Defining QoS classes based on end-to-end, fine grained metrics yields a network architecture simpler to understand and exploit. In the proportional model, ...
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 58, NO. 1, JANUARY 2010
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Flow Splitting for End-to-End Proportional QoS in OBS Networks Pablo J. Argibay-Losada, Andrés Suárez-González, Cándido López-García, and Manuel Fernández-Veiga, Member, IEEE
Abstract—In this paper, we propose probabilistic splitting of a packet stream at the edge routers as the basic method to provide end-to-end proportional QoS to packet flows carried through an OBS network, in terms of loss probability. We argue that the only requirement that the optical transport infrastructure has to satisfy is the support for two internal burst classes with wide separation between their respective service levels. Under this condition, we show how quantifiable end-to-end per-flow guarantees can be attained without diminishing network resource usage. The scheme is analyzed theoretically and evaluated through numerical simulations, both in regular topologies (ring networks) and in a mesh network. Our results suggest that a layered approach to the problem of end-to-end QoS provisioning can be very effective when a proportional service model is offered, and that the research on sophisticated scheduling algorithms at the optical switches could be too narrow-focused for that purpose. Index Terms—Optical burst-switching, quality of service, loss probability, end-to-end performance, network architecture, proportional differentiation.
I. I NTRODUCTION
O
PTICAL burst-switching (OBS [1]) is a practical switching technology for introducing high-speed optical routers in the Internet backbone. OBS networks fully separate the control and data planes. User packets with common attributes (same source-destination, for example) are assembled into larger switching units called bursts. Bursts are transmitted along the optical path, crossing bufferless optical nodes without being converted to the electronic domain. Every burst is switched as a single entity after the arrival of a small piece of control information to the next downstream node, which is used for the anticipated electronic configuration of the switching fabric. The time gap between the control header and the data (the burst) is referred to as offset. At the endpoint of the optical path, an egress router converts the burst back into an electric signal and splits it into its constituent packets. Essentially, OBS arises as a technical compromise which cuts off the need of still unavailable optical buffers, while giving more efficient bandwidth use than wavelength routing when the traffic pattern allows for statistical multiplexing gains. As OBS technology matures [2], the scalable provision of QoS is becoming an increasingly important concern that Paper approved by A. Bononi, the Editor for Optical Transmission and Networks of the IEEE Communications Society. Manuscript received January 28, 2009; revised July 3, 2009 and September 7, 2009. This work was supported by the Ministerio de Educación y Ciencia, under grant TSI2006-12507-C03-02, and is partially funded by FEDER. The authors are with the Telematics Engineering Department, University of Vigo, Spain (e-mail: {pargibay, asuarez, candido, mveiga}@det.uvigo.es). Digital Object Identifier 10.1109/TCOMM.2010.01.090058
the research community has addressed in a number of ways. Basically, the type of the solution results from the combination of a multi-class burst scheduling algorithm (possibly based on a prioritized allocation of resources, such as wavelengths, routes, time offsets) with a definition of the QoS model, as one offering either qualitative or quantitative differentiation. Relative QoS [3] refers to a scheme where a class is given only a comparative guarantee over another, but without any quantifiable control on the QoS level of each class. This means that the network declares its intent to treat some traffic class better than others (e.g., lower loss probability, less delay, greater throughput), but does not commit to specific values of these metrics. This is in contrast with absolute QoS, where strict, measurable QoS constraints are agreed between the network provider and the user. The first model is simpler, since relative differentiation is independent of the aggregated or per-class traffic demand, but may fail to provide sufficient information to the end users. The absolute QoS paradigm, however, necessitates stringent control over the users’ traffic, either with admission control, pricing schemes, or policing, in order to regulate the network load. Usually, this implies lower link utilization and complex signaling protocols. Yet, regardless of the model, the techniques proposed for resolving contention among the different burst classes have undergone increased sophistication over the years. Differentiation among the classes can be induced in many ways, such as additional offset times for high priority bursts [4], [5], burst segmentation [6], [7], deflection routing [8], or via resource allocation in the links [9], [10] in conjunction with preemptive service of high-priority bursts [11]. Other approaches have used intentional dropping of low priority bursts [12], wavelength sharing policies [13], [14] or advanced schedulers with preemption [15], [16] to enforce absolute guarantees on the burst blocking probability. The recent survey [17] thoroughly reviews most of the schemes proposed to date. Nonetheless, as the ultimate aim of these algorithms is toward minimizing the burst contention events, attaining end-toend QoS guarantees at the packet level within this framework is quite involved, since one burst drop may cause the discard of packets in many different user flows. Thus, for passing from the burst drop probability to packet loss probabilities, careful control on the burst assembling procedure is also needed, complicating the design and understanding of the QoS support. Moreover, the probability of dropping a burst is not a quantity directly measurable by users, only by the network provider, and the significance of its value may be obscure for the endhost applications.
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For these reasons, in this paper we choose end-to-end proportional packet loss probability as the performance measure of interest. Defining QoS classes based on end-to-end, fine grained metrics yields a network architecture simpler to understand and exploit. In the proportional model, introduced in [3], [18], predefined ratios between the packet loss probabilities of several classes are to be maintained, independently of the congestion level of the network if possible. Thus, the available network resources are always used, because the actual value of the overall loss probability is unconstrained, while simultaneously the users receive a concrete statement about the performance level and can exploit this information to maximize their own objectives. Moreover, the proportional QoS model is by no means a limitation, since it also can be used to offer absolute bounds on the end-to-end packet loss probability. Our basic assumption is the existence of an OBS network with two internal classes, high and low priority bursts, differing widely in their loss probabilities. Provided that such condition remains valid, the actual performance of the burst scheduling algorithm turns out to have almost no significance to achieve loss proportionality, as will be shown. Since the loss metric is multiplicative, not additive, we do not use any local tuning or coordination of the scheduling rules applied in each optical switch, thus avoiding to decompose the end-to-end QoS requirement into individual, isolated QoS values. The problem of partitioning a bound about a path metric into static individual link contributions is hard to solve optimally but, being based on a worst-case analysis, produces largely conservative strategies for resource allocation too. The approach we use to achieve end-to-end packet loss probabilities is to classify packets, according to a carefully chosen ratio, into those to be contained in high- and lowpriority bursts, and let the OBS network switch these in a manner oblivious of the QoS model seen by the users. To sum up, the main contributions of this work are: i) in showing that, by abstracting and simplifying the QoS model provided by the OBS core, proportional end-to-end QoS is possible without using overly sophisticated resource allocation strategies or contention resolution techniques inside the optical switches; ii) a novel mapping algorithm betwen packets and optical bursts that guarantees almost exact propottional service to an arbitrary number of packet classes; iii) a theoretical performance analysis for networks using ideal preemptive burst schedulers. In the context of optical networks, proportional service was previously considered in [19] by means of intentional dropping of low priority bursts at the switches. This approach results in poor utilization of the network resources. A partial preemption algorithm combined with segmentation was proposed in [20], and a refined Probabilistic Preemptive Burst Segmentation (PPBS) scheduling scheme was used by [11]. In both cases, the algorithm works locally, that is, end-to-end QoS is the result of coalescing per-hop guarantees. For the proportional loss model, it is not true that enforcing fixed loss ratios at the nodes leads to end-to-end proportionality [18], so the overall QoS objective must be decomposed into individual constraints of some other type. The optimal solution to this problem is generally unknown, and most works in this
area [14], [15] either use complex partitioning procedures, or resort to conservative approximations, such as bounding the maximal loss probability of any link in the optical path. This overestimation also causes low network usage. The authors in [7] proposed an adaptive adjustment of the offset times used for each burst class as a method to achieve directly endto-end proportionality. The adaptation rule uses measurements obtained by monitoring the network state with periodic probe packets, thus introducing extra control traffic. The proposal is a form of exchanging delay for loss. Proportional delay service has also been studied for packet switching networks. We mention briefly here the work of Dovrolis [21], who devised a class of schedulers that use delay information coded in the packet headers to ensure end-to-end proportionality. Absolute QoS support is the model which a number of works in the literature opt for [12], [13], [15]. However, the provision of absolute QoS guarantees requires a strict control of the traffic load offered to the network, usually taking the form of an admission control decisor plus traffic policing. In addition, the computation of the loss probabilities tends to be conservative, the end-to-end QoS objective is partitioned into per-hop threshold with heuristic methods, and the optical schedulers use in some cases intentional burst dropping or wavelength reservations. The overall result is a substantial reduction of the network transport capacity. The rest of the paper is organized as follows: in Section II, we present the packet mapping strategy to achieve end-toend proportional QoS assurances. Section III develops an analysis of the performance in a single node, and Section IV poses a fixed-point model for networks with ring topology. The results of several numerical experiments to illustrate the operation of the method are discussed in Section V. Finally, some concluding remarks are given in Section VI. II. P ROPORTIONAL Q O S T HROUGH PACKET C LASSIFICATION The proportional differentiation paradigm was proposed in the context of the DiffServ Internet architecture as an extended per-hop behavior (PHB) [3] to bring forth objective performance differentiation between the user classes. This idea lends itself to a natural generalization for end-to-end paths. Consider a given number of user classes, 𝑛, indexed by 𝑖 = 1, . . . , 𝑛 in decreasing order of performance, whose QoS demands are formulated in terms of their packet loss probabilities, and which are to be transmitted through the OBS network. We will use the term outer class in the remainder to refer to any one of the end-to-end service types that the ingress router to the OBS core is aware of. Define 𝑝𝑖 as the end-to-end packet loss probability suffered by flows of outer class-𝑖 in the route of interest. We consider networks in which the routing decision is taken by the provider and is static. The objective of the network provider is to ensure that (1) 𝑝𝑖 = 𝛼𝑖 𝑝𝑛 ∀𝑖 = 1, . . . , 𝑛 where 𝛼𝑛 ≜ 1 and 𝛼 = (𝛼1 , . . . , 𝛼𝑛 ) is the vector of fixed scalar coefficients specifying the relative performance of the classes. The outer class 𝑛 is used as reference, by convention. Without loss of generality, we assume that 𝛼1 < 𝛼2 < ⋅ ⋅ ⋅
0 in real networks, implying an error between the desired exact proportionality and actual performance. So, define the relative error of the received end-to-end differentiation, 𝑝𝑖 /𝑝𝑛 , with respect to the objective for outer class 𝑖, 𝛼𝑖 , as 𝑝𝑖 𝑝𝑛
− 𝛼𝑖
1 − 𝛼𝑖 𝜁, 𝑖 = 1, . . . , 𝑛 (6) 𝛼𝑖 𝛼𝑖 where we have used (3) and (5). Clearly, 𝜖𝑖 is an increasing function of the QoS requirement and of the degree of internal differentiation, 𝜁, which is common for every outer class. Hence, the maximum relative error corresponds to the most stringent outer class —class 1—, and its value is 1 − 𝛼1 𝜁 (7) 𝜖max = max 𝜖𝑖 = 𝑖 𝛼1 Therefore, exact proportionality can be approximated as closely as desired. For any 𝜖 > 0, the condition 𝜖max < 𝜖 holds when 𝛼1 𝜁< 𝜖. (8) 1 − 𝛼1 Since 𝛼1 → 0 is equivalent to impose very low loss probability for the packets in the outer class receiving best service, the inequality (9) 𝜁 < 𝛼1 𝜖 𝜖𝑖 ≜
=
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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 58, NO. 1, JANUARY 2010 100 10
−1
= 10% = 1% = 0.1%
ζ
10−2 10−3 10−4 10−5 10−6 −4 10
Fig. 2.
10−3
10−2 α0
10−1
100
Internal differentiation, external proportionality, and relative error.
is a sufficient condition to bound the maximum relative error of the proportional model. This means that the ratio of burst loss probabilities should be of the same order of magnitude as 𝛼1 𝜖 to keep the maximum relative error under 𝜖, roughly. It will be shown in the next section that (9) is achievable in a broad range of operating regimes in networks with current technology. Fig. 2 illustrates the bounding function 𝛼1 /(1 − 𝛼1 )𝜖 for three typical error tolerances, 𝜖 ∈ {0.1%, 1%, 10%}. III. A NALYSIS OF A S INGLE N ODE In this Section, we shall study the validity of the assumption 𝜁 ≪ 1 in a model of a single OBS node. To analyze the system, we need a specific model for the losses of each burst class, which in turn are determined by the scheduling algorithm. A. Link Description Assume that the OBS node transmits the bursts carrying the user packets along a single output optical link with 𝑤 wavelengths. The switch can perform full wavelength conversion on any incoming burst. Bursts are tagged with the identifier of the type or inner class they belong to, and there exist only two classes labeled 0 and 1. We consider a scheduling algorithm that gives strict preemptive priority to the transmission of class-0 bursts (high-priority) over class-1 bursts (low-priority). It is worth emphasizing that it is the transmission of a burst the entity to be preempted, not its reservation. The assumption obeys to a basic reason, namely that the scheduling algorithm does not miss any of the chances to transmit a high priority burst. Also, the subsequent analysis of the scheduler becomes more tractable. The class of preemptive schedulers has been extensively studied in the context of Multi-Protocol Label Switching [22] or OBS networks [9], [15], [16], and appears to be a robust and effective solution for providing QoS based on performance guarantees of the individual switches. The schemes proposed to date differ basically in the rules employed to identify opportunities for preemption, but their performance is very similar. In our model, a class-0 burst than encounters all wavelengths occupied can preempt one scheduled for a class-1
burst. Low-priority bursts, on the other hand, are discarded if they find all wavelengths in use, or if the arrival of a highpriority burst preempts its transmission. Note that, if resources assigned to preempted bursts are not freed, some transmission resources could be wasted by the downstream nodes. To overcome this problem, the scheduler can be combined with a signaling protocol to warn neighbor nodes when a preemption occurs [16]. Even without this aid, the impact of revoked reservations on the loss probability should be small, since preemption events should be rare in networks with moderate or low congestion. In general, with the ideal preemptive scheduler postulated in this paper, the loss probability of a high-priority burst is the same as the loss probability of an arbitrary packet contained in the burst. However, the loss probability of a low-priority burst is different than that of the conveyed packets. This discrepancy happens because longer low-priority bursts are more likely to be preempted than shorter bursts of the same type. In addition, more packets are lost when a long burst is dropped. So, the probability that a low-priority burst gets discarded does not match the exact computation of the probability that a packet marked as low priority has been dropped. Despite this theoretical remark, the difference between the burst and packet loss probabilities for the low-priority traffic class is quite small and vanishes rapidly if the overall losses in the network are low, or if the number of wavelengths of the links is not exceedingly small, which are precisely the conditions presumed in our model. Our prior work [23] addressed this issue, and allows us to conclude that the same computation is valid to estimate the packet loss probability with sufficient accuracy. Hence, in the rest of the paper, we will not make further distinction between the packet and burst loss probabilities. B. Burst Loss Probability By applying the probabilistic mapping of packets into the two burst classes and the subsequent arrangement into bursts, any set of packet flows arriving to the edge router gives rise to two burst processes sent out of it, with rates 𝜆0 and 𝜆1 (the subscript indicates the priority). Assuming that burst arrival times can be modeled as a Poisson process (see [6], [24], for example, for the justification of Poissonian statistics for aggregated traffic streams), the OBS node can be modeled as an M/G/𝑤/𝑤 loss system with two priorities and preemptive service; then, the average loss probability 𝐵 and the loss probabilities for each class, 𝐵0 and 𝐵1 , satisfy the following loss conservation law: 𝜆𝐵 = 𝜆0 𝐵0 + 𝜆1 𝐵1
(10)
where 𝜆 = 𝜆0 + 𝜆1 is just the aggregate arrival rate [11], [25]. Therefore, the blocking probability for high-priority bursts can be easily computed by 𝐵0 = 𝐸𝐵 (𝑤, 𝜆0 /𝜇),
(11)
where 𝐸𝐵 (⋅, ⋅) is the well known Erlang-B formula and 𝜇−1 is the expected transmission time of a class-0 burst. It is a simple observation that high-priority bursts are only dropped if the system is full with other bursts of the same type, that
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ARGIBAY-LOSADA et al.: FLOW SPLITTING FOR END-TO-END PROPORTIONAL QOS IN OBS NETWORKS
1
100 ζ, ρ0 /w = 0.40 ζ, ρ0 /w = 0.20 B1 , ρ0 /w = 0.40 B1 , ρ0 /w = 0.20
10−2
10−4
−6
−6
0.8
ζ
B1
10−4
10
10
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10−10 1
4
8
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32 w
64
128
256
512
0.3
10−10
is, if there are no available wavelengths in the output link and none of these holds a transmission or reservation of a low-priority burst, from which (11) follows. We also assume that the length of both burst types is exponentially distributed with mean transmission time 1/𝜇. It has been proven in this case [5] that the system’s blocking probability is given by (12)
Define the traffic intensity offered to the node as 𝜌 = 𝜆/𝜇, and similarly 𝜌0 = 𝜆0 /𝜇, 𝜌1 = 𝜆1 /𝜇. Consequently, using (10), (11) and (12), the loss probability of a low-priority burst is 𝐵1 =
𝜆𝐸𝐵 (𝑤, 𝜌) − 𝜆0 𝐸𝐵 (𝑤, 𝜌0 ) . 𝜆1
(13)
Note that 𝐵0 and 𝐵1 are coupled and the differentiation ratio 𝜁(𝑤, 𝑟0 ) =
𝐵0 (𝜆 − 𝜆0 )𝐸𝐵 (𝑤, 𝜌0 ) = 𝐵1 𝜆𝐸𝐵 (𝑤, 𝜌) − 𝜆0 𝐸𝐵 (𝑤, 𝜌0 )
(14)
is not constant, but dependent on how much traffic is routed through both burst types. We conjecture that, in large switches and under suitable traffic conditions (𝑤 large and 𝜌 = 𝑂(𝑤)), the loss probabilities 𝐵0 and 𝐵1 will be nearly separable [25], [26]. This means that the system behaves asymptotically as if a virtual partitioning of resources was being used, with a certain number of wavelengths assigned exclusively for the use of each inner class. In this case, we could write 𝐵0 = 𝐸𝐵 (𝑤, 𝜌0 ),
˜𝐵 (𝑤′ , 𝜌1 ) 𝐵1 = 𝐸
0.6
0.4
Fig. 3. 𝜁 achievable in a node with 𝑤 wavelengths with the strict priority scheduler, nominal utilization of 𝜌 = 0.50 and high-priority bursts load equal to 𝜌0 .
𝐵 = 𝐸𝐵 (𝑤, 𝜆/𝜇).
0.7
0.5
10−8
2
ρ/w = 0.10 ρ/w = 0.20 ρ/w = 0.30 ρ/w = 0.40 ρ/w = 0.50 ρ/w = 0.60 ρ/w = 0.70 ρ/w = 0.80 ρ/w = 0.90
0.9 10−2
ρ0 /ρ
100
261
(15)
˜𝐵 (⋅, ⋅) is the analytic continuation of the Erlang loss where 𝐸 function for non-integer capacities and 𝑤′ = 𝑤𝜆1 /𝜆 is the effective capacity offered to the low-priority class. Therefore, 𝐵0 and 𝐵1 would be functions of the traffic load in their class, only. The exact conditions under which this asymptotic approximation is valid are left for further study. We now collect, for reference, some elementary properties of the functions 𝐵0 and 𝜁. Lemma 1: P1 𝑥𝐸𝐵 (𝑤, 𝑥) is a positive, continuous, strictly increasing, convex and continuously differentiable function of 𝑥.
10−10
10−8
10−6
10−4 ζ
10−2
100
102
Fig. 4. Admissible region for a OBS node as a function of the required 𝜁; 𝑤 = 32.
P2 For any given 𝜆 > 0, 𝜁(𝑤, 𝜌0 ) is positive, continuous and strictly increasing in 𝜌0 . Both P1 and P2 are immediate consequences of the monotonicity of 𝐸𝐵 (𝑛, 𝑥) in 𝑛 and 𝑥, and are proven in the appendix. To illustrate numerically these properties in a general case, we have plotted in Fig. 3 the dependence of both 𝜁 and the maximum packet loss probability experienced by a flow, 𝐵1 , on the number of wavelengths 𝑤 in the link for a offered link load 𝜌 = 0.5𝑤. We pick two different values for the load of the high-priority class, namely 𝜌0 = 0.2𝑤 and 𝜌0 = 0.4𝑤. It is clear the decrease in the 𝐵1 and 𝜁 as 𝑤 increases, due to the statistical multiplexing gain. Fig. 4 shows the dependence of 𝜁 on the ratio 𝜌0 /𝜌 for a link offered load between 𝜌/𝑤 = 0.1 and 𝜌/𝑤 = 0.9, with 𝑤 = 32 wavelengths. Note that 𝜁 decays exponentially fast as 𝜌0 /𝜌 → 0, and that the packet mapping algorithm does not constrain the fraction of allowable high-priority traffic to a negligible value. For example, in a setting with total traffic intensity 𝜌 = 0.6𝑤, two external classes generating equal traffic load, with the high priority demanding as much differentiation as possible (i.e., 𝛼1 ≈ 0 and 𝜌0 = 𝜌/2), the scheduling algorithm achieves 𝜁 ≃ 10−5 . This enables the possibility of providing three orders of magnitude lower loss probability to the best outer class compared to the worst, with a 1% relative error (cf. (9)). IV. R ING N ETWORKS In this section, we extend the analysis of the packet mapping algorithm to networks with ring topology. Rings are a common configuration for optical networks, and the circular symmetry greatly simplifies the model that would show up in a general case. We continue to assume that the route followed by each flow is unique, static, and chosen by the provider. We also assume that the preferred QoS class for each flow is a user decision. So, the packet label identifying the outer class remains unchanged end-to-end. This means that the analysis of the network cannot be reduced to the analysis of a single node. A. Model and Analysis We consider a bidirectional ring with 𝑡 nodes. The links connecting every pair of consecutive nodes have 𝑤 wavelengths
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in each direction, and the route between origin node 𝑖 and destination node 𝑗 is always the minimum hop-count path. Suppose that the nodes are numbered 0, . . . , 𝑡−1 in clockwise order. Thus, the route used between the nodes (𝑖, 𝑗) has length 𝑑(𝑖, 𝑗) = min{(𝑖 − 𝑗) mod 𝑡, (𝑗 − 𝑖) mod 𝑡}, and the largest length of a route is 𝑟 = ⌊𝑡/2⌋.1 Note that, since a bidirectional ring was assumed, with one link in each direction between two consecutive nodes, and the orientation of the route between two points depends only on their distance, we may restrict our attention to the clockwise ring and the set of routes with length no greater than 𝑟. The analysis for the reverse direction would be identical. Accordingly, denote the link from node 𝑖 to node 𝑖 + 1 mod 𝑡 as link 𝑖 henceforth. As in the model for the single node, we consider the existence of two inner burst classes in the OBS network, as well as an arbitrary number of outer classes defined at the packet level. In order to compute the differentiation among these outer classes, the basic variables are the traffic load of each burst type offered to the links, and the burst loss probabilities in the 𝑡 links. (0) Therefore, let ℒ(0) = [Λ𝑖𝑗 ] be the 𝑡 × 𝑡 traffic demand matrix corresponding to class-0 (high priority) bursts, where (0) Λ𝑖𝑗 denotes the intensity of class-0 bursts generated by node 𝑖 (1) and destined to node 𝑗. Define similarly ℒ(1) = [Λ𝑖𝑗 ], the matrix of traffic demands corresponding to class-1 (low priority) (0) (1) bursts. Denote by 𝜋ℎ (resp., 𝜋ℎ ) the probability of dropping a class-0 (resp., class-1) burst) at node ℎ = 0, . . . , 𝑡 − 1. Finally, to simplify the notation, let [𝑖] ≜ 𝑖 mod 𝑡. The traffic offered to node ℎ comprises the traffic originated at nodes [ℎ − 𝑗] having a route with lengths 𝑙𝑗 = 𝑗 + 1, . . . , 𝑟, for any 𝑗 = 0, . . . , 𝑟 − 1. But the traffic along the routes [ℎ − 𝑗] → [ℎ + 𝑙𝑗 ] must be reduced by the losses occurring at links [ℎ − 𝑗], . . . , [ℎ − 1] in order to sum the aggregated load arriving to node ℎ. Therefore
𝑗], for 𝑗 = 1, . . . , 𝑟, are given by
(0)
𝜌ℎ =
𝑟−1 ∑ 𝑟 ∑ 𝑗=0 𝑙=𝑗+1
[ℎ−1]
∏
(0)
Λ[ℎ−𝑗],[ℎ−𝑗+𝑙]
(1 − 𝜋𝑠(0) )
(16)
=
𝑟−1 ∑ 𝑟 ∑ 𝑗=0 𝑙=𝑗+1
(1) Λ[ℎ−𝑗],[ℎ−𝑗+𝑙]
(1 − 𝜋𝑠(1) )
(17)
𝑠=[ℎ−𝑗]
is the corresponding quantity for high priority bursts. Now, repeating the analysis made for the case of a single node, we have, under the hypothesis of independent losses in the links, 𝜋𝑠(0) = 𝐸𝐵 (𝑤, 𝜌(0) 𝑠 ) 𝜋𝑠(1) =
(0)
(1)
(18) (0)
(1)
(0)
(0)
(𝜌𝑠 + 𝜌𝑠 )𝐸𝐵 (𝑤, 𝜌𝑠 + 𝜌𝑠 ) − 𝜌𝑠 𝐸𝐵 (𝑤, 𝜌𝑠 ) (1)
𝜌𝑠
.
(19)
Finally, the burst drop probabilities along the route ℎ → [ℎ + 1 Without loss of generality, we may assume 𝑡 odd. For 𝑡 even, only straightforward changes in the notation would be necessary.
∏
=1−
(1)
(1 − 𝜋𝑠(0) )
𝑠=ℎ [ℎ+𝑗−1]
∏
(1 − 𝜋𝑠(1) ).
𝑠=ℎ
Observe that (16)–(19) form a system of fixed-point equa(0) (1) 𝑡−1 tions for the 2𝑡 variables {(𝜋𝑠 , 𝜋𝑠 )}𝑠=0 which can be solved trivially by iterated substitutions, the existence of a solution being guaranteed by the Brouwer fixed-point theorem. The use of fixed-point methods for analyzing the performance of OBS networks is widely accepted in the literature [27], [28], [28], [29], though it is only an approximation to the exact values of the probabilities of interest. Section V validates via numerical simulation this approach. B. Traffic Demand A useful classification of current optical networks distinguishes the access segment, characterized by large asymmetry demands caused by most of its traffic destined to other networks and sent through border nodes, and the core or backbone segment, with a more balanced route demand along the nodes of the ring. To take into account this diversity in the route demands, we specialize the traffic matrices ℒ(0) and ℒ(1) to two cases, one with uniform traffic load along every route, and another for a ring with a hot spot node attracting a significant portion of the network traffic. 1) The Uniform Case: If all the routes generate the same traffic intensity, Λ, the formulae in the fixed-point system simplify notably. For both bursts classes, every link is equally loaded and shows the same loss probabilities. Therefore, we can drop the subscript of all the variables and write the equations in a more compact form. The system to solve becomes now, after some routine algebraic manipulations of (16)–(17), 𝜌(𝑘) =
𝑟−1 ∑ 𝑟 ∑
Λ(1 − 𝜋 (𝑘) )𝑗
𝑗=0 𝑙=𝑗+1
[ℎ−1]
∏
[ℎ+𝑗−1]
𝑝ℎ (𝑗) = 1 −
𝑠=[ℎ−𝑗]
is the rate of incoming low priority bursts to node ℎ. And reasoning in an analogous fashion, (1) 𝜌ℎ
(0) 𝑝ℎ (𝑗)
𝜋 (0)
( ) 𝜆 1 − 𝜋 (𝑘) − (1 − 𝜋 (𝑘) )𝑟+1 = (𝑘) 𝑟 − , 𝜋 𝜋 (𝑘) = 𝐸𝐵 (𝑤, 𝜌(0) )
𝜋 (1) =
𝑘 = 0, 1
(𝜌(0) + 𝜌(1) )𝐸𝐵 (𝑤, 𝜌(0) + 𝜌(1) ) − 𝜌(0) 𝐸𝐵 (𝑤, 𝜌(0) ) . 𝜌(1)
And the end-to-end burst loss probabilities for a route with length 𝑗 = 1, . . . , 𝑟 are given by 𝑏(0) (𝑗) = 1 − (1 − 𝜋 (0) )𝑗 ,
𝑏(1) (𝑗) = 1 − (1 − 𝜋 (1) )𝑗 . ( ) Also, it is easy to check that lim𝜋(𝑘) →0 𝜌(𝑘) = Λ 𝑟+1 2 , which is equal to the number of routes using a link. 2) The Non-Uniform Case: The case of different traffic intensities in the routes can be set up in many distinct ways, reproducing a number of causes for the asymmetry. We shall disturb the model with uniform traffic by using a single parameter. Specifically, assume that a fraction 0 < 𝑧 < 1 of the traffic between any pair of nodes is now shifted toward
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ARGIBAY-LOSADA et al.: FLOW SPLITTING FOR END-TO-END PROPORTIONAL QOS IN OBS NETWORKS
node 0, which becomes the hot spot in the ring. Node 0 itself is a sink and it does not send traffic. So, given 𝑧, the matrices ℒ(0) and ℒ(1) have entries { (1 − 𝑧)Λ, 𝑖 ∕= 𝑗, 𝑖, 𝑗 ∕= 0 (𝑘) ) 𝜆𝑖𝑗 = ( 1 + 𝑧(𝑡 − 2) Λ, 𝑗 = 0, 𝑖 ∕= 0 equally for 𝑘 = 0 and 𝑘 = 1. In other words, node 0 receives the traffic from 𝑖 ∕= 0 originally directed to it, plus a fraction 𝑧 of the other 𝑡 − 2 flows departing from 𝑖. Using (16)–(17), the traffic load at node 1 ≤ ℎ ≤ 𝑟 can be written as (𝑘)
𝜌ℎ =
=
𝑟 𝑟−1 ∑ ∑
(1 − 𝑧)Λ(1 − 𝜋 (𝑘) )𝑗
𝑗=0 𝑙=𝑗+1 𝑗∕=ℎ
( ) Λ(1 − 𝑧) 1 − 𝜋 (𝑘) − (1 − 𝜋 (𝑘) )𝑟+1 𝑟 − 𝜋 (𝑘) 𝜋 (𝑘) − Λ(1 − 𝑧)(𝑟 − ℎ)(1 − 𝜋)ℎ . (20)
Again, applying (16)–(17) for node 0 we have ( ) Λ(1 − 𝑧) 1 − 𝜋 (𝑘) − (1 − 𝜋 (𝑘) )𝑟+1 (𝑘) 𝑟− 𝜌0 = 𝜋 (𝑘) 𝜋 (𝑖) + Λ𝑧𝑟(𝑡 − 2). (21) (𝑘)
Clearly, 𝜌ℎ is decreasing in ℎ, i.e., the traffic load is larger near the hot spot. With the rates already known, we can invoke once again the single node analysis and obtain the loss probabilities for low- and high-priority bursts as (0)
(0)
𝜋ℎ = 𝐸𝐵 (𝑤, 𝜌𝑗 ) (1)
𝜋ℎ =
(0) (𝜌ℎ
+
(22)
(1) (0) 𝜌ℎ )𝐸𝐵 (𝑤, 𝜌ℎ
(1) + 𝜌ℎ ) (1) 𝜌ℎ
−𝜌
(0)
𝐸𝐵 (𝑤, 𝜌
(0)
)
(23)
for a node at position ℎ. The last step is to write down the probability of losing bursts in the route ℎ → [ℎ + 𝑗], which is simply 𝑏
(𝑘)
[ℎ+𝑗] ∏ ( ) ℎ, [ℎ + 𝑗]) = 1 − (1 − 𝜋𝑠(𝑘) ),
𝑘 = 0, 1.
𝑠=ℎ
Unlike the uniform case, the value depends not only on the length of the route, but also on the starting and ending points. In summary, the system defined by equations (20)–(23) is the fixed-point mapping whose solution determines the basic performance of the non-uniform traffic model. V. N UMERICAL RESULTS We now proceed to solve the fixed point systems of the previous section to show the performance of the packet classification algorithm in several illustrative situations. The power of the QoS differentiation algorithm has been measured, for each class of packets, by means of (7), the relative error between the desired coefficient of proportionality and the value actually observed. For simplicity, we consider in the numerical examples that follow only two packet classes. This turns out to be sufficient to discuss the main ideas, while allowing us to present the graphs more clearly. Furthermore, reducing the
263
number of end-to-end QoS classes to two can be regarded as a kind of worst case model, since the QoS levels offered are as coarse as possible. Alternatively, another interpretation could be that the results attained with the end-to-end binary model are conservative compared to a network with more QoS classes, provided the proportionality between the best and worst classes is kept constant in both cases (i.e., 𝛼1 is the same). This is so because, in the binary QoS network, those packet flows for whom the lowest priority class provides a loss probability too large would be forced to switch to the best class, whereas in a network with 𝑛 > 2 different end-to-end QoS levels —namely, a finer partition of the range [𝛼1 , 1]— some intermediate class would likely suit their requirements. The latter argument bears implicit the existence of a pricing policy in the network in charge of limiting the total demand for each QoS level, or the action of some form of congestion control responsive to packet drops (e.g., TCP). We shall not discuss in this paper the issues concerning the design of such pricing or congestion control management, though they clearly deserve further study. To sum up, in all the experiments reported below, the maximum relative error from the set ℛ of all routes 𝑝(0) 𝑟
𝜖 ≜ max
(1)
𝑝𝑟
𝑟∈ℛ
− 𝛼1
𝛼1
is the parameter taken to describe the network performance. We report in the following subsections the performance results of our proportional QoS framework in two cases: ring networks, for which we have the closed-form formulas presented above, and a network with a real topology, where the fixed-point analysis cannot be simplified. A. Ring Networks We consider first rings of variable size, where every link has 𝑤 = 64 wavelengths operating at 10 Gbps each. The traffic load is uniform with ℒ(0) = ℒ(1) , and is set such that the largest end-to-end packet loss probability along any of the routes is 1%. Logically, this value corresponds to packets traversing the longest path and also belonging to the best-effort class. 1) Effect of the Ring Size: A greater number of nodes in the ring should improve the differentiation ratio between the burst classes, since in longer routes the likelihood that a lowpriority burst gets eventually preempted increases. This effect can be clearly recognized in Fig. 5(a), where 𝜁 decreases with 𝑡 consistently for 𝛼1 = 2−2𝑘 , 𝑘 = 1, . . . , 7. On the other hand, the penalty paid by the weakest class may be excessive in terms of utilization if the network comprises many nodes, due to the augmented number of places where contention might arise. So, in large rings, the overall network utilization (i.e., the fraction of offered traffic that arrives at its destination) could be poor. This result is not so discouraging as it seems, if one keeps in mind that the rings deployed in real networks have usually few nodes. The main reason to use ring topologies is not a high degree of connectivity or route diversity, but the opposite, namely simplicity of management (static routing and automatic recovery, for example). Aside from this remark, Fig. 5(a)
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264
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 58, NO. 1, JANUARY 2010
10−7
10−2
10−8
10−4
10−9
10−6
10−10
10−8
ζ
10−11 10−12
10−10
10−13
10−12
10−14
10−14
10−15 10−16
4
α1 = 1/16384 α1 = 1/4096
8 α1 = 1/1024 α1 = 1/256
16 t
32
64
α1 = 1/64 α1 = 1/16
α1 = 1/4
102 0
10−2 10−4
10−6 10−8 10−10 10−12 10−14 10−16 10−18 8 α1 = 1/16384 α1 = 1/4096
Fig. 6.
16 α1 = 1/1024 α1 = 1/256
32 w
64 α1 = 1/64 α1 = 1/16
128 α1 = 1/4
Behavior of the relative error 𝜖 with the number of wavelengths.
shows that 𝜁 can be made negligible even in small rings, i.e., that almost exact proportionality is attainable. Fig. 5(b) depicts the maximum relative error 𝜖, showing very low values also decreasing with 𝑡. 2) Number of Wavelengths: Recall that our model assumes the possibility of full wavelength conversion at the nodes. Accordingly, allowing more wavelengths per link reduces the contention events and leads to better multiplexing gains for either class of bursts. Hence, the internal differentiation ratio should be better as 𝑤 grows, while the utilization is kept constant. This is confirmed by Fig. 6, that plots the maximum relative error as a function of the number of wavelengths in a ring of 𝑡 = 11 nodes. In this case, the main conclusion to draw is the existence of a minimum number of wavelengths 𝑤∗ for ensuring that 𝜖 is sufficiently low. This threshold 𝑤∗ depends on the traffic mix, but is increasing in the QoS gap between the best and worst burst classes, i.e., as 𝛼1 → 0. At the same time, Fig. 6 reveals two key properties of the QoS model: ∙
∙
α1 = 1/16384 α1 = 1/4096
8 α1 = 1/1024 α1 = 1/256
16 t
32
α1 = 1/64 α1 = 1/16
64 α1 = 1/4
Ring size, QoS differentiation, and proportional service.
104 10
4
(b) Maximum relative error, 𝜖, for different rings.
(a) Differentiation in the OBS network, for varying ring size. Fig. 5.
10−16
The scaling behavior of the networks. The relative error decreases exponentially fast in the number of wavelengths available in each link. The benefit of full wavelength conversion. Exact QoS proportionality can be approximated as closely as desired in DWDM networks.
3) End-to-End Proportionality: Negligible values of the maximum relative error are equivalent to almost perfect endto-end proportional differentiation. Fig. 7(a) compares the packet loss probabilities between the two given outer classes. The lines are the numerical values obtained after solving the fixed-point model, whereas the points with 95% confidence intervals were produced after averaging data drawn from 100 independent simulation runs. Both match very well and, except for a few link wavelengths (under 8-10), there is no measurable difference between the actual and the desired proportionality factors, ranging from 𝛼1 = 2−1 down to 𝛼1 = 2−6 . Fig. 7(b) shows the relative errors (theoretical and simulated) for the data in Fig. 7(a). As expected, the wider the gap between the outer classes, the larger the link capacities must be. 4) Traffic Asymmetry: Another possible concern would be whether the proportional service is affected in some way by the traffic pattern, i.e., by the spatial distribution of traffic over the ring. We have addressed this question by solving the fixed-point model for the non-uniform case described in Section IV-B2. The results for a ring with 𝑡 = 11 nodes and 64 wavelengths in each link are plotted in Fig. 8. In all cases the worst relative error is low enough to allow almost perfect proportionality in the packet loss probabilities. Surprisingly, 𝜖 is nearly insensitive to the degree of asymmetry, 𝑧. One would guess that an overloaded node would be so stressed that failed to exert the necessary differentiation between the burst types. However, under the preemptive policy this is not the case as long as the high-priority burst class is not overloaded (𝜌0 = 𝑤(1+𝑂(1))) or critically loaded (𝜌0 = 𝑤±𝑜(𝑤)), so its blocking probability begins to rise. This desirable robustness property seems to be linked to the preemptive schedulers, but suggests anyway that the algorithm will perform well in general networks, where route diversity and different traffic loads at each link are to be expected. 5) Utilization: Clearly, attaining any degree of QoS differentiation is feasible in a network if only a small amount of traffic is allowed to enter and the excess demand is blocked, in other words, by enforcing a sufficiently low utilization. This is not the case with the technique presented in the paper, as Fig. 9 shows. The results are again for a ring with 11 nodes and uniform traffic, so they refer to every link indistinctly. The
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ARGIBAY-LOSADA et al.: FLOW SPLITTING FOR END-TO-END PROPORTIONAL QOS IN OBS NETWORKS
1/2
1/2
1/4
1/4
1/8
1/8
1/16
1/16
1/32
1/32
1/64
1/64
265
7 α1 = 1/64 α1 = 1/32 α1 = 1/16 α1 = 1/8 α1 = 1/2
6
4
4
8
16
32
w α1 = 1/8 α1 = 1/16
α1 = 1/2 α1 = 1/4
1
4
8
w
16
32
(b) Relative error
10−3 10−4 10−5 10−6 10−7
2
Theoretical vs. simulation results for a ring of 11 nodes.
Fig. 7.
10−8 10−9 10−10 10−11 10−12 10−130.1
0.2
0.3
α1 = 1/16384 α1 = 1/4096
0.4
0.5
α1 = 1/1024 α1 = 1/256
z
0.6
0.7
0.8
α1 = 1/64 α1 = 1/16
0.9
1
α1 = 1/4
Maximum relative error 𝜖 and traffic asymmetry.
Fig. 8. 100
10−1
p
3
0
α1 = 1/32 α1 = 1/64
(a) End-to-end proportionality
10−2
10−3
10−4
1 16384
1 4096
1 1024
1 256
1 64
1 16
1 4
1 2
1
α1 75%
Fig. 9.
pi /pn
5
50%
25%
10%
1%
0.1%
Utilization: loss probability of the links.
total offered load to every link is set so as to make the loss probability of low-priority bursts in any link equal to a given value, ranging from low to severe congestion (namely, 0.1%, 1%, 10%, 25%, 50% and 75%). For each of these levels, and for different values of the proportionality desired by the two external classes, Fig. 9 plots the overall burst loss probability in an arbitrary link, 𝑝. Its value is virtually constant over the whole range 𝛼1 ∈ [2−14 , 2−2 ], and increases slightly when the best external class of packets demands a QoS level very similar to the other class, i.e., for 𝛼1 ≈ 1. The effect is more clearly
perceivable if the congestion level inside the OBS network is moderate or high. The significance of the values 𝛼1 ≈ 1 is that of representing the case without differentiation between the packet classes or, equivalently, a situation in which the internal differentiated scheduling is not used at all and the traffic carried by the network is maximum (for a given constraint on the maximum burst loss probability). Therefore, the nearly constant behavior of 𝑝 means that the network is able to keep the utilization of its links at a fixed level, almost independently of the gap between the QoS offered to the external packet classes. In other words, the proportional QoS framework does not introduce any noticeable penalty in link (or network) utilization, which is a significant advantage compared to the performance loss seen with the resource reservation schemes. To explain this fact, recall that our packet mapping scheme works jointly with a static two-priority preemptive burst scheduling algorithm, and that the routing decisions are independent of the congestion level inside the OBS core or of the potential contention events experienced by the bursts. So, none of the preemption opportunities for burst transmission are wasted, none of the resources (wavelengths) are reserved, and none of the contention events is solved by using additional network resources, as with the route deflection schedulers. 6) Correlations in the Input Traffic: The results from the Erlang fixed-point models and from the simulation experiments presented thus far use a Poisson process for the burst arrival/transmission instants at every optical switch. Real Internet traffic does not conform to Poisson statistics, however, so it would be wise to look at the performance of our framework when the packet or the burst stream have significant statistical correlation over long timescales. Since a fixed-point analysis with correlated traffic is in general not possible, we have resorted to simulation to investigate this issue, using a M/G/∞ process to model the exogenous burst arrivals to the ring nodes. Specifically, we tested the performance when the nodes generate high- and low-priority bursts according to two independent M/G/∞ processes with Hurst parameter 𝐻 = 0.9. The M/G/∞ queueing system models the general superposition of many independent sources. When the holding times of each customer follow a heavy-tailed distribution, their superposition
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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 58, NO. 1, JANUARY 2010
6
1/2
1/2
1/4
1/4
1/8
1/8
1/16
1/16
1/32
1/32
2
1/64
1/64
1
4 α1 = 1/2 α1 = 1/4
8
16 w α1 = 1/8 α1 = 1/16
4 3
32
0
α1 = 1/32 α1 = 1/64
4
(a) Packet loss probabilities Fig. 10.
Fig. 11.
α1 = 1/64 α1 = 1/32 α1 = 1/16 α1 = 1/8 α1 = 1/2
5
pi /pn
266
8
w
16
32
(b) Relative error
Performance with long-range dependent traffic. A ring with 11 nodes.
Abilene topology.
produces an asymptotically self-similar process with a tunable long-memory exponent. The Hurst parameter 𝐻 ∈ (0.5, 1) is a simple measure of the degree of statistical self-similarity, which is stronger for values closer to 1. The final performance of the proportional QoS model is again remarkably accurate: Fig. 10(a) shows the end-to-end packet losses and Fig. 10(b), the corresponding relative errors. In both cases, the deviations between the target and the measured values become negligible with only a moderate number of wavelengths. Overall, the long-term statistical dependence in the input traffic does not seem to perturb the algorithm’s ability of delivering loss proportionality between endhosts. Also, after comparing Figs. 7 and 10, no performance differences can be noticed between the Poissonian and the self-similar traffic loads. B. A Mesh Network We have also analyzed the performance in a mesh topology, the U.S. nationwide Abilene network (Fig. 11), as a prototypical case of an irregular network. It has 11 nodes, 110 routes (one for every ordered pair of nodes), 28 unidirectional links, and diameter equal to 5. We have used shortest path (minimum hop-count) routes. Ties are solved using the node with lower index as the first hop. The number of wavelengths in a link is varied between 4 and 32, but each wavelength always operates at 1 Gbps.
Like in previous settings, we consider only two classes of packets in every route, generating both the same traffic load. Note that, though the traffic model is uniform over the routes, a given link supports traffic from a varying number of routes, so their loads are different. In the graph of the Abilene topology, there are vertices with out degree 2 or 3 to distribute their 10 routes. The individual offered rate along a route is set such that the largest packet drop probability in the network is equal to 1%. The procedure was the following. First, the fixed-point system resulting from this graph was set up. It is composed by two sets of 28 equations (one per link and burst class) which can be derived in a straightforward way and are omitted here. Then, the system was solved and the packet loss probabilities of all the routes were computed. The computational process was iterated, applying binary search to solve for the value of the traffic rate that finally leads to the 1% bound. The largest relative error for the ratio of packet loss probabilities between the two classes —selected from within the set of all routes— is plotted in Fig. 12 for several number of wavelengths and moderate proportionality ratios, 𝛼1 = 2−1 to 𝛼1 = 2−4 . The route of interest happens to be the longest, 0 → 5, as expected, and we can easily check that the error is negligible and falls off very quickly as the number of wavelengths increases. Additionally, the Abilene graph brings the opportunity of validating over a real network topology some of the assumptions and approximations we have made so far in the analysis: 1) the use of the Erlang function to compute burst and packet drop probabilities; 2) the fixed-point approach, that is based on the condition of link independence and may have in some cases multiple convergence points; 3) the approximate identity in (4). For instance, losses in the link are generally correlated, and the approximate proportionality in (4) is plausible only if the burst differentiation algorithm performs well. For the validation, we simply compare the results obtained in the numerical solution of the fixed-point equations to the same performance measures produced by simulation. In both cases, the configuration was as follows. Each link has 𝑤 ∈ [4, 32] wavelengths in each direction, transmitting at 1 Gbps. The time needed to process a control packet at the OBS nodes is 1 𝜇s, the offset time separating the control header and the data burst is 1 ms, and the bursts transmission times
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ARGIBAY-LOSADA et al.: FLOW SPLITTING FOR END-TO-END PROPORTIONAL QOS IN OBS NETWORKS 100
number of wavelengths. If the links have 𝑤∗ ≈ 10 wavelengths or more, the two packet flows see network losses almost ideally proportional.
10−2 10−4 10−6
10−8
VI. C ONCLUDING R EMARKS
10−10 10−12 10−14 10−16 10−18 10−20
8
α1 = 1/16
Fig. 12.
16 α1 = 1/8
w
32
64
α1 = 1/4
128 α1 = 1/2
Worst case 𝜖 in the Abilene network (belonging to route 0 ↔ 5).
10−1
10−2
Theoretical; B1 Simulation; B1 , α1 = 1/2 Simulation; B1 , α1 = 1/16 Theoretical; B0 , α1 = 1/16 Theoretical; B0 , α1 = 1/2 Simulation; B0 , α1 = 1/16 Simulation; B0 , α1 = 1/2
10−3
10−4
10−5
267
4
8
w
16
32
Fig. 13. Validation results for the packet loss probabilities 𝐵0 and 𝐵1 ; results are shown for the route with highest losses (0 → 5).
along any route form a Poisson process of rate 1000 bursts per second. Routes are chosen the same as before, and there are only two classes of packets. Again, route 0 → 5 exhibits the largest packet loss probability, for it follows a 5-hop path and traverses the most loaded link, (2, 3), that is used by 20 routes. Given a configuration (𝑤, 𝛼1 ), we computed numerically the traffic load giving a 1% packet loss probability along 0 → 5, and also run 100 independent simulations. From the point samples of the simulations, we computed the 95% confidence intervals of 𝐵0 and 𝐵1 . The results appear in Fig. 13, and the values generated by simulation are close to those predicted by the fixedpoint model. As expected, the theoretical model is slightly conservative, mainly as a consequence of the link independence hypothesis that underlies the fixed-point equations. The burst loss probabilities of the high-priority class also follow accurately the values predicted numerically, 𝐵1 /2 and 𝐵1 /16, respectively. In conclusion, the analytical model posed in Sections III-IV is an accurate tool to assess the performance of our proportional differentiation framework. A final set of simulation experiments was conducted over the Abilene topology using again M/G/∞ traffic sources. Like in ring networks, the ultimate packet loss probabilities match remarkably well the proportional service model: Fig. 14(a) shows the end-to-end packet losses and Fig. 14(b), the corresponding relative errors. In both cases, the deviations between the target and the measured values vanish with an increased
We studied the provision of end-to-end proportional QoS to an arbitrary number of packet flows transmitted across an OBS network. The approach is based on abstracting the lowlayer scheduling operations of the optical switches, beyond the assumption that the optical transport network offers two classes of service. Mapping the packets into these two classes in adequate proportions turns out to be the way to attain proportional packet losses. For single nodes, we have provided an analysis of the burst and packet loss probabilities, showing that schedulers with static priorities and preemption can be used to offer two traffic classes widely separated in the optical domain. In the case of ring networks, a fixed-point model of the entire system can be written in closed form, and its performance was easily computed. The numerical experiments reported show that coupling the mapping of packets with the capabilities of the preemptive scheduler, the packet flows receive end-to-end proportional service over a wide range of traffic loads, stochastic traffic patterns and different network topologies, without affecting the utilization. The framework works remarkably well in networks with many wavelengths per link. A PPENDIX P ROOF OF L EMMA 1 For P1, let 𝑔𝑤 (𝑥) = 𝑥𝐸𝐵 (𝑤, 𝑥). Using the definition of the Erlang loss function, it is easily seen that 𝑔𝑤 (𝑥) =
𝑥𝑔𝑤−1 (𝑥) 𝑤 + 𝑔𝑤−1 (𝑥)
with 𝑔0 (𝑥) = 𝑥. Write the latter recurrence as ( ) 𝑔𝑤 (𝑥) 𝑤 + 𝑔𝑤−1 (𝑥) = 𝑥𝑔𝑤−1 (𝑥) and differentiate. This yields ( ) ( ) ′ ′ 𝑔𝑤 (𝑥) 𝑤 + 𝑔𝑤−1 (𝑥) = 𝑔𝑤−1 (𝑥) + 𝑥 − 𝑔𝑤 (𝑥) 𝑔𝑤−1 (𝑥). (24) ′ Assume that 𝑔𝑤−1 (𝑥) ≤ 1 and note that 𝑥 − 𝑔𝑤 (𝑥) ≤ 𝑤, since 𝑥 − 𝑔𝑤 (𝑥) is the expected number of busy channels in ′ ′ the system. Then 0 ≤ 𝑔𝑤 (𝑥)(𝑤 + 𝑔𝑤−1 (𝑥)) ≤ 𝑔𝑤−1 (𝑥)(𝑤 + ′ 𝑔𝑤−1 (𝑥)). Therefore, 𝑔𝑤 (𝑥) ≤ 1. To complete the induction, one can simply check that 𝑔0′ (𝑥) = 1. Differentiating (24) once again ( ) ( ) ′′ ′ ′ 𝑔𝑤 (𝑥) 𝑤 + 𝑔𝑤−1 (𝑥) = 2𝑔𝑤−1 (𝑥) 1 − 𝑔𝑤 (𝑥) + ( ) ′′ 𝑥 − 𝑔𝑤 (𝑥) 𝑔𝑤−1 (𝑥). ′ (𝑥) ≤ 1 and the induction Using 𝑔0′′ (𝑥) = 0, 𝑔𝑤 (𝑥) ≤ 𝑥, 𝑔𝑤 ′′ ′′ (𝑥) ≥ 0, i.e., the hypothesis 𝑔𝑤−1 (𝑥) ≥ 0, it follows that 𝑔𝑤 function 𝑔𝑤 (𝑥) is convex. For P2, we need to prove that the first derivative of (14) is non-negative. That derivative can be written as ( ) ′ (𝑤, 𝑥)𝐸𝐵 (𝑤, 𝜆)−𝐸𝐵 (𝑤, 𝑥) 𝐸𝐵 (𝑤, 𝜆)−𝐸𝐵 (𝑤, 𝑥) . (𝜆−𝑥)𝐸𝐵
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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 58, NO. 1, JANUARY 2010
1/2
1/2
1/4
1/4
1/8
1/8
1/16
1/16
1/32
1/32
1/64
1/64
7 α1 = 1/64 α1 = 1/32 α1 = 1/16 α1 = 1/8 α1 = 1/2
6
4
4
8
16 w α1 = 1/8 α1 = 1/16
α1 = 1/2 α1 = 1/4
32 α1 = 1/32 α1 = 1/64
(a) Packet loss probabilities Fig. 14.
pi /pn
5
3 2 1 0 4
8
w
16
32
(b) Relative error
Performance in the Abilene topology with LRD traffic.
Substituting in the latter the recurrence formulas 𝑥𝐸𝐵 (𝑤 − 1, 𝑥) 𝑤 + 𝑥𝐸𝐵 (𝑤 − 1, 𝑥) 𝑤 ′ (𝑤, 𝑥) = ( 𝐸𝐵 )2 𝑤 + 𝑥𝐸𝐵 (𝑤 − 1, 𝑥)
𝐸𝐵 (𝑤, 𝑥) =
the result is, after some algebraic manipulations, 𝑞(𝑥) = ℎ(𝑥) (𝜆𝑤(𝜆 − 𝑥)𝐸𝐵 (𝑤 − 1, 𝑥)− ( )) 𝑤𝑥𝐸𝐵 (𝑤 − 1, 𝑥) 𝜆𝐸𝐵 (𝑤 − 1, 𝜆) − 𝑥𝐸𝐵 (𝑤 − 1, 𝑥) (25) where ℎ(𝑥) = (𝑤 + 𝑔𝑤−1 (𝜆))−1 (𝑤 + 𝑔𝑤−1 (𝑥))−2 > 0. Now, ′ the convexity of 𝑔𝑤 (𝑥) implies 𝑔𝑤 (𝜆) ≥ 𝑔𝑤 (𝑥)+(𝜆−𝑥)𝑔𝑤 (𝑥). Using this fact and the condition 𝑥 ≤ 𝜆 in (25) one finally gets 𝑞(𝑥) ≥ 0, establishing the property. R EFERENCES [1] Y. Chen, C. Qiao, and X. Yu, “Optical burst switching: a new area in optical networking research,” IEEE Network, vol. 18, no. 3, pp. 16–23, May 2004. [2] I. Baldine, A. Bragg, G. Evans, M. Pratt, M. Singhai, D. Stevenson, and R. Uppalli, “Jumpstart deployments in ultra-high-performance optical networking testbeds,” IEEE Commun. Mag., vol. 43, no. 11, pp. 18–25, Nov. 2005. [3] C. Dovrolis and P. Ramanathan, “A case for relative differentiated services and the proportional differentiation model,” IEEE Network, vol. 13, no. 5, pp. 26–34, Sep. 1999. [4] M. Yoo, C. Qiao, and S. Dixit, “QoS performance of optical burst switching in IP-over-WDM networks,” IEEE J. Sel. Areas Commun., vol. 18, no. 10, pp. 2062–2071, Oct. 2000. [5] H. Vu and M. Zukerman, “Blocking probability for priority classes in optical burst switching networks,” IEEE Commun. Lett., vol. 6, no. 5, pp. 214–216, May 2002. [6] V. Vokkarane and J. Jue, “Prioritized burst segmentation and composite burst-assembly techniques for QoS support in optical burst-switched networks,” IEEE J. Sel. Areas Commun., vol. 21, no. 7, pp. 1198–1209, Sep. 2003. [7] S. Tan, G. Mohan, and K. Chua, “Feedback-based offset time selection for end-to-end proportional QoS provisioning in WDM optical burst switching networks,” Computer Commun., vol. 30, no. 4, pp. 904–921, Feb. 2007. [8] A. Pattavina, “Performance of deflection routing algorithms in IP optical transport networks,” Computer Networks, vol. 50, no. 2, pp. 207–218, 2006. [9] J. Phuritatkul, Y. Ji, and Y. Zhang, “Blocking probability of a preemption-based bandwidth-allocation scheme for service differentiation in OBS networks,” J. Lightwave Technol., vol. 24, no. 8, pp. 2986– 2993, Aug. 2006.
[10] W. Liao and C.-H. Loi, “Providing service differentiation for opticalburst-switched networks,” J. Lightwave Technol., vol. 22, no. 7, pp. 1651–1660, July 2004. [11] C. Tan, G. Mohan, and J. C.-S. Lui, “Achieving multi-class service differentiation in WDM optical burst switching networks: a probabilistic preemptive burst segmentation scheme,” IEEE J. Sel. Areas Commun., vol. 24, no. 12, pp. 106–119, Dec. 2006. [12] Q. Zhang, V. Vokkarane, J. Jue, and B. Chen, “Absolute QoS differentiation in optical burst-switched networks,” IEEE J. Sel. Areas Commun., vol. 22, no. 9, pp. 1781–1795, Nov. 2004. [13] L. Yang and G. Rouskas, “Optimal wavelength sharing policies in OBS networks subject to QoS constraints,” IEEE J. Sel. Areas Commun., vol. 25, no. 9, pp. 40–49, Dec. 2007. [14] ——, “Generalized wavelength sharing policies for absolute QoS guarantees in OBS networks,” IEEE J. Sel. Areas Commun., vol. 25, no. 3, pp. 93–104, Apr. 2007. [15] J. Phuritatkul, Y. Ji, and S. Yamada, “Proactive wavelength pre-emption for supporting absolute QoS in optical-burst-switched networks,” J. Lightwave Technol., vol. 25, no. 5, pp. 1130–1137, May 2007. [16] M. H. Phung, K. C. C. ang G. Mohan, M. Montani, and T. C. Wong, “An absolute QoS framework for loss guarantees in optical burst-switched networks,” IEEE Trans. Commun., vol. 55, no. 6, pp. 1191–1201, June 2007. [17] A. G. P. Rahbar and W. W. Yang, “Contention avoidance and resolution schemes in buferless all-optical packet-switched networks: a survey,” IEEE Commun. Surveys Tutorials, vol. 10, no. 4, pp. 94–107, 2008. [18] Y. Chen, C. Qiao, M. Hamdi, and D. Tsang, “Proportional differentiation: a scalable QoS approach,” IEEE Commun. Mag., vol. 41, no. 6, pp. 52–58, June 2003. [19] Y. Chen, M. Hamdi, and D. Tsang, “Proportional QoS over OBS networks,” in Proc. IEEE Global Commun. Conf. (GLOBECOM), 2001, vol. 3, pp. 1510–1514. [20] H. Cankaya, S. Charcranoon, and T. El-Bawab, “A preemptive scheduling technique for OBS networks with service differentiation,” in Proc. IEEE Global Commun. Conf. (GLOBECOM), 2003, pp. 2704–2709. [21] C. Dovrolis, D. Stiliadis, and P. Ramanathan, “Proportional differentiated services: delay differentiation and packet scheduling,” IEEE/ACM Trans. Networking, vol. 10, no. 1, pp. 12–26, Feb. 2002. [22] J. de Oliveira, J. Vasseur, L. Chen, and C. Scoglio, “Label switched path (LSP) preemption policies for MPLS traffic engineering,” Internet Engineering Task Force (IETF), Tech. Rep. RFC 4829, 2007. [23] P. Argibay-Losada, A. Suárez-González, M. Fernández-Veiga, R. Rodríguez-Rubio, and C. López-García, “From relative to observable proportional differentiation in OBS networks,” in Proc. ACM Conf. Emerging Network Experiment Technol. (CoNEXT), Oct. 2005, pp. 11–123. [24] T. Karagiannis, M. Molle, M. Faloutsos, and A. Broido, “A nonstationary Poisson view of internet traffic,” in Proc. IEEE INFOCOM, Mar. 2004, pp. 1558–1569. [25] Z. Zhao, S. Weber, and J. de Oliveira, “Preemption rates for a parallel link loss network,” Perf. Evaluation, vol. 66, pp. 21–46, Jan. 2009. [26] S. Zachary and I. Ziedins, “A refinement of the Hunt-Kurtz theory of large loss networks, with an application to virtual partitioning,” Ann. Adv. Prob., vol. 12, pp. 1–22, 2002. [27] Z. Rosberg, H. L. Vu, M. Zukerman, and J. White, “Performance
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ARGIBAY-LOSADA et al.: FLOW SPLITTING FOR END-TO-END PROPORTIONAL QOS IN OBS NETWORKS
analyses of optical burst-switching networks,” J. Sel. Areas Commun., vol. 21, no. 7, pp. 1187–1197, Sep. 2003. [28] Z. Rosberg, A. Zalesky, H. Vu, and M. Zukerman, “Analysis of OBS networks with limited wavelength conversion,” IEEE/ACM Trans. Networking, vol. 14, no. 5, pp. 1118–1127, Oct. 2006. [29] A. Zalesky, H. Vu, Z. Rosberg, E. Wong, and Zukerman, “Stabilizing deflection routing in optical burst switched networks,” J. Sel. Areas Commun., vol. 25, no. 6, pp. 3–19, Aug. 2007. Pablo Jesus Argibay-Losada is an assistant professor in the Department of Telematic Engineering at the University of Vigo. He received a Ph.D. degree in telecommunication engineering from the University of Vigo in 2009. His current research deals with methods to enable end-to-end quality of service in communications networks.
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Candido Lopez-Garcia is an associate professor in the Department of Telematic Engineering at the University of Vigo. He received a Ph.D. degree in telecommunication engineering from the Polytechnic University of Madrid in 1995. His research interests are in performance modeling and evaluation of communication systems, with a current focus in network quality of service.
Manuel Fernandez-Veiga is an associate professor in the Department of Telematic Engineering at the University of Vigo. He received a Ph.D. in telecommunication engineering from the University of Vigo in 2001. His research interests are in performance modeling and evaluation of communication systems, with a current focus in traffic engineering for quality-of-service enabled networks and optical networks.
Andres Suarez-Gonzalez is an associate professor in the Department of Telematic Engineering at the University of Vigo. He received a Ph.D. degree in telecommunication engineering from the University of Vigo in 2000. He is a member of ACM-SIGSIM. His current research interests include simulation methodology and analysis of stochastic systems.
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