Integer Linear Programming Network Design and Planning (2016) - PowerPoint PPT Presentation

Optimization problems Classification Optimization problem – optimization version  – Find the minimum-cost solution Optimization problem – decision version (answer is yes or no)  – Given a specific bound k , tell me if a solution x exists such that x<k  Polynomial problem – The problem in its optimization version is solvable in a polynomial time  NP problem – Class of decision problems that, under reasonable encoding schemes, can be solved by polynomial time non-deterministic algorithms  NP-complete problem – A NP problem such that any other NP problem can be transformed into it in a polynomial time – The problem is very likely not to be in P – In practice, the optimization-problem solution complexity is exponential  NP-hard problem – The problem in its decision version is not solvable in a polynomial time (is NP- complete)  the optimization problem is harder than an NP-problem – Contains an NP-complete problem as a subroutine WDM Network Design 22

Dimensions of Complexity… Routing Cost Mixed Rates Physical layer Protection Grooming Wavelength Energy Time Fiber Vendors Applications Domains WDM Network Design 23

Optimization problem solution Optimization methods  WDM network optimal design is a very complex problem. Various approaches proposed – Mathematical programming (MP) • Exact method (guarantees optimal solution) • Computationally expensive, not scalable – Heuristic methods • An alternative to MP for realistic dimension problems  According to the cost function, the problem is – Linear – Non linear WDM Network Design 24

Outline  Introduction to WDM network design and optimization  Integer Linear Programming approach  Physical Topology Design – Unprotected case – Dedicated path protection case – Shared path & link protection cases  Heuristic approach WDM Network Design 25

Optimization problem solution Mathematical programming solving  WDM-network static-design problem can be solved with the mathematical programming techniques In most cases the cost function is linear  linear programming – Variables can assume integer values  integer linear programming –  LP solution – Variables defined in the real domain – The well-known computationally-efficient Simplex algorithm is employed  ILP solution – Variables defined in the integer domain – The optimal integer solution is found by exploring all integer admissible solutions • Branch and bound technique: admissible integer solutions are explored in a tree-like search WDM Network Design 26

ILP application to WDM network design Approximate methods  A arduous challenge – NP-completeness/hardness coupled with a huge number of variables • In many cases the problem has a very high number of solutions (different virtual-topology mappings leading to the same cost-function value) – Practically tractable for small networks  Simplifications – RFWA problem decomposition: e.g., first routing and then f/w assignment – Constrained routing (route formulation, see in the following ) – Relaxed solutions (randomized rounding) – Other methodologies: • Column generation, Lagrangean relaxation, etc..  ILP, when solved with approximate methods, loses one of its main features: the possibility of finding a guaranteed minimum solution – Still ILP can provide valuable solutions WDM Network Design 27

ILP application to WDM network design ILP relaxation  Simplification can be achieved by removing the integer constraint – Connections are treated as fluid flows (multicommodity flow problem) • Can be interpreted as the limit case when the number of channels and connection requests increases indefinitely, while their granularity becomes indefinitely small • Fractional flows have no physical meaning in WDM networks as they would imply bifurcation of lightpaths on many paths – LP solution is found  In some cases the closest upper integer to the LP cost function can be taken as a lower bound to the optimal solution  Not always it works…  See [BaMu96] WDM Network Design 28

The ILP models: Notation k l  l,k : link identifiers (source and destination nodes)  F l,k : number of fibers on the link l,k   l,k : number of wavelengths on the link l,k  c l,k : weight of link l,k (es. length, administrative weight, etc.) – Usually equal to c k,l WDM Network Design 29

Cost functions Some examples  RFWA  ,  min min – Minimum fiber number M F M l k • Terminal equipment cost ( , ) l k – VWP case – Minimum fiber mileage (cost) M C   min min c F M • Line equipment [BaMu00] , , l k l k C ( , ) l k  RWA – Minimum wavelength number     min min , l k – Minimum wavelength mileage ( , ) l k  ,    min min c • [StBa99],[FuCeTaMaJa03] , l k l k C – Minimum maximal wavelength ( , ) l k   min  number on a link min [ ] max , l k MAX • [Mu97] , l k WDM Network Design 30

Outline  Introduction to WDM network design and optimization  Integer Linear Programming approach  Physical Topology Design – Unprotected case – Dedicated path protection case – Shared path & link protection cases  References WDM Network Design 31

Approaches to WDM design  Two basilar and well-known approaches [WaDe96],[Wi99] – FLOW FORMULATION (FF) – ROUTE FORMULATION (RF) WDM Network Design 32

Flow Formulation (FF) x , s d  Flow variable , l k  Flow on link ( l,k) due to a request generated by to source-destination couple ( s,d)  Fixed number of variables  Unconstrained routing s , d s , d 1 x x , , 2 2 , 1 s d , s d x x 1 2 1 , s 2 , d , s d x , s d x , 1 s , , s d s d d , 2 1 2 x x , 3 , 4 s d , , s d s d x x 3 , 1 4 , 2 , s d s , d x x , 3 s , 4 d 4 3 , s d , x s d x , , s d s d 3 , s 3 x x 4 , d , 4 4 , 3 WDM Network Design 33

ILP application to WDM network design Flow Formulation (FF)  Variables represent the amount of traffic (flow) of a given traffic relation (source-destination pair) that occupies a given channel (link, wavelength)  Lightpath-related constraints – Flow conservation at each node for each lightpath (solenoidal constraint) – Capacity constraint for each link – (Wavelength continuity constraint) – Integrity constraint for all the flow variables (lightpath granularity)  Allows to solve the RFWA problems with unconstrained routing  A very large number of variables and constraint equations WDM Network Design 34

ILP application to WDM network design Flow formulation fundamental constraints  Solenoidality constraint s d – Guarantees spatial continuity of the lightpaths (flow conservation) – For each connection request, the neat flow (tot. input flow – tot. output flow) must be: • zero in transit nodes • the total offered traffic (with appropriate sign) in s and d  Capacity constraint – On each link, the total flow must not exceed available resources 2 4 (# fibers x # wavelengths)  Wavelength continuity constraint 1 6 – Required for nodes without converters 5 3 WDM Network Design 35

Notation  c : node pair (source s c and destination d c ) having requested one or more connections  x l,k,c : number of WDM channels carried by link ( l , k) assigned to a connection requested by the pair c  A l : set of all the nodes adjacent to node i  v c : number of connection requests having s c as source node and d c as destination node  W : number of wavelengths per fiber  λ : wavelength index ( λ = {1, 2 … W }) WDM Network Design 36

Unprotected case VWP, FF   if v l d c c          Solenoidal ity if , x x v l s l c , , , , k l c l k c c c    k A k A  0 otherwise l l     Capacity ( , ) x W F l k , , , l k c l k c  integer x c,(l,k) , , l k c Integrity  integer F (l,k) , l k N.B. from now on, notation “  l,k” implies l  k WDM Network Design 37

Note, the unsplittable case!   1 if l d c          Solenoidal ity 1 if , x x l s l c , , , , k l c l k c c    k A k A  0 otherwise l l     Capacity ( , ) v x W F l k , , , c l k c l k c  binary x c,(l,k) , , l k c Integrity  integer F (l,k) , l k WDM Network Design 38

Homework  What happens in the bidirectional case? – i.e., Each transmission channel provides the same capacity λ in both directions. WDM Network Design 39

Extension to WP case  In absence of wavelength converters, each lightpath has to preserve its wavelength along its path  This constraint is referred to as wavelength continuity constraint  In order to enforce it, let us introduce a new index in the flow variable to analyze each wavelength plane  x x  , , , , , l k c l k c  The structure of the formulation does not change. The problem is simply split on different planes (one for each wavelength)  The v c traffic is split on distinct wavelengths  v c, λ  The same approach will be applied for no-flow based formulations WDM Network Design 40

Unprotected case WP, FF   if v l d  , c c           if , , x x v l s l c    , , , , , , , k l c l k c c c  Solenoidal ity   k A k A 0 otherwise  l l    v v c  , c c      Capacity ( , ), x F l k  , , , , l k c l k c   integer , x c,(l,k)  , , , l k c  Integrity integer F (l,k) , l k   integer , v c  c , WDM Network Design 41

Flow (FF) vs Route (RF) Formulation ROUTE FLOW , , s d s d 1 x x , s d , 2 2 , 1 s , d x x r 1,s,d 1 2 1 , s 2 , d s , d s d x s , d x r 3,s,d s , 1 , , s d s d 1 2 d , 2 x x , 3 , 4 s d s , d s , d x x 3 , 1 4 , 2 , s d , x s d x , 3 s , 4 d 4 3 r 2,s,d s , d x s , d x s , d s , d 3 , s 3 x x 4 , d , 4 4 , 3  Variable x l,s,d: flow on link i  Variable r p,s,d : number of associated to source-destination connections s,d routed on the couple s-d admissible path p  Fixed number of variables  Constrained routing  Unconstrained routing -k-shortest path  Sub-optimality? WDM Network Design 42

WDM mesh network design Route formulation  All the possible paths between each sd -pair are evaluated a priori  Variables represent which path is used for a given connection – r psd : path p is used by r psd connections between s and d  Path-related constraints – Routing can be easily constrained (e.g. using the K- shortest paths) • Yen’s algorithm – Useful to represent path-interference • Physical topology represented in terms of interference (crossing) between paths (e.g. i pr = 1 (0) if path p has a link in common with path r)  Number of variables and constraints – Very large in the unconstrained case, – Simpler than flow formulation when routing is constrained WDM Network Design 43

Unprotected case VWP, RF  New symbols – r c,n : number of connections routed on the n-th admissible path between source destination nodes of the node-couple c – R (l,k) : set of all admissible paths passing through link (l,k)    Solenoidal ity r v c c , c n n     Capacity ( , ) r W F l k , , c n l k  r R c , n l , k  integer ( , r c n) , c n Integrity  integer F (l,k) , l k WDM Network Design 44

Unprotected case WP, RF  Analogous extension to FF case – r c,n, λ = number of connection routed on the n-th admissible path between node pair c (source-destination) on wavelength λ     , r v c   , , , c n c Solenoidal ity n    v v c  , c c      Capacity ( , ), r F l k  , , , c n l k  r R , , c n l k   integer ( , , r c n)  , , c n  Integrity integer F (l,k) , l k   integer , v c  , c WDM Network Design 45

ILP source formulation New formulation derived from flow formulation  Reduced number of variables and constraints compared to the flow formulation  Allows to evaluate the absolute optimal solution without any approximation and with unconstrained routing  Can not be employed in case path protection is adopted as WDM protection technique – Does not support link-disjoint routing WDM Network Design 46

ILP source formulation Source Formulation (SF) fundamental constraints   , s d x x  New flow variable , , , l k s l k d – Flow carried by link l and having node s as source – Flow variables do not depend on destinations anymore  Solenodality – Source node • the sum of x l,k,s variables is equal to the total number of requests originating in the node – Transit node • the incoming traffic has to be equal to the outgoing traffic plus the nr. of lightpaths terminated in the node WDM Network Design 47

ILP source formulation An example  2 connections requests Both routing solutions are compatible with the same SF solution – 1 to 4 – 1 to 3 2 3  Solution 1 – X 1,2,1 ,X 1,6,1, X 2,3,1 , 4 X 6,5,1 ,X ,5,3,1 ,X ,3,41 = 1 – Otherwise X ,l,k,1 = 0  1° admissible solution 6 5 – L A 1-2-3-4 – L B 1-6-5-3 3  2° admissible solution – L A 1-2-3 – L B 1-6-5-3-4 WDM Network Design 48

Unprotected case VWP, SF  New symbols – x l,k,i : number of WDM channels carried by link l , k assigned connections originating at node i – C i, j : number of connection requests from node i to node j  See [ToMaPa02]      x C S i i , , , i k i i j  j k A Solenoidal ity i       , ( ) x x C i l i l , , , , , k l i l k i i l   k A k A l l     Capacity ( , ) x W F l k , , , l k i l k i  integer x i,(l,k) , , l k i Integrity  integer F (l,k) , l k WDM Network Design 49

Unprotected case WP, SF  Analogous extension to FF case     , x s i   i , k , i , i ,  k A i      s S C i  , , i i i l  Solenoidal ity l        , , , ( ) x x c i l i l    , , , , , , , , k l i l k i i j   k A k A l l     , , ( ) c C i l i l  i , l , i , l      Capacity ( , , ) x F l k  , , , , l k i l k i   integer , , x l,k i  , , , l k i  integer F l,k l , k Integrity   integer , s i  , i    integer , , , ( ) c i j i l  , , i l WDM Network Design 50

ILP source formulation Two-step solution of the optimization problem  The source formulation variables x l,k,s and F l,k – Do not give a detailed description of RWFA of each single lightpath – Describe each tree connecting a source to all the connected destination nodes (a subset of the other nodes) – Define the optimal capacity assignment (dimensioning of each link in terms of fibers per link) to support the given traffic matrix STEP 1: Optimal dimensioning computation by exploiting SF (identification problem, high computational complexity) STEP 2: RFWA computation after having assigned the number of fibers of each link evaluated in step 1 (multicommodity flow problem, negligible computational complexity) WDM Network Design 51

ILP source formulation Complexity comparison between SF and FF Formulatio n # const. # variab.    VWP source 2 2 ( 1 ) L S N L S    VWP flow 2 2 ( 1 ) L C N L C    VWP route 2 2 L C C R L        WP source ( 2 ) ( 2 ) 2 W L NS C S W S L S C L – Symbols      WP flow 2 1 2 2 W( L N C) C W C( L) L • N nodes         WP route 1 2 2 C(W ) L W C R W C W L • L links Fully connected virtual topology • R average number of paths per node pair Case FF/SF # const. FF/SF # variab. • W wavelengths per fiber     VWP O O N N • C connection node-pairs     WP O O N N • S source nodes requiring connectivity  The second step has a negligible impact on the SF computational time – F l,k are no longer variables but known terms  Fully-connected virtual topology: C = N (N-1), S=N – Worst case (assuming L << N (N-1)) WDM Network Design 52

Case-study networks Network topology and parameters NSFNET EON   N = 14 nodes N = 19 nodes   L = 22 (bidir)links L = 39 (bidir)links   C = 108 connected pairs C = 342 connected pairs   360 (unidir)conn. requests 1380 (unidir)conn. requests Seattle WA Ithaca Salt Palo Alto Ann Lake Princeton CA Arbor City UT Pittsburgh Lincoln College Boulder Pk. Champaign San Diego CO CA Atlanta Houston TX  Static traffic matrices derived from real traffic measurements  Hardware: 1 GHz processor, 460 Mbyte RAM  Software: CPLEX 6.5 WDM Network Design 53

Case-study networks SF vs. FF: variables and constraints (VWP)  The number of constraints decreases by a factor – 9 for the NSFNet – 26 for the EON  The number of variables decreases by a factor – 8.5 for the NSFNet – 34 for the EON  These simplifications affect computation time and memory occupation, achieving relevant savings of computational resources Rete/Form vincoli variabili Network/Formulation # const. # variab. NsfNet/source 284 570 NsfNet/flow 2552 4840 EON/source 517 1560 EON/flow 13650 53430 WDM Network Design 54

Case-study networks SF vs. FF: variables and constraints (WP)  In the WP case – The number of variables and constraints linearly increases with W – The gaps in the number of variables and constraints between FF and SF increase with W  The advantage of source formulation is even more relevant in the WP case E O N W P Number of variables and constraints 5 5 1 0 s o u r c e v a r i a b l e s f l o w v a r i a b l e s 5 4 1 0 s o u r c e c o n s t r a i n t s f l o w c o n s t r a i n t s 5 3 1 0 5 2 1 0 5 1 1 0 0 0 2 4 6 8 1 0 1 2 1 4 1 6 Number of wavelengths, W WDM Network Design 55

Case-study networks SF vs. FF: time, memory and convergence  The values of the cost function M source obtained by SF are always equal or better (lower) than the corresponding FF results (M flow )  Coincident values are obtained if both the formulations converge to the optimal solution – Validation of SF by induction  Memory exhaustion (Out-Of-Memory, O.O.M) prevents the convergence to the optimal solution. This event happens more frequently with FF than with SF Memory occupation Computational time W SF FF W SF FF 2 0.39MB 1.3MB 2 27m 40m 4 O.O.M O.O.M 4 28m 105m 8 5MB 42MB 8 36s 50m 16 47MB O.O.M 16 6m 10h 32 180MB O.O.M 32 19m 5h WDM Network Design 56

Outline  Introduction to WDM network design and optimization  Integer Linear Programming approach  Physical Topology Design – Unprotected case – Dedicated path protection case – Shared path & link protection cases  Heuristic approach WDM Network Design 57

Protection in WDM Networks (1) Motivations (bit rate)  Today WDM transmission systems allow the multiplexing on a single fiber of up to 160 distinct optical channels – recent experimental systems support up to 256 channels:  A single WDM channel carries from 2.5 to 40 Gb/s (ITU-T G.709)  The loss of a high-speed connection operating at such bit rates, even for few seconds, means huge waste of data !!  The increase in WDM capacity associated with the tremendous bandwidth carried by each fiber and the evolution from ring to mesh architectures brought the need for suitable protection strategies into foreground.  Example: 1ms outage for a 100G x 100Waves fiber [10Tbit/s] means 10Gbit=1.25Gbyte of data lost (n.b.: 1 cd-rom is 0.9 GByte) WDM Network Design 58

Protection in WDM Networks (2) Motivations (network dimension)  Even though fiber are very resilient, the geographical dimension of a backbone network lead to very high chances that the network is operating in a fault state.  Example: failure per 1000Km per year (2001 statistics) ≈ 2 (*) What happens on a continental network?! Hundreds of failures…. 1 11 1200 4800 12 2100 3900 1500 600 600 1500 4 8 1200 7 9 600 13 1500 5 1500 1200 2 3000 300 2700 1500 2400 14 1200 3600 3 3600 10 6 2100 (*) Source www.southern-telecom.com/AFL%20Reliability.pdf WDM Network Design 59

Immediate Causes of Fiber Cable Failures Excavation Flood Fallen Trees 1.30% Other 1.30% 1.30% 9.70% Firearm 1.30% Fire 1.90% Sabotage 2.50% Rodent Dig-ups 3.80% 58.10% Power Line 4.40% Process Error Vehicle 6.90% 7.50% Source: D. E. Crawford, “Fiber Optic Cable Dig -ups – Causes and Cures,” A Report to The Nation – Compendium of Technical Papers, pp. 1-78, FCC NRC, June 1993 WDM Network Design 60

SLA and Provisioning Customer’s concerns: Operator’s concerns: Bandwidth Resource Availability Protection Fee Penalty Network design decisions for protection are etc. very important WDM Network Design 61

3 performance metrics for protection  Resource occupation (resource overbuild)  Availability – Probability to find the service up  Protection Switching Time  Availability goes in tradeoff with the other two!! WDM Network Design 62

Dedicated Path Protection (DPP)  1+1 or 1:1 dedicated protection (>50% capacity for protection) – Both solutions are possible – Each connection-request is satisfied by setting-up a lightpath pair of a working + a protection lightpaths – RFWA must be performed in such a way that working and protection lightpaths are link disjoint  Additional constraints must be considered in network planning and optimization  Transit OXCs must not be reconfigured in case of failure  The source model can not be applied to this scenario WDM Network Design 63

“Max half” formulation (MH) Equation set (VWP case)  This formulation does not need an upgrade of unprotected flow variable  See [ToMaPa04]    2 if v i d c c           Solenoidal ity 2 if ( , ) x x v i d i c c c , , , , l k c l k c      ( , ) I ( , ) I l k l k  0 otherwise i i    Max Half , x x v l c c l , k , c k , l , c     Capacity ( , ) x W F l k , , , l k c l k c  integer , x c (l,k) , , l k c Integrity  integer F (l,k) , l k WDM Network Design 64

“Max half” formulation (MH) Limitations  Same number of variables and constraints as the unprotected flow formulation  Allows to evaluate the absolute optimal solution without any approximation and with unconstrained routing in almost all cases  Requires an a posteriori control to verify the feasibility of obtained solution  Problem: In conclusion, each unity of flow must be modelled independently, such that it can be protected independently WDM Network Design 65

Dedicated Path Protection (DPP) Flow formulation, VWP  New symbols – x l,k,c,t = number of WDM channels carried by link ( l,k) assigned to the t-th connection between source-destination couple c Rationale : for each connection request, route a link-disjoint connection   route two connections and enforce link-disjointness between them.   2 if l d c          Solenoidal ity 2 if , , x x l s l c t c , , , , , , k l c t l k c t    k A k A 0 otherwise  l l    Link - disjoint 1 , , , x x l k c t l , k , c , t k , l , c , t     Capacity , x W F l k , , , , l k c t l k ( , ) c t  binary ( ), x c,t (l,k) l , k , c , t Integrity  integer F (l,k) , l k WDM Network Design 66

Dedicated Path Protection (DPP) Route formulation (RF), VWP  New symbols – r c,n,t = 1 if the t-th connection between source destination node couple c is routed on the n-th admissible path – R (l,k) = set of all admissible paths passing through link (l,k)    Solenoidal ity 2 , r c t , , c t n n    Link - disjoint 1 , , , r c t l k , , c t n   r R R , , , , c t n l k k l     Capacity , r W F l k , , , c t n l k  r R , , , c t n l k  binary , , r c t n , , c t n Integrity  integer F l,k , l k WDM Network Design 67

Dedicated Path Protection (DPP) Route formulation (RF) II, VWP  Substitute the single path variable r c,n by a protected route variable r’ c,n (~ a cycle)  No need to explicitly enforce link disjointness  Identical formulation to unprotected case  How do we calculate the minimum-cost disjoint paths? Suurballe    Solenoidal ity ' r v c c , c n n     Capacity ' , r W F l k , , c n l k  ' ' r R c , n l , k  ' integer , r c n , c n Integrity  integer F l,k , l k WDM Network Design 68

Dedicated Path Protection (DPP) Flow formulation, WP  New symbols – x l,k,c,t , λ = number of WDM channels carried by wavelength λ on link l,k assigned to the t-th connection between source-destination couple c – V c, λ = traffic of connection c along wavelength λ   if v l d   , , c t c          if , , , ; x x v l s l c t  , , c t c   , , , , , , , , k l c t l k c t  Solenoidal ity  0 otherwise   k A k A l l    2 v c  c , t ,      Link - disjoint 1 , , , x x l k c t    , , , , , , , , , l k c t k l c t     Capacity , , x F l k  , , , , , l k c t l k ( , ) c t   binary , , x c,t l,k  , , , , l k c t  Integrity integer F l,k l , k   bynary v c,  c , t , WDM Network Design 69

Dedicated Path Protection (DPP) Route formulation (RF), WP     ( , ), r v c t  , , c t  , , , c t n Solenoidal ity n    2 , v c t  , , c t      Link - disjoint 1 ( , ), ( , ) r c t l k  , , , c t n    r  R R , , , , , c t n l k k l     Capacity ( , ), r F l k  , , , , c t n l k  r  R , , , , c t n l k   binary ( , ), , r c t n  , , , c t n  Integrity integer F l,k , l k   bynary ( ), v c,t  , , c t WDM Network Design 70

Dedicated Path Protection (DPP) Route formulation (RF) II, WP  r c,n, λ = number of connections between source-destination couple c routed on the n-th admissible couple of disjoint paths having one path over wavelength λ 1 and the other over λ 2      ' , r v c   , 1 , 2 c   1 , 2 , , c n 1 , 2 n Solen.    v v c   , c c 1 , 2   , 1 2         Cap. ' ' , , r r F l k 1     , , , , , c n c n l k 1 , 2 2 , 1    '   '  '   ' r R r R 2 , , 1 , 2 , 2 , , 2 , 1 , c n l k c n l k    ' integer , , r c n 1 , 2   , , , c n 1 2 Int.  integer F l,k , l k WDM Network Design 71

Complexity comparison between DPP RF-WP formulations  Nr of variables for r ctn λ -> R x C x T x W – R x C number of single route variables  Nr of variables for r cn λ’λ’’ - > R’ x C x W 2 – R’ x C number of protected route variables – For R=R’, this is preferable if T>W WDM Network Design 72

Outline  Introduction to WDM network design and optimization  Integer Linear Programming approach  Physical Topology Design – Unprotected case – Dedicated path protection case – Shared path & link protection cases  References WDM Network Design 73

Shared Path Protection (SPP)  Sharing is a way to decrease the capacity redundancy and the number of lightpaths that must be managed  Protection-resources sharing – Protection lightpaths of different channels share some wavelength channels – Based on the assumption of single point of failure – Working lightpaths must be link (node) disjoint  Very complex control issues – Also transit OXCs must be reconfigured in case of failure • Signaling involves also transit OXCs • Lightpath identification and tracing becomes fundamental WDM Network Design 74

How to model SPP: the Link Vector (1)  Given : – Graph G(V,E) – Set of connections L i (already routed)          e' e' { | e' E, 0 (e' )} e e e  Link vector    * e' max e e  e '  Specified in IETF (see, e.g., RSVP-TE Extensions For Shared-Mesh Restoration in Transport Networks ) WDM Network Design 75

How to model SPP: the Link Vector (2) A B e 2 (a) Sample e 1 e 3 network and e 8 e 7 C connections e 6 e 4 e 5  * e 2 e 3 e 4 e 5 e 6 e 7 e 8 e 1 e 5 ( 0, 0, 0, 0, 0, 0, 0, 0 ) 0 Initial state : e 5 ( 0, 1, 1, 0, 0, 0, 0, 0 ) 1 Connection A arrival: e 5 ( 1, 2, 1, 0, 0, 0, 0, 0 ) 2 Connection B arrival: e 5 ( 1, 2, 1, 0, 0, 0, 1, 0 ) 2 Connection C arrival: e 5 (a) Evolution of link vector for e 5 WDM Network Design 76

Two ILP approaches for SPP  Explore Shared path protection by both classical approaches a x s,a,c,1,w x a,d,c,1,w  Flow Formulation x s,a,c,1,p x a,d,c,1,p s d – Binary variables x l,k,c,t,p associated to the flow on each link l,k for each b single connection request c,t (p=w X s,b,c,1,w X b,d,c,1,w working flow, p=s protection flow) x s,b,c,1,p x b,d,c,1,p Connection request c,1  Route Formulation – 1 ° approach: integer variables r c,n r 3,s,d associated to each simple path n s joining the node pair c (e.g. s,d ). d r 1,s,d – 2 ° approach: integer variables r’ c,n r 2,s,d associated to the n-th possible working-spare route pair that joins each s-d node pair c WDM Network Design 77

How to calculate «max» in ILP   min (max ) min T Objective x (l, k) l,k,c c   v if l d c c          Solenoidal ity x x v if l s l,c k,l,c l,k,c c c    k A k A  0 otherwise l l    max T x (l,k) l,k,c c     Capacity x W F (l,k) l,k,c l,k c  integer x c,(l,k) l,k,c Integrity  integer F (l,k) l,k WDM Network Design 78

Shared Path Protection (DPP) Flow formulation, VWP   1 if l d c          1 if , , ; x x l s l c t c , , , , , , k l c t l k c t    k A k A  0 otherwise l l Solenoidal ity   1 if l d c          1 if , , ; y y l s l c t c , , , , , , k l c t l k c t    k A k A 0 otherwise  l l      Link - disjoint 1 , , , x x y y l k c t , , , , , , , , , , , , l k c t k l c t k l c t k l c t      , P x W F l k lk , , , , l k c t l k ( , ) c t Capacity    ij , ( , ) P z (l,k) i j lk lk , ct ( , ) c t    ij 1 z x y lk , ct i , j , c , t l , k , c , t  Sharing ( ), , ( , ) c,t (l,k) i j   ij ij , z x z y , , , , , , , , lk ct i j c t lk c i j c t  binary ( ), x c,t (l,k) l , k , c , t  Integrity integer F (l,k) l , k  ij binary ( ), , ( , ) z c,t (l,k) i j lk , ct WDM Network Design 79

Shared Path Protection case VWP, RF  New symbols – R l,k includes all the working-spare routes whose working path is routed on link l,k – R l’ (l,k) includes all the working-spare routes whose working path is routed on bidirectional link l′ and whose spare path is routed on link (l, k)    Solenoidal ity ' r v c c , c n n      ' ' r r W F , , , c n c n l k R R Capacity   ( , ) l k ( , ) ( , ) ( , ) c n l k c n ' l   ( , ), ' ( ) l k l l l  ' integer ( , r c n) , c n Integrity  integer F (l,k) , l k  Similar formulations can be found in [MiSa99], [RaMu99] , [BaBaGiKo99], WDM Network Design 80

Shared Path Protection case WP, RF  New symbols – Variable r c,n, λ 1, λ 2 , where λ 1 indicates the wavelength of the working path and λ 2 indicates the wavelength of the spare path.  – ( , , 2 ) l k R includes all the working-spare routes, whose working path is routed on l ' bidirectional link l’ and whose spare path is routed on link (l, k).    Solenoidal ity ' r v c c   , , c n 1 , 2 n      ' ' r r W F     , , , , , c n c n l k 1 , 2 1 , 2 R R  Capacity     ( , , 2 ) l k ( , , ) ( , , ) ( , ) c n l k c n 2 2 ' l    ( , ), ' , ( ) l k l l l 1    ' integer ( , , , r c n ) 1 2   , , c n 1 , 2 Integrity  integer F (l,k) , l k  All the previous formulations and a additional one can be found in [CoToMaPaMa03]. WDM Network Design 81

Link Protection O M S p r o t e c t i o n N o r m a l ( l i n k p r o t e c t i o n ) C o n n e c t i o n  Dedicated Link Protection (DLP) – Each link is protected by providing an alternative routing for all the WDM channels in all the fibers – Protection switching can be performed by fiber switches (fiber cross-connects) or wavelength switches – Signaling is local; transit OXCs of the protection route can be pre-configured – Fast reaction to faults – Some network fibers are reserved for protection  Shared Link Protection (SLP) – Protection fibers may be used for protection of more than one link (assuming single-point of failure) – The capacity reserved for protection is greatly reduced WDM Network Design 82

Link Protection  Different protected objects are switched 1) Fiber level 2) Wavelength level FAULT EVENT (1) (2) WDM Network Design 83

Link Protection VWP, FF (Fiber Protection Switch)  New symbols – Y (l,k),(L,q) expresses the number of backup fibers needed on link (l,k) to protect link (L,q) failure   if v i d c c  Solenoidal ity         if , x x v i s l c , , , , l k c k l c c c (working)    k A k A l l  0 otherwise   if F l L , L q  Solenoidal ity         if , ( , ) Y Y F l q l L q l , k , ( L , q ) k , l ( L , q ) L . q (spare)      ( , , ) ( , , ) k A l k L q k A l k L q l l  0 otherwise     Capacity ( , ) x W F l k , , , l k c l k c  Y , ( , ) (l,k) L q l , k , ( L , q )  Integrity integer x c,(l,k) l , k , c  integer F (l,k) , l k WDM Network Design 84

Link Protection Cost functions and sharing constraints     Cost function (dedicated case)  min F Y , , , ( , ) l k l k L q , , , l k l k L q  Cost function (shared case)    min F T , , l k l k , , l k l k   s.t. , ), ( , ) T Y (l k L q , , , ( , ) l k l k L q  OSS. Wavelength channel level protection design – Relaxing the integer constraints on Y, each channel is independently protected protected (while not collecting all the channels owing to the same fiber)  See also [RaMu99] WDM Network Design 85

Link Protection Summary results  Comparison between different protection technique on fiber needed to support the same amount of traffic  Switching protection objects at fiber or wavelength level does ot sensibly affects the amount of fibers. – This difference increase with the number of wavelength per fiber WDM Network Design 86

Traffic Grooming Definition  Optical WDM network – multiprotocol transport platform – provides connectivity in the form of optical circuits (lightpaths) ... Electronic Electronic e.g., STM1 layers connection @ 155,52 Mbps SDH ATM IP ... request Lightpath WDM Layer Optical e.g., OTN G.709 connection layers @ 2.5, 10, 40 Gbps Optical transmission provisioning A Big Difference between Electronic Traffic Requests and Optical Lightpath Capacity ! WDM Network Design 87

WDM Network Design 88

Traffic Grooming Multi-layer routing Logical Topology 2 3 Lp2 Suppose Lightpath Capacity: 10 Lp4 Gbit/sec 7 4 1 EXAMPLE CONNECTIONS ROUTING Lp1 C1 (STM1 between 4 →2 ) on Lp1 Lp3 6 5 C2 (STM1 between 4 →7 ) on Lp1 ADM ADM ADM ADM and Lp2 2 3 C3 (STM1 between 1 →5 ) on Lp3 ADM ADM ADM ADM ADM ADM C4 (STM1 between 4 →6 ) on Lp1 and Lp2 and Lp4 7 1 4 ADM ADM ADM ADM Physical Topology 6 5 WDM Network Design 89

WDM Network Design 90

Formulation (Keyao Zhu JSAC03) WDM Network Design 91

Logical Operators: AND, OR, XOR, XNOR  AND, OR, XOR etc can be expressed by using binary variables 1 0 1 0 AND OR XOR XNOR 1 0 1 0 1 0 1 1 1 1 0 1 1 0 1 1 0 0 0 0 1 0 1 0 0 1 0 0    z x z x r x y AND      z y z y 1 ( y ) t x y z x OR XOR      1   z x y z x y z t r Robert G. Jeroslow, “ Logic-based decision support: mixed integer model formulation ”, North-Holland, 1989 ISBN 0444871195 WDM Network Design 92

Some other linear operations WDM Network Design 93

Logical Operators: AND, OR, XOR, XNOR  AND, OR, XOR etc can be expressed by using binary variables 1 0 1 0 AND OR XOR 1 0 1 0 1 1 1 1 0 1 1   y x x 1 2 0 0 0 0 1 0 1 0 0   y x x 1 2   ,   y x x 1 , y x x y x x 1 2 2 2 1      1    y x x y x x 2 y x x 1 2 1 2 1 2  Negated operators (NAND, NOR, XNOR) are implemented with an additional variable z = 1 - y , where y is the “direct” operation WDM Network Design 94

Logical Operators: AND, OR, XOR, XNOR  Also, they can be extended to a general N -variable case: AND OR XOR N  y       ( [ 1 , ] x x y x i N M y x XOR XOR 1 2 i i  1 i ( ...)) x N  XOR XOR    3 ( 1 ) N y x N   i y x  Probably no simpler option i 1 i  exists 1 i or    [ 1 , ] y x i N Arbitrarily large number (at i least equal to N ) N   y x i  1 i WDM Network Design 95

References  Articles – [WaDe96] N. Wauters and P. M. Deemester, Design of the optical path layer in multiwavelength cross- connected networks , Journal on selected areas on communications,1996, Vol. 14, pages 881-891, June – [CaPaTuDe98] B. V. Caenegem, W. V. Parys, F. D. Turck, and P. M. Deemester, Dimensioning of survivable WDM networks , IEEE Journal on Selected Areas in Communications, pp. 1146 – 1157, sept 1998. – [ToMaPa02] M. Tornatore, G. Maier, and A. Pattavina, WDM Network Optimization by ILP Based on Source Formulation , Proceedings, IEEE INFOCOM ’02, June 2002. – [CoMaPaTo03] A.Concaro, G. Maier, M.Martinelli, A. Pattavina, and M.Tornatore, “QoS Provision in Optical Networks by Shared Protection: An Exact Approach,” in Quality of service in multiservice IP Networks , ser. Lectures Notes on Computer Sciences, 2601, 2003, pp. 419 – 432. – [ZhOuMu03] H. Zang, C. Ou, and B. Mukherjee, “ Path-protection routing and wavelength assignment (RWA) in WDM mesh networks under duct-layer constraints ,” IEEE/ACM Transactions on Networking, vol. 11, no. 2, pp.248 – 258, april 2003. – [BaBaGiKo99] S. Baroni, P. Bayvel, R. J. Gibbens, and S. K. Korotky, “ Analysis and design of resilient multifiber wavelength-routed optical transport networks ,” Journal of Lightwave Technology, vol. 17, pp. 743– 758, may 1999. – [ChGaKa92] I. Chamtlac, A. Ganz, and G. Karmi, “ Lightpath communications: an approach to high- bandwidth optical WAN’s ,” IEEE/ACM Transactionson Networking, vol. 40, no. 7, pp. 1172– 1182, july 1992. – [RaMu99] S. Ramamurthy and B. Mukherjee, “Survivable WDM mesh networks, part i - protection,” Proceedings, IEEE INFOCOM ’99 , vol. 2, pp. 744 – 751, March 1999. – [MiSa99] Y. Miyao and H. Saito, “Optimal design and evaluation of survivable WDM transport networks,” IEEE Journal on Selected Areas in Communications , vol. 16, pp. 1190 – 1198, sept 1999. WDM Network Design 96

References – [BaMu00] D. Banerjee and B. Mukherjee, “Wavelength -routed optical networks: linear formulation, resource budgeting tradeoffs and a reconfiguration study,” IEEE/ACM Transactions on Networking , pp. 598 – 607, oct 2000. – [BiGu95] D. Bienstock and O. Gunluk, “Computational experience with a difficult mixed integer multicommodity flow problem,” Mathematical Programming , vol. 68, pp. 213 – 237, 1995. – [RaSi96] R. Ramaswami and K. N. Sivarajan, Design of logical topologies for wavelength-routed optical networks , IEEE Journal on Selected Areas in Communications, vol. 14, pp. 840{851, June 1996. – [BaMu96] D. Banerjee and B. Mukherjee, A practical approach for routing and wavelength assignment in large wavelength-routed optical networks , IEEE Journal on Selected Areas in Communications, pp. 903- 908,June 1996. – [OzBe03] A. E. Ozdaglar and D. P. Bertsekas, Routing and wavelength assignment in optical networks , IEEE/ACM Transactions on Networking, vol. 11, no. 2, pp. 259-272, Apr 2003. – [KrSi01] Rajesh M. Krishnaswamy and Kumar N. Sivarajan, Design of logical topologies: A linear formulation for wavelength-routed optical networks with no wavelength changers , IEEE/ACM Transactions on Networking, vol. 9, no. 2, pp. 186-198, Apr 2001. – [FuCeTaMaJa99] A. Fumagalli, I. Cerutti, M. Tacca, F. Masetti, R. Jagannathan, and S. Alagar, Survivable networks based on optimal routing and WDM self-heling rings , Proceedings, IEEE INFOCOM '99, vol. 2, pp. 726-733,1999. – [ToMaPa04] M. Tornatore and G. Maier and A. Pattavina, Variable Aggregation in the ILP Design of WDM Networks with dedicated Protection , TANGO project, Workshop di metà progetto , Jan, 2004, Madonna di Campiglio, Italy WDM Network Design 97

Outline  Introduction to WDM network design and optimization  Integer Linear Programming approach to the problem  Physical Topology Design – Unprotected case – Dedicated path protection case – Shared path protection case – Link protection  Heuristic approach WDM Network Design 98

WDM mesh network design Heuristic approaches  Heuristic: method based on reasonable choices in RFWA that lead to a sub-optimal solution – Connections are routed one-by-one – In this case we will refer to an example, based on the concept of auxiliary graph  Heuristic strategies can be: – Deterministic: greedy vs. local search • Generic definitions in the following, together with a large example – Stochastic: • E.g.:simulated annealing, tabu search, genetic algorithms • Not covered in this course WDM Network Design 99

Greedy Heuristic (1) Framework  Greedy – Builds the solution step by step starting from scratch – Starts from an empty initial solution – At each iteration an element is added to the solution, such that • the partial solution is a partial feasible solution, namely it is possible to build a feasible solution starting from the partial one • the element added to the solution is the best choice, with respect to the current partial solution (the greedy is a myopic algorithm)  Features – Once a decision is taken it is not discussed anymore – The number of iterations is known in advance (polynomial) – Optimality is usually not guaranteed. WDM Network Design 100