IN GENERAL, an optical network can consist of a

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1 1 Geometric Capacity Provisioning for Wavelength-Switche WDM Networks Li-Wei Chen, Eytan Moiano Abstract In this paper, we use an asymptotic analysis similar to the sphere-packing argument in the proof of Shannon s channel capacity theorem to erive optimal provisioning requirements for networks with both static an ynamic provisioning. We consier an N-user share-link moel where W s wavelengths are statically assigne to each user, an a common pool of W wavelengths are available to all users. We erive the minimum values of W s an W require to achieve asymptotically non-blocking performance as the number of users N becomes large. We also show that it is always optimal to statically provision at least enough wavelengths to support the mean of the traffic. I. INTRODUCTION IN GENERAL, an optical network can consist of a large number of noes connecte in some arbitrary fashion (see Figure 1) an can present the network architect with a complex provisioning problem over multiple links. For simplicity, in this paper we will focus on provisioning a single share link on a backbone network. Figure 1 also shows a moel for the share colore link in the arbitrary network. We consier provisioning for traffic traveling from left to right along the link. Each wavelength on the link can be use to support one lightpath from one of the incoming fibers on the left sie of the link to one of the outgoing fibers on the right sie of the link. Broaly speaking, wavelength provisioning can be one in one of two ways. One option is to statically provision a wavelength by har-wiring the noes at the ens of the link to always route the wavelength from a given input fiber to a given output fiber. The avantage to this is that the cost of the harware require to support static provisioning is relatively low: no switching capability or intelligent ecision-making ability is require. The ownsie is a lack of flexibility in using that wavelength even if the wavelength is not neee to support a lightpath between the assigne input an output fibers, it cannot be The authors are with the Laboratory for Information an Decision Systems, Massachusetts Institute of Technology, Cambrige, MA 02139, USA ( lwchen@mit.eu, moiano@mit.eu) This work was supporte by the National Science Founation (NSF) uner Grant ANI Fig. 1. An example of a mesh optical network consisting of numerous noes an links, followe by a share-link moel base on the colore link. The otte lines enote ifferent users of the link. Since each pair of input-output fibers comprises a ifferent user, an there are 4 input fibers an 4 output fibers, there are a total of 4 4 = 16 users in this example. assigne to support a lightpath between any other pair of fibers. This shortcoming can be overcome by using ynamic provisioning. A ynamically provisione wavelength is switche at the noes on both sies of the link, allowing it to be ynamically assigne to support a lightpath between any source an estination fibers. Furthermore, this assignment can change over time as traffic emans change. This obviously imparts a great eal of aitional flexibility. The ownsie is that the ae switching an processing harware makes it more expensive to ynamically provision wavelengths. There has been much investigation of both statically provisione an ynamically provisione systems in the literature [1], [2], [3], [4]. Such approaches are wellsuite for cases where either the traffic is known a priori an can be statically provisione, or is extremely unpreictable an nees to be ynamically provisione. However, in practice, ue to statistical multiplexing it is common to see traffic emans characterize by a large mean an a small variance aroun the mean. A hybri system is well suite to such a scenario. In a hybri system, a sufficient number of wavelengths are statically provisione to support the majority of the traffic. Then, on top of this, a smaller number of wavelengths are ynamically provisione to support the inevitable variation in the realize traffic. Such an approach takes avantage of the relative preicability of the traffic by cheaply provisioning the ma-

2 2 jority of the wavelengths, but retains sufficient flexibility through the minority of ynamic wavelengths that significant wavelength overprovisioning is not necessary. After escribing the system moel use in this paper, we will use the asymptotic analysis approach from information theory incorporate in the proof of Shannon s channel capacity theorem [5] to analyze hybri networks: we allow the number of users to become large, an consier the minimum provisioning in static an ynamic wavelengths necessary to achieve nonblocking performance (i.e., to guarantee that the probability of any call in the snapshot being blocke goes to zero). We will show that it is always optimal to statically provision enough wavelengths to support the traffic mean. We also fully characterize the optimal provisioning strategy for achieving non-blocking performance with minimal wavelength provisioning. A. System Moel In the share link context, we can consier each incoming-outgoing pair of fibers to be a ifferent user of the link. Each lightpath request (which we will henceforth term a call) can therefore be thought of as belonging to the user corresponing to the incoming-outgoing fiber pair that it uses. We can similarly associate each static wavelength with the corresponing user. Uner these efinitions, a call belonging to a given user cannot use a static wavelength belonging to a ifferent user it must either use a static wavelength belonging to its own user, or employ a ynamic wavelength. Figure 2 gives a flowchart of the ecision process for amitting a call. When a user requests a new call setup, the link checks to see if a static wavelength for that user is free. If there is a free static wavelength, it is use. If not, then the link checks to see if any of the share ynamic wavelengths are free if so, then a ynamic wavelength is use. If not, then no resources are available to support the call, an it is blocke. There have been several approaches evelope in the literature for blocking probability analysis of such systems uner Poisson traffic moels [6], incluing the Equivalent Ranom Traffic (ERT) moel [7], [8], [9] an the Haywar approximation [10]. These approximations, while often able to prouce goo numerical approximations of blocking probability, are purely numerical in nature an o not provie goo intuition for guiing the imensioning of the wavelengths. Furthermore, they assume that the ynamic wavelengths must be iniviually switche, an o not consier waveban switching. In this paper, we aopt a snapshot traffic moel that leas to close-form asymptotic analysis an evelop Fig. 2. Decision process for wavelength assignment for a new call arrival. A new call first tries to use a static wavelength if it is available. If not, it tries to use a ynamic wavelength. If again none are available, then it is blocke. guielines for efficient imensioning of hybri networks. We consier examining a snapshot of the traffic eman at some instant in time. The snapshot is compose of the vector c = [c 1,..., c N ], where c i is the number of calls that user i has at the instant of the snapshot, an N is the total number of users. We moel each variable c i as a Gaussian ranom variable with mean µ i an variance σi 2. This is reasonable since each user actually consists of a collection of source-estination pairs in the larger network that all use the link from the same source fiber to the same estination fiber. In this paper, we will assume that each user has the same mean µ an variance σ 2 ; the results are extensible to general µ i an σ i but the extension is beyon the scope of this paper (see [11]). Although the traffic for each iniviual source-estination pair for the user may have some arbitrary istribution, as long as the istributions are well-behave, the sum of each traffic stream will appear Gaussian by the Central Limit Theorem. As a special case, consier the common moel of Poisson arrivals an exponential holing times for calls. Then the number of calls that woul have entere a nonblocking system at any instant in time is given by the stationary istribution of an M/M/ queue namely, Poisson with intensity equal to the loa ρ in Erlangs. For a heavy loa, this istribution is well approximate by a Gaussian ranom variable with mean ρ an variance ρ. II. WAVELENGTH-GRANULARITY SWITCHING In this section, we consier a share link, an assume that there are N users that are the source of calls on the link. Each user is statically provisione W s wavelengths for use exclusively by that user. In aition to this static provisioning, we will also provie a total of W ynamically switche wavelengths. These wavelengths can be share by any of the N users. As previously escribe, we will use a snapshot moel of traffic. The traffic is given by a vector c = [c 1,..., c N ],

3 3 where each c i is inepenent an ientically istribute as N(µ, σ 2 ). We assume that the mean µ is significantly large relative to σ that the probability of negative traffic (a physical impossibility) is low, an therefore oes not present a significant moeling concern. We will primarily be concerne with a special blocking event that we call overflow. An overflow event occurs when there are insufficient resources to support all calls in the snapshot an at least one call is blocke. We will call the probability of this event the overflow probability. From Figure 2, we see that an overflow event occurs if the total number of calls excees the ability of the static an ynamic wavelengths to support. This can be expresse mathematically as N max {c i W s, 0} > W (1) i=1 where max {c i W s, 0} is the amount of traffic from each user that excees the static provisioning; if the total amount of excess from each user excees the available pool of share ynamic wavelengths, a blocking event occurs. If we consier the N-imensional vector space occupie by c, the constraint given by (1) represents a collection of hyperplanes bouning the amissible traffic region: c i W s + W c i + c j 2W s + W, i j c i + c j + c k 3W s + W, i j k. Each constraint reflect the fact that the sum of the traffic from any subset of users clearly cannot excee the sum of the static provisioning for those users plus the entire ynamic provisioning available. Note that there are a total of N sets of constraints, where the n th set consists of C(N, n) = N! (N i)!n! equations, each involving the sum of n elements of the traffic vector c. If the traffic snapshot c falls within the region efine by the hyperplanes, all calls are amissible; otherwise, an overflow event occurs. The bole lines in Figure 3 show the amissible region for N = 2 in two imensions. A. Asymptotic Analysis We will consier the case where the number of users N becomes large, an use the law of large numbers to help us raw some conclusions. We can rewrite the call vector in the form c = µ 1 + c Fig. 3. The amissible traffic region, in two imensions, for N = 2. Three lines form the bounary constraints represente by (1). There are two lines each associate with a single element of the call vector c, an one line associate with both elements of c. The traffic sphere must be entirely containe within this amissible region for the link to be asymptotically non-blocking. where 1 is the length-n all-ones vector, an c N(0, σ 2 1) is a zero-mean Gaussian ranom vector with i.i.. components. Conceptually, we can visualize the ranom traffic vector as a ranom vector c centere at µ1. The length of this ranom vector is given by c = N c 2 i n=1 We will use an approach very similar to the sphere packing argument use in the proof of Shannon s channel capacity theorem in information theory [5]. We will show that asymptotically as the number of users becomes large, the traffic vector falls onto a sphere centere at the mean, an the provisioning becomes a problem of choosing the appropriate number of static an ynamic wavelengths so that this traffic sphere is completely containe within the amissible region. From the law of large numbers, we know that 1 N N c 2 i σ 2 n=1 as N. This implies that asymptotically, as the number of users becomes large, the call vector c becomes concentrate on a sphere of raius Nσ centere at the mean µ1. Therefore, in orer for the overflow probability to converge to zero, a necessary an sufficient conition is that the hyperplanes escribe by (1) enclose the sphere entirely. This is illustrate in Figure 3. B. Minimum Distance Constraints Next, we will erive necessary an sufficient conitions for the amissible traffic region to enclose the traffic

4 4 sphere. Our goal is to ensure that we provision W s an W such that the minimum istance from the center of the traffic sphere to the bounary of the amissible region is at least the raius of the sphere, therefore ensuring that all the traffic will fall within the amissible region. Due to the ientical istribution of the traffic for each user, the mean point µ1 will be equiistant from all planes whose escription involves the same number of elements of c. We efine a istance function f(n) such that f(n) is the minimum istance from the mean µ1 to any hyperplane whose escription involves n components of c. Lemma 1: The istance function f(n) from the traffic mean to a hyperplane involving n elements of the traffic vector c is given by f(n) = ( n W s + W ) n µ, n = 1,..., N (2) Proof: The istance can be calculate using basic geometric principles an is omitte for brevity. We efine the minimum bounary istance to be F min = min n=1,...,n f(n) A necessary an sufficient conition for the overflow probability to go to zero asymptotically with the number of users is F min Nσ We woul like to etermine the inex n such that f(n) is minimize. Unfortunately, this value of n turns out to epen on the choice of provisioning W s. Let us consier the erivative of the istance function f (n): f (n) = ( 1 2 W s W ) n n µ We can ivie W s into three regimes of interest, corresponing to ifferent ranges of values for W s an W, an characterize f(n) in each of these regions: Regime 1: If W s µ: In this region, f (n) < 0 for all n. This implies that f(n) is a ecreasing function of n, an F min = f(n), giving a minimum istance of F min = ( N W s + W ) N µ Regime 2: If µ < W s µ + W : In this region, f (n) starts out negative an ens up positive over 1 n N. This implies that f(n) is convex an has a minimum. Neglecting integrality concerns, this minimum occurs when f (n) = 0, or n = W W s µ Therefore F min = f(n ) in this regime. Substituting the appropriate values, it can be shown that the minimum istance is given by F min = 2 W (W s µ) Regime 3: If W s > µ + W : In this region, f (n) > 0 for all n. This implies that f(n) is an increasing function of n, an F min = f(1), giving a minimum istance of C. Optimal Provisioning F min = W s + W µ In the preceing section, we erive the minimum istance criteria for the hybri system. Given a fixe number of statically allocate wavelengths W s, we can use the equation F min Nσ to calculate the minimum number of ynamic wavelengths W to achieve asymptotically non-overflow performance. We can also raw a few aitional conclusions about provisioning hybri systems. Theorem 1: A minimum of µ static wavelengths shoul always be provisione per user. Proof: For W s µ, we know from Case 1 above that the minimum istance constraint is F min = ( N W s + W ) N µ Nσ W tot = NW s + W (µ + σ)n Note that the total number of wavelengths W tot = NW s + W is inepenent of W s an W in this regime, suggesting that the same total number of wavelengths are require regarless of the partitioning between static an ynamic wavelengths. Since static wavelengths are less expensive to provision than ynamic wavelengths, this shows that there is never any reason to provision less than W s = µ wavelengths. An interesting corollary to this theorem follows from the observation that the case where W s = 0 (i.e. all wavelengths are ynamic) also falls in this regime (i.e. Regime 1). Since fully ynamic provisioning is obviously the least-constraine version of this system, we can use it as a boun on the minimum number of wavelengths require by any asymptotically overflow-free system:

5 µ = 100, σ = 10 W tot (µ + σ)n We can also consier a system that is fully static, with no ynamic provisioning. This is the most inflexible wavelength partitioning, an provies us with an upper boun on the number of wavelengths require by any hybri system. Theorem 2: For a fully static system with no ynamic provisioning, the minimum number of wavelengths require is given by ( ) W tot = (µ + σ)n + N 1 Nσ Proof: Let W = 0. Then, for overflow-free operation, we obviously nee W s > µ. This puts us in Regime 3 where W s > µ + W, an the minimum istance conition gives us F min = W s + W µ > Nσ ( ) W tot = (µ + σ)n + N 1 Nσ Note that this excees the lower boun on the minimum number of wavelengths by ( N 1)Nσ. We can therefore regar this quantity as the maximum switching gain that we can achieve in the hybri system. This gain is measure in the maximum number of wavelengths that coul be save if all wavelengths were ynamically switche. Corollary: Combining the upper an lower bouns, we observe that for efficient overflow-free operation, the total number of wavelengths require by any hybri system is boune by (µ + σ)n W tot (µ + σ)n + ( N 1)Nσ D. Numerical Example Simulations were conucte to verify the accuracy of the provisioning results erive. Figure 4 verifies the results of the preceing iscussion for the case of µ = 100 an σ = 10. The rapily escening curve shows that if the theoretical minimum of W tot = (µ+σ)n wavelengths are provisione with W s = µ, then as N increases, the overflow probability rops off quickly an eventually the system becomes asymptotically non-blocking. The other two curves show that if less than W tot wavelengths are provisione, the overflow probability no longer converges to zero as the number of users increases. Note also that the convergence occurs fairly rapily if the W tot wavelengths calculate in the preceing sections are provisione. In a system with just 30 users, the overflow probability W tot 0.98 W tot 0.97 W tot users N Fig. 4. Curves show ecrease in overflow probability with increasing number of users N. The curve with the circles shows a link provisione with the theoretical minimum number of wavelengths W tot neee to achieve asymptotically non-overflowing operation. Note that if fewer than W tot wavelengths are provisione, the overflow probability no longer converges to zero as the number of users increases. overflow probability has alreay ecrease to the orer of Since the number of users is equal to the number of input-output fiber pairs, this correspons to a link with as few as 5 input fibers an 6 output fibers, for example. Therefore, the results are useful in esigning for goo network performance even when N is finite an small. REFERENCES [1] R. Ramaswami an K. N. Sivarajan, Optical Networks: A Practical Perspective, Morgan Kaufmann, [2] L. Li an A. K. Somani, Dynamic wavelength routing using congestion an neighborhoo information, IEEE/ACM Trans. Networking, vol. 7, pp , October [3] A. Birman, Computing approximate blocking probabilities for a class of all-optical networks, IEEE J. Select. Areas Commun., vol. 14, no. 5, pp , June [4] O. Gerstel, G. Sasaki, S. Kutten, an R. Ramaswami, Worstcase analysis of yanmic wavelength allocation in optical networks, IEEE/ACM Trans. Networking, vol. 7, pp , December [5] T. Cover an J. Thomas, Elements of Information Theory, Wiley- Interscience, [6] R. Guerin an L. Y.-C. Lien, Overflow analysis for finite waiting-room systems, IEEE Trans. Commun., vol. 38, pp , September [7] R. I. Wilkinson, Theories of toll traffic engineering in the u.s.a., Bell Syst. Tech. J., vol. 35, pp , March [8] R. B. Cooper, Introuction to Queueing Theory, 2n E., North Hollan, New York, [9] D. A. Garbin M. J. Fischer an G. W. Swinsky, An enhance extension to wilkinson s equivalent ranom technique with application to traffic engineering, IEEE Trans. Commun., vol. 32, pp. 1 4, January [10] A. A. Freericks, Congestion in blocking systems - a simple approximation technique, Bell Syst. Tech. J., vol. 59, pp , July-August [11] L. Chen, A Stuy on the Traeoff Between Efficient Resource Allocation an Noe Complexity in WDM Networks, Ph.D. thesis, MIT, September 2005.