Polling Models with Two-Stage Gated Service: Fairness versus Efficiency

Transcription

1 Pollng Models wth Two-Stage Gated Servce: Farness versus Effcency R.D. van der Me a,b and J.A.C. Resng c a CWI, Advanced Communcaton Networks P.O. Box 94079, 1090 GB Amsterdam, The Netherlands b Vrje Unverstet, Faculty of Scences De Boelelaan 1081a, 1081 HV Amsterdam, The Netherlands c Endhoven Unversty of Technology, Mathematcs and Computer Scence P.O. Box 513, 5600 MB Endhoven, The Netherlands October 30, 006 Abstract We consder an asymmetrc cyclc pollng system wth general servce-tme and swtchover tme dstrbutons wth so-called two-stage gated servce at each queue, an nterleavng scheme that ams to enforce farness among the dfferent customer classes. For ths model, we (1) obtan a pseudo-conservaton law, () descrbe how the mean delay at each of the queues can be obtaned recursvely va the so-called Descendant Set Approach, and (3) present a closed-form expresson for the expected delay at each of the queues when the load tends to unty (under proper heavy-traffc scalngs), whch s the man result of ths paper. The results are strkngly smple and provde new nsghts nto the behavor of two-stage pollng systems, ncludng several nsenstvty propertes of the asymptotc expected delay wth respect to the system parameters. Moreover, the results provde nsght n the delay-performance of two-stage gated pollng compared to the classcal one-stage gated servce polces. The results show that the two-stage gated servce polcy ndeed leads to better farness compared to onestage gated servce, at the expense of a decrease n effcency. Fnally, the results also suggest smple and fast approxmatons for the expected delay n stable pollng systems. Numercal experments demonstrate that the approxmatons are hghly accurate for moderately and heavly loaded systems. Keywords: traffc pollng systems, two-stage gated, watng tmes, farness and effcency, heavy 1 Introducton Ths paper s motvated by dynamc bandwdth allocaton schemes n an Ethernet Passve Optcal Network (EPON), where packets from dfferent Optcal Network Unts (ONUs) share channel capacty n the upstream drecton. An EPON s a pont-to-multpont network n A prelmnary verson of ths paper has also been presented at the Second Korea-Netherlands Conference on Queueng Theory and ts Applcatons to Telecommuncaton Systems (Amsterdam, October 4-7, 006). 1

2 the downstream drecton and a mult-pont to pont network n the upstream drecton. The Optcal Lne Termnal (OLT) resdes n the local offce, connectng the access network to the Internet. The OLT allocates the bandwdth to the Optcal Network Unts (ONUs) located at the customer premses, provdng nterfaces between the OLT and end-user network to send voce, vdeo and data traffc. In an EPON the process of transmttng data downstream from the OLT to the ONUs s broadcast n varable-length packets accordng to the 80.3 protocol 6. However, n the upstream drecton the ONUs share capacty, and varous pollng-based bandwdth allocaton schemes can be mplemented. Smple tme-dvson multplexng access (TDMA) schemes based on fxed tme-slot assgnment suffer from the lack of statstcal multplexng, makng neffcent use of the avalable bandwdth, whch rases the need for dynamc bandwdth allocaton (DBA) schemes. A dynamc scheme that reduces the tme-slot sze when there s no data to transmt would allow excess bandwdth to be used by other ONUs. However, the man obstacle of mplementng such a scheme s the fact the OLT does not know n advance how much data each ONU has to transmt. To overcome ths problem, Kramer et al. 7, 8 propose an OLT-based nterleaved pollng scheme smlar to hub-pollng to support dynamc bandwdth allocaton. To avod monopolzaton of bandwdth usage of ONUs wth hgh data volumes they propose an nterleaved DBA scheme wth a maxmum transmsson wndow sze lmt. Motvated by ths, n ths paper we analyze the effectveness of ths nterleaved DBA scheme n a queueng-theoretcal context. To ths end, we quantfy farness and effcency measures related to the expected delay fgures at each of the queues, provdng new fundamental nsght n the trade-off between farness and effcency by mplementng one-stage and two-stage gated servce polces. The two-stage gated servce polcy was ntroduced n Park et al. 10 where the authors study a symmetrc verson of the model descrbed n ths paper. A pollng system s a mult-queue sngle-server system n whch the server vsts the queues n cyclc order to process requests pendng at the queues. Pollng models occur naturally n the modelng of systems n whch servce capacty (e.g., CPU, bandwdth) s shared by dfferent types of users, each type havng specfc traffc characterstcs and performance requrements. Pollng models fnd many applcatons n the areas of computer-communcaton networks, producton, manufacturng and mantenance 9. Snce the late 1960s pollng models have receved much attenton n the lterature 1, 13. There are several good reasons for consderng heavy-traffc asymptotcs, whch have recently started to gan momentum n the lterature, ntated by the poneerng work of Coffman et al., 3 n the md 90s. Exact analyss of the delay n pollng models s only possble n some cases, and even n those cases numercal technques are usually requred to obtan the expected delay at each of the queues. However, the use of numercal technques for the analyss of pollng models has several drawbacks. Frst, numercal technques do not reveal explctly how the system performance depends on the system parameters and can therefore contrbute to the understandng of the system behavor only to a lmted extent. Exact closed-form expressons provde much more nsght nto the dependence of the performance measures on the system parameters. Second, the effcency of each of the numercal algorthms degrades sgnfcantly for heavly loaded, hghly asymmetrc systems wth a large number of queues, whle the proper operaton of the system s partcularly crtcal when the system s heavly loaded. These observatons rase the mportance of an exact asymptotc analyss of the delay n pollng models n heavy traffc. We consder an asymmetrc cyclc pollng model wth generally dstrbuted servce tmes and swtch-over tmes. Each queue receves so-called two-stage gated servce, whch works as follows: Newly ncomng customers are frst queued at the stage-1 buffer. When the server arrves at a queue, t closes a gate behnd the customers resdng n the stage-1 buffer, then serves all customers watng n the stage- buffer on a FCFS bass, and moves all customers before the gate at the stage-1 buffer to the stage- buffer before movng to the next queue. We focus on the expected delay ncurred at each of the queues. Frst, we derve a so-called pseudo-conservaton law for the model under consderaton, gvng a closed-form expresson for a specfc weghted sum of the mean watng tmes. Then, we descrbe how the ndvdual expected delays at each of the queues can be calculated recursvely by means of the so-called Descendant Set Approach (DSA), proposed for the case of standard one-stage gated and ex-

3 haustve servce n 5. The two-stage gated model ntroduces several nterestng complcatons that do not occur for the standard gated/exhaustve case. Denotng by X (k) the number of customers at Q n stage k at an arbtrary pollng nstant at Q (k = 1, ), the mean delay at Q does not only depend on the frst two margnal moments of X (), but also depends on the cross-moments E X (1) X () (see (4.3)). To ths end, we need to consder the numbers of customers at a queue n both stage 1 and stage at pollng nstants at that queue, whch leads to a two-dmensonal analyss. The man result of ths paper s the presentaton of a closed-form expresson for (1 ρ)ew, referred to as the scaled expected delay at Q, when the load tends to 1. The expresson s strkngly smple and shows explctly how the expected delay depends on the system parameters, thereby explctly quantfyng the trade-off between the ncrease n farness and decrease of effcency ntroduced by mplementng two-stage gated servce polces. In partcular, the results provde new fundamental nsght wth respect to mean watng tmes for one-stage versus two-stage gated servce polces. Furthermore, the results reveal a varety of asymptotc nsenstvty propertes, whch provde new nsghts nto the behavor of pollng system under heavy load. The valdty of these propertes s llustrated by numercal examples. In addton, the expressons obtaned suggest smple and fast approxmatons for the mean delay at each of the queues n stable pollng systems. The accuracy of the approxmatons s evaluated by numercal experments. The results show that the approxmatons are hghly accurate when the system load s sgnfcant. The remander of ths paper s organzed as follows. In secton the model s descrbed. In secton 3 we present the pseudo-conservaton law for the model under consderaton. In secton 4 we descrbe how the expected delay fgures can be obtaned by the use of the DSA. In secton 5 we present closed-form expressons for the scaled expected delay at each of the queues, and dscuss several asymptotc propertes. Fnally, n secton 6 we propose and test smple approxmatons for the moments of the watng tmes n heavy traffc, and address the practcalty of the asymptotc results. Model Descrpton Consder a system consstng of N statons Q 1,..., Q N, each consstng of a stage-1 buffer and a stage- buffer. A sngle server vsts and serves the queues n cyclc order. Type- customers arrve at Q accordng to a Posson arrval process wth rate λ, and enter the stage-1 buffer. The total arrval rate s denoted by Λ = N =1 λ. The servce tme of a type- customer s a random varable B, wth Laplace-Steltjes Transform (LST) B ( ) and wth fnte k-th moment b (k), k = 1,. The k-th moment of the servce tme of an arbtrary customer s denoted by b (k) = N =1 λb(k) /Λ, k = 1,. The load offered to Q s ρ = λ b (1), and the total offered load s equal to ρ = N =1 ρ. Defne a pollng nstant at Q as a tme epoch at whch the server vsts Q. Each queue s served accordng to the two-stage gated servce polcy, whch works as follows. When the server arrves at a queue, t closes the gate behnd the customers resdng n the stage-1 buffer. Then, all customers watng n the stage- buffer are served on a Frst-Come-Frst-Served (FCFS) bass. Subsequently, all customers before the gate at the stage-1 buffer are nstantaneously forwarded to the stage- buffer, and the server proceeds to the next queue. Upon departure from Q the server mmedately proceeds to Q +1, ncurrng a swtch-over tme R, wth LST R ( ) and fnte k-th moment r (k), k = 1,. Denote by r := N =1 r(1) > 0 the expected total swtch-over tme per cycle of the server along the queues. All nterarrval tmes, servce tmes and swtch-over tmes are assumed to be mutually ndependent and ndependent of the state of the system. Necessary and suffcent condton for the stablty of the system s ρ < 1 (cf. 4). Let W be the delay ncurred by an arbtrary customer at Q, defned as the tme between the arrval of a customer at a staton and the moment at whch t starts to receve servce. Our man nterest s n the behavor of EW. It wll be shown that, for = 1,..., N, E W = ω 1 ρ + o((1 ρ) 1 ), ρ 1. (.1) 3

4 where the lmt s taken such that the arrval rates are ncreased, whle keepng both the ratos between the arrval rates and the servce-tme dstrbutons fxed. The man result of the paper s a closed-form expresson for ω, n a general parameter settng (see secton 5). Throughout, the followng notaton s used. For each varable x that s a functon of ρ, we denote ts values evaluated at ρ = 1 by ˆx. 3 Pseudo-Conservaton Law In ths secton we present a pseudo-conservaton law (PCL) for the model descrbed above. On the bass of the prncple of work decomposton, Boxma and Groenendjk 1 show the followng result: For ρ < 1, ρ EW = ρ ρ 1 ρ =1 b () r() + ρ b (1) r + r ρ (1 ρ) =1 ρ + E M, (3.1) where M stands for the amount of work at Q at an arbtrary moment at whch the server departs from Q. Then M = M (1) + M (), (3.) where M (k) s the amount of work at stage k at a server departure epoch from Q, k = 1,. Smple balancng arguments lead to the followng expresson for E M : For = 1,..., N, E M = E M (1) + E M () = ρ 4 The Descendant Set Approach =1 r 1 ρ + ρ r 1 ρ. (3.3) For = 1,..., N, defne the two-dmensonal random varable X := X (k) ( ) X (1), X (), where s the number of stage-k customers at Q at an arbtrary pollng nstant at Q when the system s n steady state (k = 1, ), and denote the correspondng Probablty Generatng Functon (PGF) by X (z 1, z ) := E z X(1). (4.1) 1 z X() Denotng the Laplace-Steltjes Transform (LST) of the watng-tme dstrbuton at Q by W ( ), the watng-tme dstrbuton at Q s related to the dstrbuton of X through the followng expressons (cf. 10): For Re s 0, = 1,..., N, ρ < 1, W (s) = X (1 s/λ, B (s)) X (1 s/λ, 1 s/λ ). (4.) E (B (s) 1 + s/λ) X () Then t s easy to verfy that EW can be expressed n terms of the frst two (cross-)moments of X as follows: for = 1,..., N, ρ < 1, ( ρ ) EW = E X () (X () 1) + E X (1) X (). (4.3) λ E X () Note that the frst part of (4.3) s smlar to the formula for the mean delay n the model wth one-stage gated servce polcy. For ths model we have (see Takag 11) EW (one stage) E X(X 1) = (1 + ρ ), (4.4) λ E X wth X the steady-state number of customers at Q at an arbtrary pollng nstant at Q. 4

5 To derve EW, t s suffcent to obtan the frst two factoral moments of X (), and the cross-moments E X (1) X (). To ths end, note frst that straghtforward balancng arguments ndcate that the frst moments EX (k) (k = 1, ) can be expressed n followng closed form: EX (1) = EX () = λr 1 ρ. (4.5) However, n general the second-order moments can not be obtaned explctly. In the lterature, there are several (numercal) technques to obtan the moments of the delay. In ths secton we focus on the Descendant Set Approach (DSA), an teratve technque based on the concept of so-called descendant sets 5. The use of the DSA for the present model s dscussed below. The customers n a pollng system can be classfed as orgnators and non-orgnators. An orgnator s a customer that arrves at the system durng a swtch-over perod. A nonorgnator s a customer that arrves at the system durng the servce of another customer. For a customer C, defne the chldren set to be the set of customers arrvng durng the servce of C; the descendant set of C s recursvely defned to consst of C, ts chldren and the descendants of ts chldren. The DSA s focused on the determnaton of the moments of the delay at a fxed queue, say Q 1. To ths end, the DSA concentrates on( the determnaton ) of the dstrbuton of the two-dmensonal stochastc vector X 1 (P ) := X (1) 1 (P ), X () 1 (P ), where X (k) 1 (P ) s defned as the number of stage-k customers at Q 1 present at an arbtrary fxed pollng nstant P at Q 1 (k = 1, ). P s referred to as the reference pont at Q 1. The man deas are the observatons that (1) each of the customers present at Q 1 at the reference pont P (ether at stage-1 or stage-) belongs to the descendant set of exactly one orgnator, and () the evolutons of the descendant sets of dfferent orgnators are stochastcally ndependent. Therefore, the DSA concentrates on an arbtrary tagged customer whch arrved at Q n the past and on calculatng the number of type-1 descendants t has at both stages at P. Summng up these numbers over all past orgnators yelds X 1 (P ), and hence X 1, because P s chosen arbtrarly. The DSA consders the Markov process embedded at the pollng nstants of the system. To ths end, we number the successve pollng nstants as follows. Let P N,0 be the last pollng nstant at Q N pror to P, and for = N 1,..., 1, let P,0 be recursvely defned as the last pollng nstant at Q pror to P +1,0. In addton, for c = 1,,..., we defne P,c to be the last pollng nstant at Q pror to P,c 1, = 1,..., N. The DSA s orented towards the determnaton of the contrbuton to X 1 (P ) of an arbtrary customer present at Q at P,c. To ths end, defne an (, c)-customer to be a customer present ( at Q ) at P,c. Moreover, for a tagged (, c)-customer T,c at stage 1, we defne A,c := A (1),c, A(),c, where A (k),c s the number of type-1 descendants t has at stage k at P, k = 1,. In ths way, the two-dmensonal random varable A,c can be vewed as the contrbuton of T,c to X 1 (P ). Denote the jont PGF of A,c by A,c(z 1, z ) := E z A(1),c 1 z A (),c. (4.6) To express the dstrbuton of X 1 n terms of the dstrbutons of the DS varables A,c, denote by R,c the swtch-over perod from Q to ( Q +1 mmedately ) after the servce perod at Q startng at P,c. Moreover, denote S,c := S (1),c, S(),c, where S (k),c s the total contrbuton to X (k) 1 of all customers that arrve at the system durng R,c (note that, by defnton, these customers are orgnal customers), and denote the jont PGF of S,c by S,c(z 1, z ) := E z S(1),c In ths way, S,c = 1 z S (),c. (4.7) ( ) S (1),c, S(),c can be seen as the jont contrbuton of R,c to X 1. It s 5

6 readly verfed that we can wrte X 1 = ( ) X (1) 1, X() 1 = =1 c=0 ( ) S (1),c, S(),c = =1 c=0 S,c. (4.8) Note that S (1),c and S(),c are dependent f (, c) = (, c ) but ndependent otherwse. Hence we can wrte, for z 1, z 1, X 1 (z 1, z ) = N S,c(z 1, z ). (4.9) =1 c=0 Because S,c s the total jont contrbuton to X 1 of all (orgnal) customers that arrve durng R,c, the jont dstrbuton of S,c can be expressed n terms of the dstrbutons of the DSvarables A,c as follows: For = 1,..., N, c = 0, 1,..., and z 1, z 1, S,c(z 1, z ) = R ( N j=+1 λj λ ja j,c(z 1, z ) + λj λ ja j,c 1(z 1, z ) ). (4.10) To defne a recurson for the evoluton of the descendant set, note that a customer at stage-1 present at Q 1 at the pollng nstant at Q 1 durng cycle c s served durng the next cycle, whch lead to the followng relaton: For = 1,..., N, c = 0, 1,..., and z 1, z 1, ( N A,c(z 1, z ) = B λj λ ja j,c 1(z 1, z ) + λj λ ja j,c (z 1, z ) ), (4.11) j=+1 supplemented wth the bass for the recurson A, 1(z 1, z ) = z 1 I {=1}, and A, (z 1, z ) = z I {=1}, (4.1) where I E s the ndcator functon of the event E. In ths way, relatons (4.9)-(4.1) gve a complete, characterzaton of the dstrbuton of X 1. Smlarly, recursve relatons to calculate the (cross-)moments of X 1 can be readly obtaned from those equatons. 5 Results In ths secton we wll present heavy-traffc results that can be proven by explorng the use of the DSA. For compactness of the presentaton, the detals of the proofs are omtted. Theorem 1 For = 1,..., N, where lm ρ 1 δ := (1 ρ) E X () ( ) X () 1 = lm ρ 1 j=1 j=1 (1 ρ) E X (1) X () = ˆλ r + r b (), (5.1) δ b (1) ˆρ (3 + ˆρ ). (5.) =1 Proof: The result can be obtaned along the lnes smlar to the dervaton of the results for the one-stage pollng models n 14, 15. Theorem (Man result) For = 1,..., N, (3 + ˆρ ) b () r(3 + ˆρ) ω = N j=1 ˆρj(3 + ˆρj) +. (5.3) b (1) Proof: The results follows drectly by combnng (4.3), (4.5) and Theorem 1. 6

7 Theorem reveals a varety of propertes on the dependence of the lmt of the scaled mean watng tmes wth respect to the system parameters. Corollary 1 (Insenstvty) For = 1,..., N, (1) ω s ndependent of the vst order, () ω depends on the swtch-over tme dstrbutons only through r,.e., the total expected swtch-over tme per cycle, (3) ω depends on the second moments of the servce-tme dstrbutons only through b (),.e., the second moment of the servce tme of an arbtrary customer. Corollary 1 s known to be not generally vald for stable systems (.e., for ρ < 1), where the vst order, the second moments of the swtch-over tmes and the ndvdual second moments of the servce-tme dstrbutons do have an mpact on the mean watng tmes. Hence, Corollary 1 shows that the nfluence of these parameters on the mean watng tmes vanshes when the load tends to unty, and as such can be vewed as lower-order effects n heavy traffc. Let us now dscuss the trade-off between effcency and farness for the one-stage and two-stage gated servce polces, usng the exact asymptotc results presented n Theorem. To ths end, denote by ω (one stage) and ω (two stage) the heavy-traffc resdues of the mean watng tmes for the case of one-stage and two-stage gated servce at all queues, respectvely, defned n (.1). For the case of one-stage gated servce at all queues, the followng results holds (cf. 14): For = 1,..., N, ω (one stage) = (1 + ˆρ ) b () r(1 + ˆρ) N j=1 ˆρj(1 + ˆρj) +. (5.4) b (1) Also, for the one-stage gated pollng model the pseudo-conservatom law s gven by equaton (3.1), supplemented wth EM = ρ r 1 ρ. (5.5) Denote by V the amount of watng work n the system. Then usng Lttle s Law and straghtforward arguments t s readly verfed that, for ρ < 1, EV = ρ EW. (5.6) =1 Throughout, we wll use EV to be the measure of effcency: for a gven set of parameters, a combnaton of servce polces (at each of the queues) s sad to be more effcent than another combnaton of polces f the resultng value of EV s smaller. Denotng by V (one stage) and V (two stage) the amount of work n the one-stage and two-stage gated model, respectvely, the followng result follows drectly from (3.1)-(3.3), (5.5) and (5.6). Corollary (Effcency) Two-stage gated servce s less effcent than one-stage gated servce n the sense that EV (one stage) < EV (two stage). (5.7) Defnton of unfarness: For a gven pollng model, the unfarness s defned as follows: F := max,j=1,...,n EW EW 1 j. (5.8) Note that accordng to ths defnton, the hgher F, the less far s the servce polcy. Note also that the symmetrc systems are optmally far, n the sense that F = 0. The followng 7

8 result follows drectly from Theorem and (5.4). Corollary 3 (Farness) Two-stage gated servce s asymptotcally more far than one-stage gated servce n the followng sense: For, j = 1,..., N, ω (two stage) ω (two stage) j 1 = 3 + ˆρ ˆρ j < 1 + ˆρ ˆρ j = ω (one stage) ω (one stage) j 1. (5.9) Corollares and 3 gve asymptotc results on the relatve farness and effcency between one-stage and two-stage gated servce. To assess whether smlar results also hold for stable systems (.e., whth ρ < 1) we have performed extensve numercal valdaton. The results are outlned below. Consder the asymmetrc model wth the followng system parameters: N = ; the servce tmes at both queues are determnstc wth mean b (1) 1 = 0.8 and b (1) = 0.; the swtch-over tmes are determnstc wth r (1) 1 = r (1) = 1, and the arrval rates at both queues are equal. For ths model, we have calculated the expected watng tmes at both queues and the unfarness measure (5.8), for dfferent values of the load, both for one-stage and two-stage gated servce at all queues. Table 1 below shows the results. one-stage gated two-stage gated ρ EW 1 EW F EW 1 EW F Table 1. Mean watng tmes and farness for dfferent values of the load: one-stage versus two-stage gated. Note that for the model analyzed n Table 1 we have ˆρ 1 = 4/5, ˆρ = 1/5, so that t s readly seen that for the one-stage gated model F tends to 1/ as ρ goes to 1, whereas for two-stage gated F tends to 3/16 = n the lmtng case. The results shown n Table 1 show that the two-stage gated servce s ndeed more far than the one-stage gated servce for all values of the load, whch suggests that Corollary 3 s also applcable to stable systems. We suspect that ths type of results may be proven rgorously; however, a more detaled analyss of the relatve farness s beyond the scope of ths paper. 6 Approxmaton Equaton (.1) and Theorem suggest the followng approxmaton for EW n stable pollng systems: For = 1,..., N, ρ < 1, EW (app) := ω 1 ρ, (6.1) where ω s gven by Theorem. To assess the accuracy of the approxmaton n (6.1), n terms of How hgh should the load be for the approxmaton to be accurate?, we have performed numercal experments to test the accuracy of the approxmatons for dfferent values of the load of the system. The relatve error of the approxmaton of EW s defned as follows: For = 1,..., N, ( ) EW (app) EW % := abs 100%. (6.) EW For the model consdered n Table 1 above, Table shows the exact (obtaned va the DSA dscussed n secton 3) and approxmated values (obtaned va (6.1)) of EW 1 for dfferent 8

9 values of the load, both for the model n whch all queues receve one-stage gated servce, and for the model n whch all queues receve two-stage gated servce. one-stage gated two-stage gated ρ EW (app) 1 EW 1 % EW (app) 1 EW 1 % Table. Exact and approxmated values for EW 1 for dfferent values of the load. The results n Table demonstrate that the relatve error of the approxmatons ndeed tends to zero as the load tends to 1, as expected on the bass of Theorem. Moreover, the results show that the approxmaton converges to the lmt rather quckly when ρ 1. Roughly, the results are accurate when the load s 80% or more, whch demonstrates the applcablty of the asymptotc results for practcal heavy-traffc scenaros. Acknowledgment: The authors wsh to thank Onno Boxma for nterestng dscussons on ths topc. References 1 O.J. Boxma and W.P. Groenendjk (1987). Pseudo-conservaton laws n cyclc-servce systems. J. Appl. Prob. 4, E.G. Coffman, A.A. Puhalsk and M.I. Reman (1995). Pollng systems wth zero swtchover tmes: a heavy-traffc prncple. Ann. Appl. Prob. 5, E.G. Coffman, A.A. Puhalsk and M.I. Reman (1998). Pollng systems n heavy-traffc: a Bessel process lmt. Math. Oper. Res. 3, C. Frcker and M.R. Jaïb (1994). Monotoncty and stablty of perodc pollng models. Queueng Systems 15, A.G. Konhem, H. Levy and M.M. Srnvasan (1994). Descendant set: an effcent approach for the analyss of pollng systems. IEEE Trans. Commun. 4, G. Kramer, B. Muckerjee and G. Pesavento (001). Ethernet PON: desgn and analyss of an optcal access network. Phot. Net. Commun. 3, G. Kramer, B. Muckerjee and G. Pesavento (00). Interleaved pollng wth adaptve cycle tme (IPACT): a dynamc bandwdth allocaton scheme n an optcal access network. Phot. Net. Commun. 4, G. Kramer, B. Muckerjee and G. Pesavento (00). Supportng dfferentated classes of servces n Ethernet passve optcal networks. J. Opt. Netw. 1, H. Levy and M. Sd (1991). Pollng models: applcatons, modelng and optmzaton. IEEE Trans. Commun. 38, C.G. Park, D.H. Han, B. Km and H.-S. Jung (005). Queueng analyss of symmetrc pollng algorthm for DBA schemes n an EPON. In: Proceedngs Korea-Netherlands Jont Conference on Queueng Theory and ts Applcatons to Telecommuncaton Systems, ed. B.D. Cho (Seoul, Korea), H. Takag (1986). Analyss of Pollng Systems (MIT Press, Cambrdge, MA). 1 H. Takag (1990). Queueng analyss of pollng models: an update. In: Stochastc Analyss of Computer and Communcaton Systems, ed. H. Takag (North-Holland, Amsterdam),

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