### New Approach Combining Numerical Technique and Simulation for Analysis

### of

### Large Discrete Event Systems Based

### on

### Petri Nets

Ming-Hung L i n and Li-Chen

**Fu **

### Dept.

**of**C o m p u t e r Science and I n f o r m a t i o n Engineering, National Taiwan University,

Taipei, Taiwan,

### R.0.C

### E-mail:

lichen@ccms.ntu.edu.tw ABSTRACTIn this paper, me propose a new method that combines
simulation and numerical techniques and directly in-
tegrated using interval arithmetic techniques for the
analysis of large discrete event systems. **A **system is
first divided into several layers. Identifying subsys-
tems that can be modeled in isolation solves a system
for each layer. In each subsystem, the Markovian

**as- **

sumption allow-s us t o establish a set of linear equality
constraints among the espectation of state variables in
the Petri Nets, such as token numbers in the places. **A**pseudo random process is responsible for the timing of the model, event times are completely determined by simulation. Thus, linear equality constraints are com- puted according t o the pre-simulated event times, and it is possible t o know the probabilities of interactions between subsystems.

1 I n t r o d u c t i o n

In this paper, a new analysis method for

### a

discrete event system is introduced. For the analysis of a dis- crete event system, there are two different approaches can be used, simulation and numerical analysis. Jo-**hannes (21 proposes **mean value analysis for queueing

network models with intervals as input parameters.
**Shikharesh [3] presents robust bounds and throughput **
guarantees for closed multiclass queueing networks.
The generalized stochastic Petri Net modules are used
**as **basic building blocks to model and analyze complex
manufacturing system (61. Matteo [8] describes ap-
proximate mean value analysis for stochastic marked
graphs. Zhen Liu **[9] **proposed performance analysis
of stochastic timed Petri Nets using linear program-
ming approach. Numerical analysis techniques and
simulation both have their advantages and limitation.
The idea of combining analytical/numerical techniques
and simulation has been proposed several times. Pe-
ter [l] presents a method for continuous time Markov
chains. The basis of the method is the description of
a continuous-time Markov chain as **a **set of commu-
**nicating processes. It also serves as a good tutorial **
paper. However, in practice, the probleni is usually
the large size of state space and the model cannot be
tlecornposed very cleanly.

In this paper, we propose **a **new niethotl that com-

**m 2 ** **Sub-system **

Figure **1: **Hybrid model with only one layer

bines simulation and numerical techniques and directly
integrated using interval arithmetic techniques for the
analysis of large discrete event systems. **A **system is
first divided into several layers. Identifying subsys-
tems that can be modeled in isolation solves a system
for each layer. In each subsystem, the Markovian **as- **
sumption allows us t o establish a set of linear equality
constraints among the expectation of state variables in
**the Petri Nets, such as token numbers in the places. A **
pseudo random process is responsible for the timing of
the model, event times are completely determined by
simulation. Thus, linear equality constraints are com-
puted according t o the pre-simulated event times, and
it is possible t o know the probabilities of interactions
between subsystems. In this way, the state explosion
problem of numerical analysis is avoided. It is still pos-
sible to obtain more accurate results than with pure
simulation.

The organization of this paper is **as **follows. Sec-
tion **2 **describes several types

### of

combining model for performance analysis. Section**3**introduces a

### JF

queue- ing network that extends the framework of the origi- nal Jackson queueing network by adding event-driven**in special queue. Section 4 describes a bounded Petri**

**Net. Section 5 describes the development of integra-**tion using interval arithmetic technique. Section

*is conclusion.*

**6****2 ** C o m b i n i n g **Model **

* Hybrid model and Hierarchical model. The two model *
can bring together to solve the whole system. For ex-
ample, in a flexible-manufacturing cell, there are two
processes occurring in parallel. They both use a pool
of common tools. Process 1 uses rnl machines and pro-
cess

**2**uses

**m 2 machines. Fig 1 represents the hybrid**model of this system, where

p l number of process 1.
p2 number of process **2. **

m l available machines for process 1.
m2 available machines for process **2. **
t o available tools.

**Q1 **number of process **1 **a t input-buffer.
**Q3 **number of process **2 **at input-buffer.

**Q2 **number of process 1 at working-area and being

Q4 number of process **2 **at working-area and being

**Q5 **number of process 1 at output-buffer.
Q6 number of process 2 a t output-buffer.

### M 1 E

number of failure machine 1.### M 1 R

number of repaired machine 1. serviced.serviced.

### .

Notice that, transferring of a process (after service) from queue**Q2,Q4,Q5**and

**QS, a**token will be gen- erated. In this system, the machine

**2**is the back-up machine for machine 1. When machine 1 becomes fail- ure, a new token triggers the instantaneous passage of a process from queue Q1 to

**Q3.**

On the other hand, in hierarchical model with mul- tiple layers, the system performance is calculated layer by layer. The Fig 2 represents this hierarchical model. The upper layer gives a coarse analysis and the lower layer gives a fine one.

**A system is first divided into several layers. Iden- **

tifying subsystems that can be modeled in isolation
solves a system for each layer. 111 each subsystem, we
*expectation of state variables, such*

**establish a set of linear equality constraints among the****as**token numbers in the places. -4 pseudo random process is responsible for the timing of the model, event times are completely determined by simulation. Thus, linear equality con- straints are computed according to the pre-simulated event times.

The nest key goal is to know the interactions be-
tween subsystems. In nest section, we proposed a *J F *

* queueing m t w o r k that extends the framework *of the

*original . J w h o n Quevein!l Network to add ability of*event-driven.

### /----/-

*I * **I **

**System ****h y e r n + l **

Figure **2: **Hierarchical model with multiple layers

**3 **

**JF **

**Queueing Network**

**The **

**Model**

There are two types of queues, namely, joining queue and forking queue

### .

The service time of joining queue is infinite. The service time of forking queue follow some probability distribution. The service time of forking queue is zero. When### a

customer enter a joining queue, the customer will receive no service from the server and will stay in queue for blocking. In our queueing network model, we allow work both t o be moved from### a

queue t o another without service, and destroyed, un- der the effect of control customer which are either ex- ogenous, or generated by the ordinary customers after service. Before presenting our queueing model the nec- essary terminology is introduced. Let,**C **

= number of customer classes.
**C**

**I< **

### =

number of queues (resources modeled as queue-* Nkc= * number of class

**c**E (1,. .

### .

### ,

C)customers at* Vkc= *mean number of visits made by a class c ordi-

### &,=

mean number of external visits made by a class**Zk,= **

mean number of external visits made by a class
**Zk,=**

ing centers or infinite server).

queue

**k. **

**k.**

nary customer a t queue

**k. **

**k.**

ordinary customer at queue

**k. **

**k.**

control customer at queue

**k. **

**k.**

* Rkc= *mean service demand rate per visit for a class
c ordinary customer at queue

**k **

**k**

**A ** customer(contro1 customer or ordinary cus-
tomer)of class which leaves forking queue * i *(after finish
service) goes to queue

*j*

**as**a customer of class

*with*

**I***it may depart from the network with probability*

**probability p ( i , j ) ( k ,****l ) ,**

**P ( i ) ( k )**### =

**1**

### -

**E l **

**E l**

**p ( i , j ) ( k , 1) .****A**customer (control customer or ordinary customer) of class

**k**which leaves joining

**queue i (after finish service)****Figure 3: A **JF queueing model of a ordinary Jackson
queueing network

**Maddnt 1 **

Figure **4: **A

### JF

queueing model of share resource goes to queue j**as**a customer of class

*with probability zero. The arriving control customer triggers the instan- taneous passage of a customer(contro1 customer or or- dinary customer) of class*

**1***from joining queue*

**k***to class*

**i***of*

**1**

**some other queue j with probability****q(i,j)(k,l).*** With probability Q ( i ) ( k ) *= 1

### -

*C j *

**Cl **

**Cl**

*it forces a customer to leave the network. After trig- gering the instantaneous passage of a customer, the arriving control customer is destroyed. The arriving control customer triggers the instantaneous passage of a customer(contro1 customer or ordinary customer) of*

**q ( i , j ) ( k ,****l ) ,****class IC **

from **class IC**

**forking queue **

**i **

to class **i**

### 1

of some other*queue j with probability zero.*

When a customer(contro1 customer or ordinary cus-
tomer) of class * k *which leaves forking queue

*(after finish service)*

**i**### ,

it generates a new control customer from forking queue**i **

to class 1 of some other queue
**i**

*j * *with probability u ( i , j ) ( k , *I). it may not generate

**a. **new control customer into the network with prob-
* ability U ( i ) ( k ) *= 1

### -

### E, E,

*~ ( i ,*When a cus- tomer(coiitro1 customer or ordinary customer) of class

**j)(k,l)**

**k which leaves joining queue i (after finish service)**### ,

it**Figure 5: **

**A **

JF queueing model 0f.a production net-
work
generates **a **new control customer from joining queue **i **

**to class 1 of ****some other queue j with probability zero. **

**Bounds on throughput **

**As **usual the customers, in whom each queue is, define
the operation of the system in terms of the execution
of response cycles visited a certain number of times.
Let,

*Wk, *= mean waiting time(pure delay) for a class c at

**X, **

= mean throughput for class c (cycle/second).
**X,**

* Y, = *mean cycle time for a single customer in class c

**Hc = **

### h=

mean cycle rate for### a

single customer in queue*(seconds)*

**IC.**(seconds)

class c

The upper and lower bounds on the throughput for different customer classes in a open multiclass queueing network with a

**FIFO **

queueing discipline used a t each
queue (queueing center) are presented in this section.
The first type of upper bound ignores contention a t the queue and is termed the no contention upper bound. By ignoring the queueing delay a t every queue

* k *we obtain

**k = l **

for,each node of forking queue, we have **tlk ** E
*{A- *

**,...,I<} **

**,...,I<}**

On the other hand, for each node of joining queue, we
* have V k *E

*{ I , . . .>I<'}*1

*c vkc*

**Ii***= s k c*

### +

*z k c*

### +

*, v k t C 1 p ( k ' , k ) ( c ' i c )*

*=*

**k'****=U'**c'**1**

### ,

and form the*little law*we have

### .

Thus,### .

Then,Figure 6: The basic bounded

### TPPN

model are associated with the places. Denote by*t j ( k )*the time

### of

the kth firing of transition*Then, applying this notation leads t o*

**ti.***t 4 *

**(k) **

**(k)**

### =

max(min(p1### +

**t l****(k), **

**(k),**

**p l**### +

**tz****(k)}, **

**(k)},**

**p 2**### +

*t3(k*

### -

**2))**

**(3)**There are recursive equations representing the firing time of the transitions. The real potential for this ap- proach is the ability t o draw on analogies with tradi- tional control theory. It is continue t o be proven that such analogies are not only with mathematical inter- est, and that in this approach can be extended t o a large model.

However, uncertainties may be associated with

*v k c b v k c ' *

**hcWbc' **

**hcWbc'**

*VbcRkc*place. The uncertainties are based on the various rea-

### --

*I*}Petri Net parameters such as number of tokens in a

(1)
1
**I\' ****Ck=llVkC ****x'c ** _<min{

**ck=Ict **

**ck=Ict**

### ,(

*'*

**Ii***V&kC*

### +

### ,

where*b*L e m m a

**is the bottleneck queue for class c customers.****1 Consider a closed system**

*in which two cus-*

*tomers,*

**i**in**class**a and j in class b, respectively, be at*a FIFO queue*

**k . **

**k .**

*The probability*

*P ( a , *

**b) that customer****i **

**i**

*upon arrival finds customer j*

*in queue*

**or**service at*k is upper bounded by P ( a , b ) *

*5 *

min(1, ### R}

* Proof *: we omit the proof, which follows similar tech-
nique

**[3].**

**T h e o r e m 1 ***Consider a closed multiclass queueing *
*network corriposed * of *I< queues and *

*C *

*customer*

*classes. If the customers arriving at a queue are served*

*in*

**FIFO **

**FIFO**

*order, then,*

*( 2 ) *

### x:=L

*N k c*

**x ' c **

**L **

**L**

## z:=,

I,,### E:=,

*f i k , , ~ k , ,*min{l,

* Proof *:

**we**omit the proof, which follows similar tech- nique

**[3].**

**4 ** **Bounded **P e t r i Net

In order to *cast *the models of such systems in a lin-
ear system-theoretic framework, minmax algebra is ap-
plied to the Petri net models, and analogous concepts
to transfer functions, input-output models, feedback,
etc., are developed. This section will introduce some
of these basic concepts.

* Fig G *represents the basic h l i n i m a that may be

*found in an\-*TPPN model, where the time'delays

**di**sons. For example, exact values of firing time or exact
numbers of arrival process are often unknown in some
period of time. An external token visits made by a
queueing network can be occur. On the other hand, a
Petri net of another subsystem can also make external
token visits. If the uncertainty is evaluated with the
number of tokens in a place, then **Eq 3 **can be repre-
sented as Eq

**4. **

Where **k- **

**k-**

*5 *

k *5 k + . *

*[ts(k-),td(k+)J = *ma..{min([pl *i- t l ( k ) - , *

**Pl **

**Pl**

### +

*t d k ) + l ,*

### [PI+

*t z ( k ) - - , p l + t*(k)+l},*

*[p2 3- t3(k *

### -

*2 ) - , p 2*

### +

*t3(k*

### -

**2)+]}**

**(4)**

* 5 * I n t e g r a t i o n Technique

**A **data type called interval, an object represents a real
number locating between an upper and

### a

lower bound*An interval*

**which may vary during the lifetime of a computation.***x*lies between a lower bound

**written as [x-,zc]**which means that x*x-*and upper bound

*x+. *

The main principle is that any operation whose argu- ments are intervals produces an interval that is guar- anteed t o contain all possible results of the operations in interval arithmetic;

Addition: **[ x - , x + ] **

### +

**[y-,**U+]

*= [x-*

### +

**y-,z+**

### +

*Subtraction:*

**U+],***[ x - ,*

**E + ] -****[y-, g+]**

*= [x-*

### -

**y+,**z+

### -

y-1:Division: [z-, z+]/[y-, y+]

### =

[z-. r+]### *

[l/y+, lly-1, In queueing network, the input parameters**SkC **

(mean number of visits made by a class **SkC**

*ordinary customer a t queue*

**c****k) **

and **k)**

*(mean number of exter- nal visits made by a class ordinary customer a t queue*

**Z k c****k) **

can be evaluated with intervals. They can be re-
**k)**

*placed by [Sic,*

### s&]

*and [S,,*

**Szc] **

respectively. For the
queueing network bounds and Petri Yet bounds, the
interval arithmetic can be used to calculate separate
upper and lower bounds on each class throughput, giv-
ing a rectangular box circumscribed around the feasible
throughput region. Then, the next essential objective
is to integrate those two different models. It is indi-
cates that we must solve the interval parameters.
**Szc]**

In the queueing network, the external arrival tokens
can be viewed **as ***samples *over a period *At. *The period
may be short or long. In a number of instances the
sample mean, or the sample mean and sample variance,
are used to estimate the parameters of a hypothesized
distribution [14].

### If

the observations in a sample of size*n*are

**SI **

### ,

### .

### . .

### ,

**S,, **

the sample mean **S,,**

**-T **

is defined by
and the sample variance, S’, is defined by

On the other hand, we consider the marking * m k ( p ) *in
Petri Xet, or

*for simplicity. Then, assuming that transition*

**mk***t j*

**fires, j = 1 , .**### . .

**, 1 7 1****mk+l(pi) **= **mk(pi) **

### +

O(tj,pi)### -

*I ( l j i , t j )*(7)

**for all places pi****,i**### =

1,.### . . ,

*Thus, marking*

**11.***m k ( p )*

**be evaluated with intervals. The Eq 7 can be transfer **
**into Eq 8 as **following.

avoided. It is still possible to obtain more accurate results than with pure simulation.

### Our

future work is development of throughput approximation technique based on interpolation or hIV.4 (2].The interval arith- metic, however, has so-called Jependency problem### .

It is also**as**our future research.

**REFERENCES **

[l] Peter Buchholz, “A New Approach Combining Simulation and Randomization for the Analysis of Large Continuous Time Markov Chains.”

### ,

*ACM*

*Tran. on Modeling and Computer Simulation.*Vol. 8, No. 2, .4pril 1998, pp. 194-222.

[2] Johannes Luthi and Gunter Haring, ”Alean \Blue Analysis for Queueing Network models ivith inter- vals as input parameters”

### ,

*Performance Evalua-*

*tion,*Vol. 32, pp. 185-215, 1998.

[3] Shikharesh RIajumdar and C. Murray Woodside,
“Robust Bounds and Throughput Guarantees for
Closed Multiclass Queueing Networks”, *Perfor- *
*mance Evaluation, *Vol. 32, pp. 101-136, 1998.
[4] Charles Knessl and Charles Tier, “Asymptotic Ap-

proximations and Bottleneck Analysis in Prod- uct Form Queueing Networks with Large Popula- tions”

### ,

*Performance Evaluation,*Vol. 33, pp. 219- 248, 1998.

(51 K. Laevens and H. Bruneel, “Discrete-time hlulti-
server Queues with Priorities”, *Performance Eval- *
*uation, *Vol. 33, pp. 249-275, 1998.

[6] Robert Y. Al-Jaar and Alan A. Desrochers, “Per-
formance Evaluation of Automated Manufacturing
Systems Using Generalized Stochastic Petri Xets.”,
*IEEE Tran. on Robotics and Automation, *Vol. 6,
No. 6, Dec. 1990, pp. 621-638.

**[7] **Rossano Gaeta, “Efficient Discrete-Event Simula-
tion of Colored Petri Nets.”, *IEEE Tran. on *

**Soft- **

**Soft-**

*ware Eng., *Vol. 22, No. 9, Sep. 1996, pp. 629-639.
[8] Matteo Sereno,

### ‘‘

Approximate Mean Value Xnal-ysis for Stochastic Marked Graphs.”, *IEEE Tran. *

In Petri Net, the arrival external tokens can be redis- tributed from the interval irlput of queueing network. Using Quantile-Quantile Plots technique[l4], the redis- tributed problem can be solved.

[91 Zhen Liu, “perfom~ance Analysis of Stochastic
Timed Petri Sets Using Linear Programming **-\p- **

proach.”, *IEEE Tran. on Software Eng., *Vol. 24,
NO. 11, NOV. 1998, pp. 1014-1030.

**6 ** **Conclusion **

The present work introduces a new method that com- Ijines siitiulation ancl nurnerical techniques and directly integrated using interval arithmetic techniques for the analysis of large discrete event SySteIlis. In this way, the state explosion problem of numerical analysis is

[lo] Alan A. Desrochers and R.obert

### Y.

AI-Jaar, ’.-\p- plication of Petri Nets in hIanufacturing Systems: hlodeling, Control, and Performance Analysis.”,*IEEE Press.*

[ l l ] Ng Cllee Hock, ‘‘Queueing $Iodeling Fundamen-
*tals.”, John IC‘iley * *Sons Press. *

**[12] **Leonard Kleinrock, “Queueing Systems Volume 1

**[13] **Leonard Kleinrock, “Queueing Systems Volume **2 **

: Computer Applications

### .”,

*John Wiley*

**t3 **

**t3**

*Sons*

**Press. **

**[14] **Jerry Banks, John S. Carson, and Barry

**L. Nel- **

son, “Discrete-Event System Simulation.”, *Prentice *
*Hall International, Inc. Press. *