NOVEL ANALYSIS METHOD FOR OPTICAL PACKET SWITCHING NODES

Transcription

1 NOVEL ANALYSIS METHOD FOR OPTICAL PACKET SWITCHING NODES Ten Van Do Department of Telecommuncatons, Budapest Unversty of Technology and Economcs H-1111, Magyar tudósok körútja 2., Budapest, Hungary, Tel: , Emal: do@ht.bme.hu Ram Chakka Department of Computer Scence, Norfolk State Unversty, 700 Park Avenue, Norfolk, VA 23504, USA, Tel: , Emal: ramchakka@yahoo.com Zsolt Pánd Department of Telecommuncatons, Budapest Unversty of Technology and Economcs H-1111, Magyar tudósok körútja 2., Budapest, Hungary, Tel: , Emal: pand@ht.bme.hu Abstract Keywords: Packet and burst swtchng have been proposed for optcal networks because they can better accommodate bursty traffc generated by IP applcatons. In optcal packet swtchng networks the payload and the header of the same packet are conveyed n the same channel, whle burst swtchng networks allow the separate transportaton of the payload and the header of the same burst. In ths paper we consder an optcal packet swtchng node that assgns arrvng packets to channels n a lnk wth c avalable data channels (wavelengths) and a buffer of L c sze. The paper apples the novel MM P K CPP k /GE/c/L G-queue to model optcal packet swtchng nodes. It s worth emphaszng that our method can be appled to model burst swtchng nodes as well. Moreover, we show that a model prevously presented n the lterature s only the specal case of our model. Numercal results quanttatvely demonstrate that the characterstcs (e.g.: burstness) of the offered traffc have a sgnfcant mpact on the performance of optcal nodes. optcal packet swtchng, optcal burst swtchng, MM queue, Queueng theory CPP k /GE/c/L G-

2 2 ONDM 1. Introducton To effcently accommodate bursty IP data traffc two techncal solutons (packet and burst swtchng) are beng proposed for networks based on optcal technology. The fnal am s to have networks that swtch packets of constant or varable length whle the payload data stays n the optcal doman. In burst swtchng networks payload data and ts control data (header) are transported n dfferent channels, whle packet swtchng networks convey payload data and ts header n the same channel (El-Bawab and Shn, 2002; Yao et al., 2002). In ths paper we develop a new model for optcal nodes operatng n ether optcal packet swtchng or burst swtchng networks. To evaluate the performance of optcal nodes a decomposton approach s used. Namely, the performance of an optcal node s determned f we can evaluate the performance of multplexers before the transmsson lnks. That s, we consder an optcal packet (or burst) swtchng multplexer that assgns arrvng packets (or bursts) to c avalable data channels (wavelengths) and has a buffer for L c packets (or bursts). Therefore, we propose the use of the MM P K CPP k/ge/c/l G- queue to model nodes n both knds of networks (burst and packet swtchng), whch queue has been proposed recently n (Chakka et al., 2002). Ths s a homogeneous mult-server queue wth c servers, GE servce tmes and wth K ndependent customer arrval streams, each of whch s a CPP,.e. a Posson pont process wth batch arrvals of geometrcally dstrbuted batch sze. The use of the MM P K CPP k process to model packet or burst arrval process s motvated by the followng reason. Recent studes have shown that the traffc n today s telecommuncatons systems often exhbts burstness.e. batches of transmsson unts (e.g. packets) arrve together and correlaton among nterarrval tmes. As a consequence dfferent models have been proposed. These models nclude the compound Posson process (CPP) n whch the nterarrval tmes are assumed to have generalzed exponental (GE) probablty dstrbuton (Kouvatsos, 1994), the Markov modulated Posson process (MMPP) and self-smlar traffc models such as Fractonal Brownan Moton (FBM) (Mandelbrot and Ness, 1968; Norros, 1994). A CPP traffc model often gves a good representaton of burstness of the traffc from one or more sources, e.g. (Bhabuta and Harrson, 1997; Fretwell and Kouvatsos, 1999), but not of the auto-correlatons observed n real traffc. Conversely, the MMPP models can capture auto-correlaton but not burstness, e.g. (Fretwell and Kouvatsos, 1997; Meer-Hellstern, 1989). The self-smlar models such as FBM can account for both auto-correlaton and burstness, but they are analytcally ntractable n a queueng context. Often, the traffc to a node s the superposton of traffc from a number of sources complcatng the system analyss further. The MM P K CPP k captures the burstness and correlaton of the traffc, and ts parameter K can be used to model varous traffc passng optcal

3 Novel Analyss Method for Optcal Packet Swtchng Nodes 3 nodes from P dfferent sources n a flexble manner. Moreover, the Makov modulated K CPP k/ge/c/l G-queue s mathematcally tractable wth effcent analytcal soluton for the steady state probabltes wth the use of mathematcally orented transformatons (Chakka et al., 2002). To obtan the steady state probabltes and thus the performance measures ether the spectral expanson method (Chakka, 1995) or Naoumov s method (Naoumov et al., 1997) extended for QBD processes, or the matrx-geometrc soluton method (Neuts, 1995) can be used. Related to the performance analyss aspect, Turner has proposed a brth-death process to analyze a multplexer n optcal burst swtched networks (Turner, 1999). However, Turner s model has some lmtatons lke the assumpton of exponental burst arrval process and exponental servce tmes. Moreover, t consders bursts of the same sze. Obvously, our model s more general than Turner s model and t overcomes the lmtatons of Turner s model. Moreover, t can be shown and numercally demonstrated that Turner s model s a specal case of our model. The rest of the paper s organzed as follows. The proposed model s descrbed n Secton 2. Some numercal results are then presented n Secton 3. The paper concludes n Secton Model descrpton Snce we consder a multplexer before a transmsson lnk wth c avalable data channels (wavelengths) and a buffer for L c packets (or bursts), a queueng model for a multplexer has c servers and L queueng capacty 1 for packets (or bursts). In what follows we outlne the mportant characterstcs of the proposed model. 2.1 The arrval process The arrval and servce processes are modulated by a contnuous tme, rreducble Markov phase process wth N states. Let Q be the generator matrx of ths process, gven by Q = q 1 q 1;2 ::: q 1;N q 2;1 q 2 ::: q 2;N q N;1 q N;2 ::: q N ;

4 4 ONDM where q ;k ( 6= k) s the nstantaneous transton rate from phase to phase k, and q = j=1 q ;j ; q ; =0 ( =1;:::;N) Let r =(r 1 ;r 2 ;:::;r N ) be the vector of equlbrum probabltes of the modulatng phases. Then, r s unquely determned by the equatons: rq =0 ; re N =1: where e N stands for the column P vector wth N elements, each of whch s unty. K The arrval process (MM CPP k) s the superposton of K ndependent CPP arrval streams of customers 2, n a Markov modulated envronment. The customers of dfferent arrval streams are not dstngushable. The parameters of the GE nter-arrval tme dstrbuton of the k th (1» k» K) customer arrval stream n phase are (ff ;k ; ;k ). Thus, all the K arrval pont-processes are Posson, wth batches arrvng at each pont havng geometrc sze dstrbuton. Specfcally, the probablty that a batch s of sze s s (1 ;k ) s 1 ;k,n phase, for the k th stream of customers. It s obvous that ths knd of arrval processes s dfferent from the BMAP (Batch Markovan Arrval Process) by Lucanton (Lucanton, 1991) n a sense that the sojourn tmes of each phase n the BMAP are exponentally dstrbuted, whle n ths model the sojourn tmes of each phase are GE. It s worth emphaszng that our queueng model ncorporates K arrval processes and each of them s more general and complex than the arrval process presented before n the lterature. Let ff ;: ; ff ;: be the average arrval rate of customer batches and customers n phase respectvely. Let ff; ff be the overall average arrval rate of batches and customers respectvely. Then, ff ;: = ff = =1 ff ;k ; ff ;: = ff ;: r ; ff = =1 ff ;k (1 ;k ) ff ;: r Because of the superposton of many CPP s, the overall arrvals n phase can be consdered as bulk-posson (M [x] ) wth arrval rate ff ;: and wth a batch sze dstrbuton fß l= g (the probablty of batch sze beng l gven that the phase s ) that s more general than mere geometrc. The probablty that ths batch sze s l s gven by, ß l= = (1) ff ;k ff ;: (1 ;k ) l 1 ;k (2)

5 Novel Analyss Method for Optcal Packet Swtchng Nodes 5 1X l=1 The overall batch sze dstrbuton s then gven by, ß l=: = ß l= =1:0 (3) =1 r ß l= (4) Defne ß ;l as the probablty that a gven batch arrval s durng phase and s of sze l, then ß ;l = r ß l=. 2.2 The GE mult-server Each data channel wll be modelled as a server. Therefore there are c homogeneous servers n parallel, each wth GE-dstrbuted servce tmes wth parameters (μ ;ff ) n phase. The servce dscplne s FCFS and each server serves at most one customer at any gven tme. The operaton of the GE server s smlar to that descrbed for the CPP arrval processes above. L denotes the queueng capacty n all phases, ncludng the packets n servce, f any. L can be fnte or nfnte. When the number of packets s j and the arrvng batch sze of customers s greater than L j (assumng a fnte L), we assume that only L j customers are taken n and the rest are rejected. However, the batch sze assocated wth a servce completon s bounded by one more than the number of customers watng to commence servce at the departure nstant. For queues of length c» j < L +1(ncludng any packets n servce), the maxmum batch sze at a departure nstant s j c +1, only one server beng able to complete a servce perod at any one nstant under the assumpton of exponentally dstrbuted batch-servce tmes. Thus, the probablty that a departng batch has sze s s (1 ff )ff s 1 for 1» s» j c and ff j c for s = j c +1. In partcular, when j = c, the departng batch has sze 1 wth probablty one, and ths s also the case for all 1» j» c snce each packet s already engaged by a server and there are then no packets watng to commence servce. It s assumed that the frst packet n a batch arrvng at an nstant when the queue length s less than c (so that at least one server s free) never skps servce,.e. always has an exponentally dstrbuted servce tme. However, even wthout ths assumpton the methodology descrbed n ths paper s stll applcable. Note that the arrval and servce processes are modulated by the same contnuous tme, rreducble Markov phase process wth N states. Ths assumpton does not lmt the usage of the model because the case of dfferent modulatng processes for the arrval and servce can be traced back to our model (.e.: f the number of the states of the arrval phase process s N a and the number of

6 6 ONDM states of the servce phase process s N s, then we can convert ths case nto our model by consderng a jont phase process wth N = N s N a states). 2.3 Negatve customers The parameters of the GE nter-arrval tme dstrbuton of negatve customers are (ρ ;ff ) n phase. That s, the nter-arrval tme probablty dstrbuton functon s 1 (1 ff )e ρ t for the negatve customers n phase. Thus, the negatve customer arrval pont-process s Posson, wth batches arrvng at each pont havng geometrc sze dstrbuton. A negatve customer removes a postve customer n the queue, accordng to a specfed kllng dscplne. When a batch of negatve customers of sze l (1» l < j c) arrves, l postve customers are removed from the end of the queue leavng the remanng j l postve customers n the system. If l j c 1, then j c postve customers are removed, leavng none watng to commence servce (queue length equals to c). If j» c, the negatve arrvals have no effect. ρ, the average arrval rate of negatve customers n phase and ρ, the overall average arrval rate of negatve customers are gven by, ρ = ρ 1 ff ; ρ = =1 r ρ (5) Negatve customers can be used to model some phenomena n networks such as sgnal falure/loss n optcal cable, packet losses and load balancng. We show n Secton 3 how negatve customers can be used to model packet losses. 2.4 Condton for stablty When L s fnte, the system s ergodc snce the representng Markov process s rreducble. Otherwse,.e. when L = 1, the overall average departure rate ncreases wth the queue length, and ts maxmum (the overall average departure rate when the queue length tends to 1) can be determned as, μ = c =1 r μ 1 ff : (6) Hence, we conjecture that the necessary and suffcent condton for the exstence of steady state probabltes s ff<ρ + μ: (7) The above condton s obvous and ntutvely appealng. However, we have nether a rgorous proof of the same nor a sutable reference to such a proof.

7 Novel Analyss Method for Optcal Packet Swtchng Nodes The steady state balance equatons The state of the system at any tme t can be specfed completely by two nteger-valued random varables, I(t) and J(t). I(t) vares from 1 to N, representng the phase of the modulatng Markov chan, and 0» J(t) < L +1 represents the number of postve customers n the system at tme t, ncludng any n servce. The system s now modelled by a contnuous tme dscrete state Markov process, Y (Y f L s nfnte), on a rectangular lattce strp. Let I(t), the phase, vary n the horzontal drecton and J(t), the queue length or level, n the vertcal drecton. We denote the steady state probabltes by fp ;j g, where p ;j =lm t!1 Prob(I(t) =; J(t) =j), and let v j =(p 1;j ;:::;p N;j ). The process Y evolves due to the followng nstantaneous transton rates: (a) q ;k purely lateral transton rate from state (; j) to state (k; j), for all j 0 and 1» ; k» N ( 6= k), caused by a phase transton n the Markov chan governng the arrval phase process; (b) B ;j;j+s s-step upward transton rate from state (; j) to state (; j+s), for all phases, caused by a new batch arrval of sze s customers. For a gven j, s can be seen as bounded when L s fnte and unbounded when L s nfnte; (c) C ;j;j s s-step downward transton rate from state (; j) to state (; j s), (j s c +1)for all phases, caused by a batch servce completon of sze s, or a batch arrval of negatve customers of sze s; (d) C ;c+s;c s-step downward transton rate from state (; c + s) to state (; c), for all phases, caused by a batch arrval of negatve customers of sze s or a batch servce completon of sze s (1» s» L c); (e) C ;c 1+s;c 1 s-step downward transton rate, from state (; c 1+s) to state (; c 1), for all phases, caused by a batch departure of sze s (1» s» L c +1); (f) C ;j+1;j 1-step downward transton rate, from state (; j +1)to state (; j), (c 2; 0» j» c 2), for all phases, caused by a sngle departure;

8 8 ONDM where B ;j s;j = B ;j;l = (1 ;k ) s 1 ;k ff ;k (8 ; 0» j s» L 2; j s<j<l); 1X s=l j (8 ; j» L 1) ; (1 ;k ) s 1 ;k ff ;k = C ;j+s;j = (1 ff )ff s 1 cμ +(1 ff )ff s 1 ρ L j 1 ;k ff ;k (8 ; c +1» j» L 1; 1» s» L j) ; = (1 ff )ff s 1 cμ + ff s 1 ρ (8 ; j = c ; 1» s» L c) ; = ff s 1 cμ (8 ; j = c 1; 1» s» L c +1); = 0 (8 ; c 2; 0» j» c 2; s 2) ; = (j +1)μ (8 ; c 2; 0» j» c 2; s =1): Defne, B j s;j = Dag [B 1;j s;j;b 2;j s;j;:::;b N;j s;j] B s = B j s;j (j <L) = Dag " :::; (j s<j» L) ; # ff ;k (1 ;k ) s 1 ;k ;::: ± k = Dag [ff 1;k ;ff 2;k ;:::;ff N;k ] (k =1; 2;:::;K); k = Dag [ 1;k ; 2;k ;:::; N;k ] (k =1; 2;:::;K); ± = ± k ; ;

9 Novel Analyss Method for Optcal Packet Swtchng Nodes 9 R = Dag [ρ 1 ;ρ 2 ;:::;ρ N ] ; = Dag [ff 1 ;ff 2 ;:::;ff N ] ; M = Dag [μ 1 ;μ 2 ;:::;μ N ] ; Φ = Dag [ff 1 ;ff 2 ;:::;ff N ] ; C j = jm (0» j» c) ; = cm = C (j c) ; C j+s;j = Dag [C 1;j+s;j ;C 2;j+s;j ;:::;C N;j+s;j ] ; E = Dag(e 0 N ) : Then, we get, B s = B 1 = B = B L s;l = s 1 k (E k )± k ; s 1 k ± k ; (E k )± k ; C j+s;j = C(E Φ)Φ s 1 + R(E ) s 1 (c +1» j» L 1; s =1; 2;:::;L j) ; = C(E Φ)Φ s 1 + R s 1 (j = c ; s =1; 2;:::;L c) ; = CΦ s 1 (j = c 1; s =1; 2;:::;L c +1); = 0 (c 2; 0» j» c 2; s 2) ; = C j+1 (c 2; 0» j» c 2; s =1): The steady state balance equatons are, (1) For the L th row or level: LX s=1 v L sb L s;l + v L [Q C R] =0; (8)

10 10 ONDM (2) For the j th row or level: jx s=1 v j sb s + v j [Q ± C j RI j>c ]+ L j X s=1 v j+s C j+s;j =0 (0» j» L 1) ; (9) (3) Normalzaton LX j=0 v j e N =1: (10) where, I j>c =1f j >celse 0, and e N s a column vector of sze N wth all ones. Notce equatons (8, 9, 10), carefully. Each equaton has all the unknown vectors v j s. If L s unbounded, then each of these are nfnte number of equatons n nfnte number of unknowns, v j s, and each equaton s nfntely long contanng all the nfnte number of unknowns. Also, the coeffcent matrces of v j are j-dependent. It may be noted that there has been nether a soluton nor a soluton methodology to solve these equatons. In (Chakka et al., 2002), a novel methodology s developed to solve these equatons exactly and effcently. Frst these complcated equatons are transformed to a computable form by usng certan mathematcally orented transformatons. The resultng transformed equatons are of the QBD-M type (QBD wth smultaneous-multple-bounded brths and smultaneous-multple-bounded deaths) and hence can be solved by one of the several avalable methods, vz. the spectral expanson method, Bn- Men s method or the matrx-geometrc method wth foldng or block sze enlargement (Haverkort and A.Ost, 1997). 2.6 Performance measures Some performance measures can be derved as follows: Average number of packets n the system E(j) = LX j=0 jv j e: (11) Packet loss probablty LX 1X j=0 l=l j+1 0 l (L j)) v j (ß 1;l ;:::;ß N;l ) l (12)

11 Novel Analyss Method for Optcal Packet Swtchng Nodes 11 Average departure rate of postve customers where and ν n = ν 1 = LX =1 j=c+n =1 ν = X L c+1 s=1 p ;j (1 ff )ff n 1 cμ + sν s (13) p ;c+n 1ff n 1 cμ (n =2; :::; L c +1) cx =1 j=1 3. Numercal results p ;j jμ + LX =1 j=c+1 (14) p ;j (1 ff )cμ (15) Three numercal results are presented. Frst, we show that Turner s model s the specal case of our model. Next, we present the mpact of bursty traffc on the performance of the system. Note that n the frst two cases, no negatve customers are allowed n the system. Then, we show how the throughput of connectons can be determned through the presence of negatve customers. 3.1 Turner s model s the specal case of our model In ths secton we demonstrate that Turner s model for burst swtchng s the specal case of our model by lettng K =1;N =1; q ;j = 0 ; h ff;k = μ ; ;k = 0 ; ff = 1 ; μ = 0 : Fgure 1 s exactly the same as Fgure 2 n (Turner, 1999), except that the data was produced by our model wth the parameter settngs mentoned earler. In order to demonstrate the equvalence, the results were calculated and compared to 20 sgnfcant dgts usng both models for a subset of the parameter set dsplayed on Fgure 1. The calculatons were executed on a Sun Ultra 60 Workstaton, whch had a machne epslon 3 ffl =1: Table 1 summarzes the outcome. It s clear that the dfferences between the results produced by the two models are O(ffl).

12 12 ONDM packet dscard probablty e 05 1e 06 4 channels 32 channels 256 channels 0 burst stores burst stores burst stores load Fgure 1. Packet loss probablty vs load and c P K Table 1. Numercal comparson of Turner s model and the MM CPP k/ge/c/l model for c =32 load number of dentcal dgts exponent of numercal value b Impact of bursty traffc In ths secton we show the mpact of the burstness of the offered traffc on the performance of the multplexer. For the numercal study the followng system parameters were used:

13 Novel Analyss Method for Optcal Packet Swtchng Nodes 13» 0:2 0:2 q;j = 0:9 0:9» 1 2 ff;k = 2 2:5» 0:65 0:7 ;k = 0:65 0:7 ff = 5 5 μ = 0:5 0:5 where all ff ;k was scaled as approprate to set the system load to the examned values. packet dscard probablty e 05 1e 06 0 buffer sze buffer sze buffer sze 4 channels 32 channels 256 channels load Fgure 2. Packet loss probablty vs load and b Fgure 2 plots the packet dscard probablty for ths numercal example. Two mportant conclusons can be drawn by nterpretng the results, and both of them are related to the effect of batch arrvals: From the fgure t s clearly observed that batch arrvals have a sgnfcant mpact on the performance of the system, that s, performance s degraded. Batch arrvals can be better handled by ncreasng the buffer space (at the expense of some queueng delay) than by ncreasng the number of channels. The performance of 256 channels wth no buffer s worse than that of 32 channels wth a buffer for 8 packets n our example for relatve load values above 0.4.

14 14 ONDM 3.3 Impact of the connecton loss on the connecton throughput departure rate of customers r2 =0:1 r2 =0:2 r2 =0:3 r2 =0:4 r2 =0:5 r2 =0:6 r2 =0:7 r2 =0:8 r2 =0: burstness of negatve customers (ff2) Fgure 3. Effect of the negatve customer arrval process In ths secton we present an approxmaton to calculate the performance parameter (throughput) of a connecton based n the presented queueng model. We also llustrate, then, the mpact of a packet loss on the performance of a connecton. The consdered problem here s the approxmaton of the throughput of two communcatng peers n optcal networks. A prelmnary approxmaton can be proposed as follows. The throughput of two communcatng peers can be approxmated wth the queueng model of a sngle node ncorporatng the packet loss phenomena along the path. It s showed based on measurements n (Yajnk et al., 1999) that packet P loss can be modelled as a 2-state Markov chan K model. Therefore, the MM CPP k /GE/c/L G-queue can be appled n ths case, where negatve customers model the loss along the path, and the departure rate of postve customers s the performance measure related to the throughput of a connecton. In the numercal example, both postve customers and servers work ndependently of the phase; and the process of negatve customers has two phases. Therefore, two phases are consdered 4. We defned c = 32 and L = 64 to show an example wth smlar parameters to those of the former examples and set the system load to 0.8. These consderatons led to a system wth the followng parameters:

15 Novel Analyss Method for Optcal Packet Swtchng Nodes 15» 38:4 38:4 ff;k = 38:4 38:4» 0:65 0:7 ;k = 0:65 0:7 ff = 5 5 μ = 0:5 0:5 ρ = 0 4 ff = 0:0 ff2 Fgure 3 llustrates the dependency of the customer departure rate on the parameter representng the packet loss process (modelled by negatve customers). It can be observed that the correlaton of the packet losses has a sgnfcant mpact on the performance of the system. However, the parameter fttng of the negatve customer process needs further study. 4. Conclusons We have appled the new queueng model for the performance P analyss of optcal packet swtchng nodes. We have shown that the MM K CPP k/ge/c/l G-queue can be appled to overcome the lmtatons of the prevous work n the lterature by Turner. Moreover, t s shown that Turner s model s the specal case of our model. Numercal results quanttatvely demonstrate that the characterstcs (e.g.: burstness) of offered traffc have a sgnfcant mpact on the performance of optcal nodes. Thus, the model that s proposed handles large or unbounded batch szes, both n arrvals and servces, wth great computatonal effcency and hence may have defnte advantages over BMAP based models n handlng long batch szes. Acknowledgement Ths work s partly supported by the EURESCOM 1112 project. Notes 1. Includng the packets (or bursts) n servce. 2. A customer s used to denote ether a packet or a burst 3. The machne epslon s the smallest floatng pont number that bounds the roundoff n ndvdual floatng pont operatons. 4. Ths support the message that the presented queueng model s qute flexble. References Bhabuta, M. and Harrson, P. (1997). Analyss of ATM Traffc on the London MAN. In Proc. 4th Int. Conf. on Performance Modellng and Evaluaton of ATM Networks, Ilkely. Chapman and Hall. Chakka, R. (1995). Performance and Relablty Modellng of Computng Systems Usng Spectral Expanson. PhD thess, Unversty of Newcastle upon Tyne (Newcastle upon Tyne).

16 16 ONDM Chakka, R., Do, T. V., and Pand, Z. (2002). Steady state analyss and load balancng n MPLS networks usng a generalsed markovan node model. Techncal report, Norfolk Unversty and Budapest Unversty of Technology and Economcs. Submtted for publcaton. El-Bawab, T. S. and Shn, J.-D. (2002). Optcal Packet Swtchng n Core Networks: Between Vson and Realty. IEEE Communcatons Magazne, pages Fretwell, R. and Kouvatsos, D. (1997). Correlated Traffc Modellng and Batch Renewal Markov Modulated Processes. In Proc. 4th IFIP Workshop on Performance Modellng and Evaluaton of ATM Networks, pages 20 44, Ilkely. Chapman and Hall. Fretwell, R. and Kouvatsos, D. (1999). ATM Traffc Burst Lengths Are Geometrcally Bounded. In Proceedngs of the 7 th IFIP Workshop on Performance Modellng and Evaluaton of ATM & IP Networks, Antwerp, Belgum. Chapman and Hall. Haverkort, B. and A.Ost (1997). Steady State Analyses of Infnte Stochastc Petr Nets: A Comparson between the Spectral Expanson and the Matrx Geometrc Methods. In Proceedngs of the 7th Internatonal Workshop on Petr Nets and Performance Models, pages , Sant Malo, France. Kouvatsos, D. (1994). Entropy Maxmsaton and Queueng Network Models. Annals of Operatons Research, 48: Lucanton, D. (1991). New Results on the Sngle Server Queue wth a Batch Markovan Arrval Process. Commun. Statstcs / Stochastc Models. Mandelbrot, B. and Ness, J. (1968). Fractonal brownan motons, fractonal noses and applcatons. SIAM Revew, 10: Meer-Hellstern, K. (1989). The Analyss of a Queue Arsng n Overflow Models. IEEE Transactons on Communcatons, 37: Naoumov, V., Kreger, U., and Wagner, D. (1997). Analyss of a Mult-server Delay-loss System wth a General Markovan Arrval Process. In Chakravarthy, S. and Alfa, A., edtors, Matrxanalytcal methods n Stochastc models, pages Marcel Dekker. Neuts, M. (1995). Matrx-Geometrc Solutons n Stochastc Models: An Algorthmc Approach. Dover Publcatons. Norros, I. (1994). A Storage Model wth Self-smlar Input. Queueng Systems and ther Applcatons, 16: Turner, J. S. (1999). Terabt Burst Swtchng. Journal of Hgh Speed Networks, 8:3 16. Yajnk, M., Moon, S., Kurose, J., and Towsley, D. (1999). Measurement and Modelng of the Temporal Dependence n Packet Loss. In INFOCOM 99, New York. Yao, S., Xue, F., Mukherjee, B., Yoo, S. J. B., and Dxt, S. (2002). Electrcal Ingress Bufferng and Traffc Aggregaton for Optcal Packet Swtchng and Ther Effect on TCP-Level Performance n Optcal Mesh Networks. IEEE Communcatons Magazne, pages