10. Network dimensioning

Transcription

1 Partly based on slide material by Samuli Aalto and Jorma Virtamo ELEC-C7210 Modeling and analysis of communication networks 1

2 Contents Introduction Parameters: topology, routing and traffic Dimensioning for circuit switched networks Dimensioning for data networks with packet-level delay Dimensioning for data networks with elastic traffic 2

3 Role of dimensioning (1) First we need the network topology (nodes and links) Given by topology design problem, as studied in previous lecture Topology defines the connectivity between nodes but link capacities are still unknown Role of network dimensioning in network design Network topology (nodes and links), routing and offered traffic is given Provides reasonable estimates of the needed capacity to handle the offered traffic in the network at a given quality of service Models should be also simple enough so that it is not computationally too difficult to solve them! 3

4 Role of dimensioning (2) Performance analysis gives models that explain, for a given system, dependencies between: service quality of service traffic load system capacity capacity traffic In dimensioning, we use the models, fix QoS and load and ask: what is the needed capacity? 4

5 10. Network dimensioning Dimensioning as an optimization problem Typically formulated as an optimization problem min link capacities (or costs) so that offered traffic, link capacities QoS target Nonlinear optimization problems The specific form of the problem depends on network traffic assumptions and the associated performance measure 5

6 Aim of the lecture We consider three traffic models Circuit switched traffic Packet-level data traffic Elastic flow-level data traffic For end-to-end performance in the network we use the models already introduced in Network models lecture Aim is to show: The associated dimensioning problem formulations And how to solve them (numerically or analytically) 6

7 Contents Introduction Parameters: topology, routing and traffic Dimensioning for circuit switched networks Dimensioning for data networks with packet-level delay Dimensioning for data networks with elastic traffic 7

8 Input parameters to dimensioning problem Topology Nodes and the links (Simplified) routing Traffic matrix / offered traffic Traffic forecasting Busy hour traffic 8

9 Topology A data network consists of nodes and links Let N denote the set of nodes indexed with n Let J denote the set of links indexed with j Example: N < {a,b,c,d,e} J < {1,2,3,,12} a 1 2 b e 8 c d link 1 from node a to node b link 2 from node b to node a Let c j denote the capacity of link j 9

10 Routing in the network For dimensioning purposes, reasonable at this point to assume shortest path routing Exactly one route, or path, between every pair of nodes Path cost just number of hops Each path constitutes a class in our model Not an accurate assumption of actual routing but simplifies modeling Remember we aim at computationally simple models 10

11 Paths (or classes) We define a path (= route) as a set of consecutive links connecting two nodes Let P denote the set of paths indexed with p Each path p is a class in the model Example: three paths from node a to node c: 1 a 2 b e 11 8 c d red path consisting of links 1 and 3 green path consisting of links 11 and 6 blue path consisting of links 10, 8 and 6 11

12 Path matrix Each path consists of a set of links This connection is described by the path matrix A, for which element a jp < 1 if j p, that is, link j belongs to path p otherwise a jp < 0 each path is a column in the matrix each row tells which paths use that link Example: three columns of a path matrix ac1 ac2 ac

13 Need for traffic measurements and forecasts To properly dimension the network we need to estimate the traffic offered If the network is already operating, the current traffic is most precisely estimated by making traffic measurements in particular, then one estimates the busy hour traffic (recall from Traffic models lecture) Otherwise, the estimation should be based on other information, e.g. estimations on characteristic traffic generated by a subscriber estimations on the number of subscribers (from demographic data of population densities in the area) Long time-span of network investments it is not enough to estimate only the current traffic forecasts of future traffic are also needed 13

14 Traffic matrix Final result (that is, the forecast): traffic matrix describing the traffic interest between nodes (traffic areas) Traffic matrix = (t(i,j)) describes traffic interest between nodes N 2 elements (N = nr of exchanges) element t(i,i) tells the estimated traffic within node i, we assume however t(i,i) = 0 element t(i,j) tells the estimated traffic from node i to node j 14

15 Traffic matrix and offered traffic vector Since we have exactly one path for each pair (i,j), elements of traffic matrix can be represented by the offered traffic vector =,, Element gives the offered traffic on path p Unit of depends on traffic assumptions Unidirectional vs. bidirectional traffic Data traffic is unidirectional and thus each element is associated with the directed offered traffic from node i to node j. Thus, also link are directed. Circuit switched traffic is bidirectional and thus each element is associated with the sum of the offered traffic from node i to node j and from node j to node i. Thus, also links in the network and the paths are not directed. 15

16 10. Network dimensioning Link loads Then the traffic load on link j is given by the sum of offered traffic from the paths (or classes) that use link j = = () Recall from Network models lecture, R(j) is the set of paths (or classes) that use link j In matrix form, we obtain the vector of link loads r by = All our dimensioning models use link loads as traffic parameters 16

17 path links Example p ab {1} p ac {6,3} p ad {6} Consider bidirectional traffic and assume t(i,j) = 1 for all node pairs (i,j) We have 10 shortest paths, see table Thus, the corresponding link loads are: = = =4 =2 (1+1)=4 =2 ( )=8 =2 (1+1)=4 =2 (1+1)=4 =2 (1+1)=4 1 p ae {5} p bc {2} p bd {2,3} p be {1,5} p cd {3} p ce {3,4} p de {4} b 2 c 3 With unidirectional traffic all links need to be separated based on direction a 6 5 e 4 d 17

18 Contents Introduction Parameters: topology, routing and traffic Dimensioning for circuit switched networks Dimensioning for data networks with packet-level delay Dimensioning for data networks with elastic traffic 18

19 Offered traffic and performance requirement For circuit switched traffic, dimensioning is based on loss models and blocking probability Offered traffic On each path p new calls arrive at rate [1/s] Each call has an average duration h [s] Offered traffic on path p is simply their product (dimensionless erlangs) = h [erl] Recall that traffic on path p consists of the traffic from node i to node j, as well as traffic from node j to node i (bidirectional traffic!) Performance requirement End-to-end loss probability on each path < 19

20 Example: dimensioning curve for a single link For a single link we require: B req 1% Required link capacity: n = min{i = 1,2, Erl(i,a) B req } required link capacity n offered traffic a In a network, we apply the product bound (recall from Network models ) 20

21 10. Network dimensioning Dimensioning problem for circuit swítched traffic Problem formulation min so that =1 1 Erl, <, () Where the link weight parameter gives the price/capacity unit May depend, e.g., on the distance of the link This is a non-linear constrained integer optimization problem Not easy to solve numerically So called Moe s principle can be used to provide numerically efficiently a good (close to optimal) feasible solution to the problem Idea: Add capacity little by little on those links where the benefit of reduction in blocking probability is greatest until feasible solution is found 21

22 10. Network dimensioning Moe s principle in a network Let +1 denote the product bound on path p with link j having one more capacity unit, i.e., +1 =1 1 Erl + 1, 1 Erl, \ Initially, network is underdimensioned, e.g., =1on all links Iteration: Calculate gain Δ (decrease in blocking by adding one capacity unit relative to link cost) for each link. However, many paths use link j and we select the path that benefits the most and does not meet the target B req. Δ = 1 max : +1 On link j with maximum gain Δ, we increase the capacity by one unit, i.e., let = arg max Δ, and set +1 Continue until < on all paths p 22

23 Contents Introduction Parameters: topology, routing and traffic Dimensioning for circuit switched networks Dimensioning for data networks with packet-level delay Dimensioning for data networks with elastic traffic 23

24 Offered traffic and performance requirement For packet switched traffic, dimensioning is based on queuing models and the packet-level delay Offered traffic On each path p new packets arrive at rate [1/s] Each packet has an average length L [bits] Offered traffic on path p is simply their product (in bps) = [bps] Performance requirement Mean end-to-end packet delay of the network < 24

25 Dimensioning for a single link Pure queueing system Performance measure = Mean delay E[D] Formula for the mean waiting time E[D] (assuming that θ < 1): E[ D] < 1 < λ, κ L c, t Note: service rate λ = c/l and offered load parameter t = κl E[D] grows to infinity as offered load t tends to 1 25

26 Dimensioning curve for a single link Grade of Service requirement: E[D] 2*mean service time Allowed traffic θ 0.5 = 50% < 0.5 Required capacity c 2 t (blue line) required capacity c offered traffic t = L In a network, we apply the overall mean delay in the network along each path (recall from Network models ) 26

27 10. Network dimensioning Dimensioning problem for packet switched traffic Recall that the mean delay along each path is given by = Problem formulation min so that = 1 Λ () Where Λ=, i.e., total offered packet arrival rate This is a non-linear, but convex, constrained optimization problem However, this can be solved explicitly and we get a simple closed form solution for the capacities! In the optimization, the inequality constraint is obviously solved as equality 27

28 10. Network dimensioning Dimensioning problem formulation The constrained problem is converted to an unconstrained problem Include the constraint in the objective multiplied by a free parameter α = 1 Λ () Function G is so-called Lagrangian function in optimization theory For any value of α one can find the global minimum of G. Now, determine α such that the minimum satisfies the constraint. Then the global minimum of the expression G is located on the hypersurface determined by the constraint. The value of G on the hypersurface of the constraint is, for any α, the same as the original objective function, since the constraint is satisfied. Then the global minimum of G is certainly also the minimum of the function G constrained on the hypersurface. Thus, the found solution minimizes the objective function and satisfies the constraint condition and is the solution for our problem. 28

29 Performing the minimization (1) Objective function is continuous At optimum, partial derivative w.r.t. equals zero = + Λ 1 =0 = + Λ Note! Negative root not possible because of stability requirement > We get as a function of 29

30 10. Network dimensioning Performing the minimization (2) Inserting this back in the constraint, we can solve optimum α Finally, we obtain 1 Λ Λ =0 () 1 Λ = + 1 Λ Λ =0 Capacity must be at least due to stability and second term determines the additional capacity needed to satisfy our delay target! 30 = 1 Λ

31 Contents Introduction Parameters: topology, routing and traffic Dimensioning for circuit switched networks Dimensioning for data networks with packet-level delay Dimensioning for data networks with elastic traffic 31

32 Offered traffic and performance requirement For elastic data traffic, dimensioning is based on sharing models and flow-level delay or throughput For elastic data traffic we presented M/M/1-PS queue Single link with capacity C All flows share capacity perfectly fairly Under exponential assumptions same B-D process as ordinary M/M/1 FIFO queue Network model for elastic data traffic Generally called bandwidth sharing networks Not presented in the Network models lecture Reason: no simple models exist Even under Markovian assumptions we end up with complex multidimensional Markov processes with state-dependent rates 32

33 Dimensioning on a single link Throughput in M/M/1-PS queue Offered traffic = (arrival rate of flows times mean flow size) Throughput is just linear with respect to t : = Dimensioning very simple Throughput requirement Required capacity: + Excess capacity needed on top of t (stability) is just! Needed capacity as a function of offered traffic for =

34 Upper bound on throughput in bandwidth sharing networks Model On each path p new flows arrive at rate [1/s], routing matrix A Each flow has an average size S [bits] Offered traffic on path p is simply their product (dimensionless erlangs) = [bps] State gives number of on-going flows in each path On-going flows share fairly the capacity (somehow, characterized by some function of the state) Link load vector, as earlier, = Under certain conditions on the sharing function, upper bound for throughput of flows on path p is given by min ( ) Very natural interpretation: throughput bounded by throughput on bottle neck link along the path! 34

35 10. Network dimensioning Dimensioning problem for elastic traffic In a network, reasonable to require that on all paths in the network flows achieve some minimum target throughput min so that = min (), The constraints on the throughput in fact determine the solution The minimum operation can be replaced by writing the constraints separately for each link in J(p) since if the constraint is satisfied for all links it is satisfied by the minimum. Then do the same for all paths. We get many copies of the constraints per each link but the set of unique constraints are simply for all links. Thus, solution is = + That is, each link j is dimensioned as if it is an independent M/M/1-PS queue! 35

36 THE END What you should understand/remember: The concepts: network topology, traffic matrix Link loads (as determined by network topology, traffic matrix and shortest paths) Dimensioning as a nonlinear optimization problem Dimensioning of single link based on our elementary models for different traffic types Dimensioning of networks based on network level performance measure (product bound, overall packet level delay, mean flow throughput for elastic traffic) Role of dimensioning: to provide reasonable estimates for needed capacity! 36