Comparison of Mesh Protection and Restoration Schemes and the Dependency on Graph Connectivity

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1 Comparison of Mesh Protection and Restoration Schemes and the Dependency on Graph Connectivity John Doucette, Wayne D. Grover TRLabs, # Avenue, Edmonton, Alberta, Canada T5K P7 and Department of Electrical and Computer Engineering, University of Alberta, Edmonton, Alberta Contact: {doucette, Abstract We present a series of experimental designs based on different mesh survivability principles and study ho the orking and spare capacity requirements of each netork type vary ith the netork average nodal degree or notionally, the connectivity of the facilities graph of the netork. This is accompanied by an attempt at some simple theories for bounding or estimating ho capacity and redundancy ill depend on netork average nodal degree. Six types of restorable mesh netork designs are compared: + non-shared backup path protection, span restoration ith and ithout oint optimization of orking and spare paths, chain-optimized span restoration, shared backup path protection, and true dynamic path restoration. It is thought that this may be the first comprehensive comparative study of restorable mesh netork capacity requirements vieed across a range of netork degree as opposed to point comparisons of capacity on single study netorks. By shoing ho each scheme responds to the degree of connectivity of the facilities graph the ork also provides ne insights about facilities graph topology evolution. Keyords Spare Capacity Allocation, Protection & Restoration Algorithms, Netork Planning and Optimization, Transport Netorks, Netork Design Theory I. INTRODUCTION This ork provides a study of six different types of mesh-restorable netork design in terms of ho their orking and spare capacity varies ith graph connectivity. We also compare the experimental behaviour to that predicted by a loer bound and to estimators for ho netork average nodal degree affects capacity requirements. One of the loer bounds is the ell-knon /( d -) argument (here d is the average netork nodal degree) for the minimal ratio of spare to orking capacity from an end-node limited restoration standpoint. The others are ne approaches, not strictly bounds hoever, that try to take into account both the average nodal degree, and the disparity of the orking capacity values incident to each node. The ork on capacity versus nodal degree and the attempts at ne bounding or estimation theories are part of a larger ongoing effort toards optimizing the basic facilities graph topology of restorable mesh netorks []. The types of mesh-restorable netorks characterized and the short-form names used are listed here. More details of each follo in Section III. Schemes are compared on the basis of design to ensure restorability against any single complete span cut. A span is defined as the set of all orking and spare capacity beteen adacent cross-connecting nodes that undergoes a common-cause failure arising from damage to a single physical-layer entity such as a duct or cable. No changes are made to unaffected orking paths in any of these schemes. (i) + Path Protection (+ APS): These are mesh netorks on hich the shortest path route is used as a orking path and an equal amount of capacity is reserved in a non-shared manner on the next shortest fully disoint route. This is really a demand-level + APS or, in other parlance, an SNCP ring protection scheme. + APS is included as a benchmark for the highest end of the capacity consumption scale and because it seems (perhaps surprisingly) to be often assumed as the survivability scheme in different studies of mesh netorks. + APS requires only a tail-end selection sitch for protection and is therefore the simplest and fastest scheme in terms of the real time mechanism. Its simplicity is operationally attractive but e ant to make sure its capacity requirements are fully appreciated in contrast to other mesh survivability schemes. (ii) Span Restoration Spare Capacity Assignment (SCA): These are span-restorable netorks in hich demands are shortest-path routed folloed by optimal spare capacity assignment for 00% restorability against any complete span cut. The total spare capacity is minimized independently of orking capacity, hich is generated by the shortest-path demand routing. (iii) Span Restoration Joint Capacity Assignment (JCA): These are span-restorable netorks here the routing of orking paths is ointly optimized ith spare capacity assignment to minimize total capacity.

2 (iv) Meta-Mesh (M-M): These are a ne class of span-restorable netorks in hich orking and spare capacity is ointly optimized and chain-subnetorks are provided ith logical express bypass routes. Span restoration occurs logically in the meta-mesh abstraction of the complete netork (a homeomorphism of the full graph ith no degree- nodes). The logical bypass spans on chains allos a significant reduction of the otherise required loopback spare capacity ithin chains. The concept and design of these so-called meta-mesh (M-M) netorks is treated in detail in a companion paper []. Meta-mesh netorks are based on the hypothesis that span restoration on the meta-mesh graph is close to path restoration on the full graph. SCA, JCA, and M-M netorks all employ a dynamic span restoration mechanism hich constructs a replacement path-set beteen the immediate end-nodes of a failure span. Spontaneous self-organizing methods for this type of restoration, including distributed pre-planning, are ell developed and available from ork in the 990s [3]. Alternately, centralized control or OSPF-type path finding may be iterated to develop a set of k-shortest replacement paths for this type of restoration [4]. (v) Shared Backup Path Protection (SBPP): In these netorks the shortest route is used for the orking path, and a single fully-disoint route is selected for the backup path so as to permit efficient sharing of spare capacity on spans of the backup route over orking paths that are disoint. Traffic on orking paths that follo physically disoint routes over the netork ill not need the restoration capacity simultaneously, hence restoration capacity sharing is permitted. This is logically the same scheme as as proposed for ATM Backup VP restoration [5] in the special case here the maximum permissible over-subscription factor [6] is limited to.0. This approach is receiving attention in recent IETF deliberations [7] so it is timely to characterize it in this study. The restoration mechanism requires a signalling phase from each tail-end sitch to confirm availability of the backup route and to seize and cross-connect capacity to activate the backup path. (vi) True Path Restoration (Path): In a netork based on path restoration (as opposed to path protection as in + APS or SBPP), a centralized or self-organizing re-routing mechanism [8] deploys a collectively co-ordinated set of replacement paths for all affected origin-destination (O-D) node pairs in response to the specific failure that has occurred. The main difference relative to SBPP is that there is not a single predetermined restoration route for each orking path and that surviving portions of failed paths can implicitly be re-used for restoration. Ideal path restoration is equivalent to a multi-commodity maximum-flo (MCMF) solution for replacement of the failed orking paths ithin the surviving spare capacity [8]. II. BOUNDS AND ESTIMATORS FOR MESH CAPACITY EFFICIENCY In this section e develop a loer bound and to estimators for the redundancy of a span-restorable mesh netork. These are developed from arguments about the conditions for restorability local to any one isolated node ithin a spanrestorable netork. The results are only bounds or estimators because they predict necessary but not assuredly sufficient capacity conditions for the netork as a hole to be 00% restorable. On the other hand, e kno experimentally that at least for span-restorable netorks, the restoration flo is almost alays limited by the total spare capacity incident to one or the other end-nodes. Such end node limiting suggests that the conditions at the end-nodes may therefore provide the basis for a good bound or approximator, at least for the span-restorable case. Somehat similar end-ode limited conditions can also be considered for path-restorable netorks here the end nodes are the O-D nodes of each demand pair affected by a failure [9]. The three derivations all make use of Figure, hich shos a node of degree d in isolation. It makes use of a convention that the indexing of span numbers is in order of largest to smallest orking capacity, so denotes the orking capacity of the span ith the most orking capacity, is the second largest, etc. W S W W W 3 S 3 W... d... S d Figure : Basis for the nodal bound derivations.

3 A. Topological /(d -) Loer Bound Many people are already familiar ith the /( d -) result as an intuitive basis for ho redundancy should drop as connectivity increases. We have had frequent requests to see the basis for this result, hoever, so here e give the simple steps leading to it and e ill use it later in our analysis. Treating the /( d -) bound also sets a context for our attempts at to further approximating measures that follo. Consider the failure of span having orking capacity at the node of degree d in Figure. Obviously the node must have enough spare capacity on the other spans { d} to permit restoration of. Similarly (in the absence of higher netork-level considerations that may still add more spare capacity), each span i requires for its restoration that the total amount of spare capacity on surviving spans meets or exceeds the orking capacity on the failed span. It follos that in the best case from an efficiency standpoint, every span incident on this node could have i = and spare capacity ould be distributed evenly on all spans, in hich case the ratio of spare to orking capacity (hich e call the redundancy) becomes Bound : s (.. d) = d /( d ) = () d d (.. d) B. Estimators that Consider Working Capacity Disparity Effects We can no take the isolated nodal considerations a step farther to obtain to forms of estimator that also give us some insight as to ho the i values affect the achievable redundancy. Let us consider the largest span and the second largest and rite the corresponding sums of surviving spare capacity, and then add the resulting inequalities: s + s + + s s + s + + s 3 d 3 d s + s + ( s + + s ) + 3 d This tells us that + is therefore itself an estimated upper bound on the total required spare capacity at the node (because s and s are present directly in the sum and the total of all other spare capacity is accounted for tice). So for an estimator on redundancy that does not assume that all spans have i = as for the first bound, e could take the above over-estimate of the total spare capacity and combine it ith an optimistic vie of the total orking capacity at the node. The most optimistic case for the latter, here all i are not simply equal, ould be for all i : i ; i to be equal to. Then, the ratio of spare to orking capacity becomes Estimator : si + i (... d) + = (3) + + ( d ) i + ( d ) i (... d) If e assert = as a limiting case ( < is ruled out by definition of largest and second largest) this result returns /d hich is not as tight as /(d-). But in the more general case here >, Eq. (3) shos ho disparity in the i values and the topology measure d both affect the achievable efficiency. In particular it shos that as / (and by implication / i at the node) a ring-like redundancy of 00% is required regardless i of nodal degree. This is an important further understanding of hat affects efficiency in a mesh-restorable netork: it says that even ith a high nodal degree, if virtually all the orking capacity is on one span, then the spare capacity hoever it is distributed over the other spans still sums to. With no other orking capacity present to share that spare capacity, e get ring-like redundancy in the orst case. Overall this gives a characteristic behavioural model of tendencies in a span-restorable mesh netork. The netork s redundancy can only approach /( d -) if all orking quantities are balanced. And regardless of nodal degree, redundancy could range up to 00% as the disparity of orking capacity values increases. Thus, hat e are ideally seeking in a span-restorable mesh is high average nodal degree and balanced orking capacities at the nodes. Another basis for an estimator of spare capacity at a node is to start by assuming a uniform allocation of spare capacity for the restoration of span over all other spans. First consider the failure of span, having orking 3 ()

4 capacity, again ith reference to Figure. The node must have enough spare capacity on the other spans..d to permit restoration of. Therefore, in the absence of netork-level considerations that may further influence the distribution of spare capacity over these spans, let us assume an equal (fractional if need be) assignment to each other span of s = /( d ). Next e need to treat the failure of span, having... d orking capacity. In the absence of having made any assignment of spare capacity to span, the surviving spare capacity available for restoration of span is: d =. Therefore the spare capacity e need to add to span to make span restorable is: d d d d Case : Zero if d d or Case : d > d if Note the folloing interesting implication: that by making spans and fully restorable (under this uniform allocation approach) all spans at the node are also restorable. The reasoning is as follos: the surviving spare capacity at the node after span fails is adequate to restore span. The drop in available spare capacity at the node due to any other span failure is no more than /(d-). But the Case consideration (in Eq. 4) places enough extra spare capacity on span to restore span. Since span is restorable, all other spans not considered, but hich have loer orking capacity than span, and no more spare capacity than span must therefore also be restorable ithin the spare capacity on all other surviving spans including span. Another interesting implication is that depending on the nodal degree and the values of and, span may not require any spare capacity. In such a case, the total spare capacity of the node ill be exactly itself. That is, si = if d i (... d) d. Otherise d si = + = + i (... d) d d if d > d. In summary, this can be expressed as: i (... d) si = max, + d To estimate redundancy from this e can divide by the best case + + ( d ) or the orst case +. Using the former (and making parametric in / ), the ratio of spare to orking capacity becomes Estimator : (4) (5) si max, i (.. d) d i (.. d) i = + + d (6) III. STUDY METHODOLOGY We ill hold the bound and estimator results for later comparison against the detailed design results. We no document the methods for design of test netorks to characterize each of the six mesh survivability schemes. A. Test Netorks Tests ere conducted on a family of 8 progressively sparser test netorks derived from a single high-degree master netork. The series of 8 trials provides a reasonably continuous variation of d hile keeping all nodal positions and the end-to-end demand pattern common over all test netorks. The master netork had 3 nodes and 5 spans for a netork average nodal degree of d = 3.8. The 7 progressively sparser netorks ere derived from it by random removal of one span at a time subect to retaining bi-connectivity. The length and cost of each span is taken as the Euclidean distance on the plane beteen the end nodes of the span. A diagram of the master netork and samples of the progressively sparser test cases are available in a companion paper at this conference []. B. Demand Pattern All test case designs served a demand matrix that as made up according to a gravity attraction model ithout any inverse distance effect. The measure for attraction as the degree of the node in the master netork as a surrogate for demographic importance. There ere a total of C = 496 O-D node pairs, each exchanging an average demand of

5 units, for a total of 856 demand units. There ere no zero-demand pairs. The minimum demand as 5 units and the maximum demand as units. The flat distribution of demand intensities over distance is meant to reflect the type of demand pattern attributed to Internet traffic flos here transactions are as likely to be to a server halfay around the orld as to a server in the same city. In further ork not included due to space, other demand patterns have been tested but sho only second-order effects on the relative comparison of the capacity required by the different schemes. C. Mesh Netork Design and Solution Methods In this section e document the design method for each of the six types of mesh-survivable netorks compared. All netork designs ere produced through AMPL optimization models solved ith the Parallel CPLEX 6 MIP Solver (for SCA, JCA, and meta-mesh designs) and the Parallel CPLEX 7. MIP Solver (all other designs). Space does not permit detailed presentation of each design formulation, but most are documented in already published sources. The SCA and JCA designs ere produced using the formulations and solution method detailed in [0] except that for this study the modularity as one capacity unit (e.g. one avelength). The path-restorable designs ere based on the formulation given in [9]. The case used here is non-modular capacity path restoration ith stub release but ithout oint optimization of orking path routes. The non-oint path restorable version is more computationally practical and e kno from prior ork in [9] and elsehere [] that hile ointness increases the efficiency of span restoration significantly, path restoration designs benefit much less from the added aspect of oint optimization. The meta-mesh designs are basically a oint span-restorable capacity placement type of solution ith a special treatment of chains of degree- nodes to reduce the spare capacity requirement in the chains. The meta-mesh design method is documented in []. The + path protection designs do not strictly require an optimization model. They can be generated simply by routing programs that first find the shortest route and then the next shortest disoint route by temporary removal of all spans on the first route from the graph. We are not yet aare of a prior publication of the SBPP optimization model so the formulation e used is given here: SBPP: Minimize s C b Subect to: x = b R r r r Di b Rr S b r xr d s 5 (7) r D (8) (, i ) S : i (9) The obective (Eq. 7) minimizes the total cost of spare capacity for backup paths. S is the set of all spans, s is the number of spare capacity units placed on span, and C is the cost per spare unit placed on span. Constraints in Eq. 8 restrict us to only one backup route b per demand pair r. D is the set of all O-D pairs. R r is the set of all possible backup routes for demand pair r. d r is the number of demand units exchanged beteen demand pair r. Note that in this model, all d r orking paths on relation r are moved together upon failure to the same backup route. x r b is a /0 decision variable taking the value of if backup route b for demand pair r is used, and zero otherise. Constraints in Eq. 9 assign sufficient spare capacity on each span to accommodate all backup paths simultaneously crossing the span for failure of any other span. D i is the set of O-D demand pairs affected by failure of span i and R r is the set of backup routes for demand pair r hich cross span. A number of other aspects ere common to all design types and their solutions. All orking and spare capacity allocations ere integer, corresponding for example to capacity design and restoration mechanisms handling single avelengths. Any of the models can be converted to a modular design formulation as shon in [0]. For comparative studies e avoid any specific modularity assumptions hich could obscure the general underlying comparison of methods that is intended. Results are based on a full CPLEX termination or a MIPGAP under 0-4 (i.e., ithin 0.0% of optimal) ith the exception of the path restorable designs (ithin 0.% of optimal) and SBPP designs (strictly 5%, most are ithin %). All designs ere also based on an arc-path approach to capacitating the solution. This requires preprocessing steps (discussed next) to enumerate sets of eligible routes for restoration and, in the oint formulations, also for orking flo assignment. D. Route Enumeration for the Arc-Path Formulations To create the eligible route sets to populate the formulations, e used a strategy that results in a minimum target number of distinct route options for the restoration of any failure and eligible route options for routing of each orking path. A program varies the hop limit adaptively for each span failure or demand pair, finding all distinct eligible routes

6 ithin that hop limit, until at least the target number of eligible routes are found. The results are all based on representing at least 0 distinct routes for every span failure or affected demand pair, except for the SBPP designs, hich used at least 5 distinct backup routes. The JCA designs used at least 0 distinct route choices for the routing of each orking demand. In the meta-mesh designs, hen the logical chain bypass spans are added, the eligible route-sets for orking paths are updated ith an additional set of at least 0 orking routes per O-D pair to reflect ne routing options that may use the bypass spans. Similarly, for restoration of the ne bypass spans themselves, a minimum of 0 ne restoration routes are generated for each chain bypass span failure, excluding routes that ould include the associated chain itself. IV. RESULTS AND DISCUSSION A. Redundancy and Capacity Results Figures and 3 are the main presentation of the comparative design results. Figure presents results in terms of redundancy primarily so that the /( d -) bound can be overlaid for comparison. Figure 3 presents the orking and spare capacity requirements of each design. In both figures, redundancy and capacity are based on distance-eighted measures ( cost ), not ust channel counts. Figures and 3 have many interesting aspects. The first most noticeable effect in Figure is that + path protection ithout sharing is extraordinarily redundant; it is never less than 40% redundant and surpasses 00%, consistent ith the fact that + is really a form of ring-based protection. The gap in the + APS curve is due to routing infeasibilities discussed belo. 0% 00% + APS 80% Distance-Capacity Redundancy 60% 40% 0% 00% 80% SCA M-M JCA SBPP 60% Path ( d -) 40% Netork Average Nodal Degree, d Figure : Redundancy of various mesh protection and restoration schemes as a function of netork average nodal degree. In both Figures and 3, the basic ranking of SCA, JCA, and path restoration is consistent ith prior results for single-netork solutions to these problems [9]. JCA improves considerably over SCA by slight changes to the routing of orking paths that have the effect of a relative levelling out of nodal orking capacity quantities, improving the overall capacity efficiency. Also notable in Figure is ho ell SCA and JCA parallel the shape of the /( d -) loer bound. The bound is obviously loer on the scale than either the SCA or JCA curves but the similarity in shape suggests that the arguments underlying the bound also underlie the nature of these types of restorable netork in terms of ho they gain efficiency ith higher degree. In contrast the path-restoration curve is not only loer than the /( d -) bound (hich is only a bound for span restoration) but it also seems to have a different functional form hen plotted against d. It drops at a steeper rate as connectivity increases initially and is then almost flat, although still decreasing 6

7 slightly, as the netork becomes more richly connected. It is plausible that starting from a very sparse netork, path restoration can exploit increases in connectivity more rapidly than span restoration by virtue of its more sophisticated end-to-end vie of restoration re-routing problem. Note in passing the actual redundancy levels of the path restoration designs. Anyhere above d =.6 or so, they are in the 45% to 50% range. This is three to four times more efficient than + APS and almost tice as efficient as SCA. This also substantiates the idespread general appreciation that path-restoration is the most efficient possible scheme, but hopefully also clarifies that this is not at all achieved by nonshared + path protection, even though both are path schemes. Note also the slightly rising slope on segments of the redundancy curves. This is only a reflection of the fact that, as Figure 3 shos, orking capacity keeps decreasing sloly as degree rises. Since redundancy is the ratio of spare to orking, the result is that redundancy (as strictly defined) increases. The total capacity cost is nonetheless dropping. The remaining curves to discuss are SBPP and meta-mesh. The SBPP curve is punctuated by cases here the SBPP design had one or more routing infeasibilities on the given graph. The + APS designs are similarly affected for the same reason. The general issue is that illustrated beteen nodes A and B in Figure 4. As defined SBPP first takes the shortest route for the orking path as indicated, and it is then impossible to find a disoint second route. The problem can of course be overcome by instead finding the shortest cycle containing the to O-D nodes or to iteratively alter the first route choice upon discovery of the infeasibility until a disoint route exists. For present ork, hoever, e have modelled the simpler provisioning model, as e understand it is being considered in many quarters that SBPP can ork this ay. If so, e only ish to point out the issue of such routing infeasibilities and note that they can be particularly frequent in sparse physical layer transport graphs. For example in Figure 4, connections from node B to the to nodes to the right of A are similarly infeasible under the existing SBPP path provisioning logic. Setting aside the infeasibilities for SBPP (and +) e note that SBPP is generally intermediate in efficiency beteen the meta-mesh designs and true path restoration. In practice the provisioning infeasibility problem can be solved but ill add slightly to the capacity requirements of the SBPP curve. Aside from this, its relative simplicity and efficiency are attractive. Finally, the meta-mesh designs provide an interesting ne option, especially for netorks ith d around.4 to.8. In this region the meta-mesh designs are essentially as efficient as SBPP and are in fact beating the /( d -) bound, even though they require only a span restoration mechanism. This apparent contradiction is explained further in [] Work and Spare Total Capacity Costs Path Spare SBPP Spare M-M Spare JCA Spare M-M Work JCA Work S.P. Work SCA Spare Netork Average Nodal Degree, d Figure 3: Breakdon of orking and spare capacity versus netork average nodal degree. 7

8 Figure 3 shos the totals of orking and spare capacity in the designs, omitting + APS to permit closer scrutiny of the other schemes. The plot shos clearly that amongst competing mesh-restorable design types, the differences to be had are essentially all in the spare capacity. Note that the S.P. Work curve gives the shortest-path routed orking capacity used by the SCA, SBPP, and path restoration designs. In other ords all schemes use almost the same amount of orking capacity. Even here JCA and A meta-mesh employ oint optimization, the orking paths deviate very slightly in length from shortest paths. A related message that Figure 3 gives to someone ho is considering the capacity difference beteen ring and mesh schemes is that the relative benefit in orking path routing is obtained by going to any kind of mesh scheme. Only after that does the exact type of mesh scheme matter to further savings obtainable through spare capacity efficiency. The ra results behind Figures and 3 exhibit some effects that e attribute to limited routeenumeration and effects related to shortest-path B routing. It should not for instance strictly be possible Figure 4: Illustrating routing infeasibility in SBPP. for capacity to rise anyhere on the curves of Figure 3 hen going from one graph topology to a more connected one (i.e. left to right) in the progression that Figure 3 is based upon. This is because ithin the more connected graph, the routing and capacitation on a prior less connected graph alays represents a feasible default solution that could be adopted if the design cost appeared to rise in the more connected test graph. This can, hoever, happen in cases here the demands are strictly shortest-path routed in each test solution. Any changes to orking path routing, even ones that shorten the orking routes, can in principle cause increases in spare capacity requirements, especially if there is also a practical limit on the number of eligible restoration routes represented in the formulations. Having noted this computational effect in our ra data, it is not large enough to affect the relative comparison of schemes significantly. Figure 3 does, hoever, contain a fe data points adusted to minimize these anomalous effects. Several points on the SCA and M-M curves are numerically adusted ithin a fe percent to account for this. The adustments are to represent that one ould not in practice accept a higher capacity cost after finding a loer cost design on a sparser graph B. Bounds and Estimator Results Figure 5 reproduces the SCA and JCA redundancy curves overlaid by curves of the estimators developed above, hich are parametric in / as a characterization of the orking capacity disparity values. 0% Distance-Capacity Redundancy 0% 00% 90% 80% 70% 60% JCA SCA k ( Est #: k = Est #: k = 0 Est #: k = 5 Est #: k =.5 Est #: k = 5 Est #: k = 50% Est #: k =.5 ( d -) bound 40% Netork Average Nodal Degree, d Figure 5: Comparing SCA and JCA results to the estimators for span-restorable netorks. 8

9 V. CONCLUDING COMMENTS The main contribution of this ork is the systematic comparison of six mesh survivability schemes against each other and in a setting that shos ho they each respond to netork graph connectivity. All schemes ere found to have almost identical orking capacity requirements, the important differences being in spare capacity requirements. Routing infeasibilities are noted in the provisioning of the shared backup path protection scheme at lo netork degree. An interesting finding is that the chain optimized meta-mesh concept can beat the /( d -) bound on span-restorable redundancy but still employs a span restoration mechanism. As expected, path restoration outperforms all other mechanisms but is also the most complicated. Meta-mesh and SBPP are almost as efficient as true path restoration but simpler to implement (and in the meta-mesh case, easier to implement). In the ork on bounds and estimators, e find that the /( d -) bound for span restoration seems to truly explain the ay in hich span-restorable schemes react to graph connectivity. The to estimators give added insights about ho balancing orking capacity influences achievable redundancy but they do not closely track the design results for either SCA or JCA against d. Their value may be more as a stepping stone to further ork on bounds or approximators based on a coupled node-pair vie. VI. REFERENCES [] W. D. Grover, J. Doucette, Topological design of mesh-restorable transport netorks, accepted for Annals of Operations Research, Special Issue on Topological Design of Telecommunication Netorks, June 00, (5 ms). [] W. D. Grover, J. Doucette, Increasing the Efficiency of Span-Restorable Mesh Netorks on Lo-Connectivity Graphs, Proc. of 3 rd Int. Workshop on the Design of Reliable Communication Netorks (DRCN 00), Budapest, Hungary, October 00. [3] W. D. Grover, Distributed Restoration of the Transport Netork, Chapter in Netork Management into the st Century, editors T. Plevyak, S. Aidarous, IEEE / IEE Press co-publication, February 994, pp [4] D. A. Dunn, W. D. Grover, M. H. MacGregor, A comparison of k-shortest paths and maximum flo methods for netork facility restoration, IEEE Journal on Selected Areas in Communications, vol., no., January 994, pp [5] R. Kaamura, K. Sato, I. Tokizaa, Self-healing ATM netorks based on virtual path concept IEEE Journal on Selected Areas in Communications, vol., no., January 994, pp [6] W. D. Grover, Y. Zheng, VP-Based Netork Design ith Controlled Over-Subscription of Restoration Capacity, Proc. of st Int. Workshop on the Design of Reliable Communication Netorks (DRCN 998), Brugge, Belgium, May 998, O33. [7] S. Kini, M. Kodialam, T. V. Laksham, C. Villamizar, Shared backup Label Sitched Path restoration, Work in Progress, IETF Internet Draft, draft-kini-restoration-shared-backup-00.txt, November 000. [8] R.R. Iraschko, W. D. Grover, A Highly Efficient Path-Restoration Protocol for Management of Optical Netork Transport Integrity, IEEE Journal on Selected Areas in Communications, vol.8, no.5, May 000, pp [9] R.Iraschko, M. MacGregor, W. D. Grover, Optimal Capacity Placement for path restoration in STM or ATM Mesh Survivable Netorks, IEEE/ACM Transactions on Netorking, vol. 6, no. 3, June 998, pp [0] J. Doucette, W. D. Grover, Influence of Modularity and Economy-of-scale Effects on Design of Mesh-Restorable DWDM Netorks, IEEE Journal on Selected Areas in Communications, vol. 8, no. 0, October 000, pp [] Y. Xiong, L. G. Mason, Restoration Strategies and Spare Capacity Requirements in Self-Healing ATM Netorks, IEEE/ACM Transactions on Netorking, vol. 7, no., February 999, pp