Dynamic routing of connections with Known duration in WDM networks

Dynamic Routing of Connections with Known Duration in WDM Networks Diego Lucerna∗ , Massimo Tornatore† , Biswanath Mukherjee† , Achille Pattavina∗

∗ Department

of Electronics and Information, Politecnico di Milano, Via Ponzio 34-35, 20121 Milan, Italy of Computer Science, University of California, Davis, CA 95616, USA Email: {lucerna,pattavina}@elet.polimi.it {tornator,mukherje}@cs.ucdavis.edu

† Department

Abstract—Recently, new solutions for automatized management in optical networks promise to allow customers to specify on-demand the terms of the Service Level Agreement (SLA) to be guaranteed by the service provider. In this paper we show that is possible to design a highly efficient load balancing algorithm, called RABBIT, for the dynamic provisioning of connections exploiting the knowledge, among the other Service Level Specifications (SLS), of the connections duration. The core idea of RABBIT consists in routing connections based on the transient probability of future-link congestion, that can be estimated with higher precision when the knowledge of connections durations is given. So, we introduce a time-dependent link-weight assignment that evaluates future link congestions probability based on the transient analysis of the Markovian model of the link, making it computationally feasible by means of an effective approximation technique. By means of an extensive set of simulative experiments, we compare our approach to other traditional holding-time agnostic, yet efficient, dynamic routing algorithms. We consider different performance metrics, among which the Blocking Probability (BP), in a wavelength-convertible WDM mesh network scenario. For a typical US nationwide network, RABBIT obtains savings on BP of up to 20% for practical scenarios. Index Terms—Optical network, WDM, dynamic traffic, holding time, Markov chain, transient probability

I. I NTRODUCTION Optical networks provide a transport infrastructure with very high capacity, thanks to wavelength-division-multiplexing (WDM) technology. In a wavelength-routed WDM network, a lightpath must be established between a pair of source and destination nodes before data can be transferred. A lightpath is an end-to-end optical connection which may traverse multiple fiber links and optical cross-connects (OXCs): in particular, in this work we consider WDM networks where all OXCs are equipped with full wavelength conversion. Recently, in order to efficiently accommodate the lightpaths, new challenges, but also opportunities, are offered by the evolution of the prevalently static traffic towards a more dynamic traffic paradigm [1], [2]. So far, optical transport networks have been supporting connections, which are provided and leased for long period of time, e.g. weeks or months. Nowadays, new applications are emerging with requirements of large The work described in this paper was carried out with the support of the BONE-project (”Building the Future Optical Network in Europe”), a Network of Excellence funded by the European Commission through the 7th ICTFramework Programme.

bandwidth over relatively short periods of time: e.g., the video distribution of important sport or social event, or the massive data transfer for backup or storage purposes. Technology and bandwidth market are developing to provide the flexible platform the new applications are asking for. New agile OXC are emerging to create mesh-structured optical WDM backbone networks, and new control protocols such as ASON and G-MPLS allow today to provision connections on-demand [3]. Moreover, new architectures for on-demand lightpath provisioning [4] based on automatic or web-based interfaces at the the management plane (MP) will enable the on-line specification of the SLA terms to be guaranteed (with different price range) by the service provider1 . Among the other Service Level Specifications (SLS), in this paper we propose to exploit the knowledge of the connection-holding time to efficiently solve the lightpath-routing problem. Lot of strategies for efficient lightpath routing in WDM networks have been proposed in these last years. We can classify these strategies in two main categories: alternate routing (AR) and dynamic routing (DR) [5]. In AR schemes, e.g. [6], [7], [8], [9], [10], [11], the route for the required connection is chosen among a set of predefined paths, which are usually assigned to each source/destination pair individually. Since the paths of such sets are usually determined apriori by the network administrator, AR is not flexible enough to fully adapt to the variations in the network utilization. As for DR schemes, e.g. [1], [12], [13], the connection route is dynamically determined according to the present state of the network. Although the AR algorithms need less topological information and less computation time to find routing paths, the DR solutions, which will be considered in this work, are able to yield much lower connection blocking probability. Furthermore, in traditional networks, such as telephone service, where traffic could be predicted on a long-term basis, analytical models were proposed to estimate the stationary blocking performance of the traditional routing algorithms (e.g. [6], [14]). However today, the system-performance measures at steady state are becoming inadequate for the increasingly dynamic traffic scenarios, where connection durations are short and the traffic pattern is rapidly evolving. Due to 1 Note that both ASON and GMPLS use distributed real-time signaling that, in conjunction with appropriately extended versions of OSPF-TE, can take charge of the distribution of this information coming from the MP.

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the real-time dynamic nature of the traffic, it is pertinent to investigate the transient characteristic of this new problem. In this paper we provide an analytical analysis of a Markovian transient model, that estimates the close-future congestion probability of network links given the knowledge of the durations of the connections routed over it. This estimation is then used to apply an holding-time-aware link cost assignment for effective routing. This approach, called RABBIT (Route Adaptation Balanced By Information on Timing), is able to outperform existing DR approaches, achieving significant decrease of the Blocking Probability (BP). We compare our strategies to the Weighted-Shortest Path algorithm [13], which is holding-time agnostic, but has been shown to be very efficient guaranteing low network congestion. The rest of this paper is organized as follows. Section II overviews the background work on AR and DR algorithms. In Section III we formally state the DR problem in a WDM networks. In Section IV we describe two holdingtime-agnostic, yet effective, solutions for DR, namely the Shortest Path and Weighted-Shortest Path algorithms. Section V introduces the notation for our problem and then a deterministic approach (called RABBITdt and presented in our previous study [12]) is briefly defined. In Section VI a new statistical model for the link-congestion probability, which is rigorous but computationally tractable, is discussed and then applied to devise an efficient algorithm, called RABBITst , for the DR problem. Section VII evaluates by simulations the performance of RABBIT compared to the Shortest Path and Weighted-Shortest Path algorithms. Section VIII draws some conclusions. II. P RIOR WORK As mentioned above, the main objective of this work is to devise a new connection-holding-time aware procedure for DR. In this section we describe the more relevant solutions for AR and DR in literature. Effective routing algorithms have to consider not only the cost of a link, but also its utilization level, in order to avoid that excessive load is routed on congested parts of the network. The basic idea to increase network throughput in a congestion-aware scheme consists in balancing the traffic load among a set of alternate paths, so that any of those paths are lead into congestion. So, one of the major issues to be solved in congestion-aware routing is how to select the right candidate path inside an AlternatePaths Set (APS). Least Load Routing (LLR) is the most popular congestion-aware routing scheme in literature [7] and chooses, among the candidates in the APS, the path that has more free capacity available on the most-heavily-loaded link. Ref.[8] shows that selecting the path according to the optimally assigned traffic intensities can effectively reduce the BP compared with selecting the routing path according to the hop counts, such as in typical routing algorithms. Unfortunately, even though LLR balances traffic load in the network, it relies on a fixed APS which is evaluated a priori, with no relation with current network congestion. This lack of flexibility in the definition of the APS leads to two

drawbacks: (a) the congestion or failure of some links may cause the disconnection between the end nodes and (b) by precomputing the paths, you may not be able to fully exploit the knowledge of the current traffic distribution. So, more flexible strategies have been investigated. In [9] an adaptive LLR is proposed which, according to traffic distribution, periodically recomputes the route-set and dynamically switches the lightpath between the alternate routes. However, devising load balancing algorithms, care should be taken since excessive load balancing may result in paths significantly lengthened: authors in [13] propose a DR algorithm, called WeightedShortest Path (WSP), that considers both the current traffic load and the length of the paths, achieving lower blocking. Several other algorithms have been proposed in literature. We consider here only those presented above for sake of brevity, and because they are the most frequently used. III. N OTATIONS We first define the notations and then formally state the DR problem. A network is represented as a weighted, directed graph G = (V, E), where V is the set of nodes and E is the set of unidirectional fibers (referred to as links). Moreover, for each link e, Ce and We represent the link cost and the number of wavelengths on that link, respectively. We denote the set of existing lightpaths in the network at any time by L = {(si , di , li , tia , tih )}, where the quintuple (si , di , li , tia , tih ) specifies the source node, the destination node, the route, the arrival time and the holding time for the ith lightpath. We associate a link utilization level descriptor νe to each link e in the network, to identify the link occupation, that can be represented as an integer set, {νe |∀e ∈ E, 0 ≤ νe ≤ We }. Using νe , a DR procedure has to find a route for the incoming request (s, d, Ta , Th ), characterized respectively by its source node, destination node, arrival time and duration. IV. DYNAMIC ROUTING (DR) A LGORITHMS In this section, we discuss two baseline dynamic routing algorithms. We first introduce the adaptive Shortest Path (SP) algorithm based on the path length, and second a fully-adaptive version of Weighted-Shortest Path strategy (AWSP), which is able to get lower congestion than LLR exploiting the path length and the traffic load jointly. A. Shortest path strategy (SP) In the SP approach, the objective is to minimize the cost of the crossed links (e.g., the hop count, if the cost is set to unity). Let C(e) be the cost of the generic link e ∈ E:  +∞ νe = We (1) C(e) := 1 otherwise For the incoming setup request between the source/destination pair s/d, we look for the route l such that the path cost Cl is minimized:  C(e) (2) Cl := e∈l

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Finally, if a route can be found, we update the link congestion descriptors accordingly and allocate the correspondent new wavelengths; otherwise we block the incoming connection.

generic connection i is given by:  i ta + tih − Ta if (tia + tih ≤ Ta + Th ) hi := Th otherwise

B. Adaptive Weighted-Shortest Path (AWSP)

where < tia , tih > and < Ta , Th > are the pairs of the i-th of the existing connections and of the incoming connection, respectively.

The shortest-path strategy may load with heavy traffic some specific links, while others remain unloaded, therefore leading to unbalanced link utilization. To promote a more uniform network utilization (load balancing), in the LLR approach, among a predefined set of effective candidate routes, the route with less occupied channels is selected. On the other side, LLR may excessively lengthen the chosen paths, causing bandwidth waste and higher blocking rate. A trade-off solution is the WSP algorithm, which distributes the traffic load equally and reduces the resource cost as much as possible, minimizing the product between the number of occupied wavelengths on the most loaded link along the route and the length of that route. In particular, to compare our proposed algorithm with a more adaptive approach, at the instant of a incoming connection request l we dynamically define k-shortest paths between the relative pair s/d, instead of using a fixed APS. We used Yen’s algorithm [15] to compute the route-set AP S = {lk } which contains the current K shortest paths. Then, between different paths, we select the k th path with the minimum Ulk × Clk , which is the route utilization Ulk times the associated path cost Clk . The route utilization is defined as:   νe Ulk = argmaxe∈lk (3) We Finally, if a route can be found, we update the link utilization descriptor accordingly and allocate the correspondent new wavelengths; otherwise we block the incoming connection. V. H OLDING -T IME -AWARE DR: A DETERMINISTIC APPROACH

We have seen that AWSP assigns weights to the network links according to the congestion level. Actually, the link congestion level changes during the holding time of an incoming connection whenever some existing connections depart and/or some new connections arrive. For example, a link, which is considered as highly congested at the instant of provisioning of a connection, may assume a different congestion state during its lifetime, due to the deallocation of capacity because of connection departures. In this study, since we are dealing with dynamic traffic, we assume that information on future connections may not be known in advance (this further information would lead to a static scheduling problem). Nevertheless, we could exploit at least the information about the departure events, which is simply retrievable from the knowledge of the connectionholding time. So, we could modify the link-cost assignment to capture the future degree of utilization of a given link. For our approach, the residual duration of existing connections is bounded by the holding time of the incoming connection. So, the connection residual time hi for an existing

(4)

A. Time-Dependent Link Utilization In order to follow step-by-step the changes in the congestion state of the link, we introduce the new symbols νe (Δτk ) and C(e, Δτk ), which express the values of link utilization νe and link cost C(e), respectively, in the interval of time Δτk . Let us expressly define Δτk first. According to values returned by Eq. 4, the hi ’s can be ordered so that hi ≤ hi+1 , i = 1, 2, ..., |L|. As a consequence, τ = {τ0 , ..., τ|L| } = {0, h1 , h2 , ..., h|L| } will indicate the departure events in the interval Th and Δτk = τk − τk−1 expresses the time interval between two departures. Link utilization νe (Δτk ) and associated cost C(e, Δτk ) will be updated according to the k-th connection departure. In other words, we have divided the interval Th into a series of time intervals Δτ which express the distance between two departures. B. RABBITdt approach Starting from the time-dependent link utilization definition, we can now define a new holding-time aware routing, which we call RABBIT, in its deterministic version RABBITdt (see [12]). In RABBITdt , the objective is to distribute the traffic along the link considering the new time-enhanced link utilization. The cost of generic link e during the time interval Th will be evaluated in according to the following scheme:  +∞ if νe = We C(e, Th ) := (5) Th + i:e∈li hi otherwise We Th So, if on a generic link e there is at least one free wavelength, then the associated cost is equal to the sum of the existing connections residual holding time (including also the incoming connection holding time) normalized to the product between the total number of wavelengths on e and the incoming connection holding time. Hence, the new cost function would not only minimize the current wavelength utilization, but also balance the traffic along the entire holding period Th of the incoming connection. The new approach is obtained by replacing, into the SP algorithm, the holding-time-aware cost assignment in Eqn.5 to the holding-time-agnostic link cost assignment in Eqn.1. VI. H OLDING -T IME -AWARE DR: A STATISTICAL APPROACH

The efficiency of the weight function in Eqn. 5 can still be improved recurring to a statistical approach. Fig. 1 shows the utilization in time of two links e1 and e2 which accommodate four connections, respectively. The sum of the different residual-holding times of the connections over the two links is the same, 120. According to RABBITdt ’s link cost assignment

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(Eqn. 5), at the arrival time Ta = 0 of an incoming connection with Th ≥ 90, the two links have the same cost. However, a deeper look at the statistical distribution of interarrival times can easily help us to discriminate the two links: e.g., suppose for simplicity that the connections arrival processes over the two links are characterized by a constant interarrival time equals to 60; then, routing the incoming connection along link e2 would lead to better load balancing, because at time 60 all the existing connection on e2 will have left the network. Due to its deterministic nature, RABBITdt can not take into

Fig. 1. In this example, the link-cost assignments of RABBITdt In Eqn. 5 do not allow discriminate the two alternatives, while a statistical approach succeeds instead.

account information contained in the probability distributions of connection-arrival rate and connection-holding time. Note that these statistical distributions can be easily retrieved from network-historical information. In the following analysis, let us suppose that connections arrivals to the networks are Poisson distributed with mean λ and that the connection holding time is exponentially distributed with mean μ. Assuming that the statistics of link loads are mutually independent [6], the arrival rate λe on a generic link e will be a fraction of the overall λ2 . Given λe , the stationary probability Πj that j wavelengths are busy on e would be given by: Πj =

(λe /μ)j j! We (λe /μ)y y=0 y!

(6)

However, the stationary analysis is inadequate in a dynamic network scenario where the time horizon is limited by the holding time Th of the incoming connection and many connections come and go from the network on average during a connection lifetime.

The time-dependent probabilities of a Markov chain are generally very complex to calculate even in basilar birthdeath processes (see, e.g., Fig. 2) and they can be obtained by solving a hard differential-equations system. Several methods have been proposed to obtain simplified and approximated transient measures of probabilities in Markovian models: among these methods, the uniformization or Jensen’s method [16] has been often applied in these last years. We will apply this technique in our study due to its numerical robustness and its simplicity of implementation. Let X = {X(t): t ≥ 0} be the homogeneous continuoustime Markov chain in Fig.2 with transactionrate matrix Q. Let qij be the (i,j)-th element of Q, and qi = i=j qij , be the exponential rate of state i. Let Z = {Zn : n=0,1,...} be a discrete time Markov chain with the same state space of Fig. 2, but with transition probability matrix P = I + Q / Λ, where Λ = maxi {qi }. ⎡ 1 − λΛe ⎢ μ ⎢ Λ P=⎢ . ⎣ .. 0

λe Λ

··· ···

··· ···

··· 0

We μ Λ

···

0 0 .. . 1−

⎤ ⎥ ⎥ ⎥ ⎦

We μ Λ

We consider now a Poisson process N = {N (t): t ≥ 0} with rate Λ independent of Z. If the time between transition for the chain Z is exponential with rate Λ, then the residence time spent in a visit to i is exponential with mean 1/qi . Since the total residence time in i is identical in both processes as well as the probability of moving from i to j given that a transition occurs to a state different from i, we may consider X and Z equivalent processes. Let Π(t) be a vector such that the j-th element is equal to the probability that X is in state j at time t, given an initial distribution of states. After n transitions, Z will be in state j with probability vj (n), where vj (n) is the j-th entry of the vector v(n) = v(0) Pn and v(0) is the initial state probability vector. Independent of the number of transition in the interval (0, t), we obtain:

A. Jensen’s transient probability We investigate a time-dependent model to estimate the future occupation of a generic link e when the duration of existing connections is given. To obtain this estimation we have to retrieve to the transient probabilities of being in a state i in a Markov chain.

1

0

λe Λ e − μ+λ Λ

Π(t) =

∞ 

e−Λt

n=0

(Λt)n v(n) n!

(7)

If we truncate Eqn. 7 for a given values of N , the error ε(N ) of any entry of the vector Π(t) is given by: ε(N ) ≤ 1 −

N  n=0

e−Λt

(Λt)n n!

(8)

Therefore, the computational cost is proportional to Λt [17]. Fig. 2. State transition diagram about connections’ arrivals/departures on a single link 2λ e

can be estimated by means of the network history information.

B. Mean transient probability Let us now evaluate the mean transient probability. If we consider a generic time interval τ , the mean values Π(τ )

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of each elements in the vector Eqn. 7 are given by:  ∞ −Λt (Λt)n n=0 e n! v(n)dt τ = Π(τ ) = τ  ∞ v(n) e−Λt (Λt)n dt n=0 n! τ = = τ   ∞ v(n) n )n−i 1 − e−Λτ i=0 (Λτ n=0 n! (n−i)! = τ

(9)

Eqn.9 can be easily computed by recursive approach. Details are skipped for the sake of brevity. C. Transient state expected value  Since j Πj (t) = 1 in each instant time t, and that the Πj (τ ) defines the mean transient probability of the j-th element during the time interval τ , we can express the transient expected value in τ as: We j=0 Πj (τ ) ∗ j Ev(τ ) = (10) We which represents the average amount of occupied wavelengths during the generic time interval τ . D. Transient probability during a connection holding-time Now we can define the expected-mean occupation of a generic link e, evaluated in a time interval Th starting from the arrival time Ta of an incoming connection. Without loss of generality, we focus on the example in Fig. 3, where we draw the time persistence of two existing connections r1 and r2 on a generic link e (We = 4), when a connection r3 is requested. Let us suppose that λe and μ are the mean value of the connection arrival rate on e and the connections holding time, respectively. When r3 arrives into the network with

vector vk (0), the final state probability vector Π(τk ) and the transient expected value Ev(τk ). The auxiliary probability matrix  Pk of the k-th time Ik +  Qk /  Λk , where  Ik and interval is defined as  Pk =   Qk are given by the Hadamard product of the following matrix:  Qk = Q • Hk

 Ik = I • Hk

(11)

where Hk operates a filter function. Hk is composed by 0 or 1: the (i, j)-th element will be set to 1 if and only if i and j are strictly greater than the minimum number of wavelengths certainly occupied in the time interval Δτk 3 , otherwise it will k  be set to 0. Let then q ij be the (i, j)-th element of Qk , and  k k qi = i=j q ij , the exponential rate out the state i. Moreover  Λk = maxi {qi k }. The auxiliary probability matrix  Pk is utilized in place of P to obtain a Markov chain trunked in correspondence of the minimum number of connections that are certainly supported by the link during a time interval Δτk : • Δτ1 (from time 0 to 10): the minimum number of occupied wavelengths is 3 (we assume that the incoming connection r3 is routed on e). Therefore ⎤ ⎡ 0 0 0 0 0 ⎢0 0 0 0 0 ⎥ ⎥ ⎢ ⎢ 0 0 ⎥  P1 = ⎢0 0 0 ⎥ λe ⎥ ⎢0 0 0 1 − λe   ⎦ ⎣ 0

•

0

0

Λ1 4μ  Λ1

Λ1 4μ  Λ1

1−

where  Λ1 = max{λe , 4μ}. Δτ2 (from time 10 to 20): the minimum number of occupied wavelengths is 2, since r1 leaves the network at time 10. Therefore ⎤ ⎡ 0 0 0 0 0 ⎢0 0 0 0 0 ⎥ ⎥ ⎢ λe λe ⎢ 0 ⎥   P2 = ⎢0 0 1 −  ⎥ Λ2 Λ2 ⎥ ⎢0 0 3μ λe e 1 − 3μ+λ ⎣    Λ2 Λ2 Λ2 ⎦ 4μ 4μ 0 0 0 1−   Λ2

Λ2

where  Λ2 = max{3μ + λe , 4μ}. In Tab. I we report the transient state probability of the initial and final vectors during the time intervals Δτ1 and Δτ2 . TABLE I I NITIAL AND FINAL VECTORS DURING Δτ1 AND Δτ2 Fig. 3.

Example of transient analysis on a generic link e

holding time Th = 20, r1 has to linger on link e other 10 time units, so its residual holding time will be set h1 = 10. As for r2 , note that, even if r2 has to be operated other 30 time units, in our analysis its residual holding time is bounded to h2 = 20 according to Eqn. 4. Thus, the holding time Th is split into two time intervals Δτ1 and Δτ2 (see Sec.V). Let us set τ0 = Ta . For each time interval Δτk we compute: an auxiliary transition probability matrix  Pk , the initial state probability

νe v1 (0) Π(τ1 ) v2 (0) Π(τ2 )

0 0 0 0 0

1 0 0 0 0

2 0 0 Π3 (10) Π2 (20)

3 1 Π3 (10) Π4 (10) Π3 (20)

4 0 Π4 (10) 0 Π4 (20)

Note that, since connection r1 leaves deterministically the network at time τ1 = 10, the initial vector v2 (0) of the interval 3 Wavelengths are certainly occupied, if the connection holding time of the connections routed over them is not still expired.

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Δτ2 is obtained by a cyclic-left-unitary shift of the previous final state vector Π(10). Then by the Eqn.7 and Eqn.10 we compute the final state probability Π(τk ) and the transient expected value Ev(τk ) for each time interval Δτk . Finally, we can define the close Future Link Utilization (FLU) as the temporal average of the expected value Ev(τk ) within the holding time of the incoming connection Th : |L| Ev(τk ) ∗ τk FLU = k=1 (12) Th E. RABBITst approach Our proposal aim at minimizing the blocking probability by considering both the current and the close-future occupation. The cost of link e during the interval Th will be evaluated in according to the following scheme:  +∞ if νe = We C(e, Th ) := (13) αF LU + β(νe + 1) otherwise where α ≥ 0 and β ≥ 0 are the weights associated with future link occupation and current link occupation, respectively. This new cost function considers the future-link usage (FLU) along the entire connection-holding time evaluated according the methodology previous paragraphs of this section. The new algorithm obtained by replacing the deterministic holding-time-aware link-cost assignment in Eqn.5 with the new link-cost assignment in Eqn.13 will be referred from now on as statistical RABBIT, i.e. RABBITst . VII. R ESULTS We now quantitatively evaluate the performance of our proposed RABBIT algorithm compared to the baseline SP and AWSP algorithms. We simulate a dynamic network environment with the assumptions that the connection-arrival process on the network is Poisson and the connection-holding time follows a negative exponential distribution. Average connectionholding time is normalized to unity. For the illustrative results shown here, in every experiment, 106 connection requests are simulated. All the plotted values have a 95% confidence interval not larger than 5% of the plotted value. Requests are uniformly distributed among all node pairs; the example network topology with 8 wavelengths per fiber is shown in Fig. 4. We employ three performance metrics: Blocking Prob19 1 11

20

15

6 2

9

7

3

12

21

16 22

4

10 8

14 18

Fig. 4.

23

17

13 5

24

A carrier’s US nationwide backbone network topology.

ability (BP), Total Channel consumption (TCh) and standard deviation of link utilization. In the following three versions of RABBIT are evaluated: the deterministic approach RABBITdt ,

and two versions of RABBITst : RABBITst {α = 1, β = 0}, which just considers the FLU as in Eqn. 12 as link-weight, and RABBITst {α = 0.5, β = 0.5}. A. Blocking Probability The BP indicates the ratio of the blocked connections over the offered connections to the network. Tab. II shows that all versions of RABBIT block less connections than SP and AWSP. We also observe that our deterministic approach, RABBITdt , results more efficient than the pure statistical approach, RABBITst {α = 1, β = 0}. This because RABBIT based only on FLU distributes excessively the traffic network by an excess of load balancing. However, RABBITst is able to outperform RABBITdt if we include in its weight function also the current link occupation (β = 0) thus limiting the excess of load balancing. From extensive simulations, we identified the values {α = 0.5, β = 0.5} as an effective compromise for our weight function. For sake of conciseness, we report only the case {α = 0.5, β = 0.5}. With our most efficient TABLE II B LOCKING P ROBABILITY OF THE VARIOUS ROUTING ALGORITHMS AR

SP

AW SP

RABBITdt

70 80 90 100 110 120 130 140

0.0122 0.0796 0.3459 1.1007 2.6349 5.0935 8.1840 11.632

0.011 0.0712 0.3004 0.9625 2.3264 4.6315 7.6419 11.134

0.011 0.07 0.284 0.8872 2.0504 4.105 6.8627 10.251

RABBITst α = 1, β = 0 0.011 0.0704 0.2865 0.8912 2.116 4.2387 7.0597 10.501

RABBITst α = 0.5, β = 0.5 0.0108 0.0687 0.2789 . 0.865 2.0499 4.0609 6.7801 10.124

approach, RABBITst {α = 0.5, β = 0.5}, we obtained a gain percentage on BP over SP and AWSP that is up to 22% and 13%, respectively. B. Total Channel consumption, TCh TCh is the overall number of channels needed to support the offered traffic multiplied by the time interval these channels are actually used. Fig. 5 shows the RABBIT approaches tends to require more channels than SP and AWSP, since they promote load balancing, thus longer routes, in the network. In RABBITst {α = 1, β = 0}, TCh is significantly higher that for all the other approaches, especially for low loads. This large resource usage is due to an excessive recourse to load balancing, that explains why BP is higher with respect the the RABBITst {α = 0.5, β = 0.5} and even with respect to RABBITdt ; then TCh tends to decrease for increasing loads, since for higher loads, the dispersion of the routes is intrinsically limited by resources shortage. For RABBITst {α = 0.5, β = 0.5}, TCh, even if it is still higher, is closer to the other methods and stays constant for low and medium arrival rate: the weight function {α = 0.5, β = 0.5} has allowed to decrease the BP (Tab.II), without inducing excessive load balancing and requiring a reasonable resources utilization. As for RABBITdt , SP and AWSP, TCh initially tends to increase for increasing load, since for these approaches, the congestion increases the length of the routes (shortest paths are always more congested, so

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longer path are explored for increasing loads). Finally, for very high loads, where BP value goes over 4÷5%, the high number of blocked connection induces a reduction of the TCh for all the approaches.

Fig. 5.

Total channel consumption

C. Standard deviation of link utilization The main advantage of RABBIT stands in its ability to promote a proper distribution of the traffic load in the network. We focus here on the standard deviation of the link utilization (Fig. 6) to observe how the traffic loads is distributed among the links. We can observe that RABBITst returns the lowest standard deviation of link utilization, but this also implies an excessive load balancing in the version {α = 1, β = 0}, which motivates its high TCh consumption and the inefficiency with respect to RABBITdt . A still low, but more reasonable standard deviation characterizes RABBITst {α = 0.5, β = 0.5} (the link cost assignment is given by link current and link future utilization equally weighted), which has returned the best performance in BP.

Fig. 6.

Standard deviation of link utilization

VIII. C ONCLUSION In this paper we have proposed to exploit the knowledge of connection-holding time to develop a novel intelligent approach for adaptive load balancing. We have provided, for the first time, a mathematical analysis for the estimation of future link utilization given the knowledge of connectionholding time. Our holding-time-aware, dynamic, provisioning algorithm decreases network congestion and blocking probability thanks to a wiser distribution of the load in the available capacity of the network. We have observed significant savings in blocking probability by employing our new approach, called RABBIT, as opposed to others traditional approaches. The improvement in blocking probability is found to be up to 20% for a US nationwide network with typical parameters. The proposed method is applicable to other contexts as well, such as MPLS networks, for bandwidth-guaranteed connections. R EFERENCES [1] M. Tornatore, C. (Sam) Ou, J. Zhang, A. Pattavina, and B. Mukherjee, “PHOTO: an efficient shared-path protection strategy based on connection-holding-time awareness,” IEEE/OSA Journal on Lightwave Technology, vol. 23, pp. 3138–3146, October 2005. [2] M. Gagnaire, M. Koubaa, and N. Puech, “Network dimensioning under scheduled and random lightpath demands in all-optical WDM networks,” IEEE Journal on Selected Areas in Comm., vol. 25, no. 9, December 2007. [3] H. Zang, J. Jue, L. Sahasrabuddhe, R. Ramamurthy, and B. Mukherjee, “Dynamic lightpath establishment in wavelength routed WDM networks,” IEEE Comm. Magazine, vol. 39, no. 9, pp. 100–108, Sep 2001. [4] A. Iselt, A. Kirstdter, and R. Chahine, “The role of ASON and GMPLS for the bandwidth trading market,” in Proc. 1st International Conference on E-business and Telecomm. Networks (ICETE2004), August 2004. [5] H. Zang, J. Jue, and B. Mukherjeee, “A review of routing and wavelength assignment approaches for wavelength-routed optical WDM networks,” Optical Networks Magazine, January 2000. [6] A. Birman, “Computing approximate blocking probabilities for a class of all-optical networks,” IEEE Journal of Selected Areas in Communications, vol. 14, no. 5, pp. 852–857, 1996. [7] K.-M. Chan and T.P.Yum, “Analysis of least congested path routing in WDM lightwave networks,” INFOCOM ’94, vol. 2, pp. 962–969, 1994. [8] H.-C. Lin, S.-W. Wang, and C.-P. Tsai, “Traffic intensity based fixedalternate routing in all-optical WDM networks,” in Proc. of ICC ’06, vol. 6, pp. 2439–2446, June 2006. [9] X. Chu and J. Liu, “DLCR: a new adaptive routing scheme in WDM mesh networks,” in Proc. of ICC ’05, vol. 3, pp. 1797–1801, May 2005. [10] Xi Yang and B. Ramamurthy, “Dynamic routing in translucent wdm optical networks: the intradomain case,” Journal of Lightwave Technology, vol. 23, no. 3, pp. 955–971, March 2005. [11] H.-C. Lin, S.-W. Wang, and M.-L. Hung, “Finding routing paths for alternate routing in all-optical wdm networks,” Journal of Lightwave Technology, vol. 26, no. 11, pp. 1432–1444, June 2008. [12] D. Lucerna, A. Baruffaldi, M. Tornatore, and A. Pattavina, “On the efficiency of dynamic routing of connections with known duration,” accepted in ICC’09, June 2009. [13] C.-F. Hsu, T.-L. Liu, and N.-F. Huang, “Performance of adaptive routing strategies in wavelength-routed networks,” IEEE International Conference on. Performance, Computing, and Communications ’01, pp. 163–170, Apr 2001. [14] Y. Luo and N. Ansari, “A computational model for estimating blocking probabilities of multifiber WDM optical networks,” Communications Letters, IEEE, vol. 8, no. 1, pp. 60–62, Jan. 2004. [15] J. Y. Yen, “Finding K shortest loopless paths in a network,” Management Science, pp. 712–716, 1971. [16] N.M. Bhide, K.M. Sivalingam, and Fabry-Asztalos, “Markov chains as an aid in the study of markov process,” Skandinavsk Aktuarietidskrift, 1953. [17] E. de Souza e Silva and H. Gail, “The uniformization method in performability analysis,” International workshop on performability modelling of computer and communication systems, 1993.

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