Research Article On the Modelling of the Mobile WiMAX (IEEE e) Uplink Scheduler

Transcription

1 Modellng and Smulaton n Engneerng Volume 010, Artcle ID , 7 page do: /010/ Reearch Artcle On the Modellng of the Moble WMAX IEEE 80.16e) Uplnk Scheduler Darmawaty Mohd Al 1, and Kaharudn Dmyat 3 1 Department of Electrcal Engneerng, Faculty of Engneerng, Unverty of Malaya, Kuala Lumpur, Malaya Faculty of Electrcal Engneerng, Mara Unverty of Technology UTM) Shah Alam, Selangor, Malaya 3 Electrcal and Electronc Engneerng Department, Faculty of Engneerng, Natonal Defene Unverty of Malaya, Kem Sunga Be, Kuala Lumpur, Malaya Correpondence hould be addreed to Darmawaty Mohd Al, darma504@yahoo.com.my Receved 18 May 010; Reved 9 November 010; Accepted December 010 Academc Edtor: Farouk Yalaou Copyrght 010 D. Mohd Al and K. Dmyat. Th an open acce artcle dtrbuted under the Creatve Common Attrbuton Lcene, whch permt unretrcted ue, dtrbuton, and reproducton n any medum, provded the orgnal work properly cted. Packet chedulng ha drawn a great deal of attenton n the feld of wrele network a t play an mportant role n dtrbutng hared reource n a network. The proce nvolve allocatng the bandwdth among uer and determnng ther tranmon order. In th paper an uplnk UL) chedulng algorthm for the Moble Worldwde Interoperablty for Mcrowave Acce WMAX) network baed on the cyclc pollng model propoed. The model n th tudy cont of fve queue UGS, ertps, rtps, nrtps, and BE) vted by a ngle erver. A threhold polcy mpoed to the nrtps queue to enure that the delay contrant of real tme traffc UGS, ertps, and rtps) not volated makng th approach orgnal n comparon to the extng contrbuton. A mathematcal model formulated for the weghted um of the mean watng tme of each ndvdual queue baed on the peudo-conervaton law. The reult of the analy are ueful n obtanng or tetng approxmaton for ndvdual mean watng tme epecally when queue are aymmetrc where each queue may have dfferent tochatc charactertc uch a arrval rate and ervce tme dtrbuton) and when ther number large more than queue). 1. Introducton Moble WMAX a promng Broadband Wrele Acce BWA) that ha receved great nteret due to the need for data acce at all tme. The alent feature of the Moble WMAX that are attractve are the capablty of handlng the qualty of ervce QoS) and the phycal layer PHY), calablty, and the medum acce control MAC) layer, whch are excluvely degned to meet dfferent knd of traffc. IEEE 80.16e doe defne the mean and method n upportng the dfferent clae of traffc. However, t doe not pecfy how to effectvely chedule and guarantee the QoS accordng to the dfferent type of applcaton. An extenve number of chedulng cheme have been propoed by reearcher, and the mot common approach ha been by mean of mulaton. Smulaton a wdely ued technque for computng the performance meaure of all knd of model. Wth uch manner, the performance of the Moble WMAX ytem n term of delay, throughput, jtter, and packet lo evaluated. However, depte of the flexblty, mulaton may be rather neffcent n many cae and reult baed on mulaton are relatvely naccurate a compared to the mathematcal analy [1]. A number of work have been focung on evaluatng the analytcal model of the chedulng cheme n the context of Moble WMAX ytem. The author n [] put forward the dea of chedulng algorthm for the voce over IP VoIP) ervce. The algorthm clamed to olve the problem of the wate of the uplnk reource algorthm caued by Unolcted Grant Servce UGS) and MAC overhead and acce delay due to the Real-Tme Pollng Servce rtps) algorthm. The ytem model repreented a a onedmenonal Markov chan wth an on-off model for the voce traffc. From here the average number of voce uer n on-tate, the maxmum number of uer that can be ervced n one MAC frame, the ytem throughput, and the acce delay were etablhed. In [3] a novel chedulng cheme wa preented to provde QoS atfacton and ervce dfferentaton n term of delay. The tme wndow, T, of the Proportonal Farne

2 Modellng and Smulaton n Engneerng PF) manpulated to dtnguh the ervce and QoS aurance of each queue. The mean queue length and mean watng tme of the ytem derved ung the M/M/1 Markov model. The tudy n [4] concentrate on the performance analy for the pollng rtps and Non-Real-Tme Pollng Servce nrtps)) ervce baed on an approxmaton model known a the exhautve tme-lmted pollng model wth vacaton. Furthermore, the model aume that the UGS and pollng ervce PS) clae are completely parttoned whch allow for ndependent analy of the PS traffc. The blockng probablty, mean redence tme, and watng tme dtrbuton are derved from the BMAP/D/1/K model. The author n [5] formulated the analytcal model takng nto account the modulaton and codng cheme MCS) a well a the gnalng overhead n the downlnk. The VoIP packet are cheduled baed on the Frt-In-Frt-Out FIFO) polcy, and the traffc modeled a an exponentally dtrbuted on-off model. The aggregate VoIP traffc from the N ubcrber taton SS) repreented a the two-tate Markov-modulated Poon proce MMPP). The analy done ndependently for the Extended-Real Tme Pollng Servce ertps), UGS, and rtps. Though the above tudy focued on the analytcal model a part of the fndng, the model derved aumng that the traffc from dfferent clae n the Moble WMAX totally eparated whch allow for ndependent analy. However, th not alway true n the actual condton. Moble WMAX network are degned to allow for the coextence of varou type of traffc rangng from real-tme traffcuch a VoIP and vdeo conferencng to non-real-tme traffc uch a fle tranfer and e-mal ervce through varou chedulng ervce. Thu, employng a ngle chedulng ervce would lead certan applcaton to fal to functon properly. For ntance, delay the mot crtcal ue for the real-tme traffc and hould not ut wth chedulng ervce wthout the QoS requrement uch a nrtps and BE) conderng that the performance could deterorate. In th paper, dfferent chedulng ervce of the Moble WMAX are combned to obtan the weghted um of mean watng tme for each queue. Owng to the extreme complexty of the analy whch due to the amount and aymmetrc charactertc of each queue, an approxmaton model referred to a peudoconervaton law ued. In the cae of ymmetrc queue, the law gve exact expreon of the mean watng tme [6, 7]. On the contrary, an accurate approxmate model can be obtaned a n the cae of aymmetrc queue. We frt decrbe an uplnk chedulng cheme of the Moble WMAX whch baed on cyclc pollng model followed by our mathematcal model for the propoed cheme. Then the weghted um of the mean watng tme for the ndvdual queue or alo known a peudoconervaton law for the propoed chedulng algorthm obtaned. The ret of the paper organzed a follow. Secton dcue the ytem model of the propoed chedulng cheme, and Secton 3 preent the detaled decrpton. Secton 4 devoted to the dervaton of the peudoconervaton law, and fnally the paper concluded n Secton 5.. Sytem Model We conder a UL cheduler n Moble WMAX BS whch act a a erver. The cheduler erve 5 queue where chedulng ervce Unolcted Grant Servce UGS), Extended Real- Tme Pollng Servce ertps), Real-Tme Pollng Servce rtps), Non-Real-Tme Pollng Servce nrtps), and Bet Effort BE) are ymbolzed a Q 1, Q, Q 3, Q 4, and Q 5, repectvely. Each queue ha an nfnte buffer capacty any number of bandwdth requet can wat wthout lo upon arrval) to tore the watng or ncomng bandwdth requet. Infnte buffer capacty aumpton often the bet approxmaton that help to at the analy [8]. The bandwdth requet or grant arrve at all queue accordng to the ndependent Poon procee wth an arrval rate of λ 1, λ,..., λ N. The compound arrval rate gven a follow: Λ = λ. 1) Though the arrval of UGS and ertps grant modeled a a contant bt rate, n the cae of multple eon arrval, the Poon proce provde adequate accuracy [9]. Smlarly, let b be the ervce tme of Q wth the frt two moment: β = E[b ], β ) = E [ b ]. ) TheerverwtchefromQ to the next queue wth nonzero walk tme wth the frt moment,, and the econd moment, ). The frt moment of total walk tme,,gvena =. 3) The wtchover proce ndependent, and the arrval and ervce tme aumed to be ndependent and dentcally dtrbuted d). The offered traffc at the th queue ρ = λ β, and the total offered traffc ρ = ρ. 4) We conder a ngle erver ervng N queue a Q 1, Q,...Q N. The order of ervce wthn each queue frt-come-frt-erve FCFS). The queue are attended by a erver n cyclc order and ncur nonzero wtchover tme. Each queue wll be erved under exhautve ervce dcplne n whch the erver contnue to work untl the queue become empty. When Q empty, erver mmedately begn to wtch to Q +1. The propoed chedulng algorthm workafollow. The erver erve Q 1 untltherearenomorebandwdth requet avalable. When Q 1 empty, the erver mmedately begn to wtch to Q 1. Q erved next and once the queue empty, the erver wll move to Q 3.AtQ 3, when the queue empty, the erver wll check on the amount of bandwdth requet avalable at Q 4. If the amount of bandwdth requet exceed the threhold agned, n, then the erver wll carry out the ervce to Q 4 and ubequently Q 5. On the other hand, the erver wll return to ervce Q 1 f the amount of the bandwdth requet le than the threhold agned.

3 Modellng and Smulaton n Engneerng 3.1. Defnton of Key Parameter. Wedefneomeofthe mportant parameter a a reference to th model. The cycle tme, C, for Q the tme between two ucceve arrval or vt at Q by the erver. The mean cycle tme, E[C ], ndependent of do not depend on the ucceve tartng pont of the cycle) and equal to the um of total wtchover tme and total mean amount of work departng durng a cycle whch alo equal to mean amount of work arrved n a cycle, ρe[c][10]): E[C ] = + ρ E[C ]. 5) Hence, E[C] =. 6) The vt perod of Q, V, defned a the tme the erver pend ervng bandwdth requet at Q. The erver workng at a fracton ρ of the tme at Q, thu, the mean vt perod of Q, ρ E[C], equal to E[V ] = ρ. 7) Intervt tme, I, of Q the total tme the erver unavalable to Q durng a cycle. A cycle tme the um of vt perod, and the ntervt tme, E[I ], can be derved a follow: E[C] = E[I ] + ρ, E[I ] = ρ = 8)... Stablty Crtera. For a ytem to be ergodc, ρ < 1 would be the neceary condton to enure tablty for the exhautve ervce. 3. Related Work In equence to the chedulng ervce that have been degned by the WMAX forum to effcently delver broadband data ervce uch a vdeo, voce, and data, we have come to the dea that an algorthm mplemented n the Moble WMAX uplnk cheduler mut be twofold: 1) able to blend effcently wth the bandwdth requet trategy and bandwdth allocaton and ) decde on the algorthm takng nto account the mplementaton and computaton complexty. The IEEE 80.16e adopt a centralzed baed chedulng wth the ntenton that the reource can be dynamcally allocated and the cheduler can provde uperor QoS for both the UL and downlnk DL). A proper choce of good chedulng algorthm [11] for the UL cheduler crtcal a t wll enure that performance more predctable and the QoS better enforced [1]. There are fve type of chedulng ervce defned n the tandard: UGS, rtps, ertps, nrtps, and BE. UGS approprate for a fxed zed packet tranmtted at perodc nterval. Thu, bandwdth requet not requred. rtps utable for real-tme data treamng wth varable packet ze, and the MS can end bandwdth requet n repone to the bandwdth requet opportunte agned by the bae taton BS) through pollng opportunte. ertps added to the IEEE 80.16e to deal wth the varable rate of realtme applcaton. The chedulng ervce degned wth the ba of UGS and rtps, and BS wll provde uncat grant n unolcted manner to the ervce. nrtps ut for a delayed tolerant applcaton wth varable packet ze. Smlar to the rtps, nrtps wll be polled by the BS on a regular ba not neceary perodc). Fnally BE recommended for data treamng wth no throughput and delayed guarantee. MS may ue both, whch ether uncat or contenton requet. Snce the tandard doe not dtnctly defne the chedulng algorthm to be appled, we have propoed a UL chedulng algorthm baed on the cyclc pollng model. The delay property crtcal to real-tme traffc nce any volaton to the delay bound may caue the packet to be dcarded. One way of accomplhng th to agn dfferent prorte to the real-tme traffc voce,vdeo)and non-real-tme traffc. Thu, a threhold-type of ervce eem to be utable for th purpoe. In order to agn that, we explot the nrtps feature to guarantee the delay property of the real-tme traffc by mpong a threhold polcy. The erver vt poll) the UGS, ertps, and rtps n a cyclc manner, erve each queue wth exhautve ervce trategy, and contnue ervng f the requet n nrtps buffer do not exceed the predetermned threhold, n, a hown n Fgure 1. If the number of requet n nrtps greater than or equal n, the erver wll ntantly begn to ervce the requet n nrtps after ervng the rtps queue wth an exhautve ervce polcy too a hown n Fgure. Th to enure that the future opportunte for the non-real-tme traffc chedulng ervce are not tarved. To ncorporate the correlaton between each of the chedulng ervce, wtchover tme ntroduced. Swtchover tme defned a the amount of tme taken by the erver to wtch from one chedulng ervce to another. Thu, the analytcal model wll not be etablhed ndependently for each chedulng ervce, ntead, t wll be etablhed a a whole. In order to do that, the peudoconervaton law wll be ued to derve the model. 4. Peudoconervaton Law In the tuaton of zero wtchover tme, the erver work when there work n the ytem and become dle f no work preent. Thu, the conervaton law hold for the total amount of work n the ytem. The amount of work n the ytem ndependent of the ervce polcy and, hence, equal to the amount of work n an M/G/1 queue [13 16] wth the arrval rate of λ and ervce tme of β : N λ β ) E[V M/G/1 ] = ρ ), N λ β ) ρ EW = ρ ). 9)

4 4 Modellng and Smulaton n Engneerng UGS Bandwdth requet from MS Clafer ertps rtps nrtps BE Exhautve Bandwdth requet <n WMAX uplnk cheduler MAP generator Fgure 1: The Moble WMAX uplnk cheduler for threhold le than n. Exhautve UGS Bandwdth requet from MS Clafer ertps rtps nrtps BE Bandwdth requet n WMAX uplnk cheduler MAP generator Fgure : The Moble WMAX uplnk cheduler for threhold greater than or equal to n. In the cae of nonzero wtchover tme, the amount of work the weghted um of the watng tme, ρ EW.LetW be the watng tme, and let t defne tme between the arrval of bandwdth requet at Q and the moment at whch t tart to receve ervce wth mean EW ) no longer ndependent of the ervce polcy. Th called the peudoconervaton law and etablhed a follow [13 16]: V c dv M/G/1 + Y, 10) where, d the equalty n dtrbuton, V c the amount of work n a cyclc ytem at an arbtrary epoch, V M/G/1 the amount of work n a correpondng M/G/1 ytem at an arbtrary epoch, Y the amount of work n a cyclc-ervce ytem at an arbtrary epoch n a wtchng, and perod/nonervng nterval, where V M/G/1,andY are ndependent. Here, epoch degnate an ntant n tme. It follow that E[V c ] = E[V M/G/1 ] + E[Y]. 11) Moreover, E[Y] can be defned a E[Y] = E[Y ]. 1) From [15], E[Y] compoed of three term: EM 1), EM ), and ρ ) / ), where EM 1) the mean amount of work n Q at a departure epoch of the erver from Q, EM ) the mean amount of work n the ret of the ytem at a departure epoch of erver from Q,andρ ) / ) the mean amount of work arrved n the ytem n the precedng part of the wtchng nterval under conderaton: EM 1) E[Y ] = EM 1) + EM ) + ρ ). 13) totally dependent on the choce of ervce tratege appled to the queue and can only be determned when the ervce trategy dentfed. Therefore, parameter EM ) wll be nvetgated n the next ecton, that, the mean amount of work n the ret of the ytem at a departure epoch of erver from Q when the requet n nrtps are le than n and greater than or equal to n,repectvely Dervaton of EM ) When Requet n nrtps Are Le Than n. A dcued n the prevou ecton, when the bandwdth requet n Q 4 le than n, the erver wll only carry out the ervce for Q 1, Q,andQ 3 cyclcally wth an exhautve ervce polcy. In th cae, Q 4 and Q 5 wll not be attended by

5 Modellng and Smulaton n Engneerng 5 the erver a f the queue do not ext n the cheduler. Thu, a cycle n the condton when requet n nrtps are le than n cont of erver vt to Q 1, Q,andQ 3 n a cyclc order. The total wtchover tme and total offered traffc aredefned a = 3 and ρ = 3 ρ,repectvely.from7), the mean vt tme of Q E[V ] = ρ, 14) that, Q vted once n a cycle. Thu, the total amount of work at the departure epoch of erver from Q 1, Q,andQ 3 can be derved a follow: 1 EM) 1 + EM) + 3 EM) 3, 15) where ) EM ) = ρ ρ + ρ ) + ρ ρ + 16) + ρ ρ + + ρ +3 + ) ρ. Solvng 16) and expandng 15) yeld ) )] 1 [ρ + ρ ρ 1 + ρ ρ 1 + )] [ρ ρ ρ )+ρ 1 1 +ρ ) )] 3 [ρ + ρ 3 + ρ ρ + + ρ 3 17) whch can then be mplfed nto 3 EM ) = ρ h k + ρ h ρ k. 18) h <k 4.. Dervaton of EM ) When Requet n nrtps Are Greater Than or Equal to n. When the bandwdth requet n Q 4 greater than or equal to n, the erver wll vt Q 4 after the completon Q 3 empty) of Q 3 and ubequently Q 5. Q 4 and Q 5 wll be erved accordng to the exhautve ervce trategy. Thu, a cycle n the condton when requet n nrtps are greater than or equal to n cont of erver vt to Q 1, Q, Q 3, Q 4,andQ 5 n a cyclc order. The total wtchover tme and total offered traffc n the condton that requet n nrtps are greater than or equal to n are = 5 and ρ = 5 ρ,repectvely. Even though the erver vt Q 4 and Q 5 at tme when the requet of nrtps are greater than or equal to n, the amount h <k of work that arrve n a cycle tll hold for Q 4 and Q 5 n vew of the fact that the defnton of the cycle tme for Q 4 and Q 5 tll vald Q 4 and Q 5 are vted by the erver once n a cycle). A a reult, the amount of work n Q 4 can be derved a follow. Let q, n be the teady-tate probablty that the queue length of Q n when the erver arrve. Therefore, q,n = 1. N 1 n=1 q,n the probablty that n=0 le than n bandwdth requet are erved n a vt perod. 1 N 1 n=1 q,n the probablty that greater than or equal to n bandwdth requet are erved n a vt perod. λ /) the mean number of bandwdth requet arrval durng a vt perod. We obtaned the expreon of the mean number of meage erved n a vt perod whch alo equal to the mean number of meage arrvng durng a cycle: 1 1 nq,n + N 1 q,n = λ n=1 n=n. 19) n=0 The amount of work n Q 4 can be etablhed a follow: 1 1 β nq,n + β 1 q,n = β λ = ρ. n=1 n=n N n=0 0) The frt term denote the amount of work when le than n bandwdth requet arrve at Q 4 and the erver tll ervng Q 1, Q,andQ 3. The econd term reflect the amount of work when the requet greater than or equal to n. At th pont, the erver wtche to Q 4 at the completon of Q 3 and later Q 5. The thrd term repreent the amount of work that arrve n a cycle or a vt perod to Q 4. Conequently, the amount of work that arrve n Q 4 denoted a ρ 4. 1) Equaton 1) alo hold for the amount of work that arrve n Q 5. Thu the total amount of work at the departure epoch of erver from Q 1, Q, Q 3, Q 4,andQ 5 can be obtaned from 1 EM) 1 + EM) + 3 EM) EM) EM) 5. ) By notng that EM ) gven by 16) and olvng ) whch then can be mplfed to: EM) = ρ h k + ρ h ρ k. 3) h <k h <k 4.3. Dervaton of Peudoconervaton Law for the Uplnk Scheduler. Wth reference to 11), E[V c ]defneda[8, 14] E[V c ] = EW + ρ 1 λ β ). 4)

6 6 Modellng and Smulaton n Engneerng Solvng 11)from9)and4), From 1), N λ β ) ρ EW = ρ ) + E[Y]. 5) E[Y] = E[Y ], 6) whch conequently compoed of 3 E[Y ] = EM 1) + EM ) + EM ) + ρ ), 7) where EM ) derved baed on the condton of the amount of bandwdth requet n nrtps whch are le than n and greaterthanorequalton. EM 1) depend on the ervce trategy appled to the queue. For exhautve ervce, EM 1) = 0 nce there are no bandwdth requet left behnd the erver leavng the queue. Thu, E[Y] defned a follow: 3 E[Y] = ) 3 3 ρ ρ + ) ρ ρ + ρ ). 8) By nertng 8) nto 5), fnally, the weghted um of the mean watng tme for the ndvdual queue or peudoconervaton law can be wrtten a follow: 5 λ β ) 3 ρ EW = ρ ) + ) 3 3 ρ ρ + ) ρ ρ + ρ ). 9) It worth mentonng that the order of the queue n the cycle doe not nfluence the mean workload of the ytem or the weghted um of mean watng tme a n 9) along a t doe not affect and E[Y]E[Y] appear to be lnearly dependent on the mean total wtchover tme [17]. 5. Concludng Remark In th paper, we have preented the ue of the peudoconervaton law to derve the weghted um of the mean watng tme for the propoed chedulng algorthm of the Moble WMAX uplnk cheduler. The propoed chedulng algorthm baed on a cyclc pollng model wth the threhold polcy mpoed on Q 4 or nrtps and each queue erved under the exhautve dcplne. We have obtaned the weghted um of the mean watng tme for the ndvdual queue, n vew of the fact that the explct expreon for ndvdual watng tme wa n general very complcated. From the dervaton n Secton 4., ofworkatq 4 and Q 5, that, when the threhold greater than or equal to n, condton equvalent to the amount of work that arrve n a cycle that the amount of work that arrve n a cycle tll hold for Q 4 and Q 5 ). Peudoconervaton law eem to be very ueful n everal purpoe. In many complex ytem, they are the only meanngful exact relaton that can be obtaned [13]. Thu, they provde mportant nght uch a meaure of overall performance, demontraton on the effect of varou parameter on W, and a ba of approxmaton and bound [1, 18]. Note that th work dfferent from that n [14, 15] where mxed ervce tratege are condered for a dcrete-tme and contnuou-tme cyclc ervce. Reference [1] A. Roubu, Pollng ytem and ther applcaton, M.S. the, Vrje Unvertet Amterdam, 007. [] H. Lee, T. Kwon, and D. H. Cho, An effcent uplnk chedulng algorthm for VoIP ervce n IEEE BWA ytem, n Proceedng of the 60th IEEE Vehcular Technology Conference VTC 04), pp , September 004. [3] F. Hou, P.-H. Ho, X. Sherman) Shen, and A.-Y. Chen, A novel QoS chedulng cheme n IEEE network, n Wrele Communcaton and Networkng Conference, pp , Kowloon, Hong Kong, March 007. [4] H. Abu-Ghazaleh, J. Cha, and A. S. Alfa, Performance analy for pollng ervce n IEEE network under PMP mode, n Wrele Communcatn and Moble Computng Conference, pp , Crete Iland, Greece, 008. [5] J. W. So, Performance analy of uplnk chedulng algorthm for VoIP ervce n the IEEEE 80.16e OFDMA ytem, Wrele Peronal Communcaton, vol. 47, no., pp , 008. [6] O. J. Boxma and D. G. Down, Dynamc erver agnment n a two-queue model, European Operatonal Reearch, vol. 103, no. 3, pp , [7] J. F. Haye and T. V. J. Ganeh Babu, Modelng and Analy of Telecommuncaton Network, Wley-Intercence, New York, NY, USA, 004. [8] S. C. Bort, Pollng Sytem, CWI Tract, CWI Amterdam, 006. [9] F.Hou,P.H.Ho,andX.Sherman)Shen, Performanceevaluaton for unolcted grant ervce flow n network, n Proceedng of the Internatonal Wrele Communcaton and Moble Computng Conference IWCMC 06), pp , Vancouver, Canada, July 006. [10] K. Chang and D. Sandhu, Peudo-conervaton law n cyclc-erver, multqueue ytem wth a cla of lmted ervce polce, n Proceedng of the 9th Annual Jont Conference of the IEEE Computer and Communcaton Socete INFOCOM 90), pp , San Francco, Calf, USA, June [11] WMAX Forum, WMAX Sytem Evaluaton Methodology. Veron.1, July 008.

7 Modellng and Smulaton n Engneerng 7 [1] C. So-In, R. Jan, and A. K. Tamm, Schedulng n IEEE 80.16e moble WMAX network: key ue and a urvey, IEEE Journal on Selected Area n Communcaton, vol. 7, no., pp , 009. [13] H. Levy and M. Sd, Pollng ytem: applcaton, modelng, and optmzaton, IEEE Tranacton on Communcaton, vol. 38, no. 10, pp , [14] O. J. Boxma and W. P. Groenendjk, Watng tme n dcrete-tme cyclc-ervce ytem, IEEE Tranacton on Communcaton, vol. 36, no., pp , [15] O. J. Boxma and W. P. Groenendjk, Peudo-conervaton law n cyclc-ervce ytem, Appled Probablty, vol. 4, pp , [16] Y. Takahah and B. K. Kumar, Peudo-conervaton law for a prorty pollng ytem wth mxed ervce tratege, Performance Evaluaton, vol. 3, no., pp , [17] O. J. Boxma, H. Levy, and J. A. Wettrate, Effcent vt order for pollng ytem, Tech. Rep. BS-R9017, Centre for Mathematc and Computer Scence, [18] H. Takag, Analy of pollng ytem, Reearch Report and Note, Computer Sytem Sere MIT Pre, 1986.

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