From Relative to Observable Proportional Differentiation in OBS Networks

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1 Fro Relative to Observable Proportional Differentiation in OBS Networks Pablo Argibay-Losada Departaento de Enxeñería Teleática Capus Universitario s/n, E-363 Vigo, SPAIN Andrés Suárez-González Departaento de Enxeñería Teleática Capus Universitario s/n, E-363 Vigo, SPAIN Manuel Fernández-Veiga Departaento de Enxeñería Teleática Capus Universitario s/n, E-363 Vigo, SPAIN Raúl Rodríguez-Rubio Departaento de Enxeñería Teleática Capus Universitario s/n, E-363 Vigo, SPAIN Cándido López-García Departaento de Enxeñería Teleática Capus Universitario s/n, E-363 Vigo, SPAIN ABSTRACT This paper addresses the provision of proportional differentiated services to an arbitrary nuber of traffic classes in ters of the class packet loss probability easured between the ingress node and the egress node of an OBS network. Our solution relies on a key idea: OBS networks consist of bufferless nodes and can therefore be regarded as a whole like a one-hop bufferless subnetwork characterized by a collection of loss probabilities coputable with siple, approxiate circuit switching odels. Consequently, we believe it is possible to attain packet loss proportionality erely using a siple stochastic algorith to asseble two classes of bursts, provided there exists soe for of internal relative differentiation such that one of the burst classes has a uch lower loss probability than the other. In order to evaluate the algorith accuracy in attaining the proportionality, we present an analytical study of two idealized scenarios (a single link and a ulti-node syetrical network), including the effect caused by the flow dynaics of the traffic received at the ingress nodes. Our results show that, despite its siplicity, this approach is able to provide the desired proportionality over a wide range of operating conditions. Categories and Subject Descriptors G.3 [Probability and Statistics]: Queueing theory; C.. Perission to ake digital or hard copies of all or part of this work for personal or classroo use is granted without fee provided that copies are not ade or distributed for profit or coercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific perission and/or a fee. CoNEXT 5, October 7, 5, Toulouse, France. Copyright 5 ACM X/5/...$5.. [Coputer-Counication Networks]: Network Architecture and Design Network counications General Ters Algoriths Keywords Optical burst switching, proportional differentiation. INTRODUCTION The synergy between optical wavelength division ultiplexing (WDM) transission and the optical burst switching (OBS) paradig is currently being regarded as a candidate architecture for the next generation Internet backbone. The basic idea of optical switching is that of eliinating all unnecessary signal conversion inside the switches, thereby creating a ulti-hop all-optical path, not necessarily constrained by wavelength continuity. Optical burst switching is a realization of such principle in which the ato entity to switch is the burst, an aggregation of individual IP packets, and where the optical data channel is decoupled (both teporally and spatially) fro the control channel used to convey signaling essages between neighbor switches. It is also a coon view, partly otivated by the increased deployent of applications requiring quality of service (QoS), that the traditional best-effort service odel of the Internet should be enriched. QoS support can be generally enforced within two different fraeworks, naely absolute or relative assurances. Absolute QoS provision iposes a hard quantitative liit to the perforance etric of interest (e.g., delay or loss), and is ainly useful for intolerant applications or for tightly controlled network services. Relative QoS differentiation, on the other hand, defines a odel consisting of a set of service classes ordered in the QoS etrics, with the qualitative guarantee of preserving the relative

2 ordering between the but without the capability to control the quality gap fro one class to the next better one. So, class i receives better (or at least not worse) service than class i + in ters of single or coposite perforance easures, such as delay, loss or a cobination of these. Due to such elasticity, relative differentiation uses less state inforation, scales gracefully, and it is the constituent odel of the well known Differentiated Services (DiffServ) architecture [] developed for IP QoS solutions. Proportional differentiation [, 3, ] is a refineent and a quantification of relative differentiation. In this paradig, the service level offered to each class is controlled according to prespecified constant factors, and these quantitative law is kept stable even on a short tie scale. Hence, the quality of service, as seen by the user, is consistent and predictable while, as a network operator s concern, is controllable. More iportantly, it can benefit fro efficient forwarding echaniss that keep up the scalability. This paper deals with the provision of proportional differentiated services to an arbitrary nuber of traffic classes in ters of the class packet loss probability easured between the ingress node and the egress node of an OBS network. Unlike previous works in this area [5, 6, 7], we focus explicitly not on local (per-hop) differentiation, but on end-to-end quality of service, in which the endpoints are eant to be the nodes where the conversion to the optical doain is applied and reverted, and the burst assebly/disassebly perfored. As a consequence, we ove fro the liited scenario of proportional differentiation at the burst level, which has been previously addressed elsewhere, to the ore general (and probably ore useful) case of proportional differentiation directly at the packet level. Our solution relies on a key idea: OBS networks consist of bufferless nodes and can therefore be regarded as a whole, fro an ingress to an egress node, like a one-hop bufferless subnetwork characterized by a collection of loss probabilities coputable with siple, approxiate circuit switching odels. Instead of devising sophisticated scheduling or switching algoriths for the optical nodes, which are difficult to analyze and soeties show subtle behavior, we assue the existence of a siple relative global differentiation inside the optical subnetwork, and show how to transfor this qualitative odel into the quantitative proportional schee by eans of a probabilistic algorith in the ingress nodes. We should also ephasize that the proposed echanis, albeit adittedly siple, can be generalized in a straightforward anner to pure optical packet switching. The rest of the paper is structured as follows. Section explains the basic odel assuptions of our fraework. Section 3 presents the probabilistic classification algorith used in the ingress nodes to distribute the traffic aong the relative differentiated internal classes. Section introduces a dynaic odel for the throughput and briefly analyzes three possible differentiation strategies that could be applied to the external classes. With these tools, an analytical and nuerical study for single-link and ulti-node networks is provided at the end of that section. Finally, soe concluding rearks and a sketch of further work are given in Section 5.. GENERAL ASSUMPTIONS We consider an OBS core network where optical switching nodes have unrestricted wavelength conversion capabilities, eaning that any incoing burst received in a particular wavelength can be switched to any available wavelength in its output link. Therefore, wavelengths can be seen as different logical channels, and we assue of these between any pair of nodes connected directly. We also assue that the OBS subnet differentiates bursts between two internal classes (say, preiu and best-effort) on the basis of their packet loss probability, denoted by B and B, respectively, and satisfying B B. The specific scheduling algoriths, routing decisions or priorization schees needed in conjunction to ensure such condition are beyond the scope of this paper. We erely hypothesize that the echaniss already conceived to differentiate bursts [, 9, ] are able to perfor this relative service differentiation easily. We further assue that the edge routers (i.e., IP routers with a point to point link to an OBS switch, able to asseble the bursts and to convert back and forth between the electric and the optical doain) receive packets belonging to n external classes. The goal is to distribute the packets between the preiu and best-effort internal classes, during the burst assebly process, in such a way that the n external traffic classes receive proportional service differentiation in their blocking probabilities. That is, if p < p < < p n denote the packet loss probabilities of the external classes, then p i s i p n s i < s i+ i =,..., n () for a fixed set of coefficients {s,..., s n }. Note that, without loss of generality, one can assue strict inequalities between the factors, s i < s i+, since otherwise both classes could be siply regarded as one. 3. OTCD ALGORITHM The crux of the conversion fro internal relative differentiation to external proportional service is an algorith that assigns packets fro any external service class i to either the high priority (preiu) internal class of bursts, with probability h i, or to a low priority (best effort) burst, with probability h i = h i. Due to its biodal classification pattern, we will refer to the algorith as Open-Loop Two-Class Dispatcher (OTCD). In a practical setting, the differentiation is unlikely to be done on a per-packet basis, in order to aintain orderly arrivals at the egress node, but this issue can be certainly ignored for the purposes of this paper. The loss probability of external class i packets will be as long as p i = h i B + h i B h i B () h i B /h i B. The task of differentiating between the two internal classes becoes easier when high priority bursts do not carry packets fro the lowest external priority class, i.e. class n. Therefore, we arbitrarily fix h n =, and consequently h n =, so the apping fro packets in class n to (best-effort) bursts is deterinistic. In that case we can equate coefficients, because () and () iply p i h i p n. (3) Indeed, the nuber of logical channels is the su of the nuber of wavelengths in each fiber connecting the nodes, provided there is a parallel bundle of fibers in the link.

3 In this paper we will use fixed h i = s i values as deterined by the approxiation () and ai to study the algorith accuracy in attaining proportionality in the packet loss probability.. CASE STUDY In this section we apply the previous fraework to two siple but sufficiently representative scenarios: a single congested link and a ulti-node syetrical network. Our goal is to gain soe insight, both theoretical and nuerical, about the perforance of the packet classification algorith in the packet loss probability and, as a consequence, in the throughput. To that end, we first introduce a siple but realistic odel for the flow dynaics of traffic received at the ingress nodes. Specifically, we adopt here TCP-dynaics. Next, we briefly analyze how to arrange the factors s i and what effect is to be expected when coupled with the nontrivial dynaics of a congestion control algorith. Finally, a discussion about the iplications of our analytical odel and its nuerical results is given.. Flow dynaics Let us suppose that every external class reacts upon congestion inside the core OBS subnetwork reducing its offered traffic when packet losses are ore frequent. Consider, in particular, that each external class results fro the aggregation of a siilar and sufficiently large nuber of long-lived TCP Reno connections generating constant length packets. TCP is still the doinant transport traffic in Internet and its Reno ipleentation the ost widely used. Moreover, there have been proposed TCP-friendly throughput regulators for UDP traffic [], ensuring a fair sharing of the network capacity. There are well established analytical odels for the dynaic behavior of the Reno variant (see [] and references therein). Under appropriate conditions, and within the range of blocking probabilities of our interest, equation (3) in [] gives the offered rate of a single TCP session, λ(p, RTT), as p RTT p + E [W p u] + Q b (E [W u]) E[Wu] + + Q b (E [W u]) f(p) p () where p is the packet loss probability of the TCP connection, RTT is its average round trip tie, W u is the unconstrained window size, b Q (x) is the probability that a loss in a window of size x is a tieout, and f (p) = + p + p + p 3 + p + 6p 5 + 3p 6. For the validity of (), we will assue that the axiu receiver window size is arbitrarily large, so that the congestion window is never exhausted, and also that the condition discussed in [3] holds: sources are slow, i.e. there is at ost one TCP data segent fro any source in a burst, and the standard deviation of the RTT is negligible. Equation () has the following asyptotic behavior for Large enough so that the departure or the arrival of a connection does not change substantially the overall throughput of the class. eλ..... p sall values of p: Figure : TCP Reno behavior λ(p, RTT) li p λ (p, RTT) = q 3 λ (p, RTT) = RTT p. As an illustration, Figure shows the relative error e λ = λ (p, RTT) λ(p, RTT) and confirs that the approxiation λ(p, RTT) = λ(p, RTT) ( p) λ(p, RTT) λ (p, RTT), where λ(p, RTT) refers to the throughput of a single TCP session is in fact quite accurate for sall enough packet loss probabilities, such as p. which is the regie we precisely expect OBS networks to be designed for. Our ai will be to study if proportional packet loss differentiation can be achieved, and, oreover, whether proportional differentiation in the throughput is indirectly possible and to what extent. We want, (5) (6) λ i c i λ n, c i > c i+, i =,..., n, (7) with λ i the throughput of a connection of class i. Note that, in order to enforce (7) acting only on the packet loss probabilities, we are iplicitly aking the assuption that the round trip tie is approxiately the sae for all the TCP connections. Nevertheless, this hypothesis has nothing to do with constraining the actual RTTs to be constant, but it is rather a atheatical artifact that akes the analysis easier. Indeed, were the RTTs known they could be subsued into the factors c i appearing in (7) without losing generality. Then, fro (3) and (5), the packet loss probabilities in order to attain (7) ust be related by s i pn p i c i i =,..., n. () Denoting by Λ and Λ the arrival rates of packets in an edge node to the two internal classes towards a given egress node, and by N i the nuber of TCP connections of class

4 i, the ratio between the arrivals rate r Λ = Λ /Λ can be expressed using (3), (7) and () as a function of the factors c i and N i: and Xn Xn Λ j = h i j N i λ i λ n h i j N i c i, i= i= Xn Λ λ n N i, c i i= Xn Λ λ n N i r Λ = Λ Λ i= P n i= Ni ci P n i= Ni c i ci «c i. (9) We reark that the lower r Λ is, the easier will be for the OBS network to aintain the burst loss probability differentiation. But a given value of r Λ could be obtained with any different pairs (c i, N i), for i =,..., n. For analytical siplicity, fix the sae nuber of TCP connection per class, N i = N i {,..., n }. This is equivalent to say that the overall per-class throughputs are also proportional, which ay be seen as a worst case configuration. Doing so, (9) siplifies to: r Λ = Λ Λ P n i= ci P n i=. () c i. Differentiation strategies There are several possible choices for the scaling factors c i, so the behavior of r Λ is worth soe closer exaination. Consider these three arbitrary differentiation strategies: Only two external classes. Geoetric differentiation strategy: n > and c i = g n i, with g >. Linear differentiation strategy: n > and c i = n i. When the nuber of external and internal classes is two, the rate arrival ratio () becoes: r Λ = Λ Λ + c + c = c, i.e., it is approxiately equal to the throughput ultiplicative factor of the best external class, inus one. For the geoetric differentiation faily case, the ratio is: r Λ = Λ Λ P n j= gj P n j= = g n = c, g j that is, the sae approxiate result as in the syste with only two classes. And in the linear differentiation case, one gets: P n r Λ = Λ Λ i= n i = P n i= n i (n + ) n H n where H n = P n j= /j is the n-th haronic nuber. For exaple, the approxiate values of r Λ for n = 3,..., 7 are 5/.7, 9/5 = 3., 763/ , 53/ and 3557/363 9., respectively. Therefore, for each n the r Λ of the geoetric case with g is higher than for the linear sequence of factors. In addition, since the geoetric case is equivalent to the reduced scenario with two classes, only the latter will be analyzed. The perforance of any other faily is expected to be bounded by that of a two-class syste with a greater or equal r Λ..3 One link The siplest situation occurs when only one link has nonnegligible blocking probability. Following [], we assue that all traffic flows arriving at an ingress node are Poisson processes, and analyze an equilibriu situation where the offered traffic flows satisfy () λ i = λ(p i, RTT) giving rise to a total packet loss probability value B. ρ Molina B =. Erlang B =. Molina B =. Erlang B =. Molina B =. Erlang B =. Figure : Utilization factor Define the internal offered traffic intensities as: A i = Λ i S A = A + A, where S is the ean transission tie of a packet in a channel, and distinguish both ideal non-preeptive and partiallypreeptive internal scheduling schees: Non-preeptive. Only full bursts are transitted, and those of low priority only if they do not collide with high priority ones (full isolated classes). Thus, the total and the high-priority packet loss probabilities can be coputed approxiately the forer and exactly the latter through the Erlang (M/M//) odel when the length of bursts for both classes has the sae negative-exponential distribution: B = E (, A ) B B b = E (, A), where B b is the global burst loss probability (see Appendix). Partially-preeptive. The tail of a burst of low priority ay be preepted by a high priority burst, but the head is still transitted. We consider negligible the switching ties, the influence of partial packets transitted and the preeption of a burst with lower scheduled finishing tie. The length distributions of bursts is iaterial. Also, when two high 56 5

5 priority bursts collide, the non-overlapped part of the first survives. Now, the total and the high-priority packet loss probabilities can be coputed through the Molina (M/G/ ) odel [5]: A c = X i= in (i, ) Ai i! e A, M (, A) = A Ac A, B = M (, A), B = M (, A ). In both cases the low priority blocking probability is given by: B = A B A B A. () Figure shows the channel utilization factor of both odels for total blocking probabilities of.,. and.. These values define a range that, although arbitrary, sees a reasonable service regie when under TCP-copatible offered traffic flows. In the nuerical exaples that follow, we solve the Erlang and Molina odels and copute: B B 6 c = Figure 3: Internal differentiation, B = the internal loss probability ratio B /B, the relative error in the probability differentiation. e si = /s i /s i /s i es (where s i = p i/p n is the actual loss probability differentiation while s i is the desired one) and the relative error in the throughput differentiation coefficients: e ci = c i c i c i (where c i = λ n /λ i is the actual throughput differentiation while c i is the desired one),.. c = Figure : External differentiation, B = for a syste with two external classes exclusively and c =,..., 6. The ethodology used to solve both odels has been the following. We first nuerically look for the global offered traffic intensity A that produces a given packet loss probability B. Afterwards we iterate looking for such c so that the arrival rates coplies with (), starting with the desired c for the first iteration. Figures 3, and 5 plot the perforance for the Erlang odel case, fixing a global loss probability equal to.. Specifically, Figure 3 shows the internal probability differentiation, Figure shows the relative error in the probability differentiation and, finally, Figure 5 depicts the relative error in the throughput differentiation. Results for the Molina odel are not shown due to its siilar qualitative behavior to the Erlang case. Moreover, the nuerical difference is very sall. The relative difference of B B is lower than ±.% for ratios lower than (the upper liit of Figure 3). The relative difference of e s is lower than ±.3%. The difference of e c is lower than ±.3. For this reason, fro now on we focus on the Erlang odel. ec c = 6 Figure 5: External differentiation, B =

6 In suary, with B =. the goal of proportional differentiation of the packet loss probability is achieved with only a sall nuber of channels. For instance, with = 5 and c =, we have that the internal differentiation B /B 3.9 produces an external probability differentiation error e s.6%, that is, we will have p 3.66 p, /s 3.66, instead of the goal /s = c =. One can also see in Figure 5 the ipact due to the optiistic prediction of the TCP offered rate: the low external priority class suffers ore losses than the high one, hence (5) and (6) (see Figure ) overestiate ore its throughput than in the high external priority case, giving rise to e c > for sufficiently large. For instance, with = 5 and c = we already have an external throughput differentiation error e c.3% > even when /s 3.66 is clearly lower than, as stated in the previous paragraph. The behavior of the resulting configuration with global packet loss probability equal to. is depicted in Figures 6, 7 and for the Erlang odel. B B ec c = Figure : External differentiation, B =. coexist a very large nuber of TCP flows. Nevertheless, this undesirable perforance vanishes if the arbitrary condition that all classes coprise an identical nuber of TCP flows is relaxed. This behavior can be checked in the Figures 9 and (internal differentiation B /B peaks up earlier than in the Figure 6 case and so its figure is not shown here) for the Erlang odel, where the curves have been coputed for N = N. Obviously, the results will iprove accordingly for saller global packet loss probabilities c = c = Figure 6: Internal differentiation, B =. es es Figure 9: External differentiation N N =, B =... c = Figure 7: External differentiation, B =. The conclusion to draw fro these plots is that an operating regie around a global loss probability of. is too hard in order to ensure the differentiation, except for the lower c. Moreover, it is easy to see in Figure how the proposed echanis penalizes ore the low priority than the high priority flows (for > 3 and c > we get e c.5) when Multinode Network Case Next we turn to the syetrical ultinode configuration shown in Figure. Each node receives traffic flows of both external classes directed towards any other node fro the outside, and the network uses shortest path routes. In this way, any unidirectional link between two nodes carries traffic flows fro three source-destination pairs: There are two traffic flows fro two source-destination pairs with two hop paths. We will denote by a = a + a the offered traffic intensity, coposed of the internal high priority a and the internal low priority, a. There is one traffic flow fro one pair with a one hop path. We will denote siilarly by a = a + a its

7 offered traffic intensity, su of the intensities fro the high internal priority a and fro the low priority, a. ec c = Figure : External differentiation N N =, B = We will also assue that the RTT of all TCP connections is approxiately the sae, that is, it will be ainly doinated by the queue waiting ties outside the OBS network. Apart fro the external differentiation aong external classes of the sae source-destination pair, any actual network will also surely differentiate between sae class traffic of different source-destination pairs. Given that a coproise between efficiency (axiu throughput to fewer hops traffic) and equity (identical throughput to sae class traffic) is always arbitrary, in the current analysis we will siply take no special action toward any of the two, that is, we apply the OTCD algorith irrespective of the nuber of hops (h n always ). This way, the traffic with the highest nuber of hops will receive the worst service aong those of the sae external priority class. The odel so defined ay be practically solved using the Erlang fixed point approxiate ethod [6], although its convergence is not guaranteed and that is the reason explaining the absence of soe values in the figures. Given the syetry of the configuration, all of the links will have the sae packet loss probabilities B, B and B (the latter two approxiated by the respective burst loss probabilities as in previous section), and offered traffics: A = A + A B = E (, A) A = a + a + a ( B ) B = E (, A ) A = a + a + a ( B ) and B given by (). We denote the loss probability of a j i for the priority i {, } and the extra nuber of hops j {, } by b j i : b i = B i b i = ( B i) = B i B i that is, b i b i () a 5 for low enough B i values. For the two classes case we finally have: a a 3 Figure : Multinode configuration p j = b j p j = c bj + c «b j, and we solve the odel in an equilibriu situation where the offered traffic coplies with (): λ j i = λ(pj i, RTT) i, j {, } for a global packet loss probability of., while the dispatching rule is given by (7) based on the goal c i values. Actually, we solve the odel iteratively for a given a and afterwards look for the value that gives rise to the desired global packet loss probability. Figures to 5 show the external differentiation obtained for both the one and two hop traffic. Checking the differentiation between sae priority traffics in this configuration, the ratio between the one and two

8 .5..3 c =. e c. e s... c = Figure 5: Multinode, two hops traffic 5 Figure : Multinode, one hop traffic hops high external priority throughputs is in the interval [.,.3], while between the low external priority ones is in [.5,.5]. They are close to, as expected fro (5) and (). These results also explain the differences between Figures and 5. e s e c c = 6 3 Figure 3: Multinode, two hops traffic c = CONCLUSIONS We have devised a siple ethod to achieve proportional loss differentiation between packets that traverse an OBS core network. This ethod is solely based on an stochastic algorith to asseble two classes of packet bursts and erely requires that the OBS network provide soe for of internal relative differentiation such that one of the burst classes has a uch lower loss probability than the other. In order to gain soe insight, both theoretical and nuerical about the perforance of the ethod, we have presented an analytical study, by using siple, approxiate circuit switching odels, of two siple but sufficiently representative scenarios: a single congested link and a ultinode syetrical network. The results have shown the validity of the proposed algorith over a wide range of operating conditions. Nevertheless, further work reains to be done. An iportant issue is to study the ipact of variable RTTs on the behavior of the algorith. The exactness of the slow source odel in realistic network configurations ust also be assessed and a proper burst assebly algorith ust be devised in order to reduce packet loss correlation and avoid false congestion notifications. Overall, the whole odel should be validated through siulations. On the practical side, it sees interesting to extend the technique toward both ends of the counications path, including interediate IP routers, so that genuine end-to-end proportional packet loss differentiation is actually achieved Figure : Multinode, one hop traffic 5 6. ACKNOWLEDGEMENTS This work was supported by the Ministerio de Educación y Ciencia through the project TIC3-9-C3-3 of the Plan Nacional de I+D+I (partially financed with FEDER funds) and by the Secretaría Xeral de Investigación de la Xunta de Galicia through the grant PGIDTPXIC33PN.

9 APPENDIX A. PACKET LOSS PROBABILITY ERROR In the ideal non-preeptive odel sketched in Section.3, [7] shows that the overall burst loss probability B b is exactly given by (.3) when the length of bursts of both classes have the sae negative-exponential distribution (sae ean value), and hence the low priority burst loss probability can be exactly coputed through: B b = Λ Bb Λ B b Λ. with Λ = Λ + Λ. Moreover, the high priority class has the sae packet loss and burst loss probabilities, B = B b. Nevertheless, that is not the case for the low priority class, since the loss probability of a low priority burst depends on its length [] (lower probability for shorter lengths) giving rise to B > B b and B > B b. We estiate the low priority packet (traffic) loss probability through hybrid siulation-analysis in order to copute the relative error due to using B b instead of B : e B = B b /B. Figure 6 shows the estiated errors for several cases (worst and best included) incurred in the analysis results of Section.3. Their 9% confidence intervals are all narrower than ±% and therefore are not shown. For a fixed nuber of logical channels, this error increases with the ratio r Λ = Λ /Λ (higher c or lower N /N ). eb... B b =. B b =. N/N =, N/N =, c = N/N =, N/N =, c = 6 3 Figure 6: Relative error of B Nevertheless, this behavior does not invalidate our conclusions for the non-preeptive odel even when e B is not negligible, such as the axiu 3.% for N = N, B b =., c = 6 and =. Since B b underestiates B, we will have a slightly higher packet loss probability differentiation than the value coputed in our analysis, although a lower channel utilization factor and packet throughput. B. REFERENCES [] S. Blake, D. Black, M. E. Davies, Z. Wang, W. Weiss. An Architecture for Differentiated Services. RFC 75, dec. 99. [] C. Dovrolis, P. Raanathan. A Case for Relative Differentiated Services and the Proportional 56 5 Differentiation Model. IEEE Network Mag., vol. 3(5), pp. 6 3, sep [3] C. Dovrolis, D. Stiliadis, P. Raanathan. Proportional Differentiated Services: delay differentiation and packet scheduling. IEEE/ACM Trans. on Networking, vol., no., feb., pp. 6. [] Y. Chen, C. Qiao. Proportional differentiation: A scalable QoS approach. IEEE Counications Mag., jun. 3, pp [5] Y. Chen, M. Handi, D. H. K. Tsang. Proportional QoS over OBS networks. Proc. Globeco, pp [6] H. C. Cankaya, S. Charcranoon, T. S. El-Bawab. A preeptive scheduling technique for OBS networks with service differentiation. Proc. Globeco 3, pp [7] C.-W. Tan, M. Gurusay, J. C.-S. Lui. Achieving proportional loss differentiation using probabilistic preeptive burst segentation in optical burst switching WDM networks. Proc. Globeco, pp [] V. M. Vokkarane, J. P. Jue. Prioritized burst segentation and coposite burst-assebly techniques for QoS support in optical burst-switched networks. IEEE J. on Select. Areas in Co., vol., pp. 9 9, sep. 3. [9] M. Yoo, C. Qiao, S. Dixit. QoS perforance of optical burst switching in IP-over-WDM networks. IEEE J. on Select. Areas in Co., vol., pp. 6 7, oct.. [] F. Farahand, J. Jue. Supporting QoS with look-ahead window contention resolution in optical burst switched networks. Proc. Globeco 3, pp [] M. Handley, S. Floyd, J. Padhye and J. Wider. TCP Friendly Rate Control (TFRC): Protocol Specification. RFC 3, Jan. 3. [] J. Padhye, V. Firoiu, D.F. Towsley and J.F. Kurose, Modeling TCP Reno perforance: a siple odel and its epirical validation, IEEE/ACM Transactions on Networking, vol., nu., Apr., pp [3] A. Detti and M. Listanti, Ipact of Segents Aggregation on TCP Reno Flows in Optical Burst Switching Networks, Proceedings of INFOCOMM, vol. 3, pp. 3,. [] M. Izal and J. Aracil, On the Influence of Self-siilarity on Optical Burst Switching Traffic, Proceedings of GLOBECOM, vol. 3, pp. 7,. [5] Z. Rosberg, H. Vu, M. Zukeran, J. White Blocking Probabilities of Optical Burst Switching Networks Based on Reduced Load Fixed Point Approxiations. Proc. IEEE Infoco 3. [6] K. W. Ross, Multiservice Loss Models for Broadband Telecounication Networks, Springer, 995. [7] H.L. Vu and M. Zukeran, Blocking Probability for Priority Classes in Optical Burst Switching Networks, IEEE Counications Letters, vol. 6, nu. 5, pp. 6, May.. [] K. Dolzer, C. Gauger, J. Späth and S. Bodaer, Evaluation of Reservation Mechaniss for Optical Burst Switching, AEÜ Int. J. Electron. Coun., vol. 55, nu.