Structure Preserving Model Reduction for Logistic Networks

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Structure Preserving Model Reduction for

Logistic Networks

Fabian Wirth

Institute of Mathematics University of W¨urzburg

Workshop on Stochastic Models of Manufacturing Systems Einhoven, June 24–25, 2010.

joint work with: Michael Sch¨onlein

Sergey Dashkovskiy, Michael Kosmykov Thomas Makuschewitz, Bernd Scholz-Reiter

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Motivation

Queueing Systems Ranking in Graphs Heuristics

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Motivation

Sie me ns (motors) SAT (se parate rs)

Fle nde r couplings

London

Brook Hanse n (motors)

E

F B

NL UK

M e taproduct (base plate s coupling guards)

local supplier (base plate s)

D

LRVP pumps IND pumps motors Brook Hansen motors Siemens couplings Flender motors Leroy Somer baseplates and coupling guards Metaproduct, local suppliers (F, E)

SC pumps

Le roy Some r (motors)

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Queueing Systems

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

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Queueing Systems

Consider a set ofservers which are able to treat different classes of jobs. Server 2 Server 3 Server 4 Server 5 Server 1

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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 3

Server 5

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Queueing Systems

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

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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 4 Server 3

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Queueing Systems - The Maths

Classes k = 1, . . . , K .

Interarrival times: ξk(n), n = 0, 1, 2, . . . i.i.d. E(ξk(0)) < ∞ Service times: ηk(n), n = 0, 1, 2, . . . i.i.d. E(µk(0)) < ∞ routing matrixP = (pij),

pij - probability that job of class i becomes a job of class j

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Queueing Systems - Balance Equations

Classes k = 1, . . . , K

Interarrival times: ξk(n), n = 0, 1, 2, . . . i.i.d. E(ξk(0)) < ∞

Service times: ηk(n), n = 0, 1, 2, . . . i.i.d. E(ηk(0)) < ∞

routing matrix P = (pij), pij - probability that job of class i

becomes a job of class j Assumption: r (P) < 1. Q(t) = Q(0) + A(t) + PTS (t) − S (t) with Aj(t) = max ( n | n X m=0 ξk(m) ≤ t )

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.

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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.

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Ranking in Graphs

Ranking schemes try to extract information about the impor-tance/relevance of a vertex from graph properties.

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Ranking in Graphs - Eigenvector Centrality

Given a weighted adjecency matrix A corresponding to a directed graph, the following steps are performed

• Colums arerescaledto have column sum 1 (where possible). • A is madecolum stochastic by adding artificial entries in zero columns.

• A is madeirreducible e.g. by considering for some α ∈ (0, 1)

˜

A := αA + (1 − α)eeT

Then Perron-Frobenius theory says that there is an eigenvector r > 0 (unique up to scaling) such that

Ar = r

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Ranking in Graphs - Ensuring irreducibility

Given the weighted adjecency matrix A ∈ Rn×n consider the enlarged matrix

"A 0 0 0

#

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Ranking in Graphs

Consider the graph described by A as weakly coupled with a larger network. Coupling described by vectors v and w .

B ="αA + (1 − α)vne T n wneTm (1 − α)vmeTn wmeTm # Notation: e :=1 1 . . . 1T

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Ranking in Graphs

B ="αA + (1 − α)vne T n wneTm (1 − α)vmeTn wmeTm #

Proposition Assume that B is irreducible, then if x =x_nT x_mTT is an eigenvector corresponding to the eigenvalue 1 of B then xn is

an eigenvector corresponding to the eigenvalue 1 of the matrix

Aα(v , w ) := αA + (1 − α) vn+ eT_mvm 1 − eT mwm wn eT_n . Furthermore, Aα(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.

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A simple example

Frontiers in network science - advances and applications

Volkswagen Symposium: 28.-30. September 2009, Berlin 10/17

Dynamic Large-Scale Logistics Networks

10/17 Important locations of the test case 3rd-tier supplier (Production of A) 2nd-tier supplier (Production of B) 1st-tier supplier (Production of C) OEM locations (Production of D) Local warehouses (Storage and shipping of D)

Retailer (Distribution of D) v19 v20 v21 v22 v17 v18 v15 v16 v11 v12 v13 v14 v7 v8 v9 v10 v1 (35) v2 (58) v3 (47) v4 (41) v5 (44) v6 (25) (70) (70) (82) (28) (93) (47) (73) (37) (140) (110) High importance Low importance

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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.

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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 = PTλ + 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.

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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.

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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.

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