Fluid Analysis of Delay Performance for QoS Support in Wireless Networks *

Transcription

1 Fluid Aalysis of Delay Performace for QoS Support i Wireless Networks * Jeog Geu Kim ad Marwa Kruz Departmet of Electrical ad Computer Egieerig Uiversity of Arizoa Tucso, AZ fjkkim, kruzg@ece.arizoa.edu Abstract Providig quality of service (QoS) guaratees over wireless liks requires thorough uderstadig ad quatificatio of the iteractios amog the traffic source, the wireless chael, ad the uderlyig error cotrol mechaisms. I this paper, we accout for such iteractios i a etworklayer model that we use to ivestigate the delay performace for a ON/OFF traffic stream trasported over a wireless lik. The capacity of this lik fluctuates accordig to a fluid versio of Gilbert-Elliot s model. We derive the packet delay distributio via two differet approaches: uiformizatio ad Laplace trasform. Coputatioal aspects of both approaches are discussed. The delay distributio is the used to quatify the wireless effective badwidth uder a give delay guaratee. Numerical results ad simulatios are used to verify the adequacy of our aalysis ad to study the impact of error cotrol ad badwidth allocatio o the packet delay performace. keywords: Wireless etworks, QoS, delay distributio, fluid aalysis. 1. Itroductio Recet treds i wireless etworks idicate a desire to provide a flexible broadbad wireless ifrastructure that ca support emergig multimedia services as well as traditioal data services [1, 13]. I such a multi-service wireless eviromet, quality-of-service (QoS) guaratees are critical for real-time voice ad video. I cotrast to its wirelie couterpart, the provisioig of QoS guaratees over wireless liks is a more challegig problem whose difficulty stems from the eed to explicitly cosider the harsh radio-chael tras- * This research was supported by the Natioal Sciece Foudatio uder CAREER Grat ANI missio characteristics ad the uderlyig lik-layer error cotrol mechaisms. This difficulty is further compouded by host mobility ad how it impacts the available badwidth capacity. These issues idicate a clear eed for a geeral QoS framework i the wireless eviromet. QoS guaratees i wireless etworks ca be provided through a coordiatio betwee coectio-level badwidth reservatio ad packet-level schedulig. Most previous research o QoS over wireless etworks has maily focused o these issues. Levie et al. proposed the shadow cluster cocept to estimate the badwidth requiremets of a wireless coectio [9]. They idicated that this cocept ca be used i coectio admissio cotrol to provide a specified call droppig probability. Reiiger et al. idetified the high variabilityof traffic dyamics of mobile multimedia applicatios as a fuctio of time ad space, ad proposed a soft QoS cotrol which allows badwidth reegotiatio accordig to the varyig traffic coditios [1]. Capoe ad Stavrakakis ivestigated the regio of supportable QoS vectors expressed i terms of packet droppig probability [3]. Their work provided isight ito the resource maagemet aspects for hadlig diverse QoS costraits, although the study was limited to ubuffered services. Lu et al. proposed a fair schedulig algorithm with adaptatio to wireless etworks that take ito accout bursty ad locatio-depedet chael errors [11]. Although their work idetified may practical issues, it did ot address the iteractio betwee packet schedulig ad error cotrol. I [7], the authors studied the cocept of wireless effective badwidth for a guarateed packet loss rate. The geeral goals of the uderlyig work are to study the delay performace over a wireless lik ad ivestigate its implicatio o optimal badwidth allocatio uder delay guaratees. Our ivestigatios are carried out for a sigle stream that is trasported over a time-varyig wireless lik. If the lik is used to trasport more tha oe coectio, the each coectio is guarateed a costat service rate durig its active period (i.e., TDMA style). The outcome

2 of a packet trasmissio is determied by the state of the wireless chael ad the error cotrol schemes. This sceario ecompasses poit-to-poit coectios betwee mobile termials (MT) ad a base statio (BS) i cellular commuicatio systems. To achieve our goals, we follow a fluid-based approach whereby the traffic source is modeled by a o-off fluid process ad the chael is modeled by a fluid variat of Gilbert- Elliott s model. Usig fluid-flow aalysis, we compute the delay distributio for a sigle stream as a fuctio of the traffic source, the service rate, the wireless chael, ad the error cotrol schemes. To obtai this distributio, we first evaluate the queue legth distributio takig ito accout the chael behavior ad the uderlyig error cotrol schemes. The, we provide two alterative approaches for obtaiig the delay distributio via the uiformizatio ad Laplace trasform techiques. I the case of the uiformizatio approach, we are able to derive a closed-form expressio for the delay distributio. The computatioal aspects of both approaches are compared. Our aalytical results are used to obtai the wireless effective badwidth uder a give delay costrait. Note that the otio of effective badwidth has bee traditioally ivestigated i wirelie [] ad wireless [7] etworks for a give packet loss rate. We also study the optimal error cotrol strategy that miimizes the effective badwidth while guarateeig a give delay requiremet (expressed as a percetile). Extesive simulatios are coducted to verify the goodess of our aalytical results. The rest of the paper is orgaized as follows. I Sectio 2, we describe the wireless lik model. Aalysis of the delay performace is provided i Sectio 3. Numerical results ad simulatios are reported i Sectio, followed by cocludig remarks i Sectio Wireless Lik Model 2.1. Framework I order to aalyze the packet-level performace of a wireless lik, we cosider the framework show i Figure 1. This framework was used earlier to study the packet loss performace ad the correspodig effective badwidth uder the same problem settig [7]. I this framework, traffic streams from oe or more coectios are fed ito a fiite-size FIFO buffer. A costat service rate c (i packets/secod) is assiged to the wireless coectio, but the actual drai rate observed at the buffer is reduced due to retrasmissios ad FEC overhead. The actual service rate will be discussed i the followig sectio. I our study, we cosider hybrid ARQ/FEC error cotrol i which the cyclic redudacy check (CRC) code is applied first to a packet, followed by FEC. We assume that the CRC code ca aloe detect almost all bit errors i a packet. I cotrast, oly a subset of the errors ca be corrected by FEC. I additio, we impose a limit o the umber of packet trasmissios. Imposig such a limit ca be used to provide delay guaratees for real-time traffic. Oce a packet hits the limit, it will be discarded. For simplicity, we igore the overhead of the medium access cotrol (MAC) layer. The above model has three cotrol parameters: the service rate (or assiged badwidth), the FEC code rate, ad the limit o the umber of trasmissios. These parameters ca be adjusted durig coectio setup to satisfy certai QoS requiremets. From the etwork poit of view, the selectio of these parameters is very crucial ad requires thorough uderstadig of their impact o the packet-level performace. The mai theme of this study is to ivestigate the packet-level performace of a wireless lik as a fuctio of the assiged badwidth, the limit o trasmissios, ad error cotrol schemes Queueig Model I this sectio, we describe the queueig model that is used to aalyze the packet delay over a wireless lik. The source is characterized by a o-off fluid process with peak rate r. Its o ad off periods are expoetially distributed with meas 1 ad 1, respectively. The wireless chael is modeled usig a fluid versio of Gilbert-Elliott (GE) model which is ofte used to ivestigate the performace over wireless liks [6]. As explaied i Figure 2, the GE model is Markovia with two alteratig states: Good ad Bad. The bit error rates (BER) durig the Good ad Bad states are give by P eg ad, respectively, where P eg. The duratios of the Good ad Bad states are expoetially distributed with meas 1 ad 1, respectively. Good δ γ wireless chael model Traffic sources Bad C e (t) P(t) e P eg C e (t) C C g C b BER bad chael state good chael state approximate service rate Figure 2. Wireles s chael model ad corres podig s ervice rate model. The FEC capability i the uderlyig hybrid ARQ/FEC mechaism is characterized by three parameters: the umber of bits i a code block (), the umber of payload bits time time

3 chael error packet loss (buffer overflow) variable FEC ecoder variable FEC decoder Traffic Source Buffer CRC ecoder Trasmitter CRC decoder Receiver Packet Error yes Limit reached? o o yes to upper layer NAK packet loss (discard) ACK feedback (ACK/NAK) Figure 1. Framework for aalyzig the performace over a wireles s lik. (k), ad the maximum umber of correctable bits i a code block (). Note that cosists of the k payload bits ad the extra parity bits. The FEC code rate e( ) is defied as e( ) k ( ) : Assumig that a FEC code ca correct up to bits ad that bit errors durig a give chael state are idepedet, the probability that a packet cotais a o-correctable error is give by: () ( ) P c (p b ; ) p j j b (1? p b) ()?j (1) j+1 where p b is the bit error probability; p b 2 fp eg ; g. To accout for the FEC overhead, we obtai the actual service rate c e observed at the output of the buffer: c e c e( ) (2) where c is the badwidth assiged to the coectio. The exact behavior of ARQ ad FEC i the uderlyig queueig model is difficult to aalyze. To obtai aalytically tractable results, we assume that the packet departure process follows a fluid process with a service rate that is modulated by the chael state (see Fig. 2). This approximatio implies that there are two determiistic service rates: c g durig Good states ad c b durig Bad states. We assume that the feedback delay for sedig a ackowledgmet from a give receiver to the seder is smaller tha the miimum time betwee two successive trasmissios to that receiver. This assumptio is reasoable i a TDMA eviromet, where the chael capacity is beig shared by several coectios (destiatios, MTs). Each coectio is assiged oe or more slots withi a TDMA frame. Slot assigmet reflects the costat service rate that is allocated to a coectio 1. A packet is successively retrasmitted util it 1 Guarateeig a costat service rate to a coectio ca be achieved by periodic assigmet of slots i a TDMA frame. is correctly received at the destiatio or util the limit o the umber of retrasmissios is reached. I this sceario, the total time eeded to successfully deliver a packet coditioed o the chael state follows a trucated geometric distributio. Let N tr deote the umber of retrasmissios (icludig the first trasmissio) util a packet is successfully received or is discarded because it reached the limit o retrasmissios. For a give packet error probability P c ad a limit o trasmissios N l, the expected value of N tr is give by: E[N tr ] 1? P c Nl : (3) 1? P c Thus, c g ad c b correspod to the mea trasmissio rates of the trucated geometric trials with parameters (P c;g ; N l ) ad (P c;b ; N l ), respectively, where P c;g ad P c;b are the packet error probabilities i Good ad Bad states, respectively, give by (1). Formally, c g c e( ) (1? P c;g) 1? P Nl c;g () c b c e( ) (1? P c;b) : (5) 1? P Nl c;b where P c;g P c (P e;g ; ) ad P c;b P c (P e;b ; ). 3. Aalysis of Delay Performace 3.1. Queue Legth Distributio Followig the discussio i the previous sectio, we costruct a Markovia queueig system with four states as show i Figure 3. Let S deote the state space. Thus, S f(; g); (; b); (1; g); (1; b)g (6) where ad 1 deote the o ad off states of a traffic source, respectively, ad g ad b deote Good ad Bad chael states, respectively.

4 (,g) γ (,b) δ β α β α (1,g) γ (1,b) δ /1 : off/o source state g/b : good/bad chael state Figure 3. S tate tras itio diagram. Followig a stadard fluid approach (see [2], for example), the evolutio of the buffer cotet ca be described by the followig differetial equatio: d(x) D (x)m (7) dx where D diag[?c g ;?c b ; r? c g ; r? c b ], (x) [ ;g (x) ;b (x) 1;g (x) 1;b (x) ]; s (x) Prfbuffer cotet x ad the system s 2 Sg, ad M is the geerator matrix of the uderlyig Markov chai: M 2 6?( + )?( + )?( + )?( + ) Throughout the paper, matrices ad vectors are boldfaced. The solutio of (7) correspods to the solutio of the eigevalue/eigevector problem: which is geerally give by (x) : zd M (8) z i a i exp(z i x) i (9) where a i s are costat coefficiets ad the pairs (z i ; i ), i 1; 2;, are the eigevalues ad the right eigevectors of the matrix M D?1 [2, 12]. I order to solve (8), we follow the approach used i [12]. The details of how (x) is computed are give i [7]. I the followig sectio, (x) is used to obtai the delay distributio Delay Distributio I fluid queueig models with a error-free chael ad costat service rate, e.g., ATM lik, the delay distributio ca be directly obtaied from the queue legth distributio [5]. However, the sceario we cosider i this study icludes a time-varyig wireless chael that is beig approximated by a two-state Markov modulated fluid process. I this case, the packet delay distributiois much more difficult to obtai sice oe has to take ito accout the time-varyig service rate as well as the queue legth. We assume a ifiite-capacity buffer. Let D deote the delay experieced by a arrivig packet. Let C(t) deote the accumulative amout of service durig a period of legth t: C(t) Z t c(s)ds where c(s) is the service rate at time s. The chael state at time t is deoted by h(t) 2 fg; bg, where g ad b deote Good ad Bad states, respectively. The probability that the delay see by a fluid atom is less tha or equal to t is equal to the probabilitythat C(t) is greater tha or equal to the queue legth at the istat of the packet arrival Q. Thus, we have Pr[D t] Pr[C(t) Q ] r T Z 1 i2s? Pr[C(t) xji; Q x] i (x)dx Z 1? Pr[C g(t) x] 1;g (x) + Pr[C b (t) x] 1;b (x)dx (1) where T is the throughput, i is the pdf of the queue legth i a state i, i 2 S, ad C i (t) Z t c(s)ds give h() i; for i 2 fg; bg: The quatity r(x)t represets the fractio of carried flow that arrives at the queue whe its cotet is x. For a ifiite-capacity buffer, the throughput T is give by: T r(w 1;g + w 1;b ): (11) I order to obtai Pr[C i (t) x], i 2 fg; bg, we provide two methods: direct calculatio usig Laplace trasform ad uiformizatio. The equivalece of these approaches will be verified usig umerical examples. Laplace Trasform Approach For coveiece, we trasform the radom variable C i (t) to C i (t) defied by: C i (t) C i (t)? c b t; i 2 fg; bg: By this trasformatio, C i (t) is the accumulative service resultig from a ormalized chael with service rates c g? c b (durig Good states) ad (durig Bad states). Note that the miimum amout of accumulative service i a period of legth t is c b t. Thus, ( 1; if x < c b t Pr[C i (t) x] 1? Pr[ C i (t) x? c b t]; if x c b t:

5 The followig propositio gives the probabilities Pr[C g (t) x] ad Pr[C b (t) x] by solvig the partial differetial equatios (PDE) for the cosumptio rate C(t). Propositio 3.1 The probabilities Pr[C i (t) x]; i 2 fg; bg whe x c b t, are give by Pr[C g (t) x] e e?^x?(t?^x) J (2p? ^x(t? ^x))! 1 ^x + (!)?( + 1; (t? ^x)) (12) 2 Pr[C b (t) x] e?^x 1 ^x?( + 1; (t? ^x)) (13) (!) 2 where ^x (x? c b t)(c g? c b ), J (z) is the Bessel fuctio of order zero give by J (z) 1 (?1) (z2) 2 (!) 2 ; ad?(; z) is the icomplete gamma fuctio give by: Z z?(; z) e?x x?1 dx: Proof. See [8]. Equatio (13) i Propositio 3.1 is substituted ito (1) to evaluate the delay distributio. Some umerical complexity is associated with the ifiite sums i Equatio (12) ad (13). We observed that the values of these ifiite sums coverge fast for moderate values of, e.g., 2. Uiformizatio Approach As a secod approach to obtaiig Pr[C i (t) x], i 2 fg; bg, we use the uiformizatio approach. I cotiuoustime Markov chais, uiformizatio is a techique for uiformizig the trasitio rates betwee states by itroducig trasitios from a state to itself [15, 16]. Let t g ad t b deote the accumulative sojour times of Good ad Bad chael states durig a iterval of legth t, respectively. That is, Z t t g 1 fh(s)gg ds Z t t b 1 fh(s)bg ds: The, the accumulative service C(t) is give by C(t) c g t g + c b t b ; t g ; t b t: (1) Sice t t g + t b, C(t) ca be expressed as C(t) c g t g + c b (t? t g ) or C(t) c g (t? t b ) + c b t b : (15) I [8], we provide the probability distributio of t g ad t b coditioed o the chael state. The, the probability Pr[C i (t) x], i 2 fg; bg ca be directly obtaied from (15). Propositio 3.2 The probabilities Pr[C g (t) x] ad Pr[C b (t) x] are give by 1 Pr[C g (t) x] 1? e?(+)t (t) 1 k?1 ik ad Pr[C b (t) x] e?(+)t 1 1 where x?cbt (c g?c b)t. k?1 ik! i (t)! i k1 k? 1 i (1? )?i (16) k1 k? 1?i (1? ) i (17) Proof. See [8]. The equatios i Propositio 3.2 also have some umerical issues due to the presece of multiple sums. The triple sums i the equatios cause sigificat complexity. However, observig the duplicate computatio i the last sum for cosecutive idexes k s, we ca achieve a sigificat reductio i the computatio time. Up to this poit, we have discussed the alterative approaches to obtaiig the probability Pr[C i (t) x]; i 2 fg; bg. I the followig, the results from Propositio 3.2 are used to obtai the delay distributio. The followig propositio gives a closed-form expressio for delay distributio by substitutig (16) ad (17) ito (1). Propositio 3.3 Pr[D t] r T ( 1;g(c g t) + 1;b (c b t)) (t)? r T e?(+)t 1 ik (c g? c b )t 1 + r T e?(+)t 1 ik (c g? c b )t l 1 m ( + 1)! k1 k? 1 k?1 a l e zlcbt (i + 1; + 2; z l (c g? c b )t) (t) ( + 1)! a m e zmcbt k1 k? 1 k?1 (? i + 1; + 2; z m (c g? c b )t) (18)

6 P P where 1;g (x) l a le zlx, 1;b (x) m a me zmx, the idexes l; m are used to idex the egative eigevalues, ad 1 k (x; y; z) k with (a) a(a + 1) (a +? 1). (x) k z (y) k k! Proof. See [8]. Sice Equatio (18) has a similar umerical structure to the expressios i Propositio 3.2, we ca reduce the computatio time by avoidig the duplicate sums as metioed previously. We also obtaied the delay distributio usig Propositio 3.1. However, the resultig expressio cotais multiple sums, ad hece provides o advatages over the uiformizatio approach. Accordigly, to obtai the delay distributio i the first approach (Propositio 3.1), we rely o umerical itegratio. This is maily used i cotrastig our two aalysis approaches; most umerical results i Sectio are based o Propositio 3.3. Wireless Effective Badwidth The otio of effective badwidth has bee traditioally employed to provide a guarateed packet loss rate i wirelie [] ad wireless [7]. I this study, we exted this otio for a wireless coectio uder probabilistic delay costraits. We defied the wireless effective badwidth c eb uder the delay costrait Pr[delay > t] " as follows: c eb mifcjc satisfies Pr[delay > t] "g (19) where c is the service rate. I cotrast to wirelie effective badwidth, the wireless effective badwidth is cofigured alog with the optimal umber of correctable bits which miimizes the use of badwidth while providig the requested reliability at the physical lik. I Sectio, we provide some umerical examples related to this cocept ad ivestigate the characteristics of the pair of QoS parameters (c eb ; ) i more detail.. Numerical Results ad Discussio I this sectio, we preset umerical examples based o our aalytical results. We verify the adequacy of these results by cotrastig them agaist more realistic simulatios. Similar to the aalysis, the simulatio results are obtaied usig o-off traffic sources with expoetially distributed o ad off periods. The ARQ retrasmissio process is simulated i a more realistic maer, whereby a packet is trasmitted repeatedly util it is received with o errors or util it reaches the limit o the umber of trasmissios. The probability of a packet error is computed from (1) for both chael states. Trasitios betwee Good ad Bad states are assumed to occur oly at the begiig of a packet trasmissio slot. A packet is retrasmitted if it has ucorrectable errors. It is assumed that the propagatio delay is small, so that the ACK/NAK message for a packet is received at the seder before the ext attempt of trasmissio. Fially, we use a ifiite-capacity buffer i our simulatios. I our experimets, we vary the BER durig the Bad state ( ) ad fix the BER durig the Good state at P eg 1?6. We set the mea of the off period to te times that of the o period. I additio, we take the parameters related to the wireless chael from [6]. We adopt Bose-Chaudhuri- Hocqueghem (BCH) code [1] for FEC. We cosider fixed packet sizes, e.g., ATM cells. Sice we treat the CRC code as part of the payload, the FEC code is applied to 2-bit blocks (i.e., k 2 bits). I [8], a table is available to show the size, the code rate, ad the umber of correctable bits of the BCH code used i our examples. All simulatios are reported with 95% cofidece itervals. For the delay distributio, 1 7 to 1 7 samples were eeded i the simulatios. Table 1 summarizes the values of the various parameters i the simulatios ad umerical examples. For the parameters c,,, ad N l, the values i the parethesis are assumed uless specified otherwise. Figure depicts the complemetary delay distributio. We vary the service rate (c) from 8 to 12 packets/sec while fixig the other parameters at 1?2 ; 7, ad N l 1. The differece betwee the aalytical ad simulatio results is quite egligible for all service rates. We observe a slight deviatio at the tail part of the distributio. However, it is associated with the umber of samples take from the simulatio. For c 12, we geerated 1 7 packets to obtai the show results. Note that the simulatio is based o the realistic sceario i which the packet is trasmitted util it is successfully trasmitted or util it reaches the limit o the retrasmissio, whereas the aalytical results are based o the fluid approximatio. Figure 5 shows the effective badwidth as a fuctio of the umber of correctable bits () for three target delay costraits Pr[delay > "], with " :1; :5; :1. Expectedly, more badwidth is eeded to achieve a more striget delay guaratee. Iterestigly, we observe that a very large amout of badwidth is required whe oly ARQ ( ) is used for error cotrol. Thus, the use of FEC is essetial to achieve efficiet badwidth allocatio with delay guaratees. The figure clearly idicates that there is a optimal ( 7 i this example) for a give BER that satisfies a delay QoS costrait while miimizig the use of badwidth. The optimal umber of correctable bits as a fuctio of the BER of the Bad state is show i Fig. 6. The target delay costrait is fixed at Pr[delay > :1] :25. We vary the P e;b from.1 to.158. For each BER, we observe the

7 Parameter Symbol Value source peak rate r 1 Mbps (or packets/sec) service rate c 1? 8 packets/sec (1) mea o period 1 :23 sec mea off period 1 :23 sec mea Good chael period 1 :1 sec mea Bad chael period 1 :333 sec BER i Good chael state P eg 1?6 BER i Bad chael state 1?2? 1?5 (1?2 ) umber of correctable bits? 2 (7) limit o trasmissios N l 1? 1 (1) Table 1. Parameter values us ed i the s imulatios ad umerical res ults. 1 optimal umber of correctable bits. Pr[delay > t] c8 (simulatio) c8 (aalysis) c1 (simulatio) c1 (aalysis) c12 (simulatio) c12 (aalysis) Effective Badwidth (packets/sec) t (sec) Figure. Comple me tary de lay dis tributio for differet s ervice rates. Effective Badwidth (packets/sec) Pr[D >.1].1 Pr[D >.1].5 Pr[D >.1] Number of Correctable Bits (τ) Figure 5. Effective badwidth vers us for 1 target delay cos trait Pr[delay > :1] " (" :1; :5; :1) Number of Correctable Bits (τ) Figure 6. Effective badwidth vers us for differet BER's. Figure 7 shows the effective badwidth versus the target delay costraits " Pr[delay > t], for t :1; :5; :1; :1 (sec). As expected, more badwidth is required for a more striget delay requiremet, i.e., smaller Pr[delay > t] at a fixed t. Notice that the miimum ad maximum values of the effective badwidth are c 282 ad c 325 (packets/sec), respectively. The miimum value correspods to the miimum service rate satisfyig the stability coditio of the queue, whereas the maximum value is the miimum service rate satisfyig the coditio c b > r, i.e., mifcjc b > rg. Cosiderig the the source peak rate r 26:1667 (packets/sec), the maximum effective badwidth is surprisigly greater tha the source peak rate. This is i disagreemet with the wellkow fact that the wirelie effective badwidth is bouded by the source peak rate. I the uderlyig wireless sceario, the assiged service rate is reduced due to the packet errors ad FEC overhead, ad thus the priciple i wirelie effec-

8 tive badwidth caot be applied here. Aother iterestig observatio i this figure is associated with the shape of the curves with decreasig Pr[delay > t]. Cosider the case of t :1. To improve the QoS from Pr[delay > :1] 1: to Pr[delay > :1] :8, we eed to assig a extra badwidth 195. (packets/sec), whereas we eed 13. (packets/sec) for t :1. That is, the behavior of the effective badwidth depeds o the selectio of t as well as the costrait Pr[delay > t]. Effective Badwidth (packets/sec) t.1 t.1 t.5 t Pr[delay > t] Figure 7. Effe c tive badwidth ve rs us targe t delay cos traits Pr[delay > t]. 5. Coclusios I this paper, we ivestigated the delay performace for a o/off source trasported over a wireless chael. Simple yet accurate fluid models were used to capture the bursty ature of the arrivig traffic ad the chael s time-varyig error characteristics. Error cotrol schemes (ARQ ad FEC), which are essetial elemets of ay wireless packet etwork, were icorporated. We obtaied the delay distributio usig two alterative approaches: Laplace trasform ad uiformizatio. The solutio was the used to obtai the wireless effective badwidth (defied here as the miimum amout of badwidth required to satisfy a give probabilistic delay costrait), which ca be used as a valuable tool i resource allocatio ad admissio cotrol i wireless etworks. Our aalytical results were validated by cotrastig them with simulatios. It was observed that the aalytically obtaied delay distributio is quite accurate over a wide rage of parameters. I a future work, we pla to ivestigate the system capacity of the wireless lik, which is costraied by diverse QoS parameters, e.g., delay, loss, ad jitter, of multiple coectios over a shared chael. Refereces [1] A. Acampora. Wireless ATM: a perspective o issues ad prospects. IEEE Pers. Commu., 3():8 17, Aug [2] D. Aick, D. Mitra, ad M. M. Sodhi. Stochastic theory of a data hadlig system with multiple sources. Bell Syst. Tech. J., 61: , [3] J. Capoe ad I. Stavrakakis. Achievable QoS ad schedulig policies i itegrated services wireless etworks. Perform. Eval., 27/28(1):37 365, Oct [] A. I. Elwalid ad D. Mitra. Effective badwidth of geeral Markovia traffic sources ad admissio cotrol of high speed etworks. IEEE/ACM Tras. Networkig, 1(3):329 33, Jue [5] A. I. Elwalid ad D. Mitra. Statistical multiplexig with loss priorities i rate-based cogestio cotrol of high-speed etworks. IEEE Tras. Commu., 2(11): , Nov [6] N. Guo ad S. D. Morgera. Frequecy-hopped ARQ for wireless etwork data services. IEEE J. Select. Areas Commu., 12(8): , Sept [7] J. G. Kim ad M. Kruz. Effective badwidth i wireless ATM etworks. I MobiCom 98, pages , Oct [8] J. G. Kim ad M. Kruz. Fluid aalysis of delay ad packet discard performace for QoS support i wireless etworks. Techical Report CENG-TR , Departmet of ECE, Uiversity of Arizoa, Aug [9] D. A. Levie, I. F. Akyildiz, ad M. Naghshieh. A resource estimatio ad call admissio algorithm for wireless multimedia etworks usig the shadow cluster cocept. IEEE/ACM Tras. Networkig, 5(1):1 12, Feb [1] S. Li ad J. D. J. Costello. Error Cotrol Codig: Fudametals ad Applicatios. Pretice Hall, Eglewood Cliffs, NJ, [11] S. Lu, V. Bharghava, ad R. Srikat. Fair schedulig i wireless packet etworks. I SIGCOMM 97, Sept [12] D. Mitra. Stochastic theory of a fluid model of producers ad cosumers coupled by a buffer. Adv. Appl. Prob., 2:66 676, [13] D. Raychaudhuri ad N. D. Wilso. ATM-based trasport architecture for multiservices wireless persoal commuicatio etworks. IEEE J. Select. Areas Commu., 12(8):11 11, Oct [1] D. Reiiger, R. Izmailov, B. Rajagopala, M. Ott, ad D. Raychaudhuri. Soft QoS cotrol i the WATMet broadbad wireless system. IEEE Pers. Commu., 6(1):3 3, Feb [15] S. Ross. Stochastic Processes. Joh Wiley & Sos, secod editio, [16] B. Sericola. Trasiet aalysis of stochastic fluid models. Performace Evaluatio, 32:25 263, 1998.