Fixed-Alternate Routing and Wavelength Conversion in. Wavelength-Routed Optical Networks. Department of Computer Science

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1 Fixed-Alternate Routing and Wavelength Conversion in Wavelength-Routed Optical Networks S. Ramamurthy y and Biswanath Mukherjee Department of Computer Science University of California, Davis, CA 9566, U.S.A. y Corresponding Author { Tel: (530) Fax: (530) ramu@cs.ucdavis.edu March, 998 Abstract This paper considers optical networks which employ wavelength-routing switches that enable the establishment of wavelength-division-multiplexed (WDM) connections between node-pairs. In such networks, when there is no wavelength conversion, a connection is constrained to be on the same wavelength channel along its route. Alternate routing improves the blocking performance of such networks by providing multiple possible paths between node-pairs. Wavelength conversion improves the blocking performance of such networks by allowing a connection to use dierent wavelengths along its route. This paper proposes an approximate computational model that incorporates alternate routing and wavelength conversion. The model is shown to give reasonably good estimates of dierent parameters in the network including the blocking probability and link utilizations. This paper performs empirical studies based on discrete-event simulation, of the relationships between alternate routing and wavelength conversion, on three representative network topologies. The paper compares the gains in blocking probability due to alternate routing with the corresponding gains due to wavelength conversion, and demonstrates that alternate routing provides more benets than wavelength conversion, and that it is important to design alternate routes between node-pairs in an optimized fashion to exploit the connectivity of the network topology. The empirical results also indicate that xed-alternate routing with a small number of alternate routes asymptotically approaches adaptive-shortest-cost path routing in blocking performance. Keywords and Phrases: Optical network, WDM, wavelength routing, lightpath, wavelength conversion, alternate-routing, adaptive routing. S. Ramamurthy, and B. Mukherjee were supported in parts by NSF Grant No. NCR , and grants from Pacic Bell and UC-Micro.

2 Introduction: Wavelength Routing, Wavelength Conversion, and Alternate Routing Wavelength-division multiplexing (WDM) divides the tremendous bandwidth of a ber (potentially a few tens of terabits per second (Tbps)) into many nonoverlapping wavelengths (WDM channels) []. Each channel can be operated asynchronously and in parallel at any desirable speed, e.g., peak electronic speed of a few gigabits per second (Gbps). An access node may transmit signals on dierent wavelengths, which are coupled into the ber using wavelength multiplexers. An optical signal passing through an optical switch may be routed from an input ber to an output ber without undergoing optoelectronic conversion. If wavelength converters are present in a switch, the input optical signal can be translated from one wavelength channel to another wavelength channel at the output. A switch is capable of full wavelength conversion if a wavelength channel on any input port may be converted to any wavelength channel on any output port. A switch is capable of limited or sparse wavelength conversion if the switch has a limited number of wavelength conversion units, where a unit of wavelength conversion can be utilized to convert the wavelength channel of an optical signal passing through the switch. Figure illustrates a wavelength-routed optical network consisting of 6 access nodes (labeled A through F) and 6 switches (labeled through 6). B C A λ λ2 WRS λ λ2 λ WRS WRS 2 3 λ2 λ λ2 WRS 6 λ2 F λ2 WRS WRS 4 5 D E Access Node Wavelength Routing Switch Figure : Architecture of a wavelength-routed optical network. In such a network, a connection is setup by establishing a lightpath from the source node to the destination node. A lightpath is an all-optical channel which may span multiple ber links to provide a circuit-switched interconnection between two nodes. In the absence of wavelength converters, a lightpath would occupy the same wavelength on all ber links that it traverses. This is called the wavelength-continuity constraint. Two Note that, in this model, associates with a node, there is a switch and vice-versa, e.g., node A and switch ; later in this paper, for the simplicity of exposition, we will refer to the node-switch combination as an integrated unit, and continue to refer to this combination as a node. 2

3 lightpaths on a ber link must also be on dierent wavelength channels to prevent the interference of the optical signals. Figure shows two wavelength-continuous lightpaths between nodes A and C on wavelength, and between nodes A and F on wavelength 2. When wavelength converters are present at switches, a lightpath may switch between dierent wavelengths on the route from source to destination. In Fig., a wavelength-converted lightpath between nodes D and C is illustrated, where the wavelength-converted lightpath occupies wavelength on links fd,4g and f4,3g, and wavelength 2 on link f3,cg, with wavelength conversion at switch 3. The wavelength-routed optical network illustrated in Fig. may be modeled as a layered graph, in which each layer represents a wavelength, and a physical ber link has a corresponding logical link in each wavelength layer. Figure 2 illustrates the layered-graph model of the optical network in Fig.. A wavelength-continuous lightpath is a route that utilizes links in a single layer; a multiwavelength lightpath (employing wavelength conversion) is a route that utilizes links in more than one layer. λ3 Layer λ2 Layer λ Layer Wavelength Routing Switch Access Node Figure 2: Layered graph model of the optical network with three wavelength layers. Figure 3(a) illustrates a network without wavelength converters, and Fig. 3(b) illustrates a network with a wavelength converter in node 2. Assume that there are two wavelengths, and 2, on a ber link. Assume that there are two connections already established in the network: one connection utilizes wavelength on the link f2,3g, and the second connection utilizes wavelength 2 on link f,2g. Now, if a connection request arrives with source node and destination node 3, it cannot be setup on either wavelength or 2. However, if we utilize the wavelength converter at node 2, then we can establish the connection by employing 2 on the ber link between node and node 2, converting it to at node 2, and employing on the ber link between node 2 and node 3. This example illustrates that wavelength converters can enable the setup of some lightpaths which might otherwise be blocked in the absence of wavelength conversion. In Fig., the shortest-hop path from node B to node F, is fb,2,3,6,fg. We note that wavelength 3

4 Node Node 2 Node 3 λ λ λ 2 λ 2 (a) without converter Node Node 2 Node 3 λ λ λ 2 λ λ 2 (b) with converter Figure 3: Wavelength-continuity constraint in a wavelength-routed network. is utilized on link f2,3g, and wavelength 2 is employed on link f3,6g by existing connections. If there are no wavelength converters, and a maximum of two wavelengths are available on all links in the network, and if fb,2,3,6,fg is the only path available for any connection from node B to node F, then a connection that arrives at node B to destination F will be blocked. However, if we can utilize the route fb,2,5,6,fg for the connection, then we can establish the connection from B to F on on this route. This example illustrates that alternate routes can enable the setup of connections which might otherwise be blocked. From the above two illustrative examples, we observe that wavelength conversion and alternate routing are potentially benecial schemes which can alleviate the wavelength-continuity constraint in a wavelengthrouted optical network. Wavelength conversion is a \hardware/software" solution in the sense that it requires the addition of wavelength converters in the network, as well as algorithms and protocols to manage the wavelength converters. Alternate routing is a \software" solution in the sense that it needs addition of signaling, control, and management protocols that can perform alternate routing. In this paper, we will examine the interplay between alternate routing and wavelength conversion in wavelength-routed optical neworks. In the rest of this section, we provide precise algorithms for the following: (a) xed-alternate routing, (b) adaptive routing, (c) random wavelength assignment, and (d) connection setup; followed by a brief review of prior related work.. Fixed-Alternate Routing Fixed-alternate routing requires that each access node in the network have a routing table, which contains an ordered list of a limited number of xed routes to each destination node. When a connection request arrives, the source node attempts routes in sequence from the routing table, until a route with a valid wavelength assignment is found (the wavelength assignment algorithm is specied in Section.3). If no available route is found from the list of alternate routes, then the connection request is blocked and lost. Fixed-alternate 4

5 routing provides benets such as: (a) simplicity of control to setup and teardown lightpaths, and (b) fault tolerance upon link failures [2]. A direct route between a source node s and a destination node d is dened as the rst route in the list of routes to d in the routing table at s. An alternate route between s and d is any route other than the rst route in the list of routes to d in the routing table at s. The term \alternate routes" is also employed to describe all routes (including the direct route) from a source node to a destination node. As an example, Table illustrates the routing table at node A for the network shown in Fig.. In this example, each source maintains one direct route and one alternate route, for a total of two alternate routes, to each destination node. For the networks considered in this paper, the routing tables at each node are ordered by the hop distance to the destination. Therefore, the shortest-hop path to the destination is the rst route in the routing table. When there are ties in the hop distance between dierent routes, the ordering among them in the routing table is random. Destination Route Route 2 B A,,2,B A,,4,2,B C A,,2,3,C A,,4,3,C D A,,4,D A,,2,4,D E A,,4,5,E A,,2,3,5,E F A,,2,3,6,F A,,4,5,6,F Table : Routing table at node A for the network in Fig., with two alternate routes to each destination..2 Adaptive Routing In adaptive routing, the route from a source node to a destination node is chosen dynamically, depending on the network state. The network state is determined by the set of all connections that are currently setup. One form of adaptive routing which we will consider in this paper is adaptive-shortest-cost path routing under which each link in the network has a cost of unit, and each wavelength-converter link (in the layered-graph model) will have a cost of c units. When a connection arrives, we determine the shortest-cost path between the source node and the destination node in the layered-graph model. If there are multiple paths with the same distance, one of them is chosen randomly. By choosing the wavelength conversion cost c appropriately, we can ensure that wavelength-converted routes are chosen only when wavelength-continuous paths are not available (e.g., we can choose c to be the cost of the longest wavelength-continuous path in the network). In shortest-cost adaptive routing, a connection is blocked only when there is no route (either wavelength- 5

6 continuous or wavelength-converted) from the source node to the destination node in the network. Adaptive routing requires extensive support from the control and management protocols to continuosly update the routing tables at access nodes..3 Wavelength Assignment The wavelength-assignment algorithm assigns a wavelength to each link in the route, with appropriate wavelength conversion. This paper assumes the following random wavelength-assignment algorithm. Let R be the wavelength reservation parameter, which is dened implicitly in the wavelength-assignment algorithm. The wavelength reservation parameter may be used to prevent the alternate routes from consuming wavelengths that might otherwise be used by direct routes. Given a route r to which we need to assign wavelength(s), let S be the set of idle wavelengths available on the route, i.e., each wavelength w 2 S is free on each ber link of the route. Consider the following two scenarios. If there are no wavelength converters in the network: If r is a direct route, and if S is nonempty, choose a random wavelength from S. If r is a direct route, and if S is empty, the route is blocked. If r is an alternate route, and if jsj > R, then choose a random wavelength from S. If r is an alternate route, and if jsj R, then the route is blocked. If there are wavelength converters present in the network: Try to assign wavelengths without utilizing any wavelength converters, as above. If not possible, (i.e., if r is a direct route and S is empty, or if r is an alternate route and jsj R), divide the route r into subpaths, r; r2; :::; r n, depending on wavelength converter availability at intermediate nodes of the route. Let S; S2; :::; S n be the set of idle wavelengths available on subpaths r; r2; :::; r n, respectively. If r is a direct route, and if S i > 0, for i n, choose a random wavelength from each S i ; otherwise, the route is blocked. If r is an alternate route and js i j > R, choose a random wavelength from each S i ; otherwise, the route is blocked. The above algorithm is \naive" in the sense that it may utilize more wavelength converters than may be necessary to establish a lightpath. This is because the above algorithm does not exploit the possibility that certain adjacent subpaths in a lightpath may have common free wavelengths and hence a wavelength converter need not be used in going between those subpaths. However, the performance of the above algorithm provides an upper bound on the performance of any wavelength-assignment scheme. The work in [3] examines wavelength-assignment algorithms in the presence of sparse wavelength-conversion that minimize the number of wavelength converters needed to establish a wavelength-converted lightpath..4 Connection Setup The procedure for connection setup involves the following steps. 6

7 . Routing: Find a route from the source to the destination. Route nding can involve: selecting a route from a list of prespecied routes such as in xed-alternate routing; route selection can also be performed dynamically, depending on network state, as in adaptive routing. This paper focuses on xed-alternate routing, and compares empirically the performance of xed-alternate routing with adaptive-shortest-cost path routing. 2. Wavelength assignment: This paper assumes that wavelength assignment is random as described in Section Connection setup signaling: After the route selection and wavelength assignment are performed for a lightpath, connection setup involves reserving resources along the lightpath route, and then conguring the switches, and other network elements appropriately. We assume that the control and management software at the switches and access nodes implement the connection setup and teardown procedures [4]..5 Previous Work Routing strategies in wavelength-routed optical networks have been examined in [5, 6]. The work in [5] examines three routing strategies and considers their impact on the dimensioning of the network. In [6], the authors propose an analytical model for alternate routing, and examine the blocking probability of paths with dierent number of hops, and dierent wavelength-assignment policies. They also consider dynamic routing and compare the performance of alternate routing with dynamic routing. However, they do not take into account the eects of wavelength conversion. The benets of wavelength conversion have been a subject of interest in the past [7, 8, 9, 0]. It has been shown that, with xed routing, wavelength conversion provides about a moderate improvement in blocking probability, and that most of the benets can be obtained using sparse conversion. Path models for a wavelength-continuous path have been proposed in [9, ]. Least-congested routing in wavelengthrouted optical networks has been examined in [, 2]. Alternate routing has been extensively researched in circuit-switched telecommunication networks [2, 3, 4, 5, 6, 7]. Fixed-point approximation models for circuit-switched telecommunications networks with alternate routing have been studied in [2], and for state-dependent routing in [3, 4, 7]. Our work focuses on the interplay between xed-alternate routing and wavelength conversion and in this respect it is dierent from previous work. We develop a computational model for xed-alternate routing, that extends earlier models proposed in [6, ], and incorporates any degree of wavelength conversion. Using the computational model, and with simulations, we examine the relative benets of wavelength conversion and xed-alternate routing. 7

8 .6 Outline of Remaining Sections Section 2 discusses the system architecture and states our assumptions. Our analytical model is presented in Section 3. Section 4 elaborates on the approach to solve the analytical model. Section 5 presents numerical results for three representative network topologies. Section 6 concludes the paper with a discussion of the main contributions of this work and related problems for further research. 2 Network Architecture And Assumptions The network consists of nodes and links interconnected in an arbitrary mesh interconnection pattern. There are N nodes in the network, labeled ; 2; :::; N. The (unidirectional) links in the network are labeled ; 2; : : :; E. Each link can have at most C wavelengths. A lightpath r consists of a subset of ; 2; : : :; E links, that form a path, with an assignment of a wavelength to each link. A lightpath connection request is denoted by a (s; d) pair, where s is the source node, and d is the destination nodes. We label a (s; d) pair with an integer, so that there are N (N? ) possible (s; d) node-pairs in the network. Calls for node-pair i arrive according to a Poisson process with rate A i. The holding time for a call is exponentially distributed with mean, and is independent of other call arrival and holding times. The rate of calls will be denoted in units of Erlangs, where Erlang is dened to be the number of calls per unit call holding time. r i (); r i (2); :::; r i (M i ) is the ordered list of alternate routes for node-pair i. r i () is called the direct route, and r i (2); :::; r i (M i ) are called \alternate routes" for node-pair i. When a call for node-pair i arrives, routes for it are tried sequentially from r i (); r i (2); :::; r i (M i ), until a route with an available wavelength assignment is found. Wavelength assignment is performed by the algorithm presented in Section.3. Wavelengths are statistically independent, since the wavelength assignment algorithm does not distinguish between wavelengths. R is the wavelength reservation parameter, as dened in the wavelength assignment algorithm in Section.3. This work assumes that there is no access node blocking, i.e., calls cannot block because wavelengths or transceivers are not available on the ber link that connects the access node to the network. This assumption allows us to to focus on the properties of the network topology. 8

9 All trac ows are described as Poisson processes. All random processes are described by their rst moments. 2. Notation We denote the path and the network-wide parameters by upper-case letters, and link parameters by lower-case letters. Subscripts and superscripts refer to specic instances of links, node-pairs, and routes. The term \trac" means the rate of calls per unit time. The term \oered trac" denotes the trac that arrives (to the network, route or link), and \carried trac" denotes the trac that is actually setup successfully (in the network, route or link). The term \load" means the same as the term \trac". We will employ the terms \call" and \connection" interchangeably. A route r denotes a set of adjacent links. P is the network-wide blocking probability. X r is a random variable which denotes the number of idle wavelengths on route r. X j is a random variable which denotes the number of idle wavelengths on link j. B r is the blocking probability of a direct route r. Ba r is the blocking probability of an alternate route r. B r;xj =m is the blocking probability of a direct route r when link j has m idle wavelengths. Ba r;xj =m is the blocking probability of an alternate route r when link j has m idle wavelengths. A i is the oered trac for node-pair i. A i is the carried trac for node-pair i. V i r is the trac for node-pair i that is oered to route r. V i r is the trac for node-pair i that is carried on route r. V i r;x j =m is trac for node-pair i that is carried on route r when link j has m idle wavelengths. v j is the carried trac on link j. v j;m is the carried trac on link j, when there are m idle wavelengths on link j. is the network-wide average link utilization. The average link utilization for a single link is the average number of wavelengths used by lightpaths that traverse that link. 9

10 3 Analytical Model Our analysis approach consists of two main components: a) routing analysis and b) path-blocking analysis. The routing analysis consists of a set of equations that determine link-oered trac from the path-blocking probabilities. The path-blocking analysis consists of a set of equations that determine the path-blocking probabilities from the link-oered trac. An iterative method of repeated substitution [2, 4] is employed to solve the system of xed-point non-linear equations that result from the analysis. Our main contribution in the analytical model is to extend earlier analysis in [6, ] to incorporate sparse and full wavelengthconversion. 3. Overall Blocking Probability The network-wide blocking probability is the ratio of lost trac to the oered trac. P = 3.2 Carried Trac for Node-Pair i P N(N?) i= (A i? A i ) P N(N?) i= A i () The trac for node-pair i can be carried on any of the alternate routes. We express the total carried trac for node pair i, A i, as the sum of the carried tracs on the alternate routes for node-pair i, i.e., XM i 3.3 Carried Trac for Node-Pair i on Route r A i = V i r i (m) (2) m= The carried trac for node-pair i on route r can be expressed in terms of the blocking probability of the route as follows. If r is a direct route, we have If r is an alternate route, we have V i r = V i r (? B r ) (3) V i r = V i r (? Ba r) (4) 3.4 Oered Trac for Node-Pair i on Route r Figure 4 illustrates a system of alternate paths for node-pair i. By the xed-alternate routing algorithm, trac is oered to alternate path r i (k) if all the routes r i (j); j k?, are blocked. Let Pj i be the probability that the rst j alternate routes for node pair i are blocked. Then, the trac to node-pair i that is oered to route r i (k), i.e., V i r ( k), is given by V i r i (k) = Ai P i k? (5) 0

11 r() s r(2) r(3) d Link(s) shared between r(3) and r(4) r(4) where P i j and Figure 4: Illustration of alternate routes for a node pair. is dened recursively as follows: P i = B ri () (6) P i j = P i j? Prob (r i(j) is blocked j all r i (k) are blocked, k = ; 2; : : :; j? ) (7) for j 2. In this analysis, we will assume that blocking on any alternate route is independent of blocking on any other alternate route. From the assumption that alternate routes block independently, we have for j 2. Therefore, Prob (r i (j) is blocked j all r i (k) are blocked, k = ; 2; : : :; j? ) = Ba ri (j) (8) jy P i j = B ri () Ba ri (k) (9) k=2 The assumption that alternate routes block independently is reasonable because alternate routes between any node-pair are expected to be link-disjoint, i.e., alternate routes will not share links. One event when this assumption (that routes block independently) is violated is when alternate routes share links. For example, routes r(3) and r(4) in Fig. 4 share a link and therefore blocking on r(3) is related to blocking on r(4). Reference [8] presents a model that takes into account the dependencies (due to shared links) between the blocking probabilities of alternate routes. 3.5 Carried Load on a Link The carried load on link j, v j, is the sum of the carried loads on all routes on which link j is a component link, i.e., v j = N(N X?) i= X km i j2r i (k) V i r i (k) (0)

12 3.6 Blocking Model of a Wavelength-Continuous Route The blocking probability for a wavelength-continuous route is dened recursively in terms of the blocking characteristics of a basic element, which can be a single link or a two-link tandem. We utilize a single-link blocking model of a wavelength-continuous route proposed in []. We could also utilize other blocking models of a wavelength continuous path, e.g., the two-link blocking model proposed in [9] Single-Link Model In the single-link model, each link j, j E, has associated with it a random variable, X j, which indicates the number of idle wavelength on that link. We assume that the X j 's are independent. Let Y (2) be a random variable indicating the number of idle wavelengths on a two-hop path, consisting of links i and j. The conditional probability that there are k idle wavelengths given that link i has n a idle wavelengths and link j has n b idle wavelengths, P (Y (2) = kjx i = n a ; X j = n b ), is determined combinatorially as follows. Consider throwing X i blues balls at C dierent bins at random, and X j red balls at random into the same C bins (independent of the blue balls). (Recall that C equals the number of wavelengths in ber link.) Then, P (Y (2) = kjx i = n a ; X j = n b ) is the probability that there are k bins with both blue and red balls, i.e., P (Y (2) = kjx i = n a ; X j = n b ) = 8 >< >: n a k 0 A C A n C? n a n b? k A 0 otherwise if max(0; n a + n b? C) k min(n a ; n b ) For a n-hop path, r, with links l; l2; :::; l n, the probability that there are k available wavelengths on the path, P (Y (n) = k) is dened recursively as follows: CX CX P (Y (2) = k) = P (Y (2) = k)jx l = x; X l2 = y)p (X l = x)p (X l2 = y) (2) x=0 y=0 () and CX CX P (Y (n) = k) = P (Y (2) = kjy (n?) = x; X ln = y)p (Y (n?) = x)p (X ln = y) (3) x=0 y=0 The blocking probability of a wavelength-continuous direct route r, B r, is therefore determined by B r = P (Y (n) = 0) (4) and the blocking probability on a wavelength-continuous alternate route r, Ba r, is given by i=r X Ba r = P (Y (n) = i) (5) i=0 2

13 3.6.2 Distribution of Idle Wavelengths on a Link The idle wavelength distribution on a link j, P (X j = k), is determined as follows. The arrival process on a link j, when the link has m idle wavelengths, is Poisson with arrival rate v j;m. The rate at which connections are terminated when there are m idle wavelengths (and hence C? m active connections) on the link is given by C? m since the average holding time for a connection is one. Therefore, the number of idle wavelengths on the link, X j, can be described by the Markov chain in Fig. 5. Solving the Markov chain we obtain v j,c v j,c- v j,c-2 v j,2 v j, X j = C X j =C- X j =C-2 X j = X j = 0 2 C-2 C- C Figure 5: Markov chain for idle wavelength distribution on link j. P (X j = m) = Q m i=(c? i + ) Q m i= v P (X j = 0) (6) j;i P (X j = 0) = " CX Q m #? + i= (C? i + ) Q m m= i= v j;i (7) State-Dependent Arrival Rate on a Link We seek to determine v j;m, which is the carried load on link j when X j = m. From Section 3.5, we have v j;m = N(N X?) i= X km i j2r i (k) V i r(k);x j =m (8) where V i r(k);x j =m is trac from node-pair i that is carried on route r i (k) when the state of link j is X j = m. From section 3.3, if r is a direct route, we have and if r is an alternate route, we have V i r;x j =m = V i r (? B r;x j =m) (9) V i r;x j =m = V i r (? Ba r;xj =m) (20) The oered trac to route r from node-pair i, V i r, can be calculated from the analysis in Section

14 3.6.4 State-Dependent Blocking Probability of a Wavelength-Continuous Path B r;xj =m, and Ba r;xj =m can be evaluated recursively as follows. Consider a n-hop path, r, with links l; l2; : : :; j; : : :; l n. We can express path r as r = rjr3, where r = l; l2; : : : is the initial part of path r that ends in link before link j (see Fig. 6), and r3 is the rest of the path r after link j. Let U be a random l l 2 j l n r r 3 r 2 Figure 6: Decomposition of a path. variable that indicates the number of idle wavelengths in route r, when X j = m. U be a random variable that indicates the number of idle wavelengths in route r, U2 be a random variable that indicates the number of idle wavelengths in route rj, when X j = m, and U3 be a random variable that indicates the number of idle wavelengths in route r3, when X j = m. Then, CX P (U2 = k) = P (Y2 = kju = x; X j = m)p (U = x)p (X j = m) (2) x=0 CX CX P (U = k) = P (U = kju2 = x; U3 = y)p (U2 = x)p (U3 = y) (22) x=0 y=0 Therefore, we have B r;xj =m = P (U = 0) (23) and i=r X Ba r;xj =m = P (U = i) (24) i=0 3.7 Average Link Utilization The network-wide average link utilization is given by = P E j= P C m= mp (X j = C? m) E (25) 4

15 3.8 Full Wavelength Conversion Here, we assume that a given subset of nodes in the network have full wavelength conversion capabilities. We divide each route r into segments, where each segment is a path with no wavelength-conversion nodes. So, a route r can be segmented as r = rr2 : : : r k, where each r i ; i k, is a wavelength-continuous path, and nodes shared by adjacent segments have full wavelength-conversion capability. We then compute the idle wavelength distributions X i on each r i by employing the analysis presented in Section Then, the probability of blocking on a (possibly wavelength converted) direct route r is given by ky B r =? (? B ri ) (26) i= Similarly, probability of blocking on an (possibly wavelength converted) alternate route r is given by ky Ba r =? (? Ba ri ) (27) i= Here, we are assuming that the routes r i block independently which is reasonable assumption because the r i are link-disjoint. We can then compute the state-dependent blocking probability of a wavelength-converted direct path, B r;xj =m, as follows. Let j 2 r l, i.e., link j is in the l th route segment. Then, B r;xj =m =? (? B rl ;X j =m) ky i=;j62r i (? B ri ) (28) Similarly, we can compute the state-dependent blocking probability of a wavelength-converted alternate path as Ba r;xj =m =? (? Ba rl ;X j =m) 3.9 Sparse Wavelength Conversion ky i=;j62r i (? Ba ri ) (29) Here, we assume that some nodes in the network have limited wavelength-conversion capabilities. Let node j have W j number of wavelength-converter units. Each converter unit can be utilized by one lightpath that traverses the node. We assume that the requests for wavelength-converter units at a node j is a Poisson process with rate j. The number of available wavelength converters at node j, Z j, can be represented as a Markov chain illustrated in Fig. 7. We can approximate the rate at which wavelength converters are requested for use at node j, j, as the rate at which routes that go through node j are blocked. j = N(N X?) i=;j2r i () N(N X?) V i r i () B r i () + i= X 2kM i j2r i (k) V i r i (k) Ba r i (k) (30) 5

16 λ j λ j λ j λ j λ j W W- W W j - W j Figure 7: Markov chain for the number of available wavelength converters at node j. From the above Markov chain, we can determine the probability distribution of the number of available wavelength-converter units at node j, Z j, where Z j = k is the event that there are k available wavelength converters: P (Z j = k) = Q k i= (W j? i + ) ( j ) k P (Z j = 0) (3) and P (Z j = 0) = W j X m= Q 3? m i= (W j? i + ) 5 ( j ) m (32) We divide each route r into segments, where each segment is a path with no nodes with wavelengthconverter units. So, a route r can be segmented as r = rr2 : : :r k, where each r i ; i k, is a wavelengthcontinuous route. We then compute the idle wavelength distributions X i on each r i (from Section 3.6.2). Let Z i ; 2 i k?, denote the number of wavelength converters available at the i th intermediate node that contains wavelength converters in the route r. We assume that the segments r i block independently (this is reasonable since the r i are link-disjoint), and the random variables Z i are independent. Let B z be the probability that some intermediate node (with wavelength coverters) in the route r does not have any free wavelength converters (so that all the available wavelength converters at that node are utilized by current connections). Then, Y k? B z =? (? P (Z i = 0)) (33) i=2 and B rf, the probability that some segment r i has no idle wavelengths, equals ky B rf =? (? B ri ) (34) i= The probability that there are no continuous wavelengths available on route r and that each r i has at least one free available wavelength is given by B rc = P (X r = 0 and X ri ; i = ; : : :; k) (35) 6

17 which can be evaluated recursively similar to the computation of blocking probability B r in Section In the presence of sparse wavelength conversion, a route is blocked in the following two mutually-exclusive cases: (a) some segment has no available wavelengths or (b) all segments have idle wavelengths, but there is no idle wavelength on the route r and some intermediate node (with wavelengths converters) does not have a free wavelength converter. Therefore, the blocking probability of a direct path r is given by B r = B rf + B rc B z (36) In the above equation, we have assumed that the distribution of wavelength converters at intermediate nodes is independent of the idle wavelength distributions on the segments. We note that the blocking probability as computed above assumes that wavelength converters have to be available at each intermediate node. This assumption is \naive" in the sense that it may utilize more wavelength converters than may be necessary to establish a lightpath. It is possible that adjacent segments may have common free wavelengths and hence a wavelength converter may not be needed between the two segments. We compute the blocking probability of a (possibly) wavelength-converted alternate route (similar to the above computation of the blocking probability of a possibly wavelength-converted direct route) as follows. Let Ba rf be the probability that some segment r i has at most R idle wavelengths. Let Ba rc be the probability that there are no continuous wavelengths available on route r and that each r i has more than R free available wavelengths. Then, and ky Ba rf =? (? Ba ri ) (37) i= Ba rc = P (X r = 0 and X ri > R; i = ; : : :; k) (38) Therefore, the blocking probability of a (possibly) wavelength-converted alternate route equals Ba r = Ba rf + Ba rc B z (39) For state-dependent blocking probabilities of a direct (or alternate) path, i.e., the blocking probability of a (possibly) wavelength-converted direct (or alternate) path r when link j has m idle wavelengths, B r;xj =m (or Ba r;xj =m), we make the following modications to the above equations (34)-(39). Let link j be in the l th segment r l of the route r. Then, B rf;xj =m =? (? B rl ;X j =m) ky i=;i6=l (? B ri ) (40) B rc;xj =m = P (X r = 0 and X ri >= ; i = ; : : :; k; and X j = m) (4) B r;xj =m = B rf;xj =m + B rc;xj =m B z (42) Ba rf;xj =m =? (? Ba rl ;X j =m) ky i=;i6=l (? Ba ri ) (43) Ba rc;xj =m = P (X r = 0 and X ri > R; i = ; : : :; k; and X j = m) (44) Ba r;xj =m = Ba rf;xj =m + Ba rc;xj =m B z (45) 7

18 4 Solution Approach The evaluation of the blocking probability in Eqn. () requires the solution of the system of Equations ()-(45). We will utilize a iterative relaxation procedure to solve the system of non-linear equations. Initialization: Set path blocking probabilities B r = 0 and Ba r = 0 for all alternate routes r between all node-pairs i; i N(N? ). Iterate:. Route trac: For all routes r, determine V r and V r. 2. Link loads: For all links j, determine v j;xj =m. Link idle wavelength distribution: For all links j, determine P (X j = m). 3. Wavelength-continuous path-blocking probabilities: For all r, determine B r and Ba r. 4. Wavelength-continous conditional path-blocking probabilities: For all r, determine B r;xj =m and Ba r;xj =m. 5. Wavelength-converter distributions: For each node i with wavelength converters, determine P (Z i = k). 6. Possibly wavelength-converted path-blocking probabilities: For all r, determine B r and Ba r. 7. Possibly wavelength-converted conditional path-blocking probabilities: For all r, determine B r;xj =m and Ba r;xj =m. 8. Iterate k steps until jp k?p (k?) j <, where P k and P (k?) are the network-wide blocking probabilities in the k th and (k? ) th iterations, respectively. It is not clear if the system of Equations ()-(45) has an unique solution, or if the algorithm presented above will converge to a solution point. However, in practice, we observe that the above algorithm converges to a solution point for all the representative networks that we considered, and that the solution is in reasonably good agreement with simulation results. 4. Complexity and Performance The running times for each step in the above algorithm is shown in Table 2. Here, H is the average hop distance of all the alternate routes between all node-pairs, M is the number of alternate routes between node-pairs, C is the number of wavelengths, and E is the number of links. We observe in practice that the algorithm converges to within an accuracy of = 0?6 in 6 to 0 iterations, for the example network topologies examined in this work. 8

19 Step Description Running time Route loads O(N 2 M) 2 Link loads O(EN 2 MHC) 3 Path distributions O(N 2 MHC 3 ) 4 Conditional path distributions O(N 2 MH 2 C 4 ) 5 Wavelength-converter distributions O(N 3 C) 6 Wavelength-converted path distributions O(N 2 MH 2 C 3 ) 7 Wavelength-converted conditional path distribution O(N 2 MH 3 C 4 ) Table 2: Running time for each step in the algorithm. 5 Illustrative Numerical Examples and Discussion 5. Network Topologies We consider three network topologies for all our simulation and model studies: (a) a fully-connected network with 6 nodes, (b) a network of interconnected rings with 5 nodes, and (c) a bidirectional ring network with 2 nodes. The three networks show dierent levels of connectivity, in terms of average hop distance, and in terms of the number of paths between node-pairs. These networks were chosen as being indicative of realistic network topologies. The interconnected-rings network topology was provided to us by one of our project sponsors as being representative of a typical telecommunications network. 5.. Fully-Connected Network Figure 8 illustrates a network of six nodes where each node has a link to every other node. We assume that sparse wavelength conversion, when present, is present at all nodes in the network. We study ve congurations for alternate routing. The routing table at each node has one, two, three, four, or ve alternate routes respectively to each destination, in each conguration. We note that the 6-node fullyconnected network is 5-edge connected. We will employ the term \complete" network interchangeably with \fully-connected" network in the rest of this work Interconnected Rings Figure 9 illustrates a 5-node network of interconnected rings. We assume that sparse wavelength conversion, when present, is at nodes, 6, 7, and 3. We study three congurations for alternate routing. The routing table at each node has one, two, or three alternate routes respectively to each destination, in each conguration. We note that the interconnected rings network is two-edge connected, i.e., there are at least two edge-disjoint paths between each node-pair, and there is at least one node-pair with exactly two 9

20 Figure 8: A fully-connected graph on six nodes. edge-disjoint paths WC 0 WC 2 6 WC WC 2 Figure 9: Interconnected rings Bidirectional Ring Figure 0 illustrates a 2-node bidirectional ring. We assume that sparse wavelength conversion, when present, is at nodes, 4, 7, and 0. We study two congurations for alternate routing. In one conguration, the routing table at each node has at most one alternate route to each destination, and in the other conguration, the routing table at each node has two alternate routes to each destination ordered by increasing hop distance. We note that the bidirectional ring is two-edge connected, i.e., there are two edge-disjoint paths between each node-pair. 20

21 WC WC 0 WC WC Simulation and Model Parameters Figure 0: Twelve-node bidirectional ring. We have obtained results for each network with 4 and 8 wavelengths. For each simulation conguration, ve simulations runs were performed, each with a dierent seed for the random number generator, resulting in a dierent call arrival sequence for each run. Each simulation run consisted of 200; 000 calls. The reported simulation data are within the 95 percent condence interval. We assumed that each node-pair is equally loaded, i.e., the total oered load to the network is equally divided between all node-pairs. Our simulation software was developed based on the discrete-event simulation method. We utilized the Bellman-Ford algorithm for nding the shortest-cost path, to setup the xed-alternate routing tables. For adaptive routing, the simulation software performed a shortest-cost path computation for each connection setup. We considered two degrees of sparse wavelength conversion: one where selected nodes had one wavelenth converter each, and another where the selected nodes had three wavelength converters each. Unless otherwise stated, the simulation and model studies assume that the wavelength-reservation parameter R = Results We present the model and simulation results in two parts. In the rst part, we study the accuracy of dierent aspects of the analytical model by comparing the model results with the corresponding simulation results. We also highlight some observations regarding alternate routing and wavelength conversion obtained from the model results. 2

22 In the second part, we examine the simulation results and draw empirical generalizations and observations on the behaviour of alternate routing and wavelength conversion. 5.4 Model Accuracy 5.4. Model Accuracy { Alternate Routing In this section, we examine the accuracy of the alternate routing model when there is no wavelength conversion. Figures illustrates the accuracy of the model for the 4-wavelength fully-connected network, interconnected-rings network, and the bidirectional-ring network with no wavelength conversion. Results for the 8-wavelength networks are similar and hence are not shown here. We observe that the model is more accurate for the fully-connected network than the other two networks. This is because: (a) the average hop distance is one in the fully-connected network (with any alternate route it is at most two hops) whereas for the bidirectional ring, the average hop distance is more than three hops (with a maximum alternate route distance of six hops), and (b) the wavelength-continuous path blocking model is less accurate for longer paths []. We also observe that the model is more accurate at lower loads. In general, we expect the model to be more accurate for denser networks and at lower loads. Another interesting observation from the model is the following: At high loads, the model results indicate that lesser number of alternate routes is better! This may be because at high loads, alternate routes consume resources that would otherwise be used by direct routes. At high loads, the wavelength-reservation parameter R may need to be set appropriately to improve blocking performance Model Accuracy { Sparse Wavelength Conversion Figure 2 illustrates the accuracy of the model for the 4-wavelength fully-connected network, interconnectedrings network, and the bidirectional-ring network, with sparse wavelength conversion. In the sparse wavelength conversion conguration considered here, the selected nodes (refer to Section 5. for a specication of the nodes selected for sparse conversion) in each network were equipped with three wavelength-conversion units. Results for the 8-wavelength networks are similar, and hence are not shown here. We observe that the model is more accurate for the fully-connected network in comparison to the other two networks. We also observe that the model is more accurate when there is sparse wavelength conversion than when there is no wavelength conversion. This is due to the fact the wavelength converters break up long wavelength-continuous paths, and contribute to ensuring the \independence" of idle wavelength distributions on adjacent links. In general, we expect the model to be more accurate for a network with wavelength conversion than for the same network without wavelength conversion Model Accuracy { Wavelength Reservation Recall that the wavelength-reservation parameter R indicates the number of idle wavelengths that are reserved for the direct route, so that a lightpath on an alternate route can be established only when there are at 22

23 0. 0. blocking probability e-05 Model, AR= Sim, AR= Mod, AR=2 Sim, AR=2 Mod, AR=3 Sim, AR=3 Mod, AR=4 Sim, AR=4 Mod, AR=5 Sim, AR=5 blocking probability e-05 Model, AR= Sim, AR= Mod, AR=2 Sim, AR=2 Mod, AR=3 Sim, AR=3 e e (a) 4-wavelength fully-connected network (b) 4-wavelength interconnected rings 0. blocking probability e-05 Model, AR= Sim, AR= Mod, AR=2 Sim, AR=2 e (c) 4-wavelength bidirectional ring Figure : Accuracy of the alternate-routing model with no wavelength conversion. 23

24 0. 0. blocking probability e-05 Model, AR=, sparse Sim, AR=, sparse Model, AR=2, sparse Sim, AR=2, sparse Model, AR=3, sparse Sim, AR=3, sparse Model, AR=4, sparse Sim, AR=4, sparse Model, AR=5, sparse Sim, AR=5, sparse blocking probability e-05 Model, AR=, sparse Sim, AR=, sparse Model, AR=2, sparse Sim, AR=2, sparse Model, AR=3, sparse Sim, AR=3, sparse e e (a) 4-wavelength fully-connected network (b) 4-wavelength interconnected rings 0. blocking probability e-05 Model, AR=, sparse Sim, AR=, sparse Model, AR=2, sparse Sim, AR=2, sparse e (c) 4-wavelength bidirectional ring Figure 2: Accuracy of the alternate-routing model with sparse wavelength conversion. 24

25 least R + available wavelengths on the alternate route (in the absence of wavelength conversion). Figure 3 illustrates model accuracy for the 8-wavelength fully-connected network, and the interconnected-rings network respectively, with two alternate routes, no wavelength conversion, and R taking on the values 2, 4, and 6. The results for other networks and congurations are similar and hence are not shown here. We observe that the blocking probability increases with increasing values of R. This is due to the fact that, as we increase R, we prevent alternate routes from being established. We expect that, when the trac pattern is skewed, or at heavy loads, it may be benecial to set the wavelength-reservation parameter to non-zero values. 0. blocking probability e-05 Model, AR=2, R=0 Sim, AR=2, R=0 Model, AR=2, R = 2 Sim, AR=2, R = 2 Model, AR=2, R = 4 Sim, AR=2, R = 4 Model, AR=2, R = 6 Sim, AR=2, R = 6 Model, AR= e Figure 3: Accuracy of the model for the 8-wavelength fully-connected network with two alternate routes, when the wavelength-reservation parameter, R, takes on the values 2, 4, and Model Accuracy { Link Utilization In the model, the network-wide average link utilization is computed from Equation (25). In the simulation, we compute the average link utilization as follows: for each link, we compute its utilization as the time average of the number of wavelengths used on that link; the network-wide link utilization is the average value of link utilizations over all the links in the network. Figures 4 illustrates the model accuracy for the 8-wavelength fully-connected network, interconnectedrings network, and the bidirectional-ring network, with no wavelength conversion. The results for 4- wavelengths are similar and hence are not shown here. We observe that the model is accurate at low loads and tends to diverge from the simulation at high loads Model Observations In this section, we highlight some observations from the results of the model. Figures 5 illustrates the model results for the 4-wavelength fully-connected network, interconnected-rings, and the bidirectional-ring. We observe the following surprising result for all networks: at low loads, the blocking probability with two 25

26 8 8 link utilization Sim, AR= Mod, AR= Sim, AR=2 Mod, AR=2 Sim, AR=3 Mod, AR=3 Sim, AR=4 Mod, AR=4 Sim, AR=5 Mod, AR=5 link utilization Sim, AR= Mod, AR= Sim, AR=2 Mod, AR=2 Sim, AR=3 Mod, AR= (a) 8-wavelength fully-connected network (b) 8-wavelength interconnected rings Sim, AR= Mod, AR= Sim, AR=2 Mod, AR=2 link utilization (c) 8-wavelength bidirectional ring Figure 4: Accuracy of the model's average link utilization. 26

27 0. 0. blocking probability e-05 Model, AR= Model, AR=2 Model, AR=2, sparse Model, AR=2, FW Model, AR=3 Model, AR=3, sparse Model, AR=3, FW Model, AR=4 Model, AR=4, sparse Model, AR=4, FW blocking probability e-05 Model, AR= Model, AR=, sparse Model, AR=, FW Model, AR=2 Model, AR=2, sparse Model, AR=2, FW e e (a) 4-wavelength fully-connected network (b) 4-wavelength interconnected rings 0. blocking probability e-05 Model, AR= Model, AR=, sparse Model, AR=, FW Model, AR=2 Model, AR=2, sparse Model, AR=2, FW e load (c) 4-wavelength bidirectional ring Figure 5: Model results. alternate routes and no wavelength conversion is better than the blocking probability with one alternate route and full wavelength conversion. Furthermore, for the fully-connected network we observe the following: at low loads, and when the number of alternate routes is, 2, or 3, the benets in blocking probability obtained by adding an alternate route is better than the benet obtained by adding full wavelength conversion. In general, we expect that, at low loads and when the number of alternate routes between node-pairs does not fully exploit the connectivity of the network topology (i.e., the number of alternate routes between node-pairs is less than the edge-connectivity of the network), the benets in blocking probability obtained by adding an alternate route (and therefore exploiting more link-disjoint paths) may be signicantly more than the benets obtained by adding (any degree of) wavelength conversion. In the next section, we conrm these model observations by comparing them with the corresponding simulation results. 27