Research Article Performance Evaluation of Uplink Delay-Tolerant Packet Service in IEEE Based Networks

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1 Hndaw Publshng Corporaton EURASIP Journal on Wreless Communcatons and Networkng Volume 2011, Artcle ID , 12 pages do: /2011/ Research Artcle Performance Evaluaton of Uplnk Delay-Tolerant Packet Servce n IEEE Based Networks Zsolt Saffer, 1 Sergey Andreev, 2 and Yevgen Koucheryavy 2 1 Department of Telecommuncatons, Budapest Unversty of Technology and Economcs (BUTE), Magyar tudósok körútja 2, 1117 Budapest, Hungary 2 Department of Communcatons Engneerng, Tampere Unversty of Technology (TUT), Korkeakoulunkatu 10, Tampere, Fnland Correspondence should be addressed to Zsolt Saffer, safferzs@ht.bme.hu Receved 15 November 2010; Accepted 11 February 2011 Academc Edtor: Bors Bellalta Copyrght 2011 ZsoltSaffer et al. Ths s an open access artcle dstrbuted under the Creatve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted. We provde an analytcal model for effcent dynamc capacty allocaton n IEEE wreless metropoltan area network, where the nonreal-tme traffc can utlze the bandwdth unused by the real-tme traffc. We nvestgate the uplnk delay of the nrtps servce flow as a functon of the capacty allocatons for the rtps (ertps) and UGS servce flows. Uncast pollng s appled for the bandwdth reservaton of the nrtps and rtps (ertps) packets. Our analyss accounts for both reservaton and schedulng delay components. The nrtps packets arrve accordng to Posson process. The model enables asymmetrc capacty allocaton, as well as asymmetrc nrtps traffc arrval flows. The analytcal model s appled for nvestgatng the nfluence of the real-tme traffc on the delay of the nrtps servce flow. We dscuss also the determnaton of several traffc parameters under dfferent constrants, whch have potental applcatons n network control. 1. Introducton IEEE standards famly defnes an ar nterface for Broadband Wreless Access (BWA) system. As the result of a recent revson, the contemporary core standard IEEE [1] consoldates the IEEE standard wth several amendments. Accordng to [2], ths wreless nterface s recommended for Wreless Metropoltan Area Networks (WMANs). The hgh-speed ar nterface specfed by the IEEE standards famly enables multmeda servces and provdes support for several traffc types to ensure the wde range of Qualty-of-Servce (QoS) requrements of end users. The standardzaton of metropoltan-scale wreless access s an ongong actvty performed by the IEEE Workng Group for BWA wth the support of WMAX Forum [3]. The uplnk data packet scheduler, whch s out of scope of the IEEE standard, has a major mpact on ensurng QoS requrements of the end users. As a consequence, numerous research papers deal wth the problem of schedulng, lke [4 6], n whch varous frameworks are bult and analyzed to guarantee a specfed level of QoS. For nstance, the work n [7]proposedaneffcent QoS archtecture, based on prorty schedulng and dynamc bandwdth allocaton. In [8], authors compare and contrast the performance of varous reservaton schemes n the framework of the smplfed model. For a good summary on QoS n the context of IEEE , we refer to the onlne paper [9]. The majorty of the analytcal works n the lterature do not account for both the reservaton and the schedulng components of the delay. The mportance of accountng for both components to evaluate the overall delay of accesscontrol systems was emphaszed by an early fundamental theoretcal work by Rubn [10], as well as by our prevous papers [11, 12]. For a more practcal approach, we refer to [13], n whch the realstc performance measures of IEEE system are consdered by varous technques. In [13, 14], the overall system delay s approxmated and verfed. In our prevous work [15], we establshed an analytcal model for the exact overall delay of the nrtps servce flow wth uncast pollng n the IEEE system. Other pollng technques were studed n [16].

2 2 EURASIP Journal on Wreless Communcatons and Networkng IP/ATM network Subscrber staton (SS) ` ` Base staton (BS) LAN LAN VoIP Subscrber staton (SS) Subscrber staton (SS) VoIP VoD VoD Fgure 1: IEEE general archtecture. In ths paper, we contnue the works n [14, 15, 17] by extendng the analytcal model to perform an effcent dynamc capacty allocaton, n whch the nonreal-tme (delay-tolerant) traffc of each Subscrber Staton (SS) can utlze a porton of the spare bandwdth remanng after the capacty allocaton for the real-tme (delay-crtcal) traffc flows at every SS. Thus, the model ncorporates the effect of the capacty allocaton for the real-tme pollng servce (rtps), extended real-tme pollng servce (ertps), and unsolcted grant servce (UGS) flows on the overall delay of the non real-tme pollng servce (nrtps) flow. The varable nrtps capacty of the ndvdual SS s allowed to depend on real-tme capactes of every SS. The nrtps capacty of each SS s determned by means of prortes among them for ther non real-tme traffc flows. Ths prortzaton allows the realzaton of dfferent servce levels probably for dfferent prces n terms of capacty utlzaton for non realtme traffc. Ths ensures a guaranteed porton of the total avalable nrtps capacty for each SS also n the case when non real-tme traffc s saturated at one or more other SSs. The analytcal approach leads to a queueng model wth batch packet servce. The expresson for the mean packet delay s gven n terms of model probabltes, whch are computed from the equlbrum dstrbuton of a properly dentfed embedded Markov chan. The analytcal model s appled to the performance evaluaton of the uplnk nrtps traffc n the IEEE based network. Besde provdng numercal examples, we study the modeled nfluence of the real-tme traffc on the delay of the nrtps servce flow. We dscuss how to take nto accountanupperboundonmeandelayofthenrtpsservce flow at the SSs n determnng the maxmum of the sum of the real-tme capactes at every SS. Fnally, we ntroduce a cost model, whch takes nto account the QoS on delay constrant and on the real-tme capacty parameters. The dfferent aspects of ths performance analyss have potental applcatons n network control, snce they facltate the settng of the capacty parameters to the requrements of the actual applcaton scenaro. The rest of the paper s structured as follows. Secton 2 gves a bref summary of the channel allocaton schemes n IEEE In Secton 3, we provde the analytcal model ncludng the detals of the capacty allocaton and the uplnk schedulng. The analyss of the queueng model follows n Secton 4. We determne the mean overall packet delay of the nrtps servce flow n Secton 5. InSecton 6, wegve numercal examples for the performance analyss. Fnally the concluson n Secton 7 closes the paper. 2. Channel Allocaton Schemes n IEEE The mandatory centralzed pont-to-multpont (PMP) IEEE archtecture (see Fgure 1) comprses a Base Staton (BS) and one or more SSs n ts vcnty. The packets are exchanged between BS and SSs va separate channels. The downlnk (DL) channel s used for the traffc from the BS to the SSs, and the uplnk (UL) channel s used n the reverse drecton. The standard defnes two mechansms of multplexng the DL and the UL channels: Tme Dvson Duplex (TDD) and Frequency Dvson Duplex (FDD). In FDD mode, the DL and the UL channels are assgned to dfferent subband frequences. In TDD mode, the channels are dfferentated by assgnng dfferent tme ntervals to them, that s, MAC frame s dvded nto DL and UL parts. The border between these parts may change dynamcally dependng on the SSs bandwdth requrements. The SSs access the UL channel by means of Tme-Dvson Multple Access (TDMA). The structure of the MAC frame n TDD/TDMA mode s shown n Fgure 2. The current IEEE standard, as well as ts future verson IEEE m [18], specfes Orthogonal

3 EURASIP Journal on Wreless Communcatons and Networkng 3 Frame Downlnk (DL) subframe Uplnk (UL) subframe UL-MAP ndcates the startng tme slot of each uplnk burst Reservaton nterval (RI) SS 1 transmsson nterval. SS N transmsson nterval UL-MAP DL-MAP Preamble Bandwdth request (BW-req) Fgure 2: IEEE MAC frame structure n TDD/TDMA mode. Frequency-Dvson Multple Access (OFDMA) at the physcal layer. 3. Analytcal Model and Notatons In the consdered model, all the fve servce flow types are allowed at each SS (see Fgure 3), each one wth a dedcated Connecton ID () and a Servce Flow ID (SFID). For UGS, rtps, and ertps packet servce, the QoS guarantees are ensured by means of the necessary capacty allocatons. The nrtps and Best Effort (BE) servce flows utlze spare bandwdth, where the nrtps servce flow s prortzed over the BE traffc. In the evaluaton of the nrtps packet servce delay, we account also for the effects of the UGS, rtps, and ertps servce flows Restrctons of the Model. We mpose several lmtatons on the IEEE model. (R.1) The operatonal mode s PMP, and TDD/TDMA channel allocaton scheme s used. Our TDD/TDMA modeldervednthspapercanbeappledforboth OFDMA-based versons (IEEE and IEEE m). (R.2) Only the uplnk traffc s consdered, as well as uncast pollng s used for nrtps, rtps, and ertps servces. (R.3) The uplnk packet scheduler at the BS keeps an ndvdual buffer for each SS to serve the nrtps packets. (R.4) The BE traffc s assumed to be saturated. (R.5) Pggybackng s not used General Model. There are N SSs and 1 BS n the system, whch together comprse N + 1 statons. Each SS mantans separate buffers of nfnte capacty for the uplnk packets of dfferent servce flows. The nrtps packets arrve at SS accordng to the Posson arrval process wth arrval rate λ for = 1,..., N. Hence, the overall nrtps packet arrval rate s λ = N =1 λ. We call the nrtps packets arrvng to SS as -packets. The arrval processes at the dfferent SSs are mutually ndependent. The packet length s fxed and equals η 1 bt, whch ncludes data nformaton and the header wth packng/fragmentaton overhead. The transmsson rate of each channel s β bps. Therefore, the transmsson tme of adatapacketsτ = (ηβ) 1. All tme duratons are measured n seconds. T f denotes the duraton of each frame. Whle all the SSs are allowed to transmt n the uplnk of one frame, they may be grouped by the reservaton mechansm to reduce the pollng overhead [14, 19]. Accordngly, n one frame only SSs belongng to one group are polled and are allowed to send ther bandwdth request (BW-Req) messages. Then, the nonoverlappng groups are polled n consecutve frames. P denotes the number of SSs n each group, and, hence, the number of groups s L = N/P. The same SSs group s polled n every Lth frame. The mnmal perod between two consecutve pollngs of the same SSs group s called a pollng cycle. Thus, the length of a pollng cycle s LT f. The SSs groupng model s shown n Fgure 4. The duraton of the DL and the UL sub-frames are T d and T u,respectvely.t r stands for the duraton of the reservaton nterval, and T ud s the maxmum avalable duraton of the uplnk data transmsson n a frame. Therefore, T u s gven by T u = T r + T ud. The transmsson tme of a BW-Req s α.hence,t r = Pα and T ud can be expressed as T ud = T u Pα Capacty Allocaton. As the packet transmsson tme s fxed, we measure the capacty n the number of packets. Let C u denote the fxed capacty assgned for SS n a frame for the uplnk UGS traffc for = 1,..., N. Smlarly, R stands for the varable capacty assgned for SS n a frame for the uplnk rtps and ertps transmssons together. The range of the dscrete-tme random varable R s, thus, gven by R mn R R max, = 1,..., N. (1) Let H be the total remanng uplnk capacty for the nrtps packet servce of all the SSs after allocatng the necessary capacty for the above three real-tme traffc flows. Thus, H can be expressed as H = T ud τ N N C u R. (2) =1 =1 Let 0 ω 1 denote the fxed prorty weght of SS for the nrtps capacty allocaton for = 1,..., N. The varable

4 4 EURASIP Journal on Wreless Communcatons and Networkng Data traffc Subscrber staton (SS) Applcatons Connecton request Connecton response Base staton (BS) Admsson control undefned by IEEE /SFID classfcaton UGS rtps ertps nrtps BE BW-request Uplnk packet schedulng algorthm undefned by IEEE Packet scheduler UL-MAP Data packet Fgure 3: IEEE QoS archtecture. Transmsson ntervals Pollng cycle = L frames Transmsson ntervals Transmsson ntervals DL subframe SS 1 SS N DL RI SS 1 SS N DL RI SS 1 SS N P pollng slots P pollng P pollng kth frame slots (k + 1)th frame slots (k + L)th frame Fgure 4: The SSs groupng model. capacty avalable for SS n a frame for the uplnk nrtps, H, s gven by H = ω H, = 1,..., N, N ω = 1, (3) =1 where d stands for the ntegral part of d. Thus,H s gven n the dependency of the total allocated capacty for the UGS, rtps, and ertps servces of all the SSs. Usng (2)and(3)leads to the followng range of H : H mn H mn = H max = H H max, where ω T ud N τ N C u =1 ω T ud N τ N C u =1 R max 1, =1 R mn =1, = 1,..., N. Expresson (4) shows that the capacty avalable for the nrtps traffc s gven by an upper-lmted dscrete-tme random varable, whose value s at least one. Ths ensures (4) that the nrtps traffc can not be blocked by the UGS, rtps, and ertps traffcflows. Fnally, the BE servce flow utlzes the remanng capacty, whch s not used by the nrtps traffc. Ths together wth the restrcton (R.4) ensures an effcent capacty utlzaton, n whch the total avalable nonreal-tme capacty (H) s always utlzed. The descrbed capacty allocaton scheme s llustrated n Fgure 5. Summarzng, our general capacty allocaton scheme enables asymmetrc capacty allocaton for the UGS, rtps, and ertps servces, as well as asymmetrc nrtps traffcflows Model Assumptons. Let Y denote the number of actually transmtted nrtps packets of SS n a frame. In statstcal equlbrum, the mean number of transmtted nrtps packets equals the mean number of arrvng nrtps packets per frame at each SS. Ths yelds E[Y ] = λ T f, = 1,..., N. (5) The number of transmtted nrtps packets s upperlmted by the capacty avalable for them Y H, = 1,..., N. (6)

5 EURASIP Journal on Wreless Communcatons and Networkng 5 w 1 H w N H DL subframe RI UGS traffc (e)rtps traffc nrtps (BE) traffc UGS traffc (e)rtps traffc nrtps (BE) traffc C u 1 R 1 H 1 C u N R N H N Fgure 5: The capacty allocaton scheme. Below, we formulate the assumptons of our model. (A.1) Usng (5), (6), (3), and (2) mples that the followng relaton holds for the arrval rate of each SS below the stablty boundary: λ T f <E ω T ud N τ N C u R, =1 =1 (7) = 1,..., N. Ths relaton ensures the stablty of the model. (A.2) The BS uplnk scheduler processng delay s neglgble. (A.3) The channel propagaton tme s neglgble. (A.4) The transmsson channels are error free Uplnk Schedulng. ABW-ReqsentbytheSS represents the aggregated request for all nrtps packets, whch are accumulated n ts outgong buffer durng the last cycle, that s, snce the prevous BW-Req sendng. We leave the process of bandwdth requestng for rtps and ertps packets out of scope of ths paper. Furthermore, we assume that the BS knows the number of rtps and ertps packets at each SS n every frame and thus t can take them nto account calculatng the actual avalable capacty for the nrtps packets H. We note that the actual uplnk transmsson requrements represented by the rtps and ertps requests are always granted, snce they are below the avalable capacty. The fxed prorty weghts assgned to the SSs enable mutually ndependent uplnk schedulng for the nrtps servce flows of the ndvdual SSs. Thus, for the servce of the aggregated BW-Req for the nrtps packets, the BS mantans an ndvdual BS grant buffer wth nfnte capacty for each SS. Let -pollng slot stands for the ((( 1) mod P) +1)th pollng slot wthn the reservaton nterval of the frame, n whch the group of SS s polled. At the end of the - pollng slot, the BS mmedately processes the requests for the nrtps packets from SS, f any, and serves the ndvdual BS grant buffer of SS. We refer to the end of the -pollng slot as -reservaton epoch. The BS grant buffer of SS s also served at the epochs followng an -reservaton epoch by T f,2t f,...,(l 1)T f tme. Hence, all these epochs, ncludng also the -reservaton epochs,arecalled-schedulng epochs. The postons of the consdered epochs are marked n Fgure 6. Recevng a request for the nrtps packets from SS at an -reservaton epoch, an ndvdual BS grant s assgned to each nrtps data packet of that request, and then these BS grants are placed nto the correspondng ndvdual BS grant buffer of SS accordng to ther order n the request. Let the number of the BS grants n the buffer of SS be S = 0, 1,... Durng the servce of the ndvdual BS grant buffer of SS at an - schedulng epoch, the BS takes the avalable BS grants from that buffer up to the avalable capacty for the nrtps servce flow of SS (H ) and schedules them for transmsson n the UL-MAP of the followng frame. Ther number equals the number of -packets transmtted n the next frame, Y. Thus, the number of scheduled BS grants s gven by Y = mn(s, H ), (8) where mn(a, b) stands for the smallest value of the set (a, b). An example of the BS uplnk schedulng s llustrated n Fgure 7. The features of the consdered uplnk schedulng process can be summarzed as follows. (F.1) The capacty requrements of the UGS, rtps, and ertps servce flows are always satsfed. (F.2) The capacty allocaton enables prortes for the nrtps servce flows (ω at SS for 1,..., N). Ths corresponds to a weghted round-robn schedulng of the dynamcally varable capacty, whch remans avalable after ensurng the servce of the real-tme traffcflows. (F.3) The schedulng mechansm ensures effcent capacty utlzaton, snce the remanng capacty not used by the nrtps traffc flow at each SS s flled the BE traffc at ths SS. 4. Queueng System Analyss The ndvdual pollng slot for each SS n a pollng cycle and the ndependent uplnk schedulng for the ndvdual SSs together mply that the statstcal behavor of the BS grant buffer of a partcular SS s ndependent from the behavor of those of the other SSs. Therefore, we model the stochastc behavor of the BS grant buffer of a partcular SS by an ndvdual queueng system. In ths queueng system, the BS grants arrve to the BS grant buffer of SS at -reservaton epochs and they are served at -schedulng epochs The Contents of the BS Grant Buffer at -Reservaton Epochs. Let N (l) be the number of BS grants n the BS grant

6 6 EURASIP Journal on Wreless Communcatons and Networkng -reservaton epoch T f -schedulng epochs (L 1)T f DL subframe SS 1 SS N DL RI SS 1 SS N DL RI SS 1 SS N pollng slots k-th frame (k + 1)th frame (k + L 1)th frame Fgure 6: Characterstc epochs of uplnk schedulng. SS Indvdual BS grants buffer Tagged packet arrval UL-MAP formng nrtps packets transmsson UGS DL traffc (e)rtps traffc (BE) traffc DL UGS (e)rtps traffc nrtps traffc traffc BW-req for 2 packets Tagged packet overall delay (BE) traffc DL Fgure 7: Example BS uplnk schedulng for a sngle SS. buffer of SS at the lth -reservaton epoch for l > 0. The sequence {N (l), l > 0} s an embedded Markov chan on the state space {0, 1,...}.Let[Π ] j,k denote the probablty of transton from state j to state k of the Markov chan, and t s the (j, k)th element of the probablty transton matrx Π. Let H (m) be the accumulated avalable capacty for the -packets durng m consecutveframesform = 0,..., L. The dstrbuton of H (m) s gven as the m-tmes convoluton of the dstrbuton of H for m = 1,..., L. The defnton of H (0) mples that t takes the value 0 wth probablty 1. It mmedately follows that the mnmum and maxmum values of H (m) are mh mn and mh max,respectvely. Let us consder the transton from state j to state k n the above defned Markov chan. The probablty that the actual accumulated avalable capacty for the -packets durng a pollng cycle s n equals P(H (L) = n). Assumng that j n mples that the number of remanng BS grants n the BS grant buffer of SS after ts servces durng a cycle s j n, whch mples that k j n. Thus, on one hand, n j k must hold and, on the other hand, k j + n-packet arrvals occur durng ths transton. Hence, ths case contrbutes to [Π ] j,k wth the probablty j n=j k P ( H (L) = n ) ( ) k j+n λ LT f ( ) e λlt k j + n! f. (9) Now assumng that j + 1 n mples that all the j BS grants are served durng the cycle and, thus, k-packet arrvals occur durng ths transton. Thus, the contrbuton of ths case to [Π ] j,k s the probablty H (L) LH max n=j+1 P ( H (L) = n ) ( ) k λ LT f e λlt k! f. (10) Takng also nto account the lower and upper lmts of, the transton probablty [Π ] j,k can be expressed as [Π ] j,k = mn(lh max,j) n=max(lh mn,j k) ( λ LT f ) k j+n P ( H (L) = n ) ( k j + n )! e λlt f + LH max n=max(lh mn,j+1) P ( H (L) = n ) ( ) k λ LT f e λlt k! f, j, k 0, (11) where max(a, b) stands for the largest value of set (a, b). Let [π ] k denote the equlbrum probablty of the state k n the Markov chan, and t s the (k)th element of the 1 probablty vector π. Furthermore, let e be the column vector havng all elements equal to one. Then, the equlbrum probabltes of the Markov chan can be unquely determned from the followng system of lnear equatons: π Π = π, π e = 1. (12)

7 EURASIP Journal on Wreless Communcatons and Networkng 7 To keep the computaton tractable, an upper lmt K > H mn s set on the states, whch results n the fnte number of unknowns and equatons n the system of lnear equatons. An approprate value of K depends on the requred precson level, at whch the probabltes [π ] k for k > K can be neglected. These probabltes, [π ] k for k>k, are set to The Contents of the BS Grant Buffer at -Schedulng Epochs. Let [π + ] k denote the probablty that the number of BS grants n the BS grant buffer of SS at an arbtrarly chosen -schedulng epoch s exactly k, and t s the (k)th element of the 1 (K +1)probabltyvectorπ + for k = 0,..., K. The probablty that an arbtrarly-chosen -schedulng epoch s the mth after the last -reservaton epoch s 1/L for m = 0,..., L 1. Note that by defnton the 0th -schedulng epoch afterthe last -reservaton epoch s that -reservaton epoch. By defnton, the tme nstant of handlng the nrtps packet requests from SS s the -reservaton event. Smlarly, by defnton the nstants of schedulng the BS grants n the BS grant buffer of SS are the -schedulng events. The postonng of the -reservaton epoch and the -schedulng epochs (observaton epochs) relatvely to the -reservaton and -schedulng events s shown n Fgure 8. At the mth -schedulng epoch after the last -reservaton epoch, the -packets n the BS grant buffer of SS are those whch remaned after the last m servces of the BS grant buffer. Hence, the probablty [π + ] k can be establshed as L 1 [ ] π + k = 1 L m=0 L 1 [ ] π + 0 = 1 L m=0 mh max n=mh mn mh max n=mh mn P ( H (m) = n ) [π ] n+k, P ( H (m) 0 <k K, = n ) n [π ] j. j=0 (13) 4.3. The Contents of the BS Grant Buffer at an Arbtrary Epoch. At an arbtrary epoch between two consecutve -schedulng epochs, the BS grants n the BS grant buffer of SS are those whch remaned after the servce of the BS grant buffer at the last-schedulng epoch. Hence, the probablty of beng exactly k packets n the BS grant buffer of SS at an arbtrary epoch, p k,sgvenby p k = p 0 = H max n=h mn H max n=h mn P(H = n) [ π + n [ ] P(H = n) π + j=0 ]n+k, 0<k K H mn, j. (14) 4.4. The Sze of the Transmtted -Packet Batch. Let us consder the probablty of transmttng exactly n-packets n a frame for 0 n mn(h max, K ). Ths can occur n two cases. In the frst one, the actual avalable capacty for the - packetssexactlyn and there are at least n BS grants n the BS grant buffer of SS at -schedulng epoch. The probablty of ths case s K P(H = n) [ π + k=n ] k. (15) In the other case, the number of BS grants n the BS grant buffer of SS at -schedulng epoch s n, but the actual avalable capacty for the -packets, k, s greater than n. Ths has the followng probablty: H max k=n+1 P(H = k) [ π + ] n. (16) Takng also nto account the lower lmt of H, the probablty of transmttng exactly n -packets n a frame can be expressed as K P(Y = n) = P(H = n) [ π + k=n + H max k=max(h mn,n+1) 5. Overall Delay Analyss ] k P(H = k) [ π + ] n, 0 n mn ( H max, K ). (17) 5.1. Overall Delay Defnton. We defne the overall delay (W ) of the tagged -packet as the tme nterval spent from ts arrval nto the outgong buffer of SS up to the end of ts successful transmsson n the UL. It s composed of several parts W = W r + α + W s + W t + τ. (18) Here, W r s the reservaton delay, whch s defned as the tme nterval from the -packet arrval to SS untl the start of sendng a correspondng BW-Req to the BS. We defne the grant tme of the tagged -packet as the -schedulng epoch n the frame precedng the one, n whch the tagged -packet s transmtted. W s s the schedulng delay, whch s defned as the tme nterval from the end of sendng a BW-Req of the tagged -packet to ts grant tme. W t s the transmsson delay, whch s defned as the tme nterval from the grant tme of the tagged -packet to the start of ts successful transmsson n the UL sub-frame Reservaton Delay. A bandwdth request can be sent for the nrtps packets from SS n the -pollng slot of every pollng cycle. Thus, an arrvng -packet wats for the reservaton opportunty untl the end of the current cycle, and, hence, the mean reservaton delay s gven by E [ W r ] = LT f 2. (19)

8 8 EURASIP Journal on Wreless Communcatons and Networkng 1-st -schedulng event -schedulng event 2-nd -schedulng event Lth -schedulng event 0th -schedulng epoch -reservaton epoch 1-st -schedulng epoch Observaton epochs (L 1)-st -schedulng epoch Fgure 8: Postons of the observaton epochs wthn a pollng cycle Schedulng Delay. The defnton of the schedulng delay mples that the schedulng delay of the tagged -packet s exactly the sojourn tme of the BS grant assgned to the tagged -packet n the BS grant buffer of SS. Consequently, the mean schedulng delay can be determned by applyng the Lttle s law on the mean number of -packets n the BS grant buffer of SS at an arbtrary epoch. Takng also nto account the tractable computaton of π, the mean schedulng delay can be expressed as E [ W s ] k=1 kp k = = λ K H mn k=1 kp k λ. (20) 5.4. Transmsson Delay. The transmsson delay s the sum of the fxed tme from the grant tme of the tagged -packet to the start of transmsson of the -packets n the next frame and the transmsson tmes of the random number of - packets precedng the tagged -packet. Let y and y (2) be the frst two factoral moments of the number of -packets transmtted n a frame. The mean number of -packets precedng the tagged -packet s y (2) /2y (see [20]). Usng t, the defntons of the frst two factoral moments and takng nto account the range of Y, the mean transmsson delay can be expressed as E [ W t ] = T f α((( 1) mod P) +1) + Pα + C u j τ j=1 + E [ ] 1 R j τ + y j τ + y(2) τ 2y j=1 j=1 = T f + α(p (( 1) mod P) 1) + C u j τ j=1 + E [ ] 1 mn(h max,k ) R j τ + P ( Y j = k ) k τ j=1 j=1 k=1 + mn(h max,k ) k=2 P(Y = k)k(k 1) 2 mn(h max,k ) τ. k=1 P(Y = k)k (21) 5.5. Mean Overall Delay. Takng the mean of (18) and substtutng the expressons (19), (20), and (21), we obtan the expresson for the mean overall delay of the tagged - packet as E[W ] = L +2 2 T f + K H mn k=1 kp k + α(p (( 1) mod P)) + τ λ + C u j + E [ ] 1 mn(h max,k ) R j + P ( Y j = k ) k τ j=1 j=1 j=1 k=1 + mn(h max,k ) k=2 P(Y = k)k(k 1) 2 mn(h max,k ) τ. k=1 P(Y = k)k 6. Performance Evaluaton (22) In ths secton, we apply the derved analytcal model to the performance evaluaton of the uplnk nrtps packet servce n the IEEE network Numercal Examples. Here, we provde numercal examples to assess the performance of the IEEE uplnk nrtps servce flow evaluated wth the consdered analytcal model. In order to generate performance data, a smulaton program for IEEE MAC was developed. The program s an event-drven smulator that accounts for the dscussed restrctons on the consdered system model (see Secton 3). In our smulatons, we set the default values recommended by WMAX Forum [3] system evaluaton methodology, whch are also common values used n practce [21]. We assume a 10 MHz TDD system wth 5 ms frame duraton, PUSC subchannelzaton mode, and a DL : UL rato of 2 : 1. Accordng to [22], the UL sub-frame comprses 175 slots. Assumng MCS of 16 QAM 3/4, the IEEE system transmts 16 bytes per UL slot. We consder fxed packet length of 80 bytes (5 slots) for all servce flows, whch results n havng capacty to send 30 packets per UL subframe. The remanng 25 UL slots represent the necessary control overhead. For the sake of smplcty, we frstly nvestgate the case of the symmetrc system. The arrval flows have constant rate of λ = λ/n and ω = 1/N for all the SSs. Assumng fxed

9 EURASIP Journal on Wreless Communcatons and Networkng 9 Table 1: Basc evaluaton parameters. 40 Parameter Value PHY layer OFDMA Frame duraton (T f ) 5ms Subchannelzaton mode PUSC DL/UL rato 2 : 1 Channel bandwdth 10 MHz MCS 16 QAM 3/4 Packet length 80 bytes Mean nrtps delay for SS1 (ms) Number of SSs (N) 6or 2 Total capacty per frame for all SSs 30 packets UGS capacty per frame (C u ) 6 packets Normalzed arrval rate Mnmum (e)rtps capacty per frame (R mn ) Maxmum (e)rtps capacty per frame (R max ) 70 6 packets 18 packets Analyss, w 1 : w 2 = 1:5 Smulaton, w 1 : w 2 = 1:5 Analyss, w 1 : w 2 = 1:2 Smulaton, w 1 : w 2 = 1:2 Analyss, w 1 : w 2 = 1:1 Smulaton, w 1 : w 2 = 1:1 Mean nrtps delay (ms) Analyss, P = 1 Smulaton, P = 1 Analyss, P = 2 Smulaton, P = Normalzed arrval rate 0.8 Analyss, P = 3 Smulaton, P = 3 Analyss, P = 6 Smulaton, P = 6 Fgure 9: Mean nrtps packet delay n symmetrc system wth SSs groupng (N = 6). number of N = 6 SSs, we also set constant capacty-related parameters C u, R mn,andr max (see Secton 3.3). We llustrate the smplest case of the actual rtps and ertps capacty dstrbuton, that s, unform n the range [R mn, R max ]. The summary of the consdered evaluaton parameters s gven n Table 1. In Fgure 9, we plot the dependency of the mean nrtps packet delay on the arrval rate for dfferent groupngs, that s, for dfferent values of P. The next example n Fgure 10 shows the nrtps delay of SS 1 wthn the smplest asymmetrc system of 2 SSs and dfferent prorty weghts w 1 and w 2. Both Fgures 9 and 10 show very good accordance between the analytcal and the smulaton values Influence of UGS and (e)rtps Traffc onnrtpsdelay. In ths subsecton, we study the nfluence of the capacty allocaton for the UGS and the real-tme traffc on the mean 1 Fgure 10: Mean nrtps packet delay at SS 1 n asymmetrc system (N = 2). Mean nrtps delay (ms) Analyss, UGS = 0 Smulaton, UGS = 0 Analyss, UGS = Normalzed arrval rate 0.8 Smulaton, UGS = 6 Analyss, UGS = 12 Smulaton, UGS = 12 Fgure 11: Influence of the UGS traffc on the mean nrtps packet delay n symmetrc system (N = 6). packet delay of the nrtps servce flow n the symmetrc system for N = 6. In partcular, Fgure 11 demonstrates the dependency of the mean overall nrtps delay on the normalzed arrval rate for dfferent total UGS capacty values per frame. Here, the mnmum and the maxmum (e)rtps capacty per frame s set 6 and 12 packets, respectvely. It can be seen n the Fgure 11 that ncreasng the total UGS capacty per frame leads to hgher mean overall nrtps delay, as expected. Ths s due to the mpact of the total UGS capacty on the 1

10 10 EURASIP Journal on Wreless Communcatons and Networkng Mean nrtps delay, unform dstrbuton (ms) Normalzed arrval rate Mean nrtps delay, geometrc dstrbuton (ms) Normalzed arrval rate Analyss, max. (e)rtps = 12 Smulaton, max. (e)rtps = 12 Analyss, max. (e)rtps = 18 Smulaton, max. (e)rtps = 18 Analyss, max. (e)rtps = 24 Smulaton, max. (e)rtps = 24 (a) Analyss, max. (e)rtps = 12 Smulaton, max. (e)rtps = 12 Analyss, max. (e)rtps = 18 Smulaton, max. (e)rtps = 18 Analyss, max. (e)rtps = 24 Smulaton, max. (e)rtps = 24 (b) Fgure 12: Influence of (e)rtps traffc on the mean nrtps packet delay n symmetrc system (N = 6) for unform dstrbuton (a) and for truncated geometrc dstrbuton wth parameter 0.5 (b). transmsson and schedulng delays (see relatons (21) and (2)). Now, we vary the maxmum (e)rtps capacty per frame. In Fgure 12, the mean overall nrtps delay s plotted as a functon of the normalzed arrval rate for dfferent maxmum (e)rtps capacty values per frame, as well as both unform and truncated geometrc dstrbutons. Here, the UGS capacty per frame s set 0 packets, and the mnmum (e)rtps capacty per frame s 6 packets. We can observe n the fgure that the dependency on the maxmum (e)rtps capacty values for unform dstrbuton s smlar to the dependency for the total UGS capacty (see Fgure 11). However, comparng the left and the rght sdes of Fgure 12, we can conclude that the dstrbuton of the (e)rtps capacty values has an essental mpact on the mean overall nrtps delay. The postons of the curves relatvely to each other on the rght sde of Fgure 12 are the consequences of the used truncatng of the geometrc dstrbuton Enforcng an Upper Bound on Mean Delay. Our modelng can be also used to enforce specfed upper bounds on meannrtpspacketdelaysateveryssnaspecfedrangeof loads. These bounds can be dfferent for the ndvdual SSs. In ths case, the total amount of uplnk real-tme capactes n the network ( N =1 C u + N =1 R ) s maxmzed over a restrcted parameter set, whch s determned by the specfed upper bounds on mean nrtps packet delays and by the specfed range of loads. The prorty weghts of the SSs are assumed to be gven Cost Model. In case of a more general QoS requrement (delay constrant), an approprate cost model can be bult to determne the optmal parameters of the real-tme traffc flows. We developed a steady-state average cost functon F (ω), where the set of prorty weghts of the SSs ω = (ω 1,..., ω N ) s the decson varable. The parameters of the cost functon for = 1,..., N are defned as ξ cost of the mean packet delay at SS. θ reward of the UGS capacty at SS ( C u ). (23) ϑ reward of the maxmum(e)rtps capacty at SS ( R max ). Then, the optmal parameters of the real-tme traffc flows can be obtaned by mnmzng the total average system cost, whch s gven by F (ω) = ( N ξ E[W ] + θ C u =1 + ϑ ) R max. (24) The mnmum can be numercally determned as a functon of the load and the real-tme capacty parameters at every SS (C u and the dstrbuton of R for = 1,..., N), by applyng the expressons for the mean overall delay of the tagged -packet (22). 7. Concluson We presented an analytcal model for the delay of the uplnk nrtps traffc n IEEE based network, n whch () the nfluence of the real-tme (UGS and (e)rtps) capacty allocaton on the delay of the delay-tolerant (nrtps) traffcscaptured,

11 EURASIP Journal on Wreless Communcatons and Networkng 11 () the varable nrtps capacty of each SS s allowed to depend on the real-tme capactes of every SS, () the nrtps capacty at the SSs are determned by means of prortes among them. The consdered analytcal model s verfed by means of smulaton. Ths verfcaton shows an excellent accordance between the analytcal and the smulaton results n a wde range of parameter settngs. Hence, our analytcal model can be appled to model and analyze the delay of the uplnk nrtps traffc n IEEE based network. Based on the numercal examples for the performance evaluaton, the followng conclusons can be drawn. () The dependences of the mean nrtps packet delay on the total UGS capacty and on the maxmum (e)rtps capacty for unform dstrbuton show smlar tendences. () The dstrbuton of the (e)rtps capacty has essental mpact on the mean nrtps packet delay. These conclusons reman vald also n case of nonsaturated BE traffc, snce the BE traffc does not nfluence the nrtps packet delay. Ths s due to the appled capacty allocaton rule, n whch the nrtps traffc has prorty over the BE traffc at the same SS. The presented analytcal model also enables to enforce specfed upper bounds on the mean nrtps packet delays at every SS n a specfed range of loads. In ths case, the optmal value of the total amount of real-tme capactes can be determned. In case of a more general QoS requrement (delay constrant), the optmal set of prorty weghts of the SSs can be determned by usng a specfc cost model (see Secton 6.4). Acknowledgments Ths work s supported by Tampere Graduate School n Informaton Scence and Engneerng, Noka Foundaton, and HPY Research Foundaton. 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