# Parameter Extraction of complex production systems via a kinetic approach

**ABSTRACT** Continuum models of re-entrant production systems are developed that treat the flow of products in analogy to traffic flow. Specifically, the dynamics of material flow through a re -entrant factory via a parabolic conservation law is modeled describing the product density and flux in the factory. The basic idea underlying the approach is to obtain transport coefficients for fluid dynamic models in a multi-scale setting simultaneously from Monte Carlo simulations and actual observations of the physical system, i.e. the factory. Since partial differential equation (PDE) -conservation laws are successfully used for modeling the dynamical behavior of product flow in manufacturing systems, a re -entrant manufacturing system is modeled using a diffusive PDE. The specifics of the production process enter into the velocity and diffusion coefficients of the conservation law. The resulting nonlinear parabolic conservation law model allows fast and accurate simulations. With the traffic flow-like PDE model, the transient behavior of the discrete event simulation (DES) model according to the averaged influx, which is obtained out of discrete event experiments, is predicted. The work brings out an almost universally applicable tool to provide rough estimates of the behavior of complex production systems in non -equilibrium regimes.

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**ABSTRACT:**We develop continuum models of re-entrant factory production systems that treat the flow of products in analogy to traffic flow. Specifically, we model the dynamics of material flow through a re-entrant factory via a parabolic conservation law describing the product density and flux in the factory. We first extract the transport coefficients, in particular, velocity and diffusion coefficients of the particles in the production system using discrete event simulation (DES). Since PDE -conservation laws are successfully used for modeling the dynamical behavior of product flow in manufacturing systems, we model the manufacturing system using a diffusive partial differential equation (PDE). The specifics of the production process enter into the velocity and diffusion coefficient of the conservation law. The resulting nonlinear parabolic conservation law model allows fast and accurate simulations.07/2009;

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Parameter Extraction of Complex Production

Systems via a Kinetic Approach1

A.K. Unvera,∗,

aUniversity of California Los Angeles, Institute for Pure and Applied

Mathematics, 460 Portola Plaza, Los Angeles, CA 90095-7121.

C. Ringhoferb

bArizona State University, Department of Mathematics and Statistics, Tempe, AZ

85287-1804.

Abstract

Continuum models of re-entrant production systems are developed that treat the

flow of products in analogy to traffic flow. Specifically, the dynamics of material flow

through a re - entrant factory via a parabolic conservation law is modeled describing

the product density and flux in the factory. The basic idea underlying the approach

is to obtain transport coefficients for fluid dynamic models in a multi-scale setting

simultaneously from Monte Carlo simulations and actual observations of the physical

system, i.e. the factory. Since partial differential equation (PDE) - conservation

laws are successfully used for modeling the dynamical behavior of product flow

in manufacturing systems, a re - entrant manufacturing system is modeled using

a diffusive PDE. The specifics of the production process enter into the velocity

and diffusion coefficients of the conservation law. The resulting nonlinear parabolic

conservation law model allows fast and accurate simulations. With the traffic flow-

like PDE model, the transient behavior of the discrete event simulation (DES) model

according to the averaged influx, which is obtained out of discrete event experiments,

is predicted. The work brings out an almost universally applicable tool to provide

rough estimates of the behavior of complex production systems in non - equilibrium

regimes.

Key words: supply chains, re-entrant factory, conservation law

∗Corresponding author.

Email addresses: aunver@ipam.ucla.edu (A.K. Unver), ringhofer@asu.edu

(C. Ringhofer).

1This work was supported by NSF grant DMS-0204543.

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1Introduction

In recent years, factories and production systems have become larger and more

complicated. For this reason, a research endeavor has been initiated to find

time and cost efficient ways of production for such supply chains. Moreover,

a wide range of traffic flow theories and models have been developed to find

the most effective means of production. Our goal in this paper is to present a

computationally efficient way to simulate the production systems.

Understanding the behavior of large supply chains under different polices and

scenarios is a major issue for many businesses today. In large factories, no

experiments can be done involving whole supply chains. Therefore, simulation

models are developed, which substitute for the real environment. Especially

in recent years, fast scalable simulations of production flows in a supply chain

have become a very important research topic. The long term goal of a supply

chain simulation is to optimize and control the production across the whole

supply chain. Since most production deals with individual parts and the pro-

cesses that these parts undergo, discrete event simulators would be the regular

method of choice for accurate simulations.

While discrete event simulators have been highly successful to simulate single

factories, they are computationally too expensive to simulate even a moder-

ately complicated supply chain. Also, they are not scalable to a full supply

chain [1]. Alternative models that endow supply chain nodes with fixed pro-

duction capacities and fixed lead times are not accurate enough since they do

not take into account the fact that capacitated system respond nonlinearly to

increases in demand close to the limit of the production capacity.

The approach followed in this paper is to extract TPTs (Throughput Times)

and WIP (Work in Progress) levels from a discrete event simulation, replacing

actual observations of physical system for the purposes of this work, and then

to transform them into transport coefficients of a macroscopic conservation

law. Using discrete event simulations we estimate the transport coefficients;

namely velocity (convection) - ‘C’ and diffusion - ‘D’. Then, we are able to

solve the diffusive partial differential equation (PDE) numerically. The PDE

model that we use in this project is the conservation law given as follows:

∂tρ(x,t)+∂xF(x,t)=0,F(x,t)=Cρ(x,t)−D∂xρ(x,t)(1.1)

where x and t are space and time variables respectively. Here, ρ(x,t) denotes

the density of the particles and F(x,t) denotes the flux, both of which are

used to model the dynamics of material flow through the re-entrant factory.

The velocity and diffusion coefficients are extracted from observations of the

system (replaced by a discrete event simulation model for the purposes of

this paper). Since PDE models are amenable to optimization and control [10],

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we basically merge the randomness and optimization by using both DES and

PDE models.

We base our theory on the assumption that we have the following type of data

available. The production process consists of M stages. am

at which the lot number n arrives at stage m. Thus, the basic input of our

macroscopic model consists of a table of arrival times am

From this table, the throughput times τm

any part of the supply chain can be easily computed. For instance, given the

arrival times am

ndenotes the time

n, n=1:N, m=1:M.

nfor each stage as well as WIP in

nthe WIP in the m-th supplier Wm(t) is computed as

Wm(t)=

?

n

H(t−am

n)−H(t−am+1

n

)

where H denotes the usual Heaviside function. As lots/particles go through

the stages, we compute their velocities and variances depending on observed

parameters. Since each particle has a different processing time at each stage,

it has a varying velocity through out the system, so this causes a variance in

velocity for each particle. According to fluid dynamics theory, variance results

in diffusion. This results in computing TPTs (Throughput Times), means, and

variances depending on a chosen set of state parameters. Assumed data form

the times that lots have passed point in the production process. At the same

time we record some macroscopic quantity (such as the WIP) of the system

at each of the arrival times am

times, parameterized by this macroscopic quantity. We extract the transport

coefficients; namely, velocity (convection) and diffusion coefficients of the par-

ticles in the production system, using discrete event simulation (DES). De-

tailed information on all features of DES can be found in [5], [8]. The velocity

and diffusion coefficients in (1.1) are then related to the mean and variance

of this distribution. Here, the goal is to form a compact model which reorders

itself to optimization. PDE model has this important advantage, it is fast and

optimization can be applied to obtain more accurate results.

n, giving a statistical distribution of throughput

The contents of the rest of the paper are as follows: In Section 2, we present

theoretical background of traffic flow-like PDE models and how to relate TPTs

and WIPs to velocity and diffusion coefficients. In Section 3, we explain the

model hierarchies that we used to derive the flux function and how we compute

the TPT (Throughput Time) distribution and its mean and variance from the

given data. Finally, in Section 5, we apply the theories that we cover in this

paper on a model problem and give numerical results comparing the discrete

event experiments to the PDE model.

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2 Traffic Flow-like (PDE) Models

Traffic flow-like (PDE) models are continuum limits of fluid models. These

models have several advantages: They are scalable and, most important of all,

they are amenable to control and optimization [10]. There are two different

types of PDE modeling: First principle modeling and direct modeling from

observations. In the first principle modeling; on average there exists a func-

tional relation between the flux and the WIP such that the flux is given as

a function of WIP, which is known as a clearing function. Whereas in direct

modeling from observation, which we do in our PDE modeling, the clearing

function is replaced by a probability distribution function that is obtained

from the observed data (DES in our case). After we obtain the data (ran-

domly generated times) from DES, we get the TPT’s and then, we extract

the transport coefficients by using mean and variance of TPT’s. We can then

solve the PDE-conservation law given in Equation (1.1). Our ultimate goal is

to predict and optimize the transient behavior of a given system from obser-

vation without knowing details about the internal mechanisms of the system;

i.e. we always want to know how the system behaves even though the features

of the system is changed.

In our traffic flow-like PDE model, we introduce an artificial variable that

represents each product’s percentage of completion, or stage. The stage x

goes from 0 (0%) to 1 (100%). x=0 denotes the start of a product into the

factory, in other words, the product enters the factory, and x=1 denotes the

end of a product that means the product leaves the factory. The PDE model

yields a density of product as a function of time at a given stage according

to the mass conserving PDE. Also, the arrival rates of the products at stages

yield a flux function. Both the density and flux functions are very important

in PDE modeling such that we use them to model the dynamics of material

flow through the whole factory.

The PDE model that we use in this project is a parabolic conservation law. A

conservation law is a relation asserting that a specific quantity is conserved.

Basically, for a quantity, to be conserved means that whatever enters to the

system has to come out of that system after some time.

In our setting, a conservation law can be defined as a partial differential equa-

tion that expresses the fact that some physical quantity is locally conserved

in a fluid or other continuous physical system, such as energy, momentum, or

the quantity of fluid itself. Conservation law in the integral form is given by

∀a,b∈[0,1] ∂t

?b

aρ(x,t)dx=influx - outflux(2.2)

where ρ(x,t) is the conserved variable. It is the density of products with units

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[parts/space] in the system. The flux is given by F(x,t) with units [parts/time].

The integral in Equation (2.2) gives the number of products per time between

x=a and x=b.

The total number of products in the system can be found by taking the integral

of density of products ρ(x,t) over the stage variable x from 0 to 1. We get the

total WIP W(t) as a function of time as follows

W(t)=

?1

0ρ(x,t)dx. (2.3)

3Model Hierarchies for Supply Chains

We present a model hierarchy to derive the flux function F given in Equation

(1.1) for a given supply chain. The model hierarchy consists of three parts,

namely; a particle model for the trajectories of individual parts, a kinetic

model for the phase space density, and a diffusion convection equation.

3.1 Particle Models

In particle models, the evolution of a large ensemble of lots (particles) is

modeled by describing the trajectories of each individual lot in phase space

by a set of Newton equations. We introduce the trajectories as x=ξ(t) and

v=∂ξ

which we get through the TPT’s obtained from DES (detailed information

on all features of DES can be found in [5], [8]), so that we obtain a random

variable v from

dP{1

τm

n

Here, n and m denotes the numbers of lots and stages, respectively; and am

represents the times for each of these lots at each stage.

∂t. The velocity v is updated periodically from a velocity distribution V,

=v}=Vm(v,am

n)dv. (3.4)

n

Now, the position ξ(t) after a small time interval dt is given by:

ξ(t+dt)=ξ(t)+dt v(3.5)

where v is the random velocity variable.

Similarly, we have a rule for updating the velocity v according to a random

variable κ, which we calculate by flipping a coin. The probability of κ=1 equals

wdt and the probability of κ=0 is 1−wdt. Here w is the update frequency. In

Section 3.2 (Kinetic Models), we explain how to choose the update frequency

w. Now, according to this rule, if κ=0, we maintain the current velocity, and

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if κ=1, we update the velocity v from the probability distribution (3.4) given

by the formula:

v(t+dt)=(1−κ)v(t)+κv?.

Here, v?is also a random velocity variable that we get from the velocity distri-

bution V, which can be seen in Equation (3.4). Now, we describe in detail how

to compute the velocity distribution V that is obtained from discrete event

experiments.

(3.6)

As we run a simulation, or observe an actual system, where N parts pass

through M stages s1,...,sM, we record the times am

arrives at stage smand finally at the exit, which corresponds to stage sM+1.

At the same time, we record some macroscopic state variables, denoted by

Zm

nwhen part number n

n, at every time am

n.

Zm

can be variables like total WIP, the downstream WIP (at higher stages), the

upstream WIP (at lower stages), or just station index m itself. We eventually

form two tables:

n=(Z1(n,m),..,ZK(n,m)) is in general a vector of K components. They

a(1,1)a(1,2)...

a(2,1)a(2,2) ...

.....

..a(N,M +1)

Z(1,1)Z(1,2)...

Z(2,1)Z(2,2)...

.....

..Z(N,M)

(3.7)

where there are different possibilities that the macroscopic state variable

Z might depend on. Zm

on space only, or Zm

Zm

the part. From one of these possibilities, we compute a distribution for the

velocities, dependent on the global variables Z. We map the work stations

s1,...,sM on a stage interval [0,1] where the stage smcorresponds to the in-

terval (xm−1,xm) with xm=m

n entering smis given by τm

n

−am

We divide the K dimensional space of the macroscopic variables Zm

C(k) where k=(k1,...,kK) and then we define the indicator function χk(Z)

by

n=m if we want to make the velocity dependent

n=W(am

n),W>m(am

n) for dependence on the total WIP only, or

n)) for dependence on the WIP in front and behind

n=(W<m(am

M, m=0,...,M. Then, the throughput time of lot

n=am+1

n, where m=1,...M and n=1,...,N.

ninto cells

χk(Z)=

1for Z ∈C(k)

0else

The collected data give a discrete probability distribution for the velocity v,

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where v=

1

Mτ, dependent on the macroscopic state variable Z, of the form

V(v,x,Z)=

?

nδ(v−

1

Mτm

nχk(Zm

n)χk(Zm

n)

n)

?

for Z ∈C(k) and x∈[xm−1,xm), with a mean ?v?(x,Z) and a variance σ2(x,Z)

given by ?v?(x,Z)=

for Z ∈ Ck, x∈[xm−1,xm).

?

n

?

1

Mτm

nχk(Zm

nχk(Zm

n)

n)

and σ2(x,Z)=

?

n

1

M2(τm

?

n)2χk(Zm

nχk(Zm

n)

n)

−?v?(x,Z)2

For the final model used in this work, the transport coefficients needed will be

given solely in terms of these means and variances [12].

We are dealing with highly re - entrant production systems with many ma-

chines and buffers. Throughout the factory, machines with different speeds

process the products. This accounts for the fact that in a re - entrant system,

later arrivals influence the processing time of a particular product.

The equations (3.5) and (3.6) for the position and velocity define a stochastic

process for the evolution of the products. We define the randomly changing

velocity as η. So, briefly, the update algorithm for position (ξ) and velocity η

is given by

(a) ξ(t+∆t)=ξ(t)+∆tη(t),η(t+∆t)=κ(t)V(v)+(1−κ(t))η(t),(3.8)

(b) P{κ(t)=1}=ω∆t,

P{κ(t)=0}=1−ω∆t,

(c) ξ(0)=0,dP{η(0)=v}=V(v,0)dv .

dP{η(t)=v}=V(v,x)dv

As it can be seen in Equation 3.8, for every ∆t, we choose a random number

κ, which is either zero or one, to decide whether to update the velocity η or

not. We update the velocity at every time step of length ∆t with probability

ω∆t. In general we let ω depend on η(t) and t. So, (3.8)(b) will be replaced

by

P{κ(t)=1}=∆tω(η,ξ),

P{κ(t)=0}=1−∆tω(η,ξ),(3.8)(b?)

dP{η(t)=v}=V(v,x)dv .

We will deal with choosing update frequency ω in next section (Section 3.2).

For analytic purposes we would prefer to have a continuous time model, where

the function ξ(t) is not given only at discrete time intervals. This could be

achieved by the limit ω→∞, giving a stochastic process. However, in this

limit we would lose the information about the update frequency ω which is a

key parameter for controlling the stochasticity of the model.

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3.2 Kinetic Models

This section is leading to a kinetic equation for the evolution of the density of

the products at stage x with velocity v.

The update algorithm (3.8) is similar to a Monte Carlo method for the solution

of a Boltzmann equation (see [7] for an introduction). The Boltzmann equation

is the kinetic equation for the time (t) evolution of the kinetic density function

f(x,v,t) in one-particle phase space, where x and v are position and velocity,

respectively.

The function ξ(t), defined in Section 3.1, plays the role of a position which

is advanced with a randomly changing velocity η(t). It is therefore tempting

to derive a kinetic formulation. In the context of supply chains, these models

are often referred to as traffic flow models [2], [9], because of the analogy to

lots moving on a freeway from start to completion. A product arrives at the

first stage at time t=a and moves from x=0 to x=1 along a trajectory with

velocity ηa(t) in such a way that it reaches the end x=1 at time t=e, then

the velocity has to be chosen in such a way that

?e

aηa(t) dt=1 holds.

The particle model for a product that arrives at the first stage of the system

at t=a together with the phase speed model (3.8) is given as follows: The

trajectory ξa(t) satisfies the same Newton equations as in (3.8) together with

the initial conditions

ξa(a)=0,dP{ηa(a)=v}=V(v,0)dv .(3.9)

The total WIP W(t), given in terms of the WIP density ρ(x,t), and the flux

F(x,t) at any point x∈[0,1] are then given by

?

?

Here λ(t), the influx density, is concentrated on the arrival times a0

in the system, i.e.

λ(t)=

n

holds. From the definition given in (3.10), it is obvious that the WIP density

ρ and the flux F will satisfy the conservation law

ρ(x,t)=δ(x−ξa(t))λa(a) da,W(t)=

?1

0ρ(x,t) dx, (3.10)

F(x,t)=δ(x−ξa(t))λ(a)ηa(t) da .

nof parts

?

δ(t−a0

n)

∂tρ+∂xF =0. (3.11)

We make a mean field assumption for the rest of this section, that is we will

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assume that the velocity distribution V(v,x) is a given function of velocity and

stage. In reality the velocity distribution will depend on the lot trajectories ξ

through the WIP W, given by (3.10). The idea underlying this assumption is

that, for many lots present in the system, the impact of one individual lot on

the distribution V is negligible. Therefore, the lots can be treated as approxi-

mately statistically independent. This is a standard approach for a system of

many molecules in gas dynamics [7]. Making this mean field assumption, we

have a theorem as follows:

Theorem 1: Let fa(x,v,t) denote the probability density for finding ξa,ηa,

defined by (3.9), at position x and velocity v, i.e. dP{ξa(t)=x,ηa(t)=v}=

fa(x,v,t) dxdv. Then fasatisfies the initial value problem

?

(a) ∂tfa+v∂xfa=V(v,x)ω(v?,t)fa(x,v?,t) dv?−ω(v,t)fa(x,v,t), t>a

(3.12)

(b) fa(x,v,a)=δ(x)V(v,x)

in the limit ∆t→0, weakly in x,v and t.

Using Theorem 1, we can derive the corresponding initial boundary value

problem for the kinetic density f(x,v,t) of the number of lots at position x

with velocity v. The proof of Theorem 1 is given in the following.

Proof of Theorem 1:

The proof consists of essentially summing up all possibilities for the values

of κ(t),ξa(t) and ηa(t) and weighting them with their respective probabilities.

This takes the form

fa(x,v,t+∆t)=

?

dP{ξa(t)=x?,ηa(t)=v?}

=δ(x?+∆tv?−x)δ(ku+(1−k)v?−v)[∆tω(v?,t)δ(k−1)+(1−∆tω(v?,t))δ(k)]

V(u,x)fa(x?,v?,t) ds?dv?dudk .

Integrating the δ− functions with respect to k gives

?

?

and integrating with respect to x?,v?and u in the above two integrals gives

δ(x?+∆tv?−x)δ(ku+(1−k)v?−v)dP{κ(t)=k}dP{η(t)=u}

?

fa(x,v,t+∆t)=∆tω(v?,t)δ(x?+v?∆t−x)δ(u−v)V(u,x)fa(x?,v?,t) ds?dv?du

+ (1−∆tω(v?,t))δ(x?+v?∆t−x)δ(v?−v)V(u,x)fa(x?,v?,t) ds?dv?du .

fa(x,v,t+∆t)=

?

∆tω(v?,t)V(v,x)fa(x−v?∆t,v?,t) dr?+(1−∆tω(v,t))fa(x−v∆t,v,t) .

(3.13)

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where we have used the fact that

(3.13) by a test function ψ(x,v,t) and integrate over t>a giving

?V(u,x) du=1 holds. We now multiply

?

H(t−a)ψ(x,v,t)fa(x,v,t+∆t) dxdvdt

=

?

+

H(t−a)ψ(x,v,t)∆tω(v?,t)V(v,x)fa(x−v?∆t,v?,t) dv?dxdvdt

?

where H denotes the Heaviside function. Shifting the arguments in the inte-

grals gives

?

=H(t−a)ψ(x+v?∆t,v,t)∆tω(v?,t)V(v,x)fa(x,v?,t) dr?dxdrdt

?

The left hand side of the above equation can be rewritten as

H(t−a)ψ(x,v,t)(1−∆tω(v,t))fa(x−v∆t,v,t) dxdvdt ,

H(t−∆t−a)ψ(x,v,t−∆t)fa(x,v,t) dxdvdt(3.14)

?

+H(t−a)ψ(x+v∆t,v,t)(1−∆tω(v,t))fa(x,v,t) dxdvdt .

?

H(t−a)ψ(x,v,t−∆t)fa(x,v,t) dxdvdt−

?

?

?a+∆t

a

[

?

ψ(x,v,t−∆t)fa(x,v,t) dxdv] dt

=H(t−a)[ψ−∆t∂tψ)]fa(x,v,t) dxdvdt−∆t

?

?

ψ(x,v,a)fa(x,v,a) dxdv+O(∆t2)

=H(t−a)[ψ−∆t∂tψ)]fa(x,v,t) dxdvdt−∆tψ(0,v,a)V(v,a) dv+O(∆t2) .

(3.15)

Expanding the right hand side of (3.14) gives

?

H(t−a)ψ(x,v,t)∆tω(v?,t)V(v,x)fa(x,v?,t) dv?dxdvdt

?

H(t−a)ψ(x,v,t)∆tω(v,t)fa(x,v,t) dxdvdt+O(∆t2) .

Setting this equal to (3.15) and dividing by ∆t gives

+H(t−a)(ψ+v∆t∂xψ)fa(x,v,t) dxdvdt

−

?

−

?

H(t−a)[∂tψ+v∂xψ]fa(x,v,t) dxdvdt−

?

−

which for ∆t→0 is the weak formulation of (3.12).

?

ψ(0,v,a)V(v,a) dv

=H(t−a)ψ(x,v,t)ω(v?,t)V(v,x)fa(x,v?,t) dv?dxdvdt

?

H(t−a)ψ(x,v,t)ω(v,t)fa(x,v,t) dxdvdt+O(∆t) ,

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Now, we derive the corresponding initial boundary value problem for the ki-

netic density f(x,v,t). We define f by

f(x,v,t)=

?

H(t−a)fa(x,v,t)λ(a) da .(3.16)

f is not a probability density since the trajectory ξawill only exist for t>a and

the number of lots will not be constant over time. The lot density f will satisfy

the same Boltzmann equation as the probability density fawith a more natural

influx boundary condition replacing the initial condition (3.12)(b). Now, we

have Theorem 2 as follows:

Theorem 2: The lot density f, given in (3.16), satisfies the initial boundary

value problem

(a) ∂tf +v∂xf =V(v,x)

?

ω(v?,t)f(x,v?,t) dv?−ω(v,t)f(x,v,t), x>0, t>0

(3.17)

(b) f(0,v,t)=λ(t)

v

V(v,x),f(x,v,0)=0

weakly in x,v and t.

The Boltzmann equation (3.17) is a generalization of the kinetic model derived

in [2]. The random phase approach has introduced the concept of collisions,

i.e. the integral operator on the right hand side of (3.17), into the particle

model. Now, we prove Theorem 2 in the following way:

Proof of Theorem 2:

The weak form of the initial value problem (3.12) implies that for any test

function ψ(x,v,t) the relation

−

?

H(t−a)fa[∂tψ+v∂xψ](x,v,t) dxdvdt=

?

ψ(0,v,a)V(v,x) dv+ (3.18)

?

{H(t−a)ψ(x,v,t)[V(v,x)

holds, where all integrals range over R and H denotes the usual Heaviside

function. Multiplying (3.18) by λ(a) and integrating with respect to a gives

?

ω(v?,t)fa(x,v?,t) dv?−ω(v,t)fa(x,v,t)]} dxdvdt

−

?

f[∂tψ+v∂xψ](x,v,t) dxdvdt=

?

ψ(0,v,a)V(v,x)λ(a) dadv+(3.19)

?

{ψ(x,v,t)[V(v,x)

Now fa(x,v,t) vanishes identically for x<0 and λ(a) vanishes identically for

a<0. Therefore f(x,v,t) vanishes identically for x<0, t<0. Thus (3.19) is

?

ω(v?,t)f(x,v?,t) dv?−ω(v,t)f(x,v,t)]} dxdvdt .

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the same as

?

?

(3.20) is the weak formulation of the boundary value problem (3.12).

−

H(x)H(t)f[∂tψ+v∂xψ](x,v,t) dxdvdt=

?

H(t)ψ(0,v,t)T (v,t)λ(t) dtdv+

(3.20)

{H(x)H(t)ψ(x,v,t)[V(v,x)

?

ω(v?,t)f(x,v?,t) dv?−ω(v,t)f(x,v,t)]} dxdvdt .

3.2.1 Choosing the Update Frequency

Now, it remains to choose the update frequency ω(v,t). We remark that there

necessarily is some arbitrariness in this choice. The discrete event simulation

model yields one particular realization of a stochastic process. To compute

meaningful answers, one therefore will run a series of discrete event exper-

iments and compute averages. The more experiments, the closer the result

will be to the average, where a degenerate throughput time distribution (a δ

function concentrated at the mean) is used. So, the size of ω has to be chosen

in the same way as the number of realizations of the discrete event simulation

model is chosen. The guiding principle here has to be that the overall number

of random numbers generated is the same.

The basic idea of the approach presented so far is to modify the discrete

event simulation model in order to take into account the transient changes in

throughput times and influx, which will influence the outflux in the case of

re-entrant nodes. So, on the other hand, if there are no changes in influx and

throughput time, the discrete event model should correspond to a solution

of the Boltzmann equation (3.17). Translating the discrete event simulation

into kinetic model along the lines of the approach in [2] gives the following

picture. For an arrival time a we pick a velocity η according to the distribution

V(η,x) and move with the constant velocity η=1

t=a+τ. This gives the relations for the lot trajectory in phase space by

τuntil reaching x=1 at time

ξa(t)=ηa(t−a),dP{ηa=v}=V(v,0)dv .

We denote by fd

i.e. fd

as

a(x,v,t) the corresponding probability density in phase space,

a(x,v,t)dxdv=dP{ξa(t)=x,ηa(t)=v}. So, fd

acan be computed exactly

fd

a(x,v,t)=δ(v(t−a)−x)V(v,x) .

The number density fd(x,v,t) corresponding to (3.16) is then given by

fd(x,v,t)=

?

H(t−a)δ(v(t−a)−x)V(v,x)λ(a) da=1

vH(x)V(v,x)λ(t−x

v) .

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For the particle model to be equivalent to the discrete event simulation model,

fdhas to be a solution of the initial boundary value problem (3.17). Since fdis

a function of t−x

condition (3.17)(b). Therefore the residual obtained from inserting fdinto

(3.17) is given by the collision term

v, it satisfies ∂tfd+v∂xfd=0. fdalso satisfies the boundary

Q[fd](x,v,t):=V(v,x)

?ω(v?,t)

v?

V(v?,x?)λ(t−x?

v?) dv?−ω(v,t)

v

V(v,x)λ(t−x

v) ,

and Q[fd]=0 has to hold in the case that neither V nor λ depend on time,

giving

?ω(v?,t)

v?

which implies ω(v,t)=vγ(t) for some arbitrary function γ(t). This makes sense

since we update the phase speed more frequently for shorter current through-

put times, i.e. for a faster transient response of the system. So, with this choice,

the discrete event simulation and the random phase model (3.8) are equivalent

for constant V and λ and any function γ(t). In the general case, we choose γ(t)

such that we roughly compute the same number of random numbers for either

model. Using the discrete event simulation we pick ∆tλ(t) random numbers

for the velocity in the time interval ∆t. Using the random phase model, we

pick ∆tγ(t)?vV(v,x)dv in the same time interval. w and V1are now functions

of v and x. Therefore we set the update frequency ω(v,x) to

0=V(v)

V(v?) dv?−ω(v,t)

v

V(v)

ω(v,x)=

vλ0

V1(x),V1(x):=

?

vV(v,x) dv (3.21)

where λ0is a characteristic value of the influx density λ(t).

3.3Diffusion Convection Equation

In this section, we explain how we obtain the convection (velocity) and diffu-

sion coefficients. We express the flux F in terms of the density ρ via a func-

tional expansion, namely the Chapman - Enskog expansion [6]. The Chapman

- Enskog expansion (see [6] for details) is an essential asymptotic tool of the

Boltzmann equation (3.17) for a regime where average throughput time and

the time intervals between velocity updates in the particle model are small

compared to the over all time scale. Fluid equations for the simulation of

throughput times have been studied in [4] and [2], leading to simple convec-

tion equations for the density of lots as a function of DOC (degree of comple-

tion). We essentially recover these results but obtain as a generalization, an

additional diffusive term arising from the variance of the velocity distribution

V.

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A fluid dynamic approximation to the Boltzmann equation (3.17) is obtained

by taking integrals with respect to the velocity v. Defining the density ρ(x,t)

and the flux F(x,t) by

?

we obtain the conservation law

ρ(x,t)=f(x,v,t) dv,F(x,t)=

?

vf(x,v,t) dv ,(3.22)

(a) ∂tρ+∂xF =0,(b) ρ(x,0)=0,(c) F(0,t)=λ(t) . (3.23)

Here, ρ and F, defined by (3.22), represent expectation values of the position

and flux densities. The goal is now to find an expression for the flux F in terms

of the density ρ using an asymptotic solution of the Boltzmann equation (3.17)

large time scales.

We start by scaling the involved probability distributions and by bringing the

Boltzmann equation (3.17) into an appropriate dimensionless form.

From (3.17), we have that the initial - boundary value problem for the unscaled

Boltzmann equation for the part density is of the form

?

f(0,v,t)=λ(t)

v

The appropriate scale for the velocity (and therefore also the time scale) are

given by the probability distribution V(v,x). We choose the velocity scale

v0=?1

rescale the probability distribution V correspondingly:

V(v,x)=1

∂tf +v∂xf =V(v,x)ω(v?,x)f(x,v?,t) dv?−ω(v,x)f(x,v,t), x>0, t>0

V(v,0),f(x,v,0)=0,ω(v,x)=

vλ0

V1(x),V1(x)=

?

vV(v,x) dv.

0V1(x) dx, i.e. as the spatial average of the mean of the velocities, and

v0Vs(vs,x),v=v0vs

(We denote the scaled and dimensionless variables with subscript s here.) We

choose the over all time scale to be

We scale the influx density λ(t) by its characteristic value λ0and scale the

scattering frequency ω accordingly, setting

1

v0and define the scaled time as t=ts

v0.

λ(t)=λ0λs(ts),ω(v,x)=λ0ωs(vs,x)⇒ωs(vs,x)=v0vs

V1(x)

making λsand ωsdimensionless O(1) quantities. Finally, we scale the kinetic

density f(x,v,t) itself by f(x,v,t)=λ0

v2

value problem for the Boltzmann equation (3.17) now reads:

0fs(x,vs,ts). The scaled initial boundary

(a) ∂tsfs(x,vs,ts)+vs∂xfs=1

ε[Vs(vs,x)

?

ωs(v?

s,x)fs(x,v?

s,ts) dv?

s−ωs(vs,x)fs(x,vs,ts)]

(3.24)

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(b) fs(0,vs,ts)=λs(ts)

vs

Vs(vs,0),fs(x,vs,0)=0,

(c) ωs(vs,x)=

vs

V1s(x),V1s(x)=

?

vsVs(vs,x) dvs,ε=v0

λ0.

The diffusion convection equation will be derived in the limit ε→0. Note that,

according to Section 3.1 the average expectation v0is of order O(

the timescale t0=1

time through the whole chain, and ε=v0

tween arrivals in the chain are small compared to the whole TPT, i.e. there

are many parts in the whole system.

1

Mτ). Thus,

v0=O(Mτ) corresponds to the total average throughput

λ0<<1 implies that the intervals be-

Now, using the Chapman - Enskog expansion, the flux F is expressed for

ε<<1 in terms of the density ρ [6]. For notational simplicity we drop the

subscript s from here on.

In the limit ε→0 equation (3.24) will be dominated by the right hand side.

We therefore define the collision operator Q[f] as

Q[f](x,v,t)=V(v,x)

?

ω(v?,x)f(x,v?,t) dv?−ω(v,x)f(x,v,t) , (3.25)

and write the Boltzmann equation (3.24) as

∂tf +v∂xf =1

εQ[f] .(3.26)

The relation between the density ρ and the flux F in (3.22) is now obtained

via asymptotics for ε→0, in other words by a Chapman - Enskog procedure.

In general, the the Chapman - Enskog expansion consists of splitting the

dynamics of the Boltzmann equation (3.26) into a slow evolution on the kernel

of the operator Q and a fast evolution on its orthogonal complement. While

the Chapman - Enskog expansion is in general a nonlinear procedure [6], its

complexity reduces significantly in the case of linear collision operators, and

it can be written in terms of projection operators. To this end, we first define

the kernel of the operator Q in (3.25).

The definition of Q is the following: When we apply Q on a function, we

take the integral of that function and multiply by probability distribution

V(v), and then subtract that function from the multiplication obtained. The

operator Q given in (3.25) is a BGK (Bhatnagar-Gross-Krook) - type collision

operator, because the probability of the final state (vf) is independent of

the probability of the initial state (vi). BGK is a simple Computational fluid

dynamics technique used in the Boltzmann Equation. This simple form of the

collision operator allows for a relatively simple analysis of the kernel of Q.

Obviously the kernel elements are given by multiples of the function

V(v,x)

ω(v,x).

15