Robust Transient Multi-Server Queues and Feedforward Networks

Transcription

1 Submitted to Operatio Reearch maucript (Pleae, provide the maucript umber!) Robut Traiet Multi-Server Queue ad Feedforward Network Chaithaya Badi* Dimitri Bertima Nataly Youef We propoe a aalytically tractable approach for tudyig the traiet behavior of multi-erver queueig ytem ad feedforward etwork with determiitic routig. We model the queueig primitive via polyhedral ucertaity et ipired by the limit law of probability. Thee ucertaity et are characterized by parameter that cotrol the degree of coervatim of the model. Aumig the iter arrival ad ervice time belog to uch ucertaity et, we obtai cloed form expreio for the wort cae traiet ytem time i multi-erver queue ad feedforward etwork with determiitic routig. Thee aalytic formula offer rich qualitative iight o the depedece of the ytem time a a fuctio of fudametal quatitie i the queueig ytem. Moreover, we break ew groud ad preet a algorithm which appropriately average wort cae value obtaied at differet degree of coervatim. Thi methodology achieve igificat computatioal tractability ad provide good approximatio for the expected ytem time relative to imulatio. Key word : Traiet Queueig Theory, Robut Optimizatio, Heavy Tail, Relaxatio Time, Steady State 1. Itroductio The origi of queueig theory date back to the begiig of the 20 th cetury, whe Erlag (1909) publihed hi fudametal paper o cogetio i telephoe traffic. Over the pat cetury queueig theory ha foud may other applicatio, particularly i ervice, maufacturig ad traportatio idutrie. I recet year, ew queueig applicatio have emerged, uch a data ceter ad cloud computig, call ceter ad the Iteret. Thee idutrie are experiecig urgig growth rate, with call ceter ad cloud computig ejoyig repective aual growth of 20% ad 38%, accordig to the 2012 Garter ad Global Idutry Aalyt Survey. Aitat Profeor of Ecoomic ad Deciio Sciece, Kellogg School of Maagemet, Northweter Uiverity, Evato, IL 60208, c-badi@kellogg.orthweter.edu Boeig Profeor of Operatio Reearch, Co-director, Operatio Reearch Ceter, Maachuett Ititute of Techology, Cambridge, MA 02139, dbertim@mit.edu Operatio Reearch Ceter, Maachuett Ititute of Techology, Cambridge, MA 02139, youef@mit.edu 1

2 Author: Robut Traiet Multi-Server Queue ad Feedforward Network 2 Article ubmitted to Operatio Reearch; maucript o. (Pleae, provide the maucript umber!) Mot of the applicatio metioed above are characterized by time varyig arrival ad ervice patter, ad eve if they have o time-varyig uch patter, they experiece ubtatially log traiet regime, epecially uder heavy-traffic coditio, ad may ot reach teady tate withi their operatio time widow. I additio, the arrival ad ervice procee exhibit heavy-tailed behavior, a reported by Lelad et al. (1995) ad Crovella (1997) for the Iteret; by Barabai (2005) for call ceter; ad by Loboz (2012) ad Beo et al. (2010) for data ceter. I thee ituatio, teady tate i ever reached. A a reult, i may igificat applicatio, teady tate aalyi i imply ot relevat. Coequetly, cetral reearch quetio i thi cotext are maily cocered with (a) the evolutio of waitig time over time, ad (b) the time it take a queueig ytem to reach teady tate. Depite the eed for udertadig of the traiet behavior, the probabilitic aalyi of traiet queue i by ad large aalytically itractable. For M/M/1 queue, the exact aalyi of the queue legth ivolve a ifiite um of Beel fuctio ad for M/M/m queue, Karli ad McGregor (1958) obtaied the traitio probabilitie of the Markov chai decribig the queue legth a fuctio of Poio-Charlier polyomial. Bailey (1954a,b) ued double traform with repect to pace ad time to decribe the traiet behavior of a M/M/1 queue. Thi aalyi wa further exteded i a erie of paper (ee Abate ad Whitt (1987a,b), Choudhury et al. (1994), Choudhury ad Whitt (1995), Abate ad Whitt (1998)) to obtai additioal iight o the queue legth proce. Thee aalye alo provide iight o the uefule of reflected Browia motio approximatio for queue. Bertima et al. (1991) formulate the problem of fidig the ditributio of the traiet waitig time a a two-dimeioal Lidley proce ad the traform it to a Hilbert factorizatio problem. They obtai the olutio for GI/R/I, R/G/I queue, where R i the cla of ditributio with ratioal Laplace traform. Extedig thee reult, Bertima ad Nakazato (1992) ue the method of tage to tudy MGE L /MGE M /1 queueig ytem, where M GE i the cla of mixed geeralized Erlag ditributio which ca approximate a arbitrary ditributio. Maey (2002), Hamphire et al. (2006) tudy the traiet aalyi problem for

3 Author: Robut Traiet Multi-Server Queue ad Feedforward Network Article ubmitted to Operatio Reearch; maucript o. (Pleae, provide the maucript umber!) 3 proce harig markovia queue with time-varyig rate uig a techique kow a uiform acceleratio. A dicued i Odoi ad Roth (1983), there are multiple approximatio available but a tractable theory of traiet aalyi of G/G/m queue i lackig (ee alo Gro ad Harri (1974), Heyma ad Sobel (1982), ad Keilo (1979)). Further complicatig the traiet aalyi i the effect of iitial coditio, which give rie to a igificatly differet behavior a empirically ivetigated i Kelto ad Law (1985) ad Odoi ad Roth (1983). Eve umerically, the calculatio ivolve complicated itegral which do ot allow eitivity aalyi, a itegral requiremet for a ytem deiger maagig thee ytem. Give thee difficultie, a body of work ha cocetrated o developig approximate umerical olutio techique to ivetigate traiet behavior (e.g., Koopma (1972), Neut (2004), Moore (1975), Rider (1976), Grama (1977), Chag (1977), Kotiah (1978), Grama (1980), ad Rothkopf ad Ore (1979)). Newell (1971), i hi work o the diffuio approximatio of GI/G/1 queueig ytem uder heavy traffic, obtai a cloed-form expreio ad propoe a order of magitude etimate of the time required for the traiet effect to become egligible. Mori (1976), develop a umerical techique for etimatig the traiet behavior of the expected waitig time for M/M/1 ad M/D/1 queueig ytem o the bai of a recurive relatiohip ivolvig waitig time of ucceive job. All of thee approache have focued o improvig the efficiecy ad accuracy of umerical olutio techique, rather tha o uig their reult to draw cocluio o geeral attribute of traiet behavior. More recetly, baed o earlier work by Bertima ad Nataraja (2007), Oogami ad Raymod (2013) ue a emi-defiite optimizatio approach to obtai qualitative iight o the traiet behavior of queue. They derive upper boud o the tail ditributio of the traiet waitig time, ad ue it to boud the expected waitig time, for GI/GI/1 queue tartig with empty buffer for o-heavy-tailed ditributio. Xie et al. (2011) ue a exteio of the Stochatic Network Calculu framework to propoe a temporal etwork calculu approach to obtai boud o delay i iteret etwork. However, thee approache do ot tackle heavy-tailed queue ad the effect of iitial buffer coditio.

4 Author: Robut Traiet Multi-Server Queue ad Feedforward Network 4 Article ubmitted to Operatio Reearch; maucript o. (Pleae, provide the maucript umber!) Cotributio Motivated by thee challege, we propoe a aalytically tractable approach for tudyig the traiet behavior of multi-erver queueig ytem with heavy-tailed arrival ad ervice procee. Buildig upo our earlier work i Badi et al. (2012) for queue i teady tate, we firt model the queueig primitive via polyhedral ucertaity et idexed by two parameter which cotrol the degree of coervatim of the correpodig arrival ad ervice procee. We the coider a robut optimizatio perpective which yield cloed form formula for the traiet ytem time. Thee expreio offer ew qualitative iight o the depedece of the ytem time a a fuctio of fudametal quatitie i the queueig ytem. We the carry out a average cae aalyi ad break ew groud by treatig the parameter characterizig the ucertaity et a radom variable ad approximate the expected ytem time via averagig the wort cae value. Thi averagig approach achieve igificat tractability by reducig the problem of traiet aalyi to a two dimeioal itegral. Our framework combie qualitative iight via cloed form expreio ad produce accurate predictio of traiet ytem time relative to imulatio for heavy traffic queue with variou iterarrival ad ervice time ditributio, heavy tail coefficiet ad umber of erver. Furthermore, our approach i geeralizable to etwork of queue i erie (tadem queue) ad feedforward etwork with determiitic routig. The motivatio behid our idea tem from the rich developmet of optimizatio a a cietific field durig the ecod part of the 20 th cetury. From it early year (Datzig (1949)), moder optimizatio ha had the objective to olve multi-dimeioal problem efficietly from a practical poit of view. Today, may commercial code are available which ca olve truly large cale tructured (liear, mixed iteger ad quadratic) optimizatio problem. I particular, Robut Optimizatio (RO), arguably oe of the fatet growig area i optimizatio i the lat decade, provide, i our opiio, a atural modelig framework for tochatic ytem. For a review of robut optimizatio, we refer the reader to Be-Tal et al. (2009), ad Bertima et al. (2011a). The preet paper i part of a broader ivetigatio to aalyze tochatic ytem uch a market deig, iformatio theory, fiace, ad other area via robut optimizatio (ee Badi ad Bertima (2013)).

5 Author: Robut Traiet Multi-Server Queue ad Feedforward Network Article ubmitted to Operatio Reearch; maucript o. (Pleae, provide the maucript umber!) 5 The tructure of the paper i a follow. I Sectio 2, we preet our ucertaity et modelig aumptio ad motivate their cotructio via the probabilitic limit law. I Sectio 3, preet our wort cae a well a average cae aalyi for igle multi-erver queue. I Sectio 4, we exted our approach to a tadem ytem of queue. I Sectio 5, we dicu the advatage of our approach i obtaiig iight ad beig computatioally tractable. I Sectio 6, we exted the approach to aalyze feed-forward etwork with determiitic routig. Sectio 7 coclude the paper. 2. Propoed Framework I the traditioal probabilitic tudy of queue, the iterarrival time T = {T 1,...,T } ad ervice time X = {X 1,...,X } are modeled a reewal procee. I a firt-come firt-erve (FCFS) igleerver queue, the ytem time i defied by Lidley (1952) a S = (S 1 (T,X) + X T,X ) = 1 k ( X i i=k i=k+1 T i ). (1) I the evet where the queue tart it operatio with 0 0 iitial job (for which T 1,...,T 0 = 0), the ytem time recurio become S = X i T i, 1 k 0 ( X i T i ) i=k i= k i=k i=k+1 = X i T i, i=1 i= ( X i T i ). (2) k i=k i=k+1 Aalyzig the ytem time etail the udertadig of the complex relatiohip betwee the radom variable aociated with the iterarrival ad ervice time. The high dimeioal ature of the performace aalyi problem make the probabilitic aalyi by ad large itractable, epecially i the traiet domai. The tudy of multi-erver queue i eve more challegig. I thi ectio, we propoe to exted the framework itroduced i Badi et al. (2012) by modelig the ucertaity i the arrival ad ervice procee via parameterized polyhedral et, rather tha aumig probability ditributio. Thi framework ubtatially reduce the dimeioality of

6 Author: Robut Traiet Multi-Server Queue ad Feedforward Network 6 Article ubmitted to Operatio Reearch; maucript o. (Pleae, provide the maucript umber!) the ucertaity, which reult i cloed-form expreio of the wort-cae behavior of queueig ytem. Furthermore, we break ew groud by takig advatage of the ucertaity dimeioality reductio ad obtai aalytical expreio decribig the average-cae ytem behavior Ucertaity Modelig I thi paper, we coider the framework propoed by Badi et al. (2012). I particular, we cotruct polyhedral ucertaity et ipired by the geeralized Cetral Limit Theorem (CLT) reproduced below i Theorem 1. Theorem 1. Geeralized CLT (Samoroditky ad Taqqu (1994)) Let {Y 1,Y 2,...} be a equece of idepedet ad idetically ditributed radom variable, with mea µ ad udefied variace. The, the ormalized um i=1 Y i µ Y, (3) C α 1/α where Y i a table ditributio with a tail coefficiet α (1,2] ad C α i a ormalizig cotat. A i Badi et al. (2012), we cotrai the quatitie T i ad X i to take value while atifyig T i k i=k+1 Γ a, ad ( k) 1/αa i=k X i k + 1 µ Γ, k = 1,...,, ( k + 1) 1/α for ome parameter Γ a ad Γ that we ue to cotrol the degree of coervatim. Note that Γ a ad Γ are ued to cotrai the ormalized partial um for all value that the idex k ca take o. Motivated by the expreio of the ytem time i iitially oempty queue, we propoe to cotrai the total um i Eq. (2) by T i 0 i= 0 +1 γ a ad ( 0 ) 1/αa i=1 X i µ 1/α γ, for ome parameter γ a ad γ. Note that, for γ a < Γ a ad γ < Γ, the above iequalitie allow a tighter boud o performace meaure i the cae of iitially oempty queue. Coequetly, we make the followig modelig aumptio.

7 Author: Robut Traiet Multi-Server Queue ad Feedforward Network Article ubmitted to Operatio Reearch; maucript o. (Pleae, provide the maucript umber!) 7 Aumptio 1. We make the followig aumptio o the iterarrival ad ervice time. (a) The iterarrival time (T 0 +1,...,T ) belog to the parametrized ucertaity et U a = U a (γ a,γ a ) = (T 0 +1,...,T ) i= 0 +1 T i 0 T i k i=k+1 γ a ( 0 ) 1/αa Γ a ( k) 1/αa,, 0 k where 1/ i the expected iterarrival time, 0 i the iitial buffer i the queue, γ a,γ a R are parameter that cotrol the degree of coervatim, ad 1 < α a 2 model poibly heavy-tailed probability ditributio. (b) For a igle-erver queue, the ervice time (X 1,...,X ) belog to the ucertaity et U = (X 1,...,X ) i=1 X i µ γ 1/α k X i k j i=j+1 µ Γ (k j) 1/α, 0 j k where 1/µ i the expected ervice time, γ,γ R are parameter that cotrol the degree of, coervatim, ad 1 < α 2 model poibly heavy-tailed probability ditributio. (c) For a m-erver queue, m 2, we let ν be a o-egative iteger uch that ν = /m, where i the idex correpodig to the th arrivig job. We partitio the job idice ito et K i = {k k/m = i}, for i = 0,1,...,ν, i.e., K 0 = {1,...,m},K 1 = {m + 1,...,2m},...,K ν = {νm + 1,...,}. Let k i K i deote the idex that elect a job from et J i, for i = 0,...,ν. The ervice time for a multi-erver queue belog to the parameterized ucertaity et U m = (X 1,...,X ) ν i=0 X ki ν + 1 µ γ m (ν + 1) 1/α, k i K i X ki I i I µ Γ m I 1/α, k i K i, ad i I {0,...,ν}, Remark: Note that the ucertaity et that we coider i thi paper are ubet of the oe itroduced i Badi et al. (2012). Furthermore, we allow (γ a,γ a ), (γ a,γ ) ad (γ m,γ m ) to take.

8 Author: Robut Traiet Multi-Server Queue ad Feedforward Network 8 Article ubmitted to Operatio Reearch; maucript o. (Pleae, provide the maucript umber!) both poitive ad egative value. Whe thee parameter are poitive, our ucertaity et allow the um of iter arrival time to take value below the mea ad the um of ervice time to take value exceedig the mea, which yield poitive waitig time. O the other had, whe thee parameter are egative, our ucertaity et cotrai the um of the iter arrival time to take value exceedig the mea ad the um of the ervice time to take value below the mea, i which cae, the ytem yield zero waitig time Performace Aalyi Metric To udertad the performace of queueig ytem, we eek i particular a aalytical characterizatio of the expected ytem time, give by S = E T,X [S (T,X)]. (4) The above expreio i challegig to compute by modelig the primitive i a probabilitic queue via tochatic procee, due to the high dimeioality of the ucertaity ad the complex relatiohip betwee the radom variable aociated with the iterarrival ad ervice time. Uig our ucertaity modelig framework, we obtai a approximatio of the expected ytem time by (a) computig the wort cae value aumig the primitive atify Aumptio 1, the (b) averagig the reult with repect to the parameter (γ a,γ a ), (γ,γ ), ad (γ m,γ m ). We preet ext the detail of our approach. Wort Cae Behavior To characterize the wort cae behavior, we formulate the related performace aalyi quetio a a robut optimizatio problem. I particular, we eek the wort cae ytem time Ŝ i queue atifyig Aumptio 1. The wort cae aalyi ca be cat a optimizatio problem of the form Ŝ (T) = X U m S ad Ŝ = T U a Ŝ (T), (5) which give rie to a cloed form characterizatio of the wort cae ytem time. To illutrate, Theorem 2 preet the reult for a iitially empty igle-erver queue with light tailed primitive.

9 Author: Robut Traiet Multi-Server Queue ad Feedforward Network Article ubmitted to Operatio Reearch; maucript o. (Pleae, provide the maucript umber!) 9 Theorem 2. (Wort Cae Sytem Time i a Iitially Empty Sigle-Server Queue) I a igle-erver FCFS queue with 0 = 0, T U a, X U, α a = α = 2 ad ρ < 1, the wort cae ytem time for the th job i give by (Γ a + Γ ) 1 ρ + ( 1 µ + Γ ), if < 2 [(Γ a + Γ ) + ] 2 4(1 ρ) 2 Ŝ 4 [(Γ a + Γ ) + ] 2 + ( 1 1 ρ µ + Γ ), otherwie, where the otatio (a) + = (0,a). (6) The evolutio of the wort cae ytem time i characterized by two ditict tate: (a) a traiet tate where the ytem time i depedet o with the ytem time i a iitially empty queue icreaig at a order of 1/α whe Γ a + Γ > 0; ad (b) a teady tate where the ytem time i idepedet of. Whe Γ a + Γ < 0, job do ot experiece ay waitig time, ad therefore the ytem time i equal to the ervice time. The characterizatio of the wort cae waitig time bear qualitative imilarity to the boud etablihed by Oogami ad Raymod (2013) ad Kigma (1970) for the traiet ad teady tate ytem time i a GI/GI/1 queue, repectively, where e = exp(1) = E[S ] e 1 σ 2 2 a + σ 2 + µ, if < 2 (σa 2 + σ) 2 e 2 (1 ρ), 2 (σa 2 + σ) ρ µ, otherwie, Sectio 3 ad 4 preet exteio of Theorem 2 to the cae of iitially oempty multi-erver igle ad tadem queue with heavy-tailed arrival ad ervice. We ext dicu how we leverage the wort cae expreio that we obtai to predict the average ytem behavior. Average Cae Behavior Itead of takig the expectatio of the ytem time over the radom variable T ad X to aalyze the average cae behavior, we propoe to treat the parameter (γ a,γ a ), (γ,γ ) ad (γ m,γ m ) a radom variable ad compute S = E[Ŝ ]. (7)

10 Author: Robut Traiet Multi-Server Queue ad Feedforward Network 10 Article ubmitted to Operatio Reearch; maucript o. (Pleae, provide the maucript umber!) Philoophically, thi approach ditill all the probabilitic iformatio cotaied i the radom variable X i ad T i ito the the parameter {(γ a,γ a ),(γ,γ )} by allowig them to behave a radom variable. Capturig the radome of the arrival ad ervice tochatic procee via oly few radom variable allow a igificat dimeioality reductio of the ucertaity. Thi framework therefore yield a tractable aalyi of the expected traiet waitig time by reducig the problem to olvig a low-dimeioal itegral. To illutrate the averagig idea, coider agai the cae of a iitially empty igle-erver queue with light tailed primitive. We expre the boud o the wort cae ytem time Ŝ i Eq. (6) a Ŝ t (Γ a,γ ) 1 t (Γ a,γ ) + Ŝ (Γ a,γ ) 1 (Γ a,γ ), (8) where the idicator fuctio 1 t ad 1 reflect the coditio for the ytem to be i the traiet tate ad the teady tate, repectively. By treatig Γ a ad Γ a radom variable, we compute the expected value of the expreio i Eq. (8) to obtai S. Thi computatio ivolve a double itegratio, which ca be olved efficietly uig umerical itegratio techique. With a careful choice of the ditributio of the parameter {(γ a,γ a ),(γ,γ )}, we how i Sectio 3.2 ad 4.2 that thi averagig approach provide umerical output which match the imulated value accurately with mot error below 10%. 3. Aalyi of a Sigle Queue I thi ectio, we tudy the wort cae ad average behavior of a igle queue with a FCFS chedulig policy ad a traffic iteity ρ = /(mµ) < 1, where m deote the umber of erver. We alo aume that the queue begi operatio with a job buffer Wort Cae Behavior We tudy the ytem time uig the wort cae approach a propoed i Badi et al. (2012). We coider a m-erver queueig ytem with 0 iitial job. Let C deote the completio time of the th job, i.e., the time the th job leave the ytem (icludig ervice), ad C () deote the time

11 Author: Robut Traiet Multi-Server Queue ad Feedforward Network Article ubmitted to Operatio Reearch; maucript o. (Pleae, provide the maucript umber!) 11 of the th departure from the ytem. I geeral, the followig recurio decribe the dyamic i a multi-erver queue (Krivuli (1994)) C = (A,C ( m) ) + X ad S = C A = (C ( m) A,0) + X, (9) where A = i=1 T i deote the the time of arrival of the th job. It i well kow that the cetral difficulty i aalyzig multi-erver queue lie i the fact that overtakig may occur, i.e., the th departure may ot correpod to the th job arrivig to the queue. However, a oted i Badi et al. (2012), takig a wort cae approach allow u to overcome the challege of multi-erver queue dyamic ad obtai a exact characterizatio of the wort cae waitig time for the th job, for ay T. I particular, Badi et al. (2012) coider the cae of ucertaity et where γ Γ 0 ad how that the wort cae ytem time i equal to the oe achieved i a queue where o overtakig i allowed, i.e., where job leave the queue i the ame order of their arrival, yieldig ν Ŝ (T) = 0 k ν U m X r(i) i=k i=r(k)+1 T i, (10) where r(i) = (ν i)m K i, for all T. We exted thi reult to the cae where γ m Γ m ad Γ m 0 ad obtai Propoitio 1, ad the proof i preeted i Appedix A3. Propoitio 1. Give a equece of iter-arrival time T = {T 1,...,T }, the wort cae ytem time Ŝ (T) i a FCFS queue modeled by U a (γ a,γ a ), U m (γ m,γ m ), where Γ m 0, i uch that Ŝ (T) = 0 k ν Um ν i=k X r(i) i=r(k)+1 T i, (11) where r(i) = (ν i)m. To hadle the cae of Γ m < 0, we propoe a upper boud o the wort cae ytem time. We let Γ + m = (0,Γ m ), ad ice U m (Γ m ) U m (Γ + m), Ŝ (T) = S (T,X) U m (Γ m) U m (Γ + m) S (T,X) = 0 k ν U m (Γ + m) ν X r(i) i=k i=r(k)+1 T i. (12)

12 Author: Robut Traiet Multi-Server Queue ad Feedforward Network 12 Article ubmitted to Operatio Reearch; maucript o. (Pleae, provide the maucript umber!) Iitially Empty Queue We apply Aumptio 1 ad the fact that γ m Γ m, to tralate Eq. (11) ad (12) ito olvig the followig oe-dimeioal oliear optimizatio problem Ŝ {ν k Γ + m(ν k) m (ν k + 1) 1/α + Γ a [m(ν k)] 1/αa }. (13) 0 k ν µ Thi boud ca be olved efficietly for the geeral cae where α α a. Theorem 3 provide a cloed form expreio for the upper boud o the wort cae ytem time i a iitially empty queue for the pecial cae where α a = α = α. Theorem 3. (Highet Sytem Time i a Iitially Empty Multi-Server Queue) I a iitially empty m-erver FCFS queue where T U a, X U m where Γ = m 1/α Γ a + Γ + m > 0, α a = α = α ad ρ < 1, the wort-cae ytem time for all K ν i give by Γ ν 1/α Ŝ m(1 ρ) ν + ( 1 µ + Γ+ m), α 1 α α/(α 1) 1/(α 1) Γ α/(α 1) [m(1 ρ)] 1/(α 1), if ν < ( Γ/m α/(α 1) α(1 ρ) ) otherwie. (14) Proof of Theorem 3. Sice (ν k + 1) 1/α (ν k) 1/α + 1, ad give Γ + m 0, we boud Eq. (13) by Ŝ (Γ a,γ m ) {ν k 0 k ν µ + m(ν k) Γ+ m (ν k) 1/α + Γ a [m(ν k)] 1/αa } + ( 1 µ + Γ+ m), which follow from the fact that (ν k + 1) 1/α (ν k) 1/α + 1. By makig the traformatio x = ν k, where x N, we ca repreet thi imizatio problem a (β 0 x ν,x N x1/α δ x) (β 0 x ν,x R x1/α δ x), (15) where β = m 1/α Γ a +Γ + m ad δ = m(1 ρ)/ > 0, give ρ < 1. If β 0, the fuctio h(x) = β x 1/α δ x 0 for all value of x, implyig Ŝ = 1/µ + Γ + m. For β > 0, the fuctio h i cocave i x with a ucotraied imizer x = ( β αδ ) α/(α 1) = ( (Γ m + m 1/α α/(α 1) Γ a ) ). (16) αm(1 ρ)

13 Author: Robut Traiet Multi-Server Queue ad Feedforward Network Article ubmitted to Operatio Reearch; maucript o. (Pleae, provide the maucript umber!) 13 Maximizig the fuctio h( ) over the iterval [0,ν] ivolve a cotraied oe-dimeioal cocave imizatio problem whoe olutio give rie to cloed-form olutio. (a) If x [0,ν], the x i the imizer of the fuctio h over the iterval [0,ν], leadig to a expreio that i idepedet of ν, Ŝ β ( β αδ ) 1/(α 1) δ ( β αδ ) α/(α 1) + ( 1 µ + Γ+ m) = α 1 βα/(α 1) αα/(α 1) δ + ( 1 1/(α 1) µ + Γ+ m). (17) (b) If x > ν, the fuctio h i o-decreaig over the iterval [0,ν], with h(ν) h(x) for all x [0,ν], leadig to a expreio that i depedet o ν, Ŝ = β(ν) 1/α δ(ν) + ( 1 µ + Γ+ m). (18) We obtai Eq. (14) by ubtitutig β ad δ by their expreio i part (a) ad (b). Note that, for the cae where Γ = m 1/α Γ a + Γ + m 0, the fuctio i Eq. (14) i icreaig i k over the iterval k [0,ν], for ρ = /µ < 1. It i therefore imized at k = ν, which yield Ŝ = X 1 µ + Γ+ m. I thi cae, the th job doe ot experiece a waitig time before eterig ervice. Thi i due to the fact that the coditio Γ 0 ivolve typically log iter arrival time ad hort ervice time. Iitially Noempty Queue We ext aalyze the cae where 0 > 0 with T i = 0 for all i = 1,..., 0. The firt m job i the queue are routed immediately to the erver without ay delay. We are itereted i the behavior for > m. For Γ m 0 ad aumig 0 K φ, i.e., φ = 0 /m, we ca rewrite Eq. (11) a (a) for 0 Ŝ = X U m ( 0 k ν φ ν i=k X r(i) ) = X U m ( ν X r(i) ) (19) i=0 (b) for > 0 ( 0 k φ X U m Ŝ = φ<k ν X U m ν i=k X r(i) ) T U a ν X r(i) i=k T U a T i, i= 0 +1 T i i=r(k)+1. (20)

14 Author: Robut Traiet Multi-Server Queue ad Feedforward Network 14 Article ubmitted to Operatio Reearch; maucript o. (Pleae, provide the maucript umber!) Similarly to the empty cae, the wort cae ytem time ivolve oe-dimeioal oliear optimizatio problem which ca be olved efficietly. I particular, for 0, we have Ŝ (Γ m ) = X U m ( ν where r = r(0) = νm ad ν = /m. For > 0, we have i=0 X r(i) ) = ( ν µ + γ m(ν + 1) 1/α ), (21) ( ν k γ m (ν k + 1) 1/α ) 0 + γ a ( 0 ) 1/αa, µ Ŝ (Γ a,γ m ) φ<k ν (ν k Γ + m(ν k), (22) m (ν k + 1) 1/α + Γ a [m(ν k)] 1/αa ) µ A for iitially empty queue, the optimizatio problem i Eq. (22) ca be olved efficietly for the geeral cae where α a α. Theorem 4 provide a cloed form expreio for the upper boud o the wort cae ytem time for the pecial cae where α a = α = α. Theorem 4. (Highet Sytem Time i a Iitially Noempty Multi-Server Queue) I a m-erver FCFS queue with 0 K φ ad T U a, X U m uch that Γ = m 1/α +Γ + m > 0, α a = α = α ad ρ < 1, the wort cae ytem time for 0 < K ν i give by ( ν µ + γ m (ν + 1) 1/α ) 0 + γ a ( 0 ) 1/αa, Ŝ Γ(ν φ) 1/α m(1 ρ) α 1 α α/(α 1) 1/(α 1) Γ α/(α 1) [m(1 ρ)] 1/(α 1) + ( 1 µ + Γ+ m), (ν φ) + ( 1 µ + Γ+ m), if ν φ < ( Γ/m α(1 ρ) ) otherwie. α/(α 1). (23) Proof of Theorem 4. To olve boud the imizatio problem i Eq. (22), we take a imilar approach to that preeted i the proof of Theorem 3 ad cat the problem i the form β (ν φ) 1/α δ (ν φ), (β 0 x ν φ,x R x1/α δ x) = α 1 βα/(α 1) αα/(α 1) δ, 1/(α 1) if ν φ ( β αδ )α/(α 1) otherwie, where β = m 1/α Γ a +Γ + m ad δ = m(1 ρ)/. Subtitutig the term β ad φ by their repective value i the above expreio yield the deired reult.

15 Author: Robut Traiet Multi-Server Queue ad Feedforward Network Article ubmitted to Operatio Reearch; maucript o. (Pleae, provide the maucript umber!) 15 Note that, for the cae where Γ = m 1/α Γ a + Γ + m 0, the wort cae ytem time Ŝ (Γ) {( ν µ + γ m (ν + 1) 1/α ) 0 + γ a ( 0 ) 1/αa, 1 µ + Γ+ m}. I thi cae, the th job experiece a waitig time oly due to the buildup effect left by the iitial job. For big eough, thi effect become egligible ad the ytem time evetually become equal to the ervice time, tabilizig at the value 1/µ + Γ + m Average Cae Behavior To aalyze the average behavior of a queueig ytem, we treat the parameter (γ a,γ a ) ad (γ m,γ m ) a radom variable ad compute the expected value of the wort cae ytem time Ŝ. For eae of otatio, we expre the wort cae ytem time i Eq. (23) a Ŝ Ŝ Ŝ b t (Γ a,γ m ), if (γ a,γ ), m 0 m < (m 1/α Γ a + Γ + )+ αm(1 ρ) Ŝ (Γ a,γ m ), otherwie α/(α 1), (24) where Ŝ, b Ŝ, t ad Ŝ deote the quatitie aociated with the ytem time effected by the iitial buffer 0, the traiet tate ad the teady tate, repectively. We would like to rewrite the above upper boud o the wort cae ytem time a {Ŝ b (γ a,γ ), Ŝ t (Γ a,γ m ) 1 t (Γ a,γ m ) + Ŝ (Γ a,γ m ) 1 (Γ a,γ m )}, where the idicator fuctio 1 t ad 1 reflect the coditio for the ytem to be i the traiet tate ad the teady tate, repectively. Specifically, by writig the coditio i Eq. (24) a a iterval over Γ a ad Γ m, we obtai 1 t (Γ a,γ m ) = 1 if m 1/α Γ a + Γ + m > 1 (Γ a,γ m ) = 1 otherwie. αm(1 ρ) ( m 0 m ) (α 1)/α By poitig ome aumptio o the ditributio of the parameter (γ a,γ a ) ad (γ m,γ m ), we compute Ŝ via umerical itegratio. We ext dicu our choice of the ditributio of the parameter {(γ a,γ a ),(γ,γ )} ipired by the limit law of probability.

16 Author: Robut Traiet Multi-Server Queue ad Feedforward Network 16 Article ubmitted to Operatio Reearch; maucript o. (Pleae, provide the maucript umber!) Choice of Variability Ditributio From Aumptio 1, the parameter γ a ad γ ca be viewed a ormalized um of the radom variable {T 0 +1,...,T } ad {X 1,...,X }. Specifically, γ a = i= 0 +1 T i 0 ( 0 ) 1/α Z a ad γ = i=1 X i µ 1/α Z, (25) which approximately behave a a radom variable followig a limitig ditributio. I a multierver queue, ad aumig without lo of geerality that = νm, we obtai (ν+1)m X i νm ν i=1 µ γ = = 1 X m j+im ν + 1 [νm] 1/α m i=0 µ 1/α j=1 (ν + 1) 1/α 1 m m γ 1/α m, j=1 where the lat iequality i due to Aumptio 1(c). We ca therefore expre γ m a γ m = 1 m (α 1)/α γ. (a) Light-Tailed Primitive: For light tail, γ a ad γ obey the ormal ditributio, i.e., γ a N (0,σ a ) ad γ N (0,σ ), where σ a ad σ deote the tadard deviatio aociated with the iter-arrival ad ervice procee, repectively. (b) Heavy-Tailed Primitive: By Theorem 1, the ormalized um of heavy-tailed radom variable with tail coefficiet α follow a table ditributio S α (ψ,ξ,φ) with a kewe parameter ψ = 1, a cale parameter ξ = 1 ad a locatio parameter φ = 0. Therefore, γ a ad γ a expreed i Eq. (25) are uch that γ a S α ( 1,C α,0) ad γ S α (1,C α,0), where C α i a ormalizig cotat a itroduced i Eq. (3). A a cocrete example, for Pareto ditributed iter arrival ad ervice time, C α = [Γ(1 α)co(πα/2)] 1/α,

17 Author: Robut Traiet Multi-Server Queue ad Feedforward Network Article ubmitted to Operatio Reearch; maucript o. (Pleae, provide the maucript umber!) 17 where Γ( ) deote the Gamma fuctio. Note that, ulike the cae of light tail, the ditributio of γ a ad γ are aymmetrical. More pecifically, the kewe of γ a i egative ice γ a = Z a, where Z a = S α (1,C α,0). The characterizatio of the exact ditributio of the parameter (Γ a,γ,γ m ) i however challegig. Itead, we propoe a implificatio where we expre Γ a, Γ ad Γ m a liear fuctio of γ a, γ ad γ m, repectively. Specifically, we let Γ a = θ a γ a, Γ = θ γ, ad Γ m = θ m γ m, where (θ a,θ,θ m ) are ome calar which we elect a follow. (a) Light-Tailed Primitive: We chooe (θ a,θ,θ m ) are calar choe o that the average wort cae teady-tate ytem time matche the boud provided by Kigma (1970), which i particularly tight i heavy traffic. I other word, we eure that S = S for a large eough value of. To illutrate, for a multi-erver queue, we eure that 4(1 ρ) E[[(θ aγ a + θ m γ m/m + 1/2 ) + ] 2 ] = 2(1 ρ) (σ2 a + σ/m 2 2 ). (26) Let γ = θ a γ a + θ m γ + m/m 1/2 = θ a γ a + θ m γ + /m. We approximate the expectatio i Eq. (26) by E[(γ + ) 2 ] P(γ 0) E[γ 2 ] = P(γ 0) (θ 2 aσ 2 a + P(γ 0) θ 2 mσ 2 /m 2 ) By patter matchig the two expreio i Eq. (26), the parameter θ a ad θ m are uch that 2 θ a ( P(γ 0) ) 1/2 2 ad θ m ( P(γ 0) P(γ 0) ) 1/2. (27) Note that, give that γ i a ormally ditributed ditributed radom variable cetered aroud the origi, we have P(γ 0) = 1/2. Alo, P(γ 0) ca be efficietly computed umerically, with P(γ 0) = P(θ a γ a + θ m γ + /m 0) = P(γ a + 2 1/2 γ + /m 0). (b) The teady tate i heavy-tailed queue doe ot exit, which doe ot allow u to ue a imilar matchig procedure to the oe itroduced for light-tailed queue. Itead, we propoe to exted the formula i Eq. (27) to obtai α θ a ( P(γ 0) ) (α 1)/α α ad θ m ( P(γ 0) P(γ 0) ) (α 1)/α. (28)

18 Author: Robut Traiet Multi-Server Queue ad Feedforward Network 18 Article ubmitted to Operatio Reearch; maucript o. (Pleae, provide the maucript umber!) where the probabilitie ca be efficietly computed umerically. Specifically, P(γ 0) = P(θ a γ a + θ m γ + /m 0) = P(γ a + P(γ 0) (α 1)/α γ + /m 0). We ote that, by expreig Γ a ad Γ m i term of γ a ad γ, the average ytem time S ca be computed by takig expectatio with repect to γ a ad γ S = E γa,γ [{Ŝ b (γ a,γ ), Ŝ t (γ a,γ ) 1 t (γ a,γ ) + Ŝ (γ a,γ ) 1 (γ a,γ )}]. The above double itegral ca be efficietly computed uig umerical techique. We ext compare the performace of the propoed averagig techique with imulated value. Computatioal Reult We ivetigate the performace of our approach relative to imulatio ad examie the effect of the ytem parameter (traffic iteity, tail coefficiet, iitial buffer ad umber of erver) o it accuracy. Specifically, we ru imulatio for igle ad multi-erver queue with ormally ad Pareto ditributed iter arrival ad ervice time. We preet the average percet error betwee imulated expected value S ad the predictio S. I particular, we report the followig quatity Average Percet Error = N 1 N 1 =2 S S S 100%. The term N correpod to either the umber of job oberved util relaxatio i reached (oberved from imulatio) or the imum umber of job for which the imulatio wa ru (N = 5,000 for both light-tailed ad heavy-tailed ditributio). Table 1 ad 2 report the error for multi-erver queue with ormally ad Pareto ditributed iter arrival ad ervice time, repectively. Figure 1 compare our approximatio (dotted lie) with imulatio (olid lie) for queue with ormally ditributed iter arrival ad ervice time with m = 1 (top pael) ad m = 20 (bottom pael). Figure 2 preet a graphical aphot of our approximatio (dotted lie) i compario to imulatio (olid lie) for queue with Pareto ditributed primitive with m = 1 (top pael) ad m = 20 (bottom pael).

19 Author: Robut Traiet Multi-Server Queue ad Feedforward Network Article ubmitted to Operatio Reearch; maucript o. (Pleae, provide the maucript umber!) 19 Table 1 σa = σ = 2.5 σa = σ = 4.0 Error relative to imulatio for multi-erver queue with ormally ditributed primitive. 1 Server 10 Server 20 Server ρ 0 = 0 0 = 5 0 = 20 0 = 0 0 = 20 0 = 50 0 = 0 0 = 50 0 = AverageSytemTime Simulatio Predictio Job Job Job (a) (b) (c) AverageuSytemuTime Simulatio Predictio Job (d) Job (e) Job (f) Figure 1 Simulated (olid lie) veru predicted value (dotted lie) for a igle queue with ormally ditributed primitive (σ a = σ = 4.0) ad ρ = Pael (a) (c) correpod to a itace with m = 1 ad 0 = 0,5,10. Pael (d) (f) correpod to a itace with ρ = 0.99 ad 0 = 0,50,100.

20 Author: Robut Traiet Multi-Server Queue ad Feedforward Network 20 Article ubmitted to Operatio Reearch; maucript o. (Pleae, provide the maucript umber!) αa = α = 1.6 αa = α = 1.7 Table 2 Error relative to imulatio for multi-erver queue with Pareto ditributed primitive. 1 Server 10 Server 20 Server ρ 0 = 0 0 = 50 0 = = 0 0 = 50 0 = = 0 0 = 50 0 = AveragePSytemPTime Simulatio Predictio Job Job Job (a) (b) (c) AverageSytemTime Simulatio Predictio Job Job Job (d) (e) (f) 100 Figure 2 Simulated (olid lie) veru predicted value (dotted lie) for a igle queue with Pareto ditributed primitive (α a = α = 1.6) ad ρ = Pael (a) (c) correpod to a itace with m = 1 ad 0 = 0,50,200. Pael (d) (f) correpod to a itace with m = 20 ad 0 = 0,50,200.

21 Author: Robut Traiet Multi-Server Queue ad Feedforward Network Article ubmitted to Operatio Reearch; maucript o. (Pleae, provide the maucript umber!) 21 Our approach geerally yield error withi 10% for multi-erver queue with ormally ditributed iter arrival ad ervice time. Error for heavy-tailed multi-erver queue eem to icreae with the umber of erver, with magitude withi 15% for 10-erver queue ad 30% for 20-erver queue. However, we till capture the geeral behavior of the ytem time a how i Figure 2. Furthermore, a how by imulatio ad empirical tudie performed by Odoi ad Roth (1983) o light-tailed queueig ytem, the expected traiet ytem time ha broadly four differet behavior depedig o the iitial job. Our averagig approach i capable of capturig thee behavior. (a) The firt behavior occur whe the ytem i iitially empty. The average ytem time fuctio i mootoic ad cocave i. Thi behavior i detected i Figure 1(a),(d). (b) The ecod behavior occur whe the umber of iitial job i mall creatig a iitial ytem time S 0 that i below the teady tate value. The ytem time i thi cae iitially decreae ad ubequetly icreae util reachig teady tate, a ee i Figure 1(b). (c) The third behavior occur whe the umber of iitial job create a iitial ytem time S 0 that i higher tha the teady tate value. I thi cae, the average ytem time i covex i ad decreae expoetially util reachig teady tate, a detected i i Figure 1(c). (d) The fourth behavior occur whe the iitial buffer create a iitial ytem time S 0 that i ubtatially larger tha the teady tate value. The iitial decreae i approximately liear with job leavig the ytem at the rate of µ, a ee i Figure 1(e),(f). 4. Aalyi of Queue i Serie I thi ectio, we exted our aalyi of igle queue to the aalyi of tadem queue. Coider a etwork of J queue i erie ad let X (j) = {X (j) 1,...,X (j) } deote the ervice time at the j th queue. We make the followig aumptio. Aumptio 2. We make the followig aumptio for the ervice time at the j th queue i a tadem etwork. Let 1/µ j be the expected ervice time, Γ (j) R a parameter that cotrol the degree

22 Author: Robut Traiet Multi-Server Queue ad Feedforward Network 22 Article ubmitted to Operatio Reearch; maucript o. (Pleae, provide the maucript umber!) of coervatim, ad 1 < α (j) 2 modelig poibly heavy-tailed probability ditributio. (a) For a igle-erver queue j, the ervice time belog to the ucertaity et U j = {(X (j) 1,...,X (j) l ) X i l k i=k+1 Γ (j) µ j (j) 1/α (l k), 0 k l }, (b) For a m-erver queue j, the ervice time belog to the ucertaity et U m j = {(X (j) 1,...,X (j) ) X (j) k i I i I Γ m (j) µ j I 1/α(j), k i K i, ad i I {0,...,ν} }. We further aume that the iter arrival time T = (T 1,...,T ) to the tadem etwork atify the ucertaity et U a, a decribed i Aumptio 1. We coider, for the purpoe of the dicuio, a tadem etwork with J igle-erver queue. The ytem time of the th job at the j th queue i uch that S (j) = 0 k j X (j) i i=k j T (j) i i=k j +1 where T (j) = (T (j) 1,...,T (j) ) deote the iter arrival time to queue j. Note that T (j) i exactly the vector of iter departure time D (j 1) from queue j 1, which ca be cat a T (j) i=k j +1 i = D (j 1) i = i=k j +1 i=k j +1 T (j 1) i, + S (j 1) S (j 1) k j. Recurively, the iter arrival time to queue j ca be expreed a a fuctio of the iter arrival time T to the etwork ad the ervice time X (1) through X (j 1). For a iolated queue, Badi et al. (2012) how that the iterdeparture time belog to the iter-arrival ucertaity et U a. However, thi characterizatio i oly tight uder teady-tate coditio. Obtaiig a exact traiet characterizatio of the iter-departure proce i challegig. Itead of goig thi route, we propoe to ue the recurive formula that defie the dyamic i a etwork of queue i erie to tudy the overall ytem time S = S (1) +...S (J). Bertima et al. (2011b) obtai a exact characterizatio of the ytem time i a tadem etwork of igle-erver queue where S = 1 k 1... k J k 2 X (1) i + i=k 1 k 3 i=k 2 X (2) i X (J) i i=k J T i i=k (29)

23 Author: Robut Traiet Multi-Server Queue ad Feedforward Network Article ubmitted to Operatio Reearch; maucript o. (Pleae, provide the maucript umber!) 23 Similarly to the aalyi of a igle queue, we propoe a aalyi of the wort cae overall ytem time uder Aumptio 2 which provide cloed form boud. We the leverage the aalytic expreio of the wort cae ytem time to udertad the average behavior of tadem queueig etwork with multiple erver Wort Cae Performace Uder the wort cae approach, ad applyig the adverarial ervice time at each queue, the wort cae ytem time of the th job for ay realizatio of T i give by Ŝ (T) = 1 k 1... k J U 1 k 2 X (1) i i=k 1 + U 2 k 3 X (2) i i=k U J X (J) i i=k J T i i=k (30) Theorem 5 provide a imilar reult for multi-erver queue i erie, uder the aumptio that each queue act adverarially i view of imizig it ytem time, for all T. Theorem 5. (Wort Cae Sytem Time i a Tadem Queue with Multiple Server) I a etwork of J multi-erver queue i erie with iter arrival time T = {T 1,...,T }, the overall ytem time of the th job i give by Ŝ (T) = 0 k 1... k J ν U 1 k 2 X (1) r(i) i=k 1 + U 2 k 3 X (2) r(i) i=k U J X (J) r(i) i=k J i=r(k 1 )+1 T i, (31) where r(i) = (ν i)m. The proof i imilar to the proof preeted i Badi et al. (2012), ad preeted i Appedix A1. By miimizig the partial um of the iterarrival time, we obtai a exact characterizatio of the wort cae ytem time i a tadem queue a Ŝ = 0 k 1... k J ν U 1 k 2 X (1) r(i) i=k U J X (J) i=k J T U a i=r(k 1 )+1 r(i) T i. (32)

24 Author: Robut Traiet Multi-Server Queue ad Feedforward Network 24 Article ubmitted to Operatio Reearch; maucript o. (Pleae, provide the maucript umber!) Iitially Empty Queue i Tadem By Aumptio 1, the wort cae ytem time i bouded by Ŝ (Γ a,γ (1),...,Γ (J) ) 0 k 1... k J ν J j=1 k j+1 k j Γ (j) m µ j m(ν k 1) (k j+1 k j + 1) 1/α(j) + Γ a [m(ν k 1 )] 1/αa, (33) which ivolve a J-dimeioal oliear optimizatio problem. Theorem 6 provide a cloed form upper boud o the wort cae ytem time i a iitially empty etwork of J idetical queue i tadem, with µ 1 =... = µ J ad α a = α (1) =... = α (J) = α. Theorem 6. (Highet Sytem Time i a Iitially Empty Tadem Queue) I a iitially empty etwork of J multi-erver queue i erie with T U a, X (j) U j, for all j = 1,...,J, α a = α (1) =... = α (J) = α, ad Γ = m 1/α Γ a + Γ m > 0, where Γ m = ( J (Γ (j)+ j=1 α 1/α m ) α/α 1 ) the wort-cae ytem time for µ 1 =... = µ J ad ρ < 1 i give by, (34) Γ ν 1/α Ŝ m(1 ρ) ν + J J µ + Γ (i)+ i=1 Γ m, if ν + J [ αm(1 ρ) ] α 1 α 1/(α 1) Γ α/(α 1) α/(α 1) [m(1 ρ)] + J J 1/(α 1) µ + Γ (i)+ m, otherwie. i=1 α/(α 1) (35) Proof of Theorem 6. From Eq. (33), we have that the wort cae ytem time i give by Ŝ = J µ + 0 k 1... k J ν [Γ (1)+ m (k 2 k 1 + 1) 1/α Γ m (J)+ (ν k J + 1) 1/α ]+ Γ a [m(ν k 1 )] 1/α m(1 ρ). (ν k 1 ) Furthermore, ice (k j+1 k j + 1) 1/α (k j+1 k j ) 1/α + 1, for all j=1,..., J, we obtai Ŝ J J µ + Γ (j)+ m + j=1 0 k 1... k J ν We will iolate the problem of imizig [Γ (1)+ m [Γ m (1)+ (k 2 k 1 ) 1/α Γ m (J)+ (ν k J ) 1/α ]+ Γ a [m(ν k 1 )] 1/α m(1 ρ). (ν k 1 ) (k 2 k 1 ) 1/α Γ m (J)+ (ν k J ) 1/α ] for fixed value

25 Author: Robut Traiet Multi-Server Queue ad Feedforward Network Article ubmitted to Operatio Reearch; maucript o. (Pleae, provide the maucript umber!) 25 of k 1,ν, ad make the traformatio x 1 = k 2 k 1,...,x J = ν k J, where x j N, for all j = 1,...,J. With thee traformatio, the optimizatio problem implifie to 0 k 1 ν,k 1 N m 1/α Γ a (ν k 1 ) 1/α m(1 ρ) (ν k 1 ) + [Γ (1)+ m x 1/α Γ (J)+ m x 1/α.t. x x J = ν k 1 x j N, j = 2,...,J J ]. (36) It i eay to ee, baed o firt order optimality coditio (ee Appedix B), that the optimal olutio to the ier optimizatio problem atifie Γ (1)+ m (x 1) 1/(α 1) = Γ (2)+ m (x 2) 1/(α 1) =... = Γ (J)+ m (x J) 1/(α 1). Uig the additioal coditio that J j=1 x j = 1, we obtai leadig to a optimal value of x k = (ν k 1)(Γ (k)+ m J (Γ (j)+ j=1 ) α/(α 1) Γ (1)+ m (x 1) 1/α Γ (J)+ m (x 1) 1/α = ( m ) α/(α 1) k = 1,2,...,J, J (Γ (j)+ j=1 (α 1)/α m ) α/(α 1) ) Subtitutig the optimal olutio of the ier problem i Eq. (36), the performace aalyi reduce to olvig the followig oe-dimeioal optimizatio problem. 0 k 1 ν J m1/α Γ a + [ j=1 (α 1)/α m ) α/(α 1) ] (ν k 1) 1/α (Γ (j)+ m(1 ρ) (ν k 1 ), (37) which ca be cat i the form of the optimizatio problem i Eq. (15), with β = m 1/α Γ a + ( J (Γ (j)+ j=1 (α 1)/α m ) α/(α 1) ) ad δ = m(1 ρ). Referrig to the proof of Theorem 3, the olutio to Eq. (37) i J x ν+j β ν 1/α δ ν, β x 1/α δ x = α 1 βα/(α 1) αα/(α 1) δ, 1/(α 1) if ν + J ( β αδ )α/(α 1) otherwie. We obtai the deired reult by ubtitutig β ad δ by their repective value.

26 Author: Robut Traiet Multi-Server Queue ad Feedforward Network 26 Article ubmitted to Operatio Reearch; maucript o. (Pleae, provide the maucript umber!) The cae where Γ = m 1/α Γ a + Γ m 0 arie whe Γ a < 0, ice Γ m > 0 a defied i Eq. (34). Thi ceario i characterized by log iter-arrival time yieldig zero waitig time. The wort cae ytem time therefore reduce to J Ŝ = j=1 X (j) J J µ + j=1 Γ (j)+ m. Note that thi ceario become le likely with a icreaed umber of queue i erie. Iitially Noempty Queue i Tadem We ext aalyze the cae where 0 > 0. The firt m job i the queue are routed immediately to the erver of the firt queue without ay delay. We are itereted i the behavior for 0 > m. Sice T i = 0 for all i = 1,..., 0, we ca rewrite Eq. (32) a (a) for 0 Ŝ = 0 k 1... k J ν γ U 1 k 2 X (1) r(i) i=k U J X (J) r(i) i=k J (38) (b) for > 0 Ŝ = 0 k 1... k J ν k 1 γ γ<k 1... k J ν U 1 U 1 k 2 X (1) r(i) i=k 1 k 2 X (1) r(i) i=k U J U J X (J) r(i) i=k J X (J) i=k J T U a r(i) i= 0 +1 T U a i=r(k 1 )+1 T i,.(39) T i By Aumptio 1, the wort cae ytem time ivolve olvig J-dimeioal oliear optimizatio problem. Theorem 7 provide a cloed form boud o the wort cae ytem time i a iitially oempty etwork of J idetical queue i tadem, with µ 1 =... = µ J ad α a = α (1) =... = α (J) = α. Theorem 7. (Highet Waitig Time i a Iitially Noempty Tadem Queue) I a iitially oempty etwork of J multi-erver queue i erie with 0 > m, T U a, X (j) U j, for all j = 1,...,J, α a = α (1) =... = α (J) = α, ad Γ = m 1/α Γ a + Γ m > 0, where Γ m i defied i Eq. (34), the wort-cae ytem time for µ 1 =... = µ J ad ρ < 1 i give by

27 Author: Robut Traiet Multi-Server Queue ad Feedforward Network Article ubmitted to Operatio Reearch; maucript o. (Pleae, provide the maucript umber!) 27 ν + J µ + Γ m (ν + J) 1/α ( 0) + + γ a [( 0 ) + ] 1/α, Ŝ Γ[(ν φ) + ] 1/α m(1 ρ) α 1 α α/(α 1) 1/(α 1) Γ α/(α 1) [m(1 ρ)] 1/(α 1), (ν φ) +, if (ν φ) + < [ Γ/m α(1 ρ) ] otherwie α/(α 1). (40) The proof i imilar to the proof of Theorem 6, ad preeted i Appedix A2. Note that, for the cae where Γ = m 1/α Γ a + Γ m 0, the wort cae ytem time Ŝ ( ν + J µ + Γ m (ν + J) 1/α ( 0) + + γ a [( 0 ) + ] 1/α, J J µ + j=1 Γ (j)+ m }. I thi cae, the th job experiece a waitig time oly due to the buildup effect left by the iitial job. For big eough, thi effect become egligible ad the ytem time evetually become equal to um of the ervice time Average Cae Behavior To aalyze the average behavior of a queueig ytem, we treat the parameter (γ a,γ a ) ad (γ (j) m,γ (j) m ) a radom variable ad compute S, the expected value of the wort cae ytem time. Similarly to the cae of a igle queue, we expre S a S = E[{Ŝ b (γ a,γ m ), Ŝ t (Γ a,γ m ) 1 t (Γ a,γ m ) + Ŝ (Γ a,γ m ) 1 (Γ a,γ m )}], where Ŝ b, Ŝ t, ad Ŝ deote the quatitie aociated with the ytem time effected by the iitial buffer 0, the traiet tate ad the teady tate, repectively, ad Γ m i a fuctio of Γ (j) m, for j = 1,...,J, a depicted i Eq. (34). Alo the idicator fuctio 1 t ad 1 reflect the coditio for the ytem to be i the traiet tate ad the teady tate, repectively, with 1 t (Γ a,γ m ) = 1 if m 1/α Γ a + Γ m > 1 (Γ a,γ m ) = 1 otherwie. αm(1 ρ) ( m 0 m ) (α 1)/α By poitig ome aumptio o the ditributio of the parameter {(γ a,γ a ),(γ m,γ m )}, we compute Ŝ via umerical itegratio.

28 Author: Robut Traiet Multi-Server Queue ad Feedforward Network 28 Article ubmitted to Operatio Reearch; maucript o. (Pleae, provide the maucript umber!) Choice of Variability Ditributio For a etwork of J queue i erie, we propoe to expre the parameter Γ a = θ a γ a, Γ (j) = θ γ (j) ad Γ (j) m = θ m γ (j) γ (j) m = θ m m, (α 1)/α where γ a ad γ (j) follow limitig ditributio a defied i the cae of a igle queue, for j = 1,...,J. More pecifically, γ a N (0,σ a ) ad γ (j) N (0,σ ) for light-tailed primitive, γ a S α ( 1,C α,0) ad γ (j) S (1,C α,0) for heavy-tailed primitive. Note that the effective parameter Γ m i captured a a fuctio of Γ (j) m, for j = 1,...,J. Specifically, by Eq. (34), Γ m = ( J (Γ (j)+ j=1 α 1/α m ) α/α 1 ) θ m = m (α 1)/α γ+ where γ + = ( J (γ (j)+ j=1 α 1/α ) α/α 1 ). (41) We propoe a approximatio of the ditributio of γ + by fittig geeralized extreme value ditributio to the ampled ditributio with a hape parameter ψ, cale parameter ξ ad a locatio parameter φ. Table 3 ummarize the parameter defiig the geeralized extreme value ditributio for light-tailed ervice time with σ = 1 ad heavy-tailed queue for J = 10,25 ad 50. Figure 3 how that thi fit provide a good approximatio of the ampled ditributio for the example of J = 25 queue i erie. Table 3 Geeralized extreme value ditributio for γ + for light (σ = 1) ad heavy-tailed ervice. 10 Queue 25 Queue 50 Queue Parameter α = 2 α = 1.6 α = 1.7 α = 2 α = 1.6 α = 1.7 α = 2 α = 1.7 α = 1.6 ψ ξ φ Thi tep allow u to reduce the computatioal effort to obtai Ŝ from olvig a (J + 1)- dimeioal itegral with repect to γ a ad γ (j) to a double itegral with repect to γ a ad γ +. We ext take a imilar approach to chooe (θ a,θ,θ m ) a i the cae of a igle queue. (a) Light-Tailed Queue: The overall expected ytem time i a tadem etwork of idetical queue S = S (1) S (J) = J S (j) J 2(1 ρ) (σ2 a + σ 2 /m 2 ),