Chapter 3: Technology Related Modelling Examples

Transcription

1 Chapter 3: Technology Related Modelling Examples Deep Medhi University of Missouri Kansas City DRAFT Version August 2005 Note: These slides are based on material from the book: M. Pióro and D. Medhi, Routing, Flow, and Capacity Design in Communication and Computer Networks, Morgan Kaufmann Publishers/Elsevier, July For brevity, no attribution or referencing is done here; please consult the book for referenced models, attributions, and the list of references. Slides are still evolving; please check back the course web-site for updated slides. Report any errors to Many thanks to Jayesh Kumaran for his help in preparation of these slides. Routing, Flow, and Capacity Design in Communication and Computer Networks p.1/16

2 IP Networks Intra-Domain Traffic Engineering (3.1.1) p = 1, 2,..., P d candidate paths for flows realizing demand d c e capacity of link e δ edp = 1, if link e belongs to path p realizing demand d; 0, otherwise h d volume of demand d w e metric of link e, w = (w 1, w 2,..., w E ) (primary) x dp (w) y e (w) flow allocated to path p of demand d determined by the link system w load of link e determined by the link system w r maximum link utilization variable, r = max y e=1,...,e e (w)/c e The problem of minimizing the maximum link utilization can be formulated as minimize w,r F = r p x dp(w) = h d d p δ edpx dp (w) c e r r continuous w e non-negative integers. d = 1, 2,..., D e = 1, 2,..., E Routing, Flow, and Capacity Design in Communication and Computer Networks p.2/16

3 MPLS Networks Tunneling Optimization (3.2.1 ) p = 1, 2,..., P d number of possible tunnels for demand d c e capacity of link e δ edp = 1, if link e belongs to tunnel p realizing demand d; 0, otherwise h d volume of demand d x dp fraction of the demand volume d carried over tunnel p ε lower bound on fraction of flow on a tunnel (path) u dp = 1, to denote selection of a tunnel if the lower bound is satisfied; 0, otherwise r maximum number of tunnels over all links. (Contd.) Routing, Flow, and Capacity Design in Communication and Computer Networks p.3/16

4 MPLS Networks Tunneling Optimization (3.2.1) (Contd.) The problem of minimizing the number of tunnels on each MPLS router/link and load balancing in an MPLS network can be formulated as minimize x,u,r F = r p x dp = 1 d = 1, 2,..., D d h d p δ edpx dp c e e = 1, 2,..., E εu dp h d x dp d = 1, 2,..., D p = 1, 2,..., P d x dp u dp d = 1, 2,..., D p = 1, 2,..., P d d p δ edpu dp r e = 1, 2,..., E x dp continuous and non-negative binary, r integer. u dp Routing, Flow, and Capacity Design in Communication and Computer Networks p.4/16

5 ATM Networks Virtual Path Design (3.3.2) p = 1, 2,..., P d number of possible paths for a virtual path (VP) for demand d δ edp = 1, if VP path p for demand d uses link e; 0, otherwise h d volume of demand d (Mbps) ξ e unit cost of a 155 Mbps link(lcu) on link e M capacity unit of an ATM link (in terms of the the number of modules) u dp = 1, if path p for demand d is selected link e; 0, otherwise y e capacity of link e (expressed in 155Mbps modules) The problem of determining the link capacity so that the total cost is minimized given that the ATM virtual path demand requirement and link capacity is in modular units (155 Mbps) minimize u,y F = e ξ e y e p u dp = 1, d h d p δ edpu dp My e, u dp binary, y e integers. d = 1, 2,..., D e = 1, 2,..., E Routing, Flow, and Capacity Design in Communication and Computer Networks p.5/16

6 Digital Circuit-Switched Telephone Networks Single Busy-Hour Network Dimensioning (3.4.2) p = 1, 2,..., P d constants number of available routes for demand d δ edp = 1, if link e belongs to path p realizing demand d; 0, otherwise h d volume of demand d in Erlangs(Erl) ξ e unit modular capacity cost of link e b e call blocking probability for link e to maintain a certain grade of service M modular capacity (e.g., in T1 24 voice circuits, or, E1 30 voice circuits) x dp flow allocated to path p of demand d y e capacity of link e expressed as number of modules M. The problem of determining the modular capacity needed in a network so that offered traffic is carried with an acceptable grade -of-service can be formulated as minimize x,y F = e ξ e y e p x dp = h d, F e ( d p δ edpx dp ) My e, x dp continuous, non-negative y e integers d = 1, 2,..., D e = 1, 2,..., E where F e (a) = C(a; b e ). The function C(a; b) is the inverse of the Erlang blocking Deep Medhi/Version formula Augustfor 2005 offered load a and blocking b. Routing, Flow, and Capacity Design in Communication and Computer Networks p.6/16

7 Digital Circuit-Switched Telephone Networks Multi Busy-Hour Network Dimensioning (3.4.3) p = 1, 2,..., P d number of available routes for demand d t = 1, 2,..., T number of traffic hour partitions h dt volume of demand d in Erlangs(Erl) for time partition (hour) t δ edpt = 1, if link e belongs to path p demand d for time partition t; = 0, otherwise call blocking probability for link e for time partition t b et ξ e x dpt unit modular capacity cost of link e flow allocated to path p of demand d for time partition t y e capacity of link e expressed as number of modules M. The problem of determining the modular capacity given that traffic volume is different for different hours of a day minimize x,y F = e ξ e y e p x dpt = h dt, d = 1, 2,..., D t = 1, 2,..., T F et d p δ edptx dpt x dpt continuous, non-negative My e, e = 1, 2,..., E t = 1, 2,..., T y e integers, where F et (a) = C(a; b et ). The function C(a; b) is the inverse of the Erlang blocking formula for offered load a and blocking b. Routing, Flow, and Capacity Design in Communication and Computer Networks p.7/16

8 SONET/SDH Transport Networks Capacity and Protection Design (3.5.1) p = 1, 2,..., P d number of available routes for demand d δ edp = 1, if link e belongs to path p realizing demand d; 0, otherwise h d volume of demand d in term,s of VC-12s ξ e cost of one LCU(STM-1 system ) on link e M = 63 (Each STM-1 module can carry 63 VC-12 containers) x dp flow allocated to path p of demand d capacity of link e (expressed in STM-1 modules) y e The SDH transport network capacity design problem can be formulated as follows : minimize x,y F = e ξ e y e p x dp = h d, d p δ edpx dp My e, x dp, y e non-negative integers d = 1, 2,..., D e = 1, 2,..., E Routing, Flow, and Capacity Design in Communication and Computer Networks p.8/16

9 SONET/SDH Transport Networks Capacity and Protection Design (3.5.2) p = 1, 2,..., P d number of available routes for demand d δ edp = 1, if link e belongs to path p realizing demand d; 0, otherwise h d volume of demand d in term,s of VC-12s ξ en cost of one transmission system STM-n realized on link e M n = 63 n, the modularity of STM-n system x dp flow allocated to path p of demand d number of STM-n systems realized on link e y en The SDH transport network capacity design problem that differentiates costs of STM modules can be formulated as follows : minimize x,y F = e n ξ en y en p x dp = h d, d p δ edpx dp n M ny en, x dp, y en non-negative integers d = 1, 2,..., D e = 1, 2,..., E Routing, Flow, and Capacity Design in Communication and Computer Networks p.9/16

10 SONET/SDH Transport Networks Capacity and Protection Design (3.5.5) q = 1, 2,..., Q e list of restoration paths available for link e ξ e unit cost of link e h d volume of demand d in Erlangs(Erl) c e capacity of link e β feq =1, if link f belongs to path q restoring link; 0, otherwise y e protection capacity of link e capacity restored by path q that restores link f z fq The problem of minimizing the cost of the necessary link protection capacity can be formulated as follows : minimize z,y F = e ξ e y e q z eq = c e, e = 1, 2,..., E e q β fek z eq = y f, f = 1,..., E, e = 1, 2,..., E, f e z eq, y e non-negative integers. Routing, Flow, and Capacity Design in Communication and Computer Networks p.10/16

11 SONET/SDH Rings Ring Bandwidth Design (3.6.1) v = 1, 2,..., V nodes e = 1, 2,..., E segments h vw demand volume between nodes v and w, with v < w M Modularity of the STM system δ evw = 1, if v e < w ; 0, otherwise u vw flow on the clockwise path from w to v flow on the clockwise path from v to w z vw The problem of determining the minimal number and type of (parallel) rings needed can be formulated as minimize u,z,r r u vw + z vw = h vw, v, w = 1, 2,..., V, v < w δ evw u vw + (1 δ evw )z vw Mr, e = 1, 2,..., E u vw, z vw, r non-negative integers. Routing, Flow, and Capacity Design in Communication and Computer Networks p.11/16

12 WDM Networks Restoration Design with Optical Cross-Connects (3.7.1) c = 1, 2,..., C colors v = 1, 2,..., V nodes s = 0, 1,..., S failure situations h ds (d = 1, 2,..., D) volume of demand d to be realized in situation s, ξ e (e = 1, 2,..., E) cost of one LCU (i.e., optical fibre) on link e α es = 0 if link e is failed in situation s ; 1, otherwise δ edp = 1 if link e belongs to path p realizing demand d, ; 0, otherwise θ dps =0 if path p of demand d is failed in situation s ; 1, otherwise x dpc flow (number of light-paths) realizing demand d in color c on path p z ce number of times the color c is used on link e y e capacity of link e expressed in the number of fibers The optimization problem for the OXCs without wavelength conversion can be formulated as minimize x,z,y F = e ξ e y e p θ dps c x dpc h ds, d = 1, 2,..., D s = 0, 1, 2,..., S d p δ edpx dpc = z ce, c = 1,..., C, e = 1, 2,..., E y e z ce, c = 1,..., C, e = 1, 2,..., E. x dpc, z ce, y es non-negative integers Routing, Flow, and Capacity Design in Communication and Computer Networks p.12/16

13 WDM Networks: Restoration Design with Optical Cross-Connects (3.7.2) c = 1, 2,..., C colors v = 1, 2,..., V nodes s = 0, 1,..., S failure situations h ds volume of demand d to be realized in situation s, ξ e cost of one LCU (i.e., optical fibre) on link e κ e link opening cost for the link e α es = 0 if link e is failed in situation s ; 1, otherwise δ edp = 1 if link e belongs to path p realizing demand d, ; 0, otherwise θ dps =0 if path p of demand d is failed in situation s ; 1, otherwise x dpc flow (number of light-paths) realizing demand d in color c on path p z ce number of times the color c is used on link e y e capacity of link e expressed in the number of fibers =0 if the link e is installed; 1, otherwise u e (Contd.) Routing, Flow, and Capacity Design in Communication and Computer Networks p.13/16

14 WDM Networks Restoration Design with Optical Cross-Connects (3.7.2) (Contd.) The optimization problem for the OXCs without wavelength conversion that takes in to account the link opening costs can be formulated as minimize x,z,y,u F = e (ξ e y e + κ e u e ) p θ dps c x dpc h ds, d = 1, 2,..., D s = 0, 1, 2,..., S d p δ edpx dpc = z ce, c = 1,..., C, e = 1, 2,..., E y e z ce, c = 1,..., C, e = 1, 2,..., E y e Mu e, e = 1, 2,..., E. x dpc, z ce, y es non-negative integers M is a large number Routing, Flow, and Capacity Design in Communication and Computer Networks p.14/16

15 IP Over SONET: Combined Two-Layer Design (3.8.1) q = 1, 2,..., Q e list of candidate paths for link e c g capacity of link g in the SONET network expressed in OC-48 modules δ edp = 1, if link e belongs to path p realizing demand d; 0, otherwise h d volume of demand d ξ e link termination cost by the cost of the OC-3 interfaces at the end routers of link e ζ eq routing cost at the SONET layer ρ link utilization coefficient M =size of the link capacity in IP network(e.g., Mbps) N =size of the link capacity in SONET network (e.g., 2, Mbps) w e metric of link e, w = (w 1, w 2,..., w E ) x dp (w) flow allocated to path p of demand d determined by the link system w y e modular capacity of the IP layer link e z eq flow allocated to path q realizing capacity link e γ geq =1, if path q on the transport layer for demand e uses link g ; and 0, otherwise (Contd.) Routing, Flow, and Capacity Design in Communication and Computer Networks p.15/16

16 IP Over SONET: Combined Two-Layer Design (3.8.1) (Contd.) The problem to determine the capacity required for the IP links, and the routing of these links in the SONET network in an integrated manner to minimize the IP network cost can be formulated as minimize w,y,z e ξ e y e + e q ζ eq z eq p x dp(w) = h d, d p δ edpx dp (w) ρmy e, q c z eq = y e, e M q γ geq z eq Nc g, w e non-negative integer y e, z eq non-negative integer. d = 1, 2,..., D e = 1, 2,..., E e = 1, 2,..., E g = 1, 2,..., G Routing, Flow, and Capacity Design in Communication and Computer Networks p.16/16