A Markov-based Reservation Algorithm for Wavelength. Assignment in All-optical Networks

Transcription

1 A Markov-based Reservation Algorithm for Wavelength Assignment in All-optical Networks Wenhao Lin, Richard S. Wolff, Brendan Mumey Montana State University, Bozeman, MT (Phone) ( Abstract: Most routing and wavelength assignment (RWA) algorithms for all-optical networks do not consider the potential problem of wavelength reservation confliction, which can happen even if the network is only lightly loaded. In this paper, we propose a new reservation protocol MBR (Markov-based backward reservation) based on Markov modeling of the optical links. Our simulations show that this new protocol can improve network performance as measured by decreased blocking probability by resolving wavelength reservation confliction. Keywords: Optical communication, Networks, Communication system routing, Algorithms, Access protocols, Markov processes, Communication system traffic. 1

2 1. Introduction In recent years, the demand for bandwidth has increased exponentially. Dense wavelength division multiplexing (DWDM) offers enormous bandwidth expansion, and is a promising technology that can meet such demand. Effective utilization of DWDM requires methodologies to assign paths and wavelengths dynamically to meet changing end user connectivity requirements. For reasons of cost effectiveness and operational efficiency, all-optical routing (e.g., no wavelength conversion) is ideal for WDM networks. The routing and wavelength-assignment problem (RWA problem) for all-optical network has received intensive interest in the literature [1-12, 14-17]. The constraints in the RWA problem for all-optical networks include wavelength continuity, physical impairments, and traffic engineering considerations. In this paper, we assume no wavelength conversion; all physical link and network components are ideal; and that user initiated connection requests arrive randomly. The major consideration of the proposed algorithm is to decrease the network blocking probability for a fixed number of links and wavelengths. The RWA problem is usually addressed by a two step process, first choose a path, and then choose a free wavelength on the selected path. Routing algorithms are used to choose a path for a connection request, while wavelength assignment algorithms are used to pick a free wavelength on the selected path. Routing algorithms can be divided into two classes: static and adaptive. In static routing (Fixed routing [1], Fixed-Alternative routing [1], et.al.) algorithms, one or several paths are pre-computed for each pair of source-destination nodes. Static routing can decrease the time of provisioning, however, static approaches can t respond to dynamic traffic changes in network. Adaptive routing algorithms (such as, Shortest-Path [1], Shortest-Cost-Path [2], Least-Congested-Path [3]) usually use Dijkstra s algorithm to compute a minimal cost path from source to destination. The link cost function is critical for such algorithms. There are also several wavelength assignment algorithms in the literature. In the Random algorithm [1], one free wavelength is randomly selected from among the unused wavelengths available on the source-destination path. In the First-Fit algorithm [1], the free wavelength with the smallest index is selected. In the Most-Used algorithm [1], the free wavelength which is used most often in the network is selected. In the Least-Used algorithm [1], the free wavelength which is used least in the network is selected. The Min-Product [4], Least-loaded [5], Max-Sum [6], Relative-Capacity-Loss [7] algorithms have been proposed for multi-fiber networks. The performances of different wavelength assignment algorithms were compared in [2]. It was reported that when the network is lightly loaded, the Most-Used algorithm is the best and the Least-Used algorithm is the worst. When the network is heavily loaded, there are no big differences between the wavelength assignment algorithms. We notice that in the Most-Used algorithm, free wavelengths in network are less segmentated than with other algorithms. Hence connections with large hop count have a larger chance of being accepted. Two routing assignment architectures can be used in an all-optical network: centralized and distributed. In a centralized architecture, a controlling node monitors the network state and controls all resource allocation. Upon receiving a connection request, an edge node sends a message to the controlling node. The controlling node executes the routing algorithm and the wavelength assignment algorithm. Upon deciding a path and free wavelength, the controlling node will reserve resources on all nodes along the path. This architecture poses problems such as, 2

3 performance bottlenecks, single point of failure and scalability. In a distributed control architecture, information about network state is broadcast periodically and each edge node can compute the path upon receipt of a connection request. Distributed control is more scalable and robust. The discussion in this paper will be on the use of a distributed control architecture. Upon receiving a connection request, an edge node first executes a routing algorithm to compute a path. It then starts a path and wavelength reservation protocol. The wavelength assignment algorithm can be executed by either the destination or source node to pick a free wavelength. Both approaches have been previously discussed [1]. There are two different wavelength reservation protocols: Forward Reservation and Backward Reservation. In Forward Reservation, a resv message (specifying path/ wavelength) is sent from source to destination on the specified path. When an intermediate node receives resv, it executes the resource reservation operation. When the destination node receives resv, a conf messaged is sent upstream. In Backward Reservation, a prob message (specifying path) is sent from source to destination. The message collects wavelength usage information on the relevant optical links as it propagates on the specified path. After receiving the prob message, the destination node picks one free wavelength and a resv message is sent upstream to finish the actual resource reservation. Figure 1 and Figure 2 illustrate the Forward Reservation and Backward Reservation processes respectively. resv prob resv prob cnf cnf resv resv Figure 1, Forward Reservation Figure 2, Backward Reservation In [8], modifications of Forward and Backward Reservation are defined. In its Forward Reservation, the source node sends out a resv message (only specifying path). Upon receiving resv, each intermediate node locks all free wavelengths on the chosen output link that are also free on all preceding links upstream, and sends a release message upstream to release all wavelengths that are busy on the output link but are free on all links upstream (these wavelengths were locked by upstream nodes for this connection request). In the Backward Reservation protocol, the same mechanism is used. It was reported that Backward Reservation is more efficient than Forward Reservation [8], because the Forward Reservation protocol tends to lock more free wavelengths unnecessarily. As a result, other connection requests arriving later are more likely to be blocked. The performance of RWA algorithms is generally measured in terms of connection request blocking probability. But most algorithms in the literature try to decrease blocking probability caused by lack of resources (free wavelengths), without considering blocking caused by reservation confliction. When paths of several connection requests share a common link, confliction will happen if more than one destination node picks the same free wavelength. 3

4 Figure 3 illustrates a confliction scenario in Backward Reservation. We assume in this simple network, Shortest-Path and First-Fit algorithms are used for routing and wavelength assignment C A B Figure 3, an example of reservation confliction D Let s assume all wavelengths on all links are free initially. The prob message of connection request C1 (A->B->C) arrives at C first. Node C will choose λ 0 according to the assumption (First-Fit). Before C sends a resv message upstream, the prob message of connection request C2 (A->B->D) arrives at B, which will still find that all wavelengths on link A->B and B->D are free. When the prob message associated with C2 arrives at D, node D will also choose λ 0. One connection request will be blocked depending on whose resv message arrives at B first. Such reservation confliction can always happen no matter which routing algorithm, wavelength assignment algorithm and reservation protocol are used in a network. This can especially be a problem when the traffic is bursty. When the network is not heavily loaded, reservation confliction is the major cause of blocking. In Forward Reservation the wavelength is decided by the source node (usually based on some broadcast information). When a resv message arrives at a node, it is possible that the selected wavelength is already being used by another connection. In Backward Reservation, the wavelength is chosen by the destination node based on information in the prob message, and the time window for reservation confliction is smaller than in Forward Reservation. There are not many algorithms reported in the literature that take into account reservation confliction. In [9], a new model was proposed to predict the blocking probability considering both the lack of free wavelengths and reservation confliction. Their assumption (random wavelength selection) and computation method (fixed-point) make it impossible to use this model in a real time routing algorithm. In [10] parallel reservation was used to decrease the reservation confliction probability. The source node uses Forward Reservation, but will send resv messages to all nodes on the selected path at the same time. In optical burst networks, Delayed-Reservation, Delayed- Reservation-without-Void-Filling and Delayed-Reservation-with-Void-Filling algorithms are used to solve confliction under the assumption that a switch node knows the start time and duration of each burst [11]. In [12], a wavelength priority table is created in each node. Each time a connection is accepted using a free wavelength, the priority of this wavelength is increased; resulting in connection requests from different nodes tending to use different wavelengths. Although the reservation confliction probability can be decreased, blocking probability due to a lack of free wavelengths will increase, because it will be more difficult to find a free wavelength for a large hop-count connection. In this paper, we propose a new reservation protocol to decrease the reservation confliction probability. The new protocol is a modification of Backward Reservation. Our basic idea is to 4

5 guess the wavelengths that other competing connection requests will use. Then when a destination node chooses a free wavelength, it can avoid selecting wavelengths possibly chosen by other connection requests. The paper is organized as follows. In section 2.1, we first explain the network model and related assumptions. In section 2.2, we explain how to build a continuous time Markov chain to model and predict wavelength utilization behavior of an optical link. In section 2.3, we explain the data structure created in each node to help a connection request detect interfering requests. In section 2.4, we describe the functions used in the new reservation protocol. Simulation results and comparisons with other reservation protocols are given in section 2.5. In section 2.6, we propose two heuristics which can be used with the new reservation protocol to improve its performance. Conclusions are given in section Markov-based Backward Reservation 2.1 Network model An optical network can be presented as a directed or undirected graph. Figure 4 illustrates the topology of the NSF network. Figure 4, topology of NSF network Nodes in the graph are optical switches and are connected by multi-wavelength optical links. We assume in this paper that optical fiber is directed, so each optical link between two adjacent nodes A, B includes two fibers (A->B and B->A). However, we will still use optical link to refer to optical fiber afterwards for simplicity. There are N indexed wavelengths on each fiber λ 0,..., λ N 1 (different fibers can have different number of wavelengths, our assumption is used to simplify the model and implementation ). Workstations attached to the switches send out connection requests randomly according to a Poisson process with arrival rateγ, using source routing. The Duration of connection is an exponential distribution with mean µ. Each node in the network will periodically broadcast wavelength usage information of adjacent links every T seconds. Source nodes apply the Shortest-Path routing algorithm and use the Backward Reservation protocol to create a connection. The source node sends out a prob message to destination node. prob message includes the following fields. < source id, destination id, path_info, wave_map, connection id > path_info is just the ordered list of nodes on the selected path and wave_map is an array indicating the availability/unavailability of each wavelength. After receiving a prob message, an intermediate node will update the wave_map field by marking busy wavelengths on the input link 5

6 and chosen output link as Busy. Upon receiving a prob message, the destination node picks a free wavelength (by First-Fit, Random, et.al.) based on information in the wave_map field, and sends a resv message upstream. All nodes along the path will lock the wavelength specified in the resv message. The resv message includes the following fields. < connection id, selected wavelength > When a connection finishes, the source node sends a release message downstream to release the used wavelength. Upon receiving the release message, the destination node sends a release conf message upstream to the source. 2.2 Markov modeling of an optical link The proposed reservation protocol (MBR) works as follows. First, the source node calls a routing algorithm (We use Shortest-Path, but other routing algorithms can also be used with MBR) to compute a path from source to destination. The source node then starts the MBR protocol by sending a prob message to the destination node. The destination node picks a free wavelength according to the First-Fit algorithm. To decrease the probability of reservation confliction, First-Fit used in this paper will avoid selecting free wavelengths that are possibly used by other competing connection requests. For a given connection request, intermediate nodes can guess the wavelength assignment decision of the destination node if they know wavelength usage information on links along the path of this connection request. If the guess result is stored in intermediates nodes, then all competing connection requests arriving later could avoid selecting this wavelength. However, traffic on all links is dynamic. Although each node broadcasts wavelength usage information every T seconds, the broadcast information is not necessarily correct during the period between two successive broadcast moments. Instead of using the broadcast information directly, we build a continuous time Markov chain (C-T Markov chain) for each link and use the C-T Markov chain to predict wavelength usage at any arbitrary time. Each C-T Markov chain corresponds to one link in the network. There are two ways to define its state space. One method is to use a bit array of length N to present the state of each wavelength (0: Free, 1: Busy). Each possible combination of the bit array defines one state. The size of the N state space is 2. Such a state space can be used with any wavelength assignment algorithm. But it can only be used in a network with few wavelengths because the size of the state space increases exponentially with the number of wavelengths. Another method can be used if the First-Fit wavelength assignment algorithm is used. In First-Fit, the destination node will always use the free wavelength with the smallest index. So, if the destination node chooses λ i, then λ 0,, λ i 1 all are busy. In this situation, each state of the C-T Markov chain can be defined as the number of used wavelengths on a link (state i means i wavelengths are used). The size of such a state space is N+1. We use this definition of state space in this paper. So in MBR, the destination node will use the First-Fit wavelength assignment algorithm. Figure 5 illustrates the state space of a chain. 6

7 α 0 α N N-1 N β 0 β N 1 Figure 5, the C-T Markov chain of a link with N wavelengths. α, β are transition rates i i The parameters of a chain (transition rate between states) are obtained by network monitoring. For an arbitrary link A->B, we call A the controlling node of the link and the controlling node will do the monitoring. For each link in the network, during a monitoring period, its controlling node will record how many times transition (k-> k-1) and transition (k-> k+1) occur, and the sum of transition times that these transitions have used. The transition time of transition (k-> k+1) and (k-> k-1) is defined as the interval from the moment when the chain enters state k until the moment when the chain enters state k+1 or state k-1 respectively. We denote the transition (i-> j) as Tr(i,j) and define count (i, j) as the number of times that Tr(i, j) happened in a monitoring period. Let sum (i, j) denote the summation of transition time of Tr(i,j). The transition rate R(i, j) of Tr (i, j) is defined as follows. count(, i i+ 1) + count(,-1) i i count(, i j) Ri (, j) = * sum(,-1) i i + sum(, i i+ 1) count(, i i+ 1) + count(,-1) i i, while i-j = 1; = 0, otherwise. (1) R(i,j) is the transition rate from state i to state j. The R(i,j)s totally describe the dynamics of an optical link. The controlling node of an optical link will broadcast these parameters periodically or only when changes happen depending on network control policy. Other nodes will store these parameters in their database as the properties of an optical link. We define the transition rate R(i) of a state i as: R( i) count( i, i+ 1) + count( i, i-1) = sum ( i, i-1) + sum ( i, i+ 1) (2) This is the transition rate of state i to other states (leaving rate). Every T seconds, nodes will broadcast wavelength usage of links. Each node then knows the exact wavelength allocation on all links at time 0T, 1T,.., KT,. At an arbitrary time KT+ µ between KT and (K+1)T, a node can use the C-T Markov chain to guess the free wavelength distribution on links which are not adjacent to itself. The transient analysis of a C-T Markov chain is given in [13]. Given that a chain is at state i at time 0, then the probability Pi,j that the chain will be at state j at time t is: ( vt) n P = e ( P, ) (3) n vt i, j i j n = 0 n! Formula (3) is called the Uniformization method in C-T Markov chain transient analysis. A C-T 7

8 Markov chain can be looked at as a Poisson process plus a discrete Markov chain. However, the Poisson process has different rates when the chain is at different states. The Uniformization method transforms the original chain into an equivalent chain with a uniform rate. The uniform rate ν is defined as ν = max { R(i) i =0, 1,, N} (4) The transition probability P, (the probability the embedded discrete chain will transit to state j i j when it leaves state i) is defined as P, = R(i,j)/ ν ; while i-j =1; i j = (ν -R(i,j))/ν ; while i = j; = 0 ; else. (5) Using formula (3), we can compute the state distribution (the set of probabilities that the chain will be at each state) of a chain at time KT+ µ, given that it is at state i at time KT. Figure 6 illustrates an example. The traffic of a link in this example is 2 Erlangs. Given that the chain is at state 5 at t=0, we want to predict which state the chain will be at after 5, 50 and 100 seconds. From the figure, we see that the chain will converge to its stable state given enough time (2 wavelengths are used in this example). Figure 6, prediction of the behavior of a C-T Markov chain Based on the First-Fit assumption in MBR, the probability that wavelengthλ is free on link l at time KT+ t, given i wavelengths are used at time KT, is computed by P ( λ free on l) = (6) k j = k P i, j j = 0, where P i, j is defined in (3). Using the assumption that usage of wavelengths on different links are independent, we can k 8

9 compute the probability that a wavelength λ is free on path R ( λ is free on all links of R) as k k P ( λk free on R) = l R P ( λ free on l). (7) k While the independence assumption is not completely true in our routing algorithm whenever two routing paths share some edges, we will use this calculation as an approximation for the probability that a wavelength is free and introduce an element of randomization to mitigate some of the dependence in routing decisions made for conflicting paths. 2.3 Detecting interfering connection requests When a connection request arrives at a node, other requests may have already arrived at the node earlier, but the node still doesn t know their wavelength assignment decision (because it has not received the resv messages from the destination nodes). We call these requests Ongoing requests. If these earlier arriving requests share one link with the request, we call them interfering requests. The Interfering Set of a connection request includes all interfering requests which the request will find when it is propagated from source to destination. Notice the Interfering Set of a request only includes requests arriving earlier than this request. Figure 7 illustrates request C4 and its interfering requests C1, C2 and C3. F C1 C2 A B C D E C3 C4 G Figure 7, example of interfering connection requests A table (Ongoing table) is created at each node in the network. When a connection request arrives at a node, the node will store in the Ongoing table information about the connection request. A record of the Ongoing table includes the following fields. <source id, destination id, connection id, pri hop id, next hop id, time, wheel_game> Interfering connection requests are those which have the same pri hop id as the connection request under consideration. By checking the Ongoing table in each node, a connection request will find its interfering connections. We will explain field wheel-game later. 2.4 Markov-based Backward Reservation protocol (MBR) The new reservation protocol is a modification to the Backward Reservation protocol. Only 9

10 small changes to the prob message are needed. When the prob message of a connection request (A->E, in figure 7) arrives at an intermediate node (C), the node first updates the wave_map field in the message, marking busy wavelengths on input and output links as Busy. If the node finds no interfering requests in its Ongoing table, the probabilities that the connection request will use different wavelengths according to First-Fit are computed first (this means we try to guess the wavelength assignment decision by the destination node of the connection request). Code 1 in the appendix illustrates the pseudo code to compute those probabilities. The input parameter wave_flag is only a copy of the wave_map field in the prob message in the case that no interfering requests exist. When another connection request arrives at node C later, we also need to compute these probabilities. The computation has to be conditioned on possible wavelength assignment by this earlier connection request. To keep the computation tractable, we use a heuristic method. After computing the probabilities that the connection will use different wavelengths, we call function wheel_of_fortune, listed as Code 2 in the appendix. Node C then stores a record about the request in the Ongoing table, setting field wheel_game to the result of function wheel_of_fortune, which is the wavelength node C guesses that the destination of this connection will select. At last, the prob message is sent to downstream nodes. If the request arrives at node C and finds interfering requests in the Ongoing table, it will call function solve_interference, listed as code 3 in the appendix. In function solve_interference we don t mark the wavelength possibly to be used by interfering requests with the same next hop as Guessed Busy, because the chance of solving their interference still exists in downstream nodes, and the downstream nodes should have more precise knowledge of the wavelength assignment of those requests. Function solve_interference will update the filed wave_map in prob mssage to solve interference from competing connection requests. The updated prob message will be sent to downstreams. The destination node will find three sets of wavelengths in field wave_map of the received prob message, Free, Busy and Guessed Busy. The destination node runs the First-Fit algorithm on the set of Free wavelengths. If no Free wavelength exists, a release message is sent upstream to the source node. 2.5 Simulation experiment To evaluate this approach we have run MBR on the NSF network. The benchmark algorithms include Backward Reservation with First-Fit (FF), and Backward Reservation with Random assignment (RND, which will randomly select one wavelength from the first several free wavelengths). We use these two algorithms because they are easy to implement and have comparatively good performance. In our experiments, we assume no wavelength converters exist in the network. Workstations attached to nodes randomly send connection requests. The source node uses the Shortest-Path routing algorithm to select the path. Each workstation introduces 1 Erlang of traffic to the network. We increase the traffic load by adding more workstations to the network. To make the simulation result more obvious, each destination node will delay 6 seconds upon receiving a prob message before sending resv message. In the first experiment, each optical link has 10 wavelengths. As we increase the load from 1 to 15 Erlangs, the network is still lightly loaded. Almost all blocking is caused by reservation confliction. Figure 9 provides a comparison of the results. 10

11 Figure 9, the first comparison experiment result (each link has 10 wavelengths) From these results, we see that MBR has very good performance in decreasing reservation confliction when the network is lightly loaded. In the second experiment, each optical link has 6 wavelengths. As we increase the traffic load, the blocking can be caused by reservation confliction and lack of free wavelengths. Figure 10 shows the comparison results. Figure 10, the second comparison experiment result (each link has 6 wavelengths) From these results, we see that MBR still outperforms the other two protocols when the traffic load is high. However, the improvement becomes marginal when the network is heavily loaded. This is consistent with the conclusion in [2]. In the last experiment, we fix the traffic load at 15 Erlangs, and increase the number of wavelengths per fiber. The result is shown in Figure

12 Figure 11, fixed traffic and varying number of wavelengths From our simulations, we can draw the following conclusion: the improvement obtained with the proposed protocol is most obvious when the network is lightly loaded. In this situation, the principle cause of blocking is reservation confliction, which is a major consideration in the design of MBR. When the network is heavily loaded, the performance difference of all algorithms becomes marginal. A possible routing scheme for an all-optical network could be that when the network is lightly loaded, apply effective algorithms such as MBR to decrease the blocking probability; and when the network is heavily loaded, apply simple algorithms such as backward reservation + First-Fit. 2.6 Heuristics to improve the performance of MBR We can use the following heuristics to improve the performance of MBR. (H1): In MBR, the size of the state space of each chain is Ο (N). Notice that the states which a chain can visit during a monitoring period may be only a subset of the whole state space. We can further decrease the size of the state space by constructing a chain including only those visited states. This would be very helpful in a DWDM network with many wavelengths. (H2): Under the wavelength continuity constraint, the free wavelength picked for a connection request according to First-Fit is decided by links with high traffic load. So, we need to model only these busy links using a Markov chain. 3. Conclusion In this paper, we describe a wavelength reservation protocol that uses a continuous time Markov chain to predict the future behavior of wavelength assignment on optical links. Based on such knowledge, we designed new functions to solve the confliction problem among competing connection requests. Compared to other reservation protocols, the MBR protocol has two advantages: first, unlike the protocols described in [8], in the proposed MBR algorithm, switch nodes will only lock one selected wavelength after receiving resv message; second, MBR explicitly use an algorithm to solve the confliction among interfering connections. Our simulation 12

13 results show the improvement in decreasing blocking probability. The disadvantages are that: the parameters of the C-T Markov chain need to be broadcast, which is not supported by deployed networks and standards and that the proposed protocol is computation intensive. This will put a burden on the switch nodes and the control system when the network is heavily loaded. References [1] C.Murthy and M.Gurusamy, WDM optical networks, concepts, design and algorithms, Prentice Hall, 2001, pp , [2] H.Zang, J.P.Jue and B.Mukheriee, A review of routing and wavelength assignment approaches for wavelength-routed optical WDM networks, Optical Networks Magazine, vol. 1, no. 1, Jan [3] K,Chan and T.P.Yum, Analysis of least congested path routing in WDM lightwave networks, Proc. IEEE INFOCOM 94, vol.2, pp , Toronto, Canada, Apr [4] G.Jeong and E.Ayanoglu, Comparison of wavelength-interchanging and wavelength-selective cross-connects in multiwavelength all-optical networks, Proc. IEEE INFOCOM 96, vol.1, pp , San Francisco, CA, Mar [5] E.Karasan and E.Ayanoglu, Effects of wavelength routing and selection algorithm on wavelength conversion gain in WDM optical networks, IEEE/ACM Trans. Networking., vol.6, no.2, pp , Apr [6] S.Subramaniam and R.A.Barry, Wavelength assignment in fixed routing WDM networks, Proc. ICC 97, vol.1, pp , Montreal, Canada, Jun [7] X.Zhang and C.Qiao, Wavelength assignment for dynamical traffic in Multi-Fiber WDM networks, Proc. 7 th International Conference on Computer Communications and Networks, pp , Lafayette, LA, Oct [8] F.Fang, X.Zheng and H.Zhang, Performance study of distributed wavelength reservation protocols within both single and Multi-Fiber WDM networks, Photonic Network Communications, vol. 6, no. 2, Sep [9] K.Lu, G.Xiao and I.Chlamtac, Analysis of blocking probabilities for distributed lightpath establishment in WDM optical networks, IEEE/ACM Trans. Networking., vol.13, no.1, Feb, [10] R. Ramaswami and A. Segall, Distributed network control for optical networks, IEEE/ACM Trans. Networking., vol. 5, no. 6, pp , Dec [11] J.Teng, G.N.Rouskas, A comparison of the JIT, JET, and Horizon wavelength reservation schemes on a single OBS node, Proc. The First International Workshop on Optical Burst Switching, Dallas, Texas, Oct, [12] X.Wang, H.Morikawa, and T.Aoyama, Priority-based wavelength assignment algorithm for burst switched photonic networks, IEICE Trans. Communications., vol.e86-b, no.5, pp , May [13] H.C.Tijms, A first course in stochastic models, Wiley publisher, 2003, pp [14] H.Zang, J. P. Jue, L. Sahasrabuddhe, R. Ramamurthy and B.Mukherjee, Dynamic lightpath establishment in wavelength-routed WDM networks, IEEE Communications Magazine, pp , Sep [15] H. Zang, L. Sahasrabuddhe, J. P. Jue, S. Ramamurthy, and B. Mukherjee, Connection management for wavelength-routed WDM networks, Proc. IEEE GLOBECOM 99, vol. 2, 13

14 pp , Rio de Janeiro, Brazil, Dec [16] J.S. Choi, N. Golmie, F. Lapeyrere, F. Mouveaux and D. Su, Classification of routing and wavelength assignment schemes in DWDM networks: static case, 7th International Conference on Optical Communications and Networks, Jan [17] L.Li and A. Somani Dynamic wavelength routing using congestion and neighborhood information, IEEE/ACM Trans. Networking., vol.7, no.5, pp , Oct Wenhao Lin received a B.S. degree in computer science from Huazhong University of Science and Technology, Wuhan, China, in 1995 and an M.S. degree in computer science from Montana State University, Bozeman, Montana, in He is currently working toward the Ph.D. degree in electrical engineering at Montana State University Richard S. Wolff is the Gilhousen Chair in Telecommunications and professor of Electrical Engineering at Montana State University, Bozeman. His research interests are in novel applications of emerging technologies in telecommunications systems. Prior to joining MSU, he spent 25 years in telecommunications research at Telcordia, Bellcore and Bell Labs, and taught physics at Columbia University. He earned a BS in Engineering Physics at the University of California, Berkeley and a Ph. D. in Physics at Columbia University. He has published over 50 papers, has been awarded two patents, and is a senior member of the IEEE. Brendan Mumey is an associate professor of Computer Science at Montana State University, Bozeman. His research interests are in algorithm design and computational biology. He earned a B.Sc. in Mathematics at the University of Alberta, a M.Sc. in Computer Science at the University of British Columbia and a Ph.D. in Computer Science at the University of Washington. He has published over 20 papers and has been awarded one patent. Appendix: pseudo codes Code 1, function to guess the probabilities that a connection will use different wavelengths function guess_wavlength_selection_probs ( prob message, wave_flag) // input: // prob : The received prob message, // wave_flag: array indicates which wavelengths are busy. //output: // guessed_probs: array stores the guessed probabilities, each element // guessed_probs [i] is the probability that the connection request will // select wavelengthλ i. R = the path from the node to destination; for each wavelength λ marked as Busy in wave_flag i guessed_probs [i] = 0; // can t choose these wavelengths because they are busy 14

15 end; S = 1; F = { λ i0, λ i1,, λ in }; // the probability that lower indexed wavelengths are all busy // the set of wavelengths marked as Free in wave_flag for each wavelengths λ ik in F P( the connection will use λ ik by First-Fit) = S * P ( λk free on R); stores the probability in array guessed_probs; S = S*(1- P ( λ free on R)); end; return guessed_probs; function end k Code 2, Wheel_of_fortune function. function wheel_of_fortune(guessed_probs) //Input: // guessed_probs[0: N-1]: returned result by function guess_wavlength_selection_probs //Output: // guessed_wave: a wavelength index which the function guesses the connection request will use F = { λ i0, λ i1,, λ in }; // the set of wavelengths whose guessed_probs value is not 0 N = sizeof (F) +1; Divide interval [0-1] into N subintervals, A[0], A[1],.., A[N-1]; // the last subinterval // corresponds to the possibility that the connection request will be blocked. for each wavelength λ ik in F set length of subinterval A[k] = guessed_probs ( index of λ ik ); end; N 2 k = 0 set the length of subinterval A[N-1] = 1- length ( A[ k ]) ; generates a random number RND between 0 and 1; S = {A[0], A[1],,A[N-2]}; if RND falls into some subinterval A[k] in S guessed_wave = index of wavelength λ ik ; // return the index of wavelength corresponding else // to chosen subinterval guessed_wave = -1; // otherwise, wheel_of_fortune guess the connection request will be blocked end; return guessed_wave; function end 15

16 Code 3, solve_interference function function solve_interference ( prob message) // input: // prob: the received prob message //output: // prob: prob message with field wave_map being updated wave_flag = field wave_map in prob message; for each interfering request Ci stored in the Ongoing table guessed_wave = field wheel_game of Ci record; marks λ guessed _ wave in wave_flag as Busy ; if Ci.next_hop!= next_hop of of this connection request wave_map[ λ guessed _ wave ] of prob message = Guessed Busy ; end; end; guessed_probs = guess_wavlength_selection_probs ( prob message, wave_flag); wheel_game_ret = wheel_of_fortune (guessed_probs); creates a record about this request in the Ongoing table, setting field wheel_game = wheel_game_ret ; return prob message; function end 16