W 1 − V 1 ) P 1 − µ ( H 1 + A 1 + W 1 ) − 1 e 1 ( P 0 − g 0 ) , which implies that k P 1 − g 1 k ≤ k ( H 1 + A 1 + W 1 ) − 1 kk ( W 1 − V 1 ) P 1 k + λ k ( H 1 + A 1 + W 1 ) − 1 kk P 0 − g 0 k 2 and hence k P 1 − g 1 k is dominated by a term which is O (exp( ε 5 / 2 exp( − α ( a ∧ b ) 2 / (2 σ 2 ε 2 ))) when ε → 0.  5.2. P o wer series expansion of the solution g . 5.2.1. Notation. W e a ssume that the s olution g to Equation (30) can be uniquely decomp osed a s a p ow er series expansion o f the form (36) g = g (0) + εg (1) + ε 2 g (2) + ...., where g ( i ) ∈ L 2 ( R , n ; ℓ 2 ( ρ )) for i ≥ 0. In addition, to facilitate the computations, we shall consider the generating function g u ( x ) = P ∞ j =0 g j ( x ) u j ρ j , which is an element of L 2 ( R , n ; ℓ 2 (1 /ρ )). Indeed, if g is wr itten in the form g = ∞ X j =0 c j ( u ) h j ( x ) , where c j ( u ) = P ∞ m =0 c j,m u m with ( c j,m , m ≥ 0) ∈ ℓ 2 ( ρ ), then g u ( x ) can b e written as g u ( x ) = ∞ X j =0 C j ( u ) h j ( x ) , PER TURBA TION ANAL YSIS OF AN M / M / 1 QUEUE 17 with C j ( u ) = P ∞ m =0 C j,m u m = c j ( ρu ). Since ( c j,m , m ≥ 0) ∈ ℓ 2 ( ρ ), ( C j,m , m ≥ 0) ∈ ℓ 2 (1 /ρ ). Finally , we have k g k 2 = P ∞ j =0 k c j k 2 ρ = P ∞ j =0 k C j k 2 1 /ρ . The generating function g u ( x ) will b e expa nded as (37) g u ( x ) = g (0) u ( x ) + εg (1) u ( x ) + ε 2 g (2) u ( x ) + · · · , where g ( i ) u ∈ L 2 ( R , n ) for all i ≥ 0. The function g (0) u ( x ) c orresp onds to the cas e ε = 0 and is given by (38) g (0) u ( x ) = c ( g ) 1 − ρu , where c ( g ) is the nor malizing constant. In the follo wing, we pr ov e that the ele men ts g ( i ) hav e to satisfy a r ecurrence relation of the fo r m g ( i ) = Θ ( xg ( i − 1) ) for i ≥ 1 and for some line a r op era to r Θ whose no rm is finite. In the following, we assume that the expansion (37) is v alid and we inv estigate the conditions which hav e to b e s a tisfied by the ele ments g ( i ) . In a fir st step, we prov e the following prop erty satisfied by the functions ( g ( i ) u ( x )). Lemma 9. F or i ≥ 0 , the ve ctor g ( i ) is in L 2 i ( R , n ; ℓ 2 ( ρ )) , wher e L 2 i ( R , n ; ℓ 2 ( ρ )) is the sub-sp ac e of L 2 ( R , n ; ℓ 2 ( ρ ) c omp ose d of those elements ( f j ( x )) such that f j ( x ) ∈ span( h 0 , . . . , h i ) for al l j ≥ 0 , the functions h j b eing define d by Equation (11) ; the function g ( i ) u ( x ) in Exp ansion (37) henc e satisfies for N > i (39) lim x →±∞ 1 x N g ( i ) u ( x ) = 0 . Pr o of. The pro o f is by mathematical induction. The result is true for i = 0 since g (0) = c ( g ) e ( e b eing the vector with all c ompo nents equal to 1). If the res ult is true for i . F rom Equation (30), we hav e ( H + A ) g ( i +1) = − W g ( i ) . By using the recurr ence relation satisfied by Her mite p olyno mials [10] (40) H j +1 ( x ) − 2 xH j ( x ) + 2 i H j − 1 ( x ) = 0 , it is easily chec k ed that the ima ge by the multiplication by x of span( h 0 , . . . , h i ) is span( h 0 , . . . , h i +1 ). Therefore, since by a ssumption g ( i ) belo ngs to L 2 i ( R , n ; ℓ 2 ( ρ )), we immediately deduce from the uniqueness o f the decomp osition on the basis ( h i e j , i ≥ 0 , j ≥ 0) of the Hilb ert space L 2 ( R , n ; ℓ 2 ( ρ )) and the s elfadjointn ess of the o per ator H + A , that g ( i +1) is in L 2 i +1 ( R , n ; ℓ 2 ( ρ )) and the res ult follows.  5.2.2. First or der term. In a first step, we pay s pecia l a tten tion to the der iv ation of the first or de r term b ecause it gives the bas ic arg umen ts to derive higher o rder terms. Moreover, the explicit for m of the first order term will b e us ed to ex amine the v a lidit y of the reduced service r ate appr oximation (see Theorem 1). On the basis o f the domination prop erty given by Lemma 9, we explicitly com- pute the function g (1) u ( x ). F r om E q uation (30), it is easily check ed that the function 18 CHRISTINE FRICKER, F ABRICE GUILLEMIN, AND PHILIPP E ROBER T g (1) u ( x ) satisfies the eq uation σ 2 2 ∂ 2 g (1) u ∂ x 2 − α ( x − m ) ∂ g (1) u ∂ x + αν ( u ) g (1) u ( x ) (41) = µ  1 u − 1   g (1) 0 ( x ) − x ( g (0) 0 ( x ) − g (0) u ( x ))  = µ  1 u − 1   g (1) 0 ( x ) + x ρuc ( g ) (1 − ρu )  where the co ns tant ν ( u ) is given by ν ( u ) = µ (1 − u )(1 − ρu ) αu . In a fir st step, we sear ch for a particular solution to the ordinary differential equation σ 2 2 ∂ 2 ξ u ∂ x 2 − α ( x − m ) ∂ ξ u ∂ x + αν ( u ) ξ u ( x ) = x ρµ (1 − u ) c ( g ) (1 − ρu ) of the form ξ u ( x ) = a ( u ) + b ( u ) x. Straightforw ard manipulations s how that b ( u ) = ρµ (1 − u )(1 − ρ ) α ( ν ( u ) − 1 ) c ( g ) and a ( u ) = − m ν ( u ) b ( u ) . Noting that ξ 0 ( x ) ≡ 0, it follows that if we write g (1) u ( x ) = ξ u ( x ) + ψ u ( x ), then the function ψ u ( x ) is solutio n to the equation (42) σ 2 2 ∂ 2 ψ u ∂ x 2 − α ( x − m ) ∂ ψ u ∂ x + αν ( u ) ψ u ( x ) = µ  1 u − 1  ψ 0 ( x ) . By using the domina tion prop erty o f Lemma 9, we can deter mine the form o f the function ψ 0 ( x ). Lemma 10. The function ψ 0 ( x ) is given by ψ 0 ( x ) = c 0 + c 1 √ α ( x − m ) σ for some c onstants c 0 and c 1 . Pr o of. By introducing the function k u ( x ) defined by (43) k u ( x ) = exp  − α ( x − m ) 2 2 σ 2  ψ u ( x ) and then the change of v a riable (44) z = √ α ( x − m ) σ , Equation (42 ) b ecomes (45) ∂ 2 k u ∂ z 2 + (2 ν ( u ) + 1 − z 2 ) k u ( z ) = 2 µ α  1 u − 1  k 0 ( z ) . The homog eneous equation reads ∂ 2 k u ∂ z 2 + (2 ν ( u ) + 1 − z 2 ) k u = 0 , PER TURBA TION ANAL YSIS OF AN M / M / 1 QUEUE 19 which so lutio ns are parab olic cylinder functions (see Leb edev [10] for details). Two independent solutions v 1 ( u ; z ) a nd v 2 ( u ; z ) o f this homogeneo us equation ar e g iven in terms of Hermite functions as (46) v 1 ( u ; z ) = e − z 2 / 2 H ν ( u ) ( z ) and v 2 ( u ; z ) = e z 2 / 2 H − ν ( u ) − 1 ( iz ) . The W ro nskian W of these tw o functions is given by W ( z ) = e − ( ν +1) π i/ 2 . By using the metho d of v ariatio n of para meters, the solution to Equation (45) is given by k u ( z ) = γ 1 ( u ) v 1 ( u ; z ) + γ 2 ( u ) v 2 ( u ; z ) − 2 µ α  1 u − 1  e ( ν +1) π i/ 2 Z z 0 [ v 1 ( u ; y ) v 2 ( u ; z ) − v 1 ( u ; z ) v 2 ( u ; y )] k 0 ( y ) dy , where γ 1 ( u ) and γ 2 ( u ) are constants, which dep end up on u . The function ψ u ( x ) enjo ys the same domination pro per ty as function g (1) u ( x ), given by Lemma 9. Hence, for N > 1 (47) lim z →±∞ 1 z N e z 2 / 2 k u ( z ) = 0 . F rom Leb edev [10], we have the following asymptotic estimates (48) H ν ( z ) ∼ (2 z ) ν " n X k =0 ( − 1) k k ! ( − ν ) 2 k (2 z ) − 2 k + O ( | z | − 2 n − 2 ) # when | z | → ∞ and | arg z | ≤ 3 π / 4 − δ fo r so me δ > 0. Moreover, when z → −∞ H ν ( z ) ∼          √ π Γ( − ν ) | z | − ν − 1 e z 2 " n X k =0 1 k ! ( ν + 1) 2 k (2 z ) − 2 k + O ( | z | − 2 n − 2 ) # , ν / ∈ N (2 z ) ν , ν ∈ N . The ab ov e asy mptotic estimates and Lemma (9) imply that for u ∈ (0 , 1) such that ν ( u ) ∈ N with ν > 1, we hav e γ 1 ( u ) = − 2 µ α  1 u − 1  e ( ν +1) π i/ 2 Z ∞ 0 v 2 ( u ; y ) k 0 ( y ) dy = − 2 µ α  1 u − 1  e ( ν +1) π i/ 2 Z −∞ 0 v 2 ( u ; y ) k 0 ( y ) dy and γ 2 ( u ) = 2 µ α  1 u − 1  e ( ν +1) π i/ 2 Z ∞ 0 v 1 ( u ; y ) k 0 ( y ) dy = 2 µ α  1 u − 1  e ( ν +1) π i/ 2 Z −∞ 0 v 1 ( u ; y ) k 0 ( y ) dy . The latter eq ua tion implies that for all n > 1 (49) Z ∞ −∞ e − y 2 / 2 k 0 ( y ) H n ( y ) dy = 0 , where H n ( x ) is the n th Her mite p olynomial. By Prop erty (47), the function y → exp( y 2 / 2) k 0 ( y ) is in L 2 ( R , exp( − y 2 ) dy ). Since Hermite polyno mials for m 20 CHRISTINE FRICKER, F ABRICE GUILLEMIN, AND PHILIPP E ROBER T an orthogo nal bas is in this Hilb ert s pace, Eq uation (49) entails that the function y → exp( y 2 / 2) k 0 ( y ) is orthogona l to all Hermite p olynomials H n with n > 1 and then tha t this function b elongs to the vector space spanned by H 0 and H 1 . Hence, function k 0 ( z ) sho uld b e of the form k 0 ( z ) = ( c 0 + c 1 z ) e − z 2 / 2 for so me consta n ts c 0 and c 1 and the r esult follows.  By using the ab ov e lemma , we ar e now able to establish the expressio n o f g (1) u ( x ). Prop ositio n 5. The function g (1) u ( x ) is given by (50) g (1) u ( x ) = c ( g )( u 1 − 1)( ˜ u 1 − ρu )( u − 1 ) u 1 ˜ u 1 (1 − ρ )( u − ˜ u 1 )(1 − ρu 1 )(1 − ρu ) 2 m + c ( g )(1 − u ) ( u − ˜ u 1 )(1 − ρu 1 )(1 − ρu ) x, wher e u 1 and ˜ u 1 ar e t he two r e al solutions to the quadr atic e quation ρu 2 −  1 + ρ + α µ  u + 1 = 0 with 0 < u 1 < 1 < ˜ u 1 Pr o of. By taking into account Lemma 10, the function K u ( z ) defined by K u ( x ) = g (1) u ( x ) exp  − α ( x − m ) 2 2 σ 2  and the change of v ariable (44), s atisfies the equa tion (51) ∂ 2 K u ∂ z 2 + (2 ν ( u ) + 1 − z 2 ) K u ( z ) = 2 µ α  1 u − 1   c 0 + c 1 z + ρuc ( g ) 1 − ρu  σ z √ α + m  e − z 2 / 2 . W e search fo r a particular solution of the fo r m K u ( z ) = ( a ( u ) + b ( u ) z ) e − z 2 / 2 . Straightforw ard computations y ield a ( u ) = 1 (1 − ρu )  c 0 + ρuc ( g ) 1 − ρu m  , b ( u ) = (1 − u ) ρ ( u − u 1 )( u − ˜ u 1 )  c 1 + ρσ uc ( g ) √ α (1 − ρu )  . It follows that the general solution to the ab ove equation can be wr itten a s (52) K u ( z ) = ( a ( u ) + b ( u ) z ) e − z 2 / 2 + γ 1 ( u ) v 1 ( u ; z ) + γ 2 ( u ) v 2 ( u ; z ) , where the functions v 1 and v 2 are defined by Equation (46) a nd the cons ta n ts γ 1 ( u ) and γ 2 ( u ) depend up on u . By differentiating once Eq ua tion (52) with resp e c t to z and using the fact that the W ronskian of the functions v 1 ( u ; z ) and v 2 ( u ; z ) is exp[( ν ( u ) + 1 ) π i/ 2], we ca n easily express γ 1 ( u ) and γ 2 ( u ) by means of K u ( z ), a ( u ), and b ( u ). This shows that γ 1 ( u ) and γ 2 ( u ) ar e ana ly tic in the op en unit disk deprived o f the p oints 0 and u 1 . F rom the asymptotic prop er ties satisfied by the functions v 1 and v 2 , we know that PER TURBA TION ANAL YSIS OF AN M / M / 1 QUEUE 21 γ 1 ( u ) = 0 and γ 2 ( u ) = 0 for u such that ν ( u ) > 1. It follows that γ 1 ( u ) ≡ γ 2 ( u ) ≡ 0 for | u | < 1. By using the fact that g (1) u ( x ) ha s to b e a nalytic in v ar iable u in the unit disk, we necess arily have c 1 = − ρσ u 1 c ( g ) √ α (1 − ρu 1 ) and then, b ( u ) = σ c ( g )(1 − u ) √ α ( u − ˜ u 1 )(1 − ρu 1 )(1 − ρu ) . Moreov er, s ince g (1) 1 ( x ) ≡ 0, we hav e c 0 = − ρc ( g ) m 1 − ρ and then, (53) a ( u ) = ρ ( u − 1) c ( g ) (1 − ρ )(1 − ρu ) 2 m. By using the express ions of a ( u ) a nd b ( u ), the result follows.  5.2.3. Higher or der t erms. W e a ssume that g ( i ) u ( x ) can b e expressed as (54) g ( i ) u ( x ) = i X j =0 c i,j ( u ) h j ( x ) , where the function h j is defined by Equa tion (11) and the co efficients c i,j are analytic functions in v ariable u . This ass umption will be justified a po steriori. F rom previous sections , this representation is v a lid for i = 0 , 1. If it is v a lid for i − 1, then the function g ( i ) u ( x ), i ≥ 1, s a tisfies the equation (55) σ 2 2 ∂ 2 g ( i ) u ∂ x 2 − α ( x − m ) ∂ g ( i ) u ∂ x + αν ( u ) g ( i ) u ( x ) = µ  1 u − 1   g ( i ) 0 ( x ) − x ( g ( i − 1) 0 ( x ) − g ( i − 1) u ( x ))  . First note that b y using the r ecurrence re la tion (40) satisfied b y Hermite poly- nomials, it is easily chec ked that x ( g ( i − 1) u ( x ) − g ( i − 1) 0 ( x )) = i X j =0 d i,j ( u ) h j ( x ) , where d i,i ( u ) = σ √ i 2 √ α ( c i − 1 ,i − 1 ( u ) − c i − 1 ,i − 1 (0)) , and for 0 ≤ j ≤ i − 1 , d j,i ( u ) = σ √ j 2 √ α ( c i − 1 ,j − 1 ( u ) − c i − 1 ,j − 1 (0)) + m ( c i − 1 ,j ( u ) − c i − 1 ,j (0)) + √ j + 1 σ √ α ( c i − 1 ,j +1 ( u ) − c i − 1 ,j +1 (0)) . By using the ab ov e no tation, we have the following result. 22 CHRISTINE FRICKER, F ABRICE GUILLEMIN, AND PHILIPP E ROBER T Prop ositio n 6. The c o efficients c i,j app e aring in the r epr esentation (54) of g ( i ) u ( x ) ar e re cursively define d as fol lows: we have c 0 , 0 ( u ) = 1 − ρ 1 − ρu , and for i ≥ 1 , c i, 0 ( u ) = d i, 0 ( u ) − d i, 0 (1) 1 − ρu , c i,j ( u ) = µ α  1 u − 1  d i,j ( u ) − d i,j ( u j ) ν ( u ) − j 1 ≤ j ≤ i, wher e for j ≥ 1 , u j and ˜ u j ar e the two r e al solutions to the quadr atic e quation ν ( u ) = j , i.e. ρu 2 −  1 + ρ + j α µ  u + 1 = 0 with 0 < u j < 1 < ˜ u j . Pr o of. As in the previous section, we firs t se a rch for a s olution to the equation σ 2 2 ∂ 2 ξ ( i ) u ∂ x 2 − α ( x − m ) ∂ ξ ( i ) u ∂ x + αν ( u ) ξ ( i ) u ( x ) = µ  1 u − 1  x ( g ( i − 1) u ( x ) − g ( i − 1) 0 ( x )) . Assuming that the function ξ ( i ) u ( x ) is of the form ξ ( i ) u ( x ) = i X k =0 δ i,j ( u ) h j ( x ) , we have, by using the fact that the functions h j ( x ) are eigenfunctions o f the op er- ator H asso ciated with the eigenv alues − αj and that these functions are linearly independent, for j = 0 , . . . , i , δ i,j = µ α  1 u − 1  d j,k ( u ) ν ( u ) − j . It is easily chec ked that ξ ( i ) 0 ( x ) ≡ 0. W e ca n then decomp ose g ( i ) u ( x ) as a s g ( i ) u ( x ) = ψ ( i ) u ( x ) + ξ ( i ) u ( x ) , where the function ψ ( i ) u ( x ) is solution to the equation σ 2 2 ∂ 2 ψ ( i ) u ∂ x 2 − α ( x − m ) ∂ ψ ( i ) u ∂ x + ν ( u ) ψ ( i ) u ( x ) = µ  1 u − 1  ψ ( i ) 0 ( x ) . By using the same arg umen ts as in the pro of of Lemma 10, we can easily show that ψ ( i ) 0 ( x ) has the for m ψ ( i ) 0 ( x ) = i X j =0 c j h j ( x ) , PER TURBA TION ANAL YSIS OF AN M / M / 1 QUEUE 23 where the co efficients c j ∈ C for j = 0 , ..., i . It follows that the function g ( i ) u ( x ) is solution to the ordinar y differential equation σ 2 2 ∂ 2 g ( i ) u ∂ x 2 − α ( x − m ) ∂ g ( i ) u ∂ x + αν ( u ) g ( i ) u ( x ) = µ  1 u − 1  i X j =0 ( c j + d i,j ( u )) h j ( x ) . By using the same a rguments as in the pro of of P rop osition 5 , we come up with the co nclusion that g ( i ) u ( x ) is of the form (54) with the co efficients c i,j ( u ) given by c i,j ( u ) = µ α  1 u − 1  c j + d i,j ( u ) ν ( u ) − j . Since the function g ( i ) u ( x ) has to be ana lytic in the ope n unit disk, w e have for j ≥ 1 c j = − d i,j ( u j ) In addition, since g ( i ) 1 ( x ) ≡ 0, we hav e c 0 = − d i,j (1).  The normalizing co nstant c ( g ) is chosen such that Z ∞ −∞ g 1 ( x ) n ( x ) dx = 1 . F rom the a bove analy sis, we see that g ( i ) 1 ( x ) ≡ 0 for i ≥ 1 so that c ( g ) = 1 − ρ . 5.2.4. R adius of c onver genc e. In this section, we examine under whic h conditions the expa nsion (36) defines an element of L 2 ( R , n, ℓ 2 ( ρ )). In a firs t step, note that as a conse q uence of Prop ositio n 6, the function g ( i ) u ( x ) can b e written as g ( i ) u ( x ) = x Θ  g ( i − 1) u ( x )  = Θ  xg ( i − 1) u ( x )  where the o per ator Θ is defined in L 2 ( R , n ; ℓ 2 (1 /ρ )) as follo ws: for a n ele men t f ∈ L 2 ( R , n ; ℓ 2 (1 /ρ )) represented a s f u ( x ) def = ∞ X j =0 c j ( u ) h j ( x ) , the ele ment F = Θ f is defined by F u ( x ) = ∞ X j =0 µ  1 u − 1  c j ( u ) − c j ( u j ) ν ( u ) − j h j ( x ) , where we set u 0 = 1 and ˜ u 0 = 1 /ρ . It is ea s ily c heck ed that for j ≥ 1, 0 < u j < 1 < 1 / √ ρ < ˜ u j . Mor eov er, the function c j ( u ) app earing in the expres sion of f u is analytic in the disk D ρ = { z : | z | < 1 / √ ρ } and contin uous in the closed disk D ρ = { z : | z | ≤ 1 / √ ρ } for j ≥ 0. Similarly , for all j ≥ 0, the function u → µ α  1 u − 1  c j ( u ) − c j ( u j ) ν ( u ) − j is ana lytic in D ρ and contin uous in D ρ . With the a bove no tation, we can state the main res ult of this section. Prop ositio n 7. The op er ator Θ is b ounde d and if ε < 1 / ( m k Θ k ) , wher e k Θ k denotes the norm of Θ , then the se quenc e define d by Equat ion (36) (or e quivalently by Equation (37) ) is in L 2 ( R , n ; ℓ 2 ( ρ )) . 24 CHRISTINE FRICKER, F ABRICE GUILLEMIN, AND PHILIPP E ROBER T Pr o of. Let f ∈ L 2 ( R , n ; ℓ 2 (1 /ρ )) b e defined by the function f u ( x ) = ∞ X j =0 c j ( u ) h j ( x ) . F or ( c j ) ∈ ℓ 2 (1 /ρ ) asso cia ted with the gener ating function c ( u ) = ∞ X j =0 c j u j , we have k c k 2 1 /ρ = 1 2 π Z 2 π 0     c  1 √ ρ e iθ      2 dθ. Let us more over define the sequence (˜ c j ) a sso ciated with the genera ting function ˜ c ( u ) = µ α  1 u − 1  c ( u ) − c ( u j ) ν ( u ) − j . Assume first that j ≥ 1, then ˜ c ( u ) = 1 ρ (1 − u ) 1 u − ˜ u j c ( u ) − c ( u j ) u − u j . and then k ˜ c k 2 1 /ρ ≤ 1 ρ 2  1 + 1 √ ρ  2 1 ( ˜ u j − 1 / √ ρ ) 2 1 2 π Z 2 π 0     c ( e iθ / √ ρ ) − c ( u j ) e iθ / √ ρ − u j     2 dθ. Simple ma nipulations show that 1 2 π Z 2 π 0     c ( e iθ / √ ρ ) − c ( u j ) e iθ / √ ρ − u j     2 dθ ≤ k c k 2 ρ 1 (1 / √ ρ − u j ) 2 1 + s 1 1 − ρu 2 j ! 2 . It follows that k ˜ c k 1 /ρ ≤ κ j k c k 1 /ρ , whe r e κ j = 1 ρ  1 + 1 √ ρ  1 ( ˜ u j − 1 / √ ρ )(1 / √ ρ − u j ) 1 + s 1 1 − ρu 2 j ! = 1 + √ ρ (1 − √ ρ ) 2 + nα µ 1 + s 1 1 − ρu 2 j ! . It is easily chec ked that the s equence ( κ j ) for n ≥ 1 is decrea sing. When j = 0, we define ˜ c ( u ) = µ α  1 u − 1  c ( u ) − c (1) ν ( u ) = c ( u ) − c (1) 1 − ρu . It is then easily chec ked that k ˜ c k 1 /ρ ≤ κ 0 k c k 1 /ρ , whe r e κ 0 = 1 1 − √ ρ  1 + r 1 1 − ρ  . Define κ = max { κ 0 , κ 1 } . The ab ov e computations show that for all f ∈ ℓ 2 (1 /ρ ), k Θ f k ≤ κ k f k . It follows that the ope r ator Θ is bo unded; its no rm is denoted by k Θ k def . = inf { c > 0 : ∀ f ∈ H , k Θ f k ≤ c k f k} . T he a bove computations show that (56) k Θ k ≤ 1 + √ ρ (1 − √ ρ ) 2  1 + r 1 1 − ρ  . PER TURBA TION ANAL YSIS OF AN M / M / 1 QUEUE 25 F rom the a bove computations, we deduce that k g ( i ) k ≤ k Θ k i k c (0) ∗ i k where the seq uence c (0) ∗ i is as s o ciated with the function 1 − ρ 1 − ρu x i . Straightforw ard computations s how that (57) k c (0) ∗ i k 2 =  σ 2 √ α  2 i H 2 i  √ αm σ  , where H i ( x ) is the i th Hermite po lynomial. Using the asymptotic estimate (48), we have k c (0) ∗ i k ∼ m i when i → ∞ . It follows that k c ( i ) k ≤ a i with a i ∼ ( k Θ k m ) i as i tends to infin- it y . It follows that the sequence defined b y the expans io n (37) is con vergen t in L 2 ( R , n ; ℓ 2 ( ρ )) if ε k Θ k m < 1.  5.3. Pro of of Theorem 1. W e assume that Conditions (26) a nd (29) are sa tisfied. By obs erving that E  u L ( t )  = ( P, U ), we hav e for u ≤ 1    E  u L ( t )  − ( g , U )    ≤ k P − g kk U k = 1 1 − ρu k P − g k ≤ 1 1 − ρu D ( ε ) where U is the vector of L 2 ( R , n, ℓ 2 ( ρ )) with the j th co mpo nen t equal to u j and D ( ε ) is defined by Equation (34). F rom the p ower s eries expansio n of g , we hav e    ( g , U ) − ( g (0) , U ) − ε ( g (1) , U )    ≤ ∞ X j =2 ε j k g ( i ) kk U k ≤ 1 1 − ρu ∞ X j =2 κ ( j ) ε j , with κ ( j ) =  1 + √ ρ (1 − √ ρ ) 2  1 + r 1 1 − ρ  j  σ 2 √ α  2 i H 2 i  √ αm σ  , where we hav e used Equations (5 6 ) a nd (57). W e clear ly hav e ( g (0) , U ) = Z ∞ −∞ g (0) u ( x ) n ( x ) dx = 1 − ρ 1 − ρu ( g (1) , U ) = Z ∞ −∞ g (1) u ( x ) n ( x ) dx = a ( u ) = − ρ (1 − u ) (1 − ρu ) 2 m, where a ( u ) is defined by E quation (53). Theorem 1 then follows. T o complete the analy sis, note that the g enerating function of L ε , the stationar y nu m be r of customers in an M / M / 1 with input rate λ and service ra te µ (1 − mε ), is for | u | < 1 a nd when ε < (1 − ρ ) /m E  u L ε  − ( g (0) , U ) − ( g (1) , U ) ε = ρ (1 − u ) m 2 (1 − ρ − mε )(1 − ρ ) 2 ε 2 26 CHRISTINE FRICKER, F ABRICE GUILLEMIN, AND PHILIPP E ROBER T Hence, by gathering these relations a nd by taking u = 1, we obtain an uniform bo und for the difference b et ween E  u L ( t )  and E  u L ε  for u ∈ [0 , 1]. (58) sup 0 ≤ u ≤ 1    E  u L ( t )  − E  u L ε     ≤ E B def. = 1 1 − ρ D ( ε ) + 1 1 − ρ ∞ X j =2 κ ( j ) ε j + 2 ρm 2 (1 − ρ − mε )(1 − ρ ) 2 ε 2 Below ar e some numerical ex p er iences on the r ole of ε a nd σ on the b ound E B for a = 1 / 2, b = 1 m = 1, λ = 7, α = 1 and µ = 10. It is reaso nably lo w for small v alues o f ε , it seems to be quite sensitive on the v a lues of the para meter σ as the figures b elow show. 0 0.01 0.02 0.03 0.04 0.05 0.06 0.0002 0.0004 0.0006 0.0008 0.001 σ = 2 σ = 3 σ = 4 ε E B Figure 1. The b ound ε → E B of Relation (58) for σ = 2, 3, 4 . 0 0.5 1 1.5 2 2.5 10 20 30 40 50 x = 1 x = 2 x = 3 x = 5 σ E B Figure 2. The b ound σ → E B of Rela tion (58) for ε = x · 10 − 4 with x = 1 , 2 , 3 , 5. PER TURBA TION ANAL YSIS OF AN M / M / 1 QUEUE 27 6. Conclud ing remarks The pe r turbation analy s is p erformed in this pap er has a llow ed us to prov e the v alidity of the so-called r educed ser v ice ra te approximation for the system consid- ered under so me sp ecific conditions. Such a n approximation is v ery imp ortant fro m a practica l po int of view bec ause ea ch type of tr affic can b e conside r ed in iso lation, the impa c t of unresp onsive traffic o n ela stic tra ffic is only via the mea n v alue. The results presented in this pap er have b een obtained for a par ticular form of the per turbation function φ ( x ). Of co urse, the same appr oach c o uld b e extended to more complica ted p er turbation functions o f the for m Φ ( x ) = 1 − εp ( x ) for s o me function p ( x ). The key p oint consists of determining how the op erator co rresp ond- ing to the multiplication by p ( x ) acts on the basic functions h j ( x ) for j ≥ 0 (defined by Equation (11)). F or computing explicit expressions, how ever, the main difficult y is in solving the differential equations satisfied by the co efficie nts of the expansion. When p ( x ) is a p olynomial, a par ticula r so lution to the equations similar to Equa- tions (41 ) a nd (55) is obtained in the form o f a po lynomial times the function exp( − α ( x − m ) 2 /σ 2 ) and in that case, explicit co mputations can b e car ried out. The pe rturbation function φ ( x ) defined by Equation (3) corres po nds to the case when unresp onsive flows ha ve a pe ak bit rate ε muc h smaller than the transmission capacity of the link. The results of this pap er show tha t the re duce d ser vice rate approximation yields in some conditions accur ate results for the perfo rmance of elastic flows. Appendix A. P r oof of Proposition 3 T o prove Pr op osition 3, we pro c e e d as for the pro of of Prop osition 2. W e first show that the o per ator ( H 1 + A 1 ) − 1 W 1 is b ounded, where W 1 is the restriction to ℓ 2 1 ( ρ ) o f the oper ator W . (Note that the ope r ator asso c ia ted with the m ultiplicatio n by Φ( x ) − 1 is not b ounded in L 2 ( R , n ).) F o r this purp ose we use the fact that an element of f = ( f j ( x )) ∈ L 2 ( R , n ; ℓ 2 1 ( ρ )) can b e decomp os ed as f = ∞ X j =1 ∞ X k =0 c j,k h k ( x ) e j . and the squar ed norm is k f k 2 = ∞ X j =1 ∞ X k =0 c 2 j,k ρ j . By using the a b ove decomp osition, we have B f = ∞ X j =1 ∞ X k =0 ( − µc j,k + λc j +1 ,k ) h k ( x ) e j By using the recurr ence relation satisfied by Her mite p olyno mials (59) xH k ( x ) = 1 2 H k +1 ( x ) + k H k − 1 ( x ) , we deduce that xh k ( x ) = σ √ 2 α r k + 1 2 h k +1 ( x ) + r k 2 h k − 1 ( x ) ! + mh k ( x ) . 28 CHRISTINE FRICKER, F ABRICE GUILLEMIN, AND PHILIPP E ROBER T and then W 1 f = ε ∞ X j =1 ∞ X k =0 C j,k h k ( x ) e j , where C j,k = m ( µc j,k − λc j +1 ,k ) + σ √ 2 α 1 { k> 0 } r k 2 ( µc j,k − 1 − λc j +1 ,k − 1 ) + r k + 1 2 ( µc j,k +1 − λc j +1 ,k +1 ) ! . F rom the a bove rela tions, we hav e (60) ( H 1 + A 1 ) − 1 W 1 f = ε   ∞ X j =1 C j, 0 ( H 1 + A 1 ) − 1 h 0 ( x ) e j + ∞ X j =1 ∞ X k =1 C j,k ( H 1 + A 1 ) − 1 h k ( x ) e j   F or the first term in the right hand s ide o f Equatio n (60), we hav e       ∞ X j =1 C j, 0 ( H 1 + A 1 ) − 1 h 0 ( x ) e j       ≤   ( H 1 + A 1 ) − 1         ∞ X j =1 C j, 0 h 0 ( x ) e j       =   ( H 1 + A 1 ) − 1   v u u t ∞ X j =1 C 2 j, 0 ρ j ≤ µ (1 + √ ρ )  m + σ 2 √ α    ( H 1 + A 1 ) − 1   k f k , since ∞ X j =1 C 2 j, 0 ρ j ≤   m v u u t ∞ X j =1 ( µc j, 0 − λc j +1 , 0 ) 2 ρ j + σ 2 √ α v u u t ∞ X j =1 ( µc j, 1 − λc j +1 , 1 ) 2 ρ j   2 together with the inequalities v u u t ∞ X j =1 ( µc j, 0 − λc j +1 , 0 ) 2 ρ j ≤   µ v u u t ∞ X j =1 c 2 j, 0 ρ j + p λµ v u u t ∞ X j =1 c 2 j +1 , 0 ρ j +1   ≤ µ (1 + √ ρ ) k f k , and v u u t ∞ X j =1 ( µc j, 1 − λc j +1 , 1 ) 2 ρ j ≤ µ (1 + √ ρ ) k f k . F or the s econd term in the right hand side of Eq uation (60), s ince the op erator H is inv ertible o n the spa ce span( h k , k ≥ 1), we hav e ∞ X j =1 ∞ X k =1 C j,k ( H 1 + A 1 ) − 1 h k ( x ) e j = ∞ X j =1 ∞ X k =1 C j,k ( I + H − 1 1 A 1 ) − 1 H − 1 1 h k ( x ) e j and then, by using the fact that k ( I + H − 1 1 A 1 ) − 1 k ≤ 1, we obtain       ∞ X j =1 ∞ X k =1 C j,k ( H 1 + A 1 ) − 1 h k ( x ) e j       ≤ v u u t ∞ X j =1 ∞ X k =1  C j,k k  2 ρ j . PER TURBA TION ANAL YSIS OF AN M / M / 1 QUEUE 29 F rom the ineq uality v u u t ∞ X j =1 ∞ X k =1  C j,k k  2 ρ j ≤ m v u u t ∞ X j =1 ∞ X k =1 ( µc j,k − λc j +1 ,k ) 2 k 2 ρ j + σ 2 √ α v u u t ∞ X j =1 ∞ X k =1 ( µc j,k − 1 − λc j +1 ,k − 1 ) 2 k ρ j + σ 2 √ α v u u t ∞ X j =1 ∞ X k =1 ( k + 1) ( µc j,k +1 − λc j +1 ,k +1 ) 2 k 2 ρ j we deduce that v u u t ∞ X j =1 ∞ X k =1  C j,k k  2 ρ j ≤ µ (1 + √ ρ )  m + σ √ α  k f k Finally , b y using the fact that k ( H 1 + A 1 ) − 1 k ≤ 1 / ( µ (1 − √ ρ ) 2 ), we come up with the co nclus ion that (61) k ( H 1 + A 1 ) − 1 W 1 f k ≤ 2 ε (1 + √ ρ ) (1 − √ ρ ) 2  m + σ √ α  k f k . The op era to r ( H 1 + A 1 ) − 1 W 1 is hence b ounded. Under Condition (29), the norm of this op era tor is le s s than 1 and we can adapt word b y word the pro of o f Pro po - sition 2. References [1] E. Altman, K. Avrachenk ov, and R. R. N ´ u˜ nez-Queija. Perturbation analysis for denumer- able marko v chains wi th application to queueing models. Adv ances in Appli ed Probability , 36(3):839– 853, 2004. [2] N. Ant unes, C. F rick er, F. Guillemin, and P . Robert. Int egration of streaming services and TCP data transmi ssion in the In ternet. Pe rformance Ev aluation , 62(1-4):263–277, October 2005. [3] N. Antun es, C. F ri c k er, F. Guil lemin, and P . Robert. P erturbation analysi s of a v ariable M / M / 1 queue: A pr obabili stic approac h. Adv ances in Applied Probability , 38(1):263–283, 2006. [4] A. A. Boro vk ov. Limit laws for queueing processes in multic hannel systems. Sibirsk. M at. ˇ Z., 8:983–1004, 1967. [5] F. Delcoigne, A . Prouti ` ere, and G. R´ egni ´ e. Mo delling integ ration of streaming and data traffic. In ITC speciali st seminar on IP traffic , W ¨ urzburg, Germany , July 2002. [6] D. Iglehart. W eak conv ergence of compound sto c hastic pr ocess. I. Stochastic Processes Appl. , 1:11–31; corrigendum, ibid. 1 (197 3), 185–186, 1973. [7] S. Karli n and J.L. McGregor. Many server queueing pr ocesses with Poisson i nput and exp o- nen tial service times. Pa cific J. Math. , 8:87–118, 1958. [8] S. Karli n and J.L. M cGregor. Random walks. Illinois J. Math , 3:66–81, 1959. [9] S. Karli n and J.L. M cGregor. Ehrenfest urn mo del. J. Appli. Probab. , 2:352–376, 1965. [10] N. N . Leb edev. Special f unctions and their appli cations . Dov er Publications, 1972. [11] L. Massouli ´ e and J. Rob erts. Bandwidth sharing: Ob jectives and algorithms. In INF OCOM’99. Eighteen th Annual Joint Conference of the IEEE Computer and Commun ications Societies , pages 1395–1 403, 1999. [12] S. M eyn and R . Twee die. Marko v chains and stochastic stability . Communica tions and con trol engineering series. Springer, 1993. [13] R. N´ u ˜ nez-Queija. So journ times in a processor sharing queue with service interruptions. Queueing Systems, 34:351–3 86, 2000. 30 CHRISTINE FRICKER, F ABRICE GUILLEMIN, AND PHILIPP E ROBER T [14] R. N´ u ˜ nez-Queija. So journ times in non-homogeneous QB D processes with pro cessor sharing. Stoch. Mod. , pages 61–92, 2001. [15] R. N ´ u ˜ nez-Queija and O. J. Box ma. Analysis of a mult i-server queueing model of ABR. J. Appl. Math. Stoc h. An. , 11:339–354, 1998. [16] M. Reed and B. Simon. Methods of M odern Mathematical P hysics, V ol. 2: F ourier Analysis, Self-Adjointness. Academic Press, New Y or k, 1975. [17] P . R ob ert. Stochastic Net works and Queues , vo lume 52 of Stochast ic Mo delling and Applied Probability Series . Springer, New-Y ork, June 2003. [18] G. Sansigre and G. V alen t. A large family of semi-classical p olynomials: the perturb ed Cheby - shev. J. Comput. Appl. Math. , 57:271–281, 1995. [19] E. Sc haum burg. Estimation of Marko v pr ocesses with L´ evy type generators. Prepri n t av ail able at ht tp://www.k ellogg.north w estern.edu/facult y/scha umbu rg/h tm/ResearchP ap ers/ml e.p d, 2005. (Christine F rick er) INRIA P a ris — Rocquencour t, Dom aine de Voluceau, 78153 Le Ches- na y, France E-mail addr ess : Christine.Fri cker@inria.fr (F abrice Guil lemin) France Telecom R&D, CORE/CPN, 223 00 Lannion, France E-mail addr ess : Fabrice.Guill emin@orange-ftgroup.com (Philipp e Rob ert) INRIA P aris — Ro cquencour t, Doma ine de Volucea u, 78153 Le Ches- na y, France E-mail addr ess : Philippe.Robe rt@inria.fr URL : http://ww w-rocq.i nria.fr/~robert