On the Transient and Steady-State Analysis of a Special Single Server Queuing System with HOL Priority Scheduling

Transcription

1 96 On the Transent and Steady-State Analyss of a Specal On the Transent and Steady-State Analyss of a Specal Sngle Server Queung Syste wth HOL Prorty Schedulng Faou Kaoun Duba Uversty College, College of Inforaton Technology, PO Box 443 Duba, Uted Arab Erates faoun@ducacae Abstract In ths paper, we consder a specal dscrete-te queung syste wth two head-of-lne (HOL prorty queues and a x of correlated and uncorrelated arrvals The arrval process to the hgh prorty queue s correlated and conssts of a tran of a fxed nuber of fxed-length pacets, whle the low prorty traffc conssts of batch arrvals that are ndependent and dentcally dstrbuted fro slot-to-slot We derve an expresson for the functonal equaton descrbng the transent evoluton of ths prorty queung syste Ths functonal equaton s then apulated and transfored nto a atheatcal tractable for By applyng the fnal-value theore, the correspondng exact expressons for the steady-state argnal pgfs are derved We also show how the transent analyss provdes nsghts nto the dervaton of the syste s busy perod dstrbuton Fnally, we llustrate our soluton techque wth soe nuercal exaples, whereby we deonstrate the negatve effect of correlaton (n the hgh-prorty queue on the perforance of the low-prorty queue Introducton ATM and next generaton coucatons networs are beng bult around the otvaton of havng a sngle cost-effectve pacet-based networ that s capable of supportng dverse classes of servces, each wth ts own QoS requreent To acheve ths goal, varous pacet servce dscplnes that deterne the order by whch pacets are served have been proposed Aong the splest prorty schedulng dscplnes, the non-preeptve head-of-lne (HOL prorty schedulng has been proposed to provde dfferentated servces to the hgh-prorty traffc class Under ths schee, whenever the server s dle, t always schedules the delay-senstve traffc frst (f present In the lterature, there have been varous contrbutons towards the perforance analyss of HOL-based prorty systes (see for exaple [], [], [3], [4] Ths paper focuses on a specal sngle-server dscrete-te queung syste wth two head-of-lne (HOL prorty queues The arrval process to the hgh prorty queue s correlated and conssts of a tran of a fxed nuber of cells, whle the low prorty traffc conssts of batch arrvals that are ndependent and dentcally dstrbuted fro slot-to-slot Our wor departs fro the prevous contrbutons, cted above n the lterature, n at least four aspects: cgests-oct5

2 GESTS Int l Trans Coputer Scence and Engr, Vol8, o 97 Frst, unle prevous studes that erely focused on the steady-state behavor of prorty systes, our analyss as towards explorng the transent behavor of such systes as well As such, the present wor can serve as a bass towards the transent analyss of other prorty systes, under dfferent arrval odels and servce-te dstrbutons Second, whle the effect of correlaton on the hgh-prorty queue (wth correlated arrvals has been thoroughly nvestgated (see for exaple [5], ts pact on the behavor of the low-prorty queue (fed by uncorrelated pacet arrvals has not been addressed It s one of the obectves of ths wor to explore ths portant perforance ssue Thrd, we show how the transent analyss leads to the characteraton of the dstrbuton of the busy perod of the syste, whch s another dstnctve contrbuton of ths paper Fnally, our proposed approach whch s purely based on probablty generatng functons s entrely analytcal, does not nvolve any atrx concepts and provdes a ufed fraewor to extract transent as well as steady-state perforance easures The reang of ths paper s orgaed as follows: In sectons and 3, we descrbe the queung odel and present the functonal equaton for the ont pgf of the state vector In secton 4, the resultng functonal equaton s transfored nto a new for that s atheatcally tractable In sectons 5 and 6, we present the exact transent analyss The correspondng steady-state results are derved n sectons 7 and 8 In sectons 9, we charactere the dstrbuton of the busy perod of the prorty syste uercal results that provde nsghts nto the behavor of the queung syste are provded n secton Fnally, a suary of the an fndngs of the paper and recoendatons for future research are provded n secton Analytcal Model Descrpton In ths paper, we consder a dscrete-te queung syste wth nfte buffer capactes and a sngle (FCFS deterstc server The te axs s dvded nto equal length slots and pacet transsson s synchroed to occur at the slot boundares Here a slot s the te perod requred to transt exactly one pacet fro the buffers We consder two types of prorty traffc, naely a hgh-prorty and a low prorty traffc (thereafter referred to as type- and type, respectvely The schedulng s based on a HOL te-prorty schee, whereby type- pacets have absolute nonpreventve prorty over type- pacets Under ths prorty schee, the server wll always serve type- pacets (f any based on a FCFS bass If there are no type- pacets n the hgh-prorty queue, then type- pacets (f any wll be served; agan based on a FCFS rule We odel the arrval process to the hgh-prorty queue by the correlated real-te traffc coposed of a fxed-length pacet-tran arrval process Correlated tran arrvals are encountered n varous applcatons whereby custoers are essages (eg Fraes or ubo pacets that consst of ultple fxed-length pacets; see eg [6] More precsely, the hgh-prorty queue s fed wth nput lns, recevng fxedlength essages of pacets, each These essages enter the syste at a fxed rate of one pacet per slot In addton, traffc on dfferent nput lns s assued to be GESTS-Oct5

3 98 On the Transent and Steady-State Analyss of a Specal ndependent and wth the sae statstcal characterstcs On any nput ln, the probablty that the frst (leadng pacet of a essage enters the buffer n any gven slot s q f the frst pacet of the prevous essage on ths ln dd not enter the buffer durng the prevous ( - slots and t s ero otherwse Further, let {c ; } be a seres of ndependent and dentcal Bernoull rando varables wth pgf C ( q+ q ew essage arrvals to the low-prorty queue are assued to be d fro slot-toslot and charactered by the sae pgf B ( b E, ndependent of For splcty, t s assued that that each type- essage arrves n a batch of pacets; though our approach can be easly extended to handle varable-length essages Further, we defne the load offered by class- pacets as ρ (,, whle we denote by + the total load to the syste For stablty, we assue that ρ < T T 3 Syste Evoluton and Prelnares The prorty queung odel can be forulated as a dscrete-te Marov chan The state of the syste s defned by ( u,, u,, a,, a,, a, where u, are the syste contents of class (, queue at the end of the th slot, whle an, ( n s the nuber of nput lns havng sent the n th pacet of a essage to type- buffer n slot The total syste content ( u, + u wll be denoted by u, T, Clearly [5]: a a, n ( n, + n, I ( a c ; I a, + n, n The evoluton of the syste contents s descrbed by the followng syste equatons +, +, [, ] + n, + n u u u a u, + + u, [ u, ] + b+ f u, u, + b+ f u, > (3 where [ x ] + s a rando varable whch taes the value f x > and otherwse ext, defne the ont generatng functon of the prorty syste at the end of the th slot as follows: Q x Q x x x E x x u, u, a, a, (,, (,,,, Fro the above and usng (-3, we can easly derve the followng functonal equaton relatng the ont pgf of the syste between two consecutve slots: cgests-oct5

4 GESTS Int l Trans Coputer Scence and Engr, Vol8, o 99 Q(,, ϒ(, x, ϒ(, x Q + (,, x B( [ C( x ] ( Q (,,, + ( Q (,, + (4 x where ϒ (, x and x for notatonal purposes Cx ( Fro the above, t s clear that the presence of the ϒ(, x ters n the rght-hand sde of (4 coplcates the analyss For nstance, tang the lt as on both sdes of (4 does not help n extractng the steady-state ont pgf of the syste In the sequel, we show how to handle the above functonal equaton to extract not only the steadystate but also the transent ont pgf of the queung syste 4 Transforng the Functonal Equaton nto a ew For In ths secton, we show how to transfor the functonal equaton (4 of the syste nto a new for that wll lead tself to a soluton Wthout any loss of generalty, we wll assue ero tal condtons, whereby the syste s tally epty, wth all lns beng n an dle state, e Q (,, x Further, because of the Marovan property of the syste, the steady-state behavor s ndependent of the tal condtons Wth ero tal condtons and by expandng Q + (,, x n (4 for the frst few values of, we can prove by sple nducton the followng aor result whch enables us to express Q (,, x n a ore sutable for 4 Theore : Transent Jont PGF of the Prorty Syste Under ero tal condtons, the functonal equaton (4 descrbng the queung odel under consderaton can be wrtten as follows: Q (,, x where: B( ( [ J, x ( ] + [ J, x ( ] ( + [ J, x ( ] B( Q (,,, B( Q (,,, (5 GESTS-Oct5

5 On the Transent and Steady-State Analyss of a Specal, x ( C, x ( λ ( J (6 + q x λ ( C, x ( λ ( (7 λ ( ( ( q and λ s (,, are the dstnct roots of the characterstc equaton: ( λ ( ( q λ( q (8 The proof of ths theore s by nducton (n a slar way as n [7] and wll not be gven here Further, fro (8, t s obvous that one of the roots has the property that λ ( Ths partcular root s thereafter denoted by λ ( 5 Deternaton of the Transent Boundary Ters To derve the transent boundary ters Q (,, and Q (,, appearng n (5, we proceed as follows: Frst, let us defne the followng three transfors ( w < : Q (,, xw, Q(,, xw ; Pw ( Q(,, w ; R (, w Q(,, w (9 By substtutng Q (,, x fro (5 nto Q (,, xw, as defned above, and usng (6, we derve the followng expresson for the w-transfor of the transent ont pgf of the syste: Q(,, x, w ( R(, w + ( +! n! n! n! wb n + n + + n P( w B( w ( C, x ( ( λ ( n + n + + n! n! n! n! wb( ( C, x ( λ ( n λ ( The above expresson wll not only be useful to derve the two transent boundary ters Q (,, and Q (,,, appearng n (5, but the steady-state results related to our queung odel, as well ( cgests-oct5

6 GESTS Int l Trans Coputer Scence and Engr, Vol8, o 5 Deternaton of the Transent Boundary Constants The transent probabltes of an epty syste Q (,, can be readly obtaned fro ( as follows: Frst settng n ( yelds the followng expresson for the w-transfor of the transent ont pgf of the total syste contents ut, and the nuber of actve nput lnes Q(,, x, w n + n + + n + ( P( w B(! n! n! n! wb w ( C, x( ( n + n + + n λ ( (! n! n! n! wb( ( C, x( λ ( λ ( ext we deterne Pwfro ( the above expresson by nvong the analytc property of ( nsde the polyds ( ; w < For ths purpose, let H λ ( and * ( denote by Y (, w the uque root nsde the ut crcle of the equaton: w B( H ( ( Then, followng a slar approach as n [7], we can apply Rouchés theore to show that P( w (3 * Applyng Lagrange s theore to (-3, allows us to express the boundary functon P(w as follows: whch ples that: [ B( H ( ] w d P ( w + (4! d ( B ( H ( d Q (,, (! d ( or equvalently ( usng Leb s rule for the th dervatve of a product : (5 GESTS-Oct5

7 On the Transent and Steady-State Analyss of a Specal { ( ( } d B ( H Q (,, (6! d 5 Deternaton of the Transent Boundary Functons The boundary ters Q (,, appearng n (5 can be evaluated by explotng the analytcal property of ( nsde the polyds ( ; w < (,, whereby whenever the denonator of Q (,, xw, vashes, the nuerator ust also vash at the sae values We frst derve the w-transfor R(, w of ths boundary ter, then tae the nverse w-transfor to obtan Q (,,, Agan, Let H λ ( * and for a gven < denote by X (, the uque ( root nsde the ut crcle < of the equaton: Fro Rouchés theore, t can be shown that R(, w w B( H ( X (, w( P( w X (, w w (7 (8 The transent boundary functons Q (,, can now be obtaned by tang the nverse w-transfor of (8, gvng: Where: Q (,,, Γ ( + ( Γ( Q (,, (9 ( B ( ( d H Γ (! d and Q (,, are as defned n (6 Fro the above, t s clear that equatons (5-7, (6 and (9- fully charactere the transent ont pgf Q (,, x of the queung syste at the end of the th slot We also note that snce type- traffc s not nfluenced by the low-prorty traffc, the hgh-prorty queue could have been analyed n solaton fro the class- queue A detaled analyss of the resultng sngle-class queue can be found n [7] Our transent and steady-state results related to the hgher- prorty queue, whch are otted heren ( cgests-oct5

8 GESTS Int l Trans Coputer Scence and Engr, Vol8, o 3 due to lac of space, are also n agreeent wth the correspondng results derved n [7] Thus, n the sequel, our analyss wll anly focus on the lower-prorty queue 6 Transent Analyss of Class- (Low-Prorty Queue The argnal pgf P, ( of class- syste contents of at the end of the th slot s readly obtaned fro (5: P, ( ( [ B( ] + [ B( ] [ Q (,,, Q (,,] ( where the boundary ters Q (,, and Q (,,, are as prevously defned n (6 and (9-, respectvely Fro the above argnal pgf, te-dependent perforance easures can be derved For nstance, let, denote the ean queue length of class- queue at the end of the th slot Then fro ( we can show that: dp ( B'( Q (,,, Q (,, (,, 7 Steady-State Jont PGF of the Syste The steady-state ont pgf of the syste s readly obtaned by applyng the fnal-value theore to (: Q (,, x l ( wq (,, xw, (3 or equvalently: Q(,, x ( Q(,,, + ( w [ Q(,, ] ( C x B, ( ( (! n n n n n n + + +!!! B( λ ( λ (4 The only unnowns n the above expresson are the boundary ters Q(,,, and Q (,, These can be readly deterned fro ther transent counterparts, as shown below The frst boundary ter Q(,, s derved by applyng the fnal value-theore to (3: GESTS-Oct5

9 4 On the Transent and Steady-State Analyss of a Specal or equvalently: Q w (,, l( w P( w l w w Y (, w dw Q(,, B'( q * d w + ( q Ths also ples that the total load of the syste s q ( q ρ T B '( + + ext, the second boundary ter Q(,,, appearng n (4 s derved by applyng the fnal value-theore to (8: Q X ( ( (5 (6 (,,, l( w R(, w ( ρ (7 T w X ( where, for gven <, X s the uque root nsde the ut crcle < of the ( ( equaton: B H ( ext, by substtutng (7 nto (4, we obtan the followng equvalent expresson for the steady-state ont pgf of the syste: Q(,, x ( ρ ( B( T! n! n! n n + n + + n X ( X ( ( C, x ( λ (! B( λ ( (8 8 Steady-State Analyss of Class- (Low- Prorty Queue The argnal pgf P ( of class- syste contents s readly obtaned by settng x (, - n (8; gvng: P ( X ( B( ( ρ ( (9 ( X ( ( B( T Alternatvely, by settng x (, - n (4, we can express ( P n ters of the two boundary ters, as follows: cgests-oct5

10 GESTS Int l Trans Coputer Scence and Engr, Vol8, o 5 Q(,,, (,, ( ( ( Q B P (3 B( Applyng the noralaton condton, P ( to (3, yelds ρ q and ρ ρ ρ B'( T + ( q ext let denote the ean buffer occupancy of the low-prorty queue n steadystate By dfferentatng (3 twce wth respect to and settng n the resultng expresson, we get after few nteredate steps: ρ H''( H ''( B''( ( ρ ρ ρt ( ρt ( ρ ( ρt ( ρt (3 where: ( ρ ρ + ρ H''( ( 9 Busy Perod Analyss of the Prorty Queung Syste In order to charactere the probablty dstrbuton of the busy perod of the prorty syste, we wll use the techque descrbed n [] Specfcally, let: Ω ( pr b slots (,,, denote the probablty that the syste s busy for slots Assung that the last te the prorty-syste was epty at the end of the (- th slot, we can express the transent probabltes of an epty syste as: Q (,, Ω ( Q (,, ( > (3 ext, defne the followng w-transfor: Bw ( Ω ( w By substtutng for Q (,, as n (3 nto P(w as gven n (9 we get: Pw ( Bw ( (33 wp( w Snce n general we defne the busy perod of a queung syste as the te between two consecutve dle perods, then the busy perod ust consst of at least one slot (e tated by at least one arrval, whch occurs wth probablty B(( q Un- GESTS-Oct5

11 6 On the Transent and Steady-State Analyss of a Specal der ths conventon, let the rando varable b denote the length of an arbtrary busy perod n nuber of slots and let Bwbe ( the correspondng pgf Clearly: Ω ( ( Ω ( probablty[ b slots] B(( q ( It follows that: * wb(( q Bw ( w B(( q where * s as defned n ( Fro the above pgf, we can use Lagrange s theore to derve the expresson for the correspondng probablty ass functon Ω ( : (34 (35 d B( H( Ω ( ( > B(( q ( +! d + (36 uercal Exaples and Dscussons of the Results In ths secton, we llustrate our approach through soe nuercal exaples, and probe further nto the nterplay between the correlaton of type- traffc and the perforance of class- queue All the nuercal results presented were obtaned usng Maple [8] coputatonal software We have assued that the nuber of essage arrvals to the low-prorty (type- queue follows a Geoetrc Batch arrval process, ρ charactered by the pgf B ( ( + ( In fgure, we plot the transent ean queue length, (, of type- buffer, as well as the transent ean of the total syste content T, In partcular, we note that whle the exponental rse behavor n the transent eante curve of type- queue, s typcal n any other queung systes, the transent ean-te curve of type- queue exhbts a sharper lnear growth behavor Ths can be explaned by the earler observaton that all of the 4 pacets of a type- essage enter ther buffer durng the sae slot, whereas type- essages enter ther buffer at the fxed rate of one pacet/slot cgests-oct5

12 GESTS Int l Trans Coputer Scence and Engr, Vol8, o 7 Transent ean of queue length T,,, Te (nuber of slots Fg Transent eans of buffer occupances ( ρ 8; ρ ; 4; ; 8 ext to nvestgate the effect of type- correlaton on the behavor of type- queue, we plot n fgure, the steady-state ean buffer content of type- queue versus the total load ρt ρ+ ρfor dfferent values of type- correlaton ndcator ( Here, we assued 5, 5, and fxed ρ Steady-state ean buffer occupancy Total load Fg Steady-state ean buffer occupancy of the low-prorty queue As ay be seen fro fgure, the correlaton factor (assocated wth the hghprorty buffer has a sgfcant nfluence on the ean buffer occupancy of type- queue For the sae traffc loads and traffc paraeters, ncreases rapdly as ncreases The above leads us to conclude that the correlaton of type- traffc has a drect negatve pact not only on the hgh-prorty queue t s feedng, but also on the low-prorty queue Therefore pror studes whch gnored ths correlaton effect should be revsted as they ght lead to naccurate buffer densong, adsson and congeston control polces GESTS-Oct5

13 8 On the Transent and Steady-State Analyss of a Specal Conclusons and Suggestons for Further Research In ths paper, we have proposed an alternate soluton techque towards the transent and steady-state analyss of a specal non-preeptve HOL prorty queung syste Our wor revealed that n a HOL-based prorty syste, the correlaton of pacet arrvals to the hgh-prorty queues affects not only these queues, but the lowerprorty queues as well Ths wor can be further explored n any drectons For exaple, varants to our queung odel, such as dfferent arrval processes, fte buffers and ult-server cases can be nvestgated These are left for future research References [] J Walraevens, and H Bruneel, Perforance Analyss of a Sngle-Server ATM Queue wth Prorty Schedulng, Coputers and Operatons Research, Vol 3, o, 3, pp [] M Mehet Al and X, Song, A Perforance Analyss of a Dscrete-Te Prorty Queung Syste wth Correlated Arrvals, Perforance Evaluaton, o 57, 4, pp [3] K Laevens, and H Bruneel, Dscrete-Te Mult-server Queues wth Prortes, Perforance Evaluaton, o 33, 998, pp [4] R Jafar and K Sohraby, Perforance Analyss of a Prorty Based ATM Multplexer wth Correlated Arrvals, IEEE Infoco 99, 999, pp [5] Y Xong and H Bruneel, Perforance of Statstcal ultplexers wth Fte uber of Inputs and Tran Arrvals, n proceedngs of IFOCOM, 99, pp [6] H Bruneel, Pacet Delay and Queue Length of Statstcal Multplexers wth Low- Speed Access Lnes, Coputer etwors, Vol 5, 993, pp [7] F Kaoun, Perforance Analyss of a Dscrete-Te Queung Syste wth a Correlated Tran Arrval Process, Perforance Evaluaton, In Press [8] Maple s a regstered tradear of Maplesoft ( Bography Faou Kaoun PO Box 443 Duba UAE Holds a PhD degree n Electrcal and Coputer Engneerng fro Concorda Uversty, and an MBA degree n Manageent fro McGll Uversty, Canada He oned the College of IT at Duba Uversty College n Septeber, as an Assstant Professor Pror to that, he has been wth ortel etwors (Canada snce 995, where pror to hs departure n, he was a Seor Techcal Advsor n the H-CAP Optcal etwors Dvson Hs research nterests are n the odelng and perforance analyss of coucatons networs Tel: E-al: faoun@ducacae cgests-oct5