Structure Preserving Model Reduction for Logistic Networks

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1 Structure Preserving Model Reduction for Logistic Networks Fabian Wirth Institute of Mathematics University of Würzburg Workshop on Stochastic Models of Manufacturing Systems Einhoven, June 24 25, joint work with: Michael Schönlein Sergey Dashkovskiy, Michael Kosmykov Thomas Makuschewitz, Bernd Scholz-Reiter

2 Motivation Queueing Systems Ranking in Graphs Heuristics Conclusions

3 Motivation UK SAT (separaters) Brook Hansen (motors) Flender couplings London B NL D Metaproduct (baseplates coupling guards) Siemens (motors) F local supplier (baseplates) Leroy Somer (motors) E local supplier (baseplates) baseplates and coupling guards Metaproduct, local suppliers (F, E) motors Leroy Somer motors Siemens motors Brook Hansen couplings Flender LRVP pumps IND pumps SC pumps

4

5

6 Queueing Systems Queueing systems provide a stochastic framework for the modelling of logistic systems

7 Queueing Systems Consider a set of servers which are able to treat different classes of jobs. Server 1 Server 2 Server 5 Server 3 Server 4

8 Queueing Systems Without loss of generality each class only receives service at one given server. Unserved jobs wait in a queue. Server 1 Server 2 Server 5 Server 3 Server 4

9 Queueing Systems In open systems jobs arrive from the outside according to some stochastic process. Server 1 Server 2 Server 5 Server 3 Server 4

10 Queueing Systems After service jobs leave the network or with a certain probability they go to another station to receive service there. Server 1 Server 2 Server 5 Server 3 Server 4

11 Queueing Systems - The Maths Classes k = 1,..., K. Interarrival times: ξ k (n), n = 0, 1, 2,... i.i.d. E(ξ k (0)) < Service times: η k (n), n = 0, 1, 2,... i.i.d. E(µ k (0)) < routing matrix P = (p ij ), p ij - probability that job of class i becomes a job of class j Assumption: r(p) < 1.

12 Queueing Systems - Balance Equations Classes k = 1,..., K Interarrival times: ξ k (n), n = 0, 1, 2,... i.i.d. E(ξ k (0)) < Service times: η k (n), n = 0, 1, 2,... i.i.d. E(η k (0)) < routing matrix P = (p ij ), p ij - probability that job of class i becomes a job of class j Assumption: r(p) < 1. with Q(t) = Q(0) + A(t) + P T S(t) S(t) A j (t) = max { n } n ξ k (m) t m=0 S(t) is the service process - depends on the service discipline. The state space X is very often countable, but also depends on the service discipline.

13 Why Model Reduction For large scale manufacturing processes a complete description of the process as a queueing system can become too large to be accessible to analysis or reliable simulation. Thus reduced order models are of interest, that capture some of the essential features of the original network. We argue that in logistic networks structure of the network is an essential feature so classical reduction techniques from control theory do not really help.

14 Ranking in Graphs Ranking schemes try to extract information about the importance/relevance of a vertex from graph properties.

15 Ranking in Graphs - Eigenvector Centrality Given a weighted adjecency matrix A corresponding to a directed graph, the following steps are performed Colums are rescaled to have column sum 1 (where possible). A is made colum stochastic by adding artificial entries in zero columns. A is made irreducible e.g. by considering for some α (0, 1) Ã := αa + (1 α)ee T Then Perron-Frobenius theory says that there is an eigenvector r > 0 (unique up to scaling) such that Ar = r The entries of r quantify the importance of the nodes.

16 Ranking in Graphs: Eliminating zero columns

17 Ranking in Graphs: Eliminating zero columns

18 Ranking in Graphs - Ensuring irreducibility Given the weighted adjecency matrix A R n n consider the enlarged matrix [ A ] R (n+m) (n+m)

19 Ranking in Graphs Consider the graph described by A as weakly coupled with a larger network. Coupling described by vectors v and w. [ ] αa + (1 α)vn e T n w n e T m B = (1 α)v m e T n w m e T m Notation: e := [ ] T

20 Ranking in Graphs B = [ αa + (1 α)vn e T n w n e T m (1 α)v m e T n w m e T m Proposition Assume that B is irreducible, then if x = [ xn T xm T ] T is an eigenvector corresponding to the eigenvalue 1 of B then x n is an eigenvector corresponding to the eigenvalue 1 of the matrix ( ) A α (v, w) := αa + (1 α) v n + et mv m 1 e T w n e T n. mw m Furthermore, A α (v, w) is irreducible. Interpretation: The added vector describes a weak coupling of the network under consideration to a larger network, that has been neglected. ]

21 Dynamic Large-Scale A simple example Logistics Networks Important locations of the test case 3rd-tier supplier (Production of A) v 1 (35) v 2 (58) v 3 (47) v 4 (41) v 5 (44) v 6 (25) 2nd-tier supplier (Production of B) v 7 (70) v 8 (70) v 9 (82) v 10 (28) 1st-tier supplier (Production of C) v 11 (93) v 12 (47) v v 13 (73) 14 (37) OEM locations (Production of D) v 15 (140) v 16 (110) Local warehouses (Storage and shipping of D) v 17 v 18 High importance Low importance Retailer (Distribution of D) v 19 v 20 v 21 v 22 Frontiers in network science - advances and applications Volkswagen Symposium: September 2009, Berlin 10/17

22 Ranking in Queueing Networks How can we evaluate if a reduced order model is sufficiently close to the original one? The invariant probability distributions should be close to the original one.

23 Particular case: Jackson Networks In Jackson networks each server serves exactly one class of jobs. The arrival process is Poisson and the service times are exponentially distributed. Vector of external arrivals: α Traffic equation for effective load of a server: λ = P T λ + α. Without the embedding in a larger network λ is the ranking vector. λ determines the stationary probability distribution. So in this case there is no problem.

24 Heuristics A straightforward reduction approach is now: 1. Compute rank vector. 2. Delete unimportant nodes from the network. 3. Use this as the basis for simulation. Error estimates for the load of the reduced order networks can be obtained, but are conservative. A more refined method can be described: Theorem The following procedures for reduction do not change the rank of unaffected nodes. Interpretation: The load of the untouched nodes remains the same, so reduced order model is more useful.

25 Heuristics - Parallel Servers

26 Heuristics - Consecutive Servers

27 Heuristics - Subnetworks

28 Conclusions We have discussed reduction techniques for queueing networks, based on eigenvector centrality, which quantifies load of nodes of the network. There are other graph ranking algorithms. The benefits or disadvantages of these need to be investigated in this context. Approximation properties concerning the dynamics of the reduced order model still needs to be investigated. In particular, stability perserving model reduction is of interest - this is still an open question.

29 Thank you

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