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Part 2 & 3Performance evaluation

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Goals

• Understand the complex behavior of systems subject to "random phenomena"

• Develop intuitive understanding of the behaviors of stochastic systems

• Learn performance evalation methods and tools

• Able to model real-life systems for analysis of both qualitative behaviors and quantitative performances

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Stochastic ?

•Stochastic: from Greek stokhastikos(conjectural), meaning results of hasard

•Stochastic phenomena : which is not deterministic

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Performance evaluation

SystemSystem

ModelsModels

PerformancesPerformances

ModelingModeling

Performance Performance evaluationevaluation

Analysis of the Analysis of the resultsresults

! Attention: the results are performances of the model ! Attention: the results are performances of the model and not those of the system!and not those of the system!

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A possible model

Ta : time between two consecutive arrivals

ga: probability density of Ta

Ts : Service time

gs: probability density of Ts

Server

Queue

N(t)N(t) : : nb of customers in the queue

Customer arrival

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Performance measures

•4 important performance indicators of queueing systems

–Throughput rate X (or TH)

–Number of customers Q

–Resource utilisation ratio U

–Response time R

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Performance evaluation methods

•Discrete event simulation

–A very general approach

–Long computation time

–Difficulty of results analysis

•Analytical methods

–Limited to simple models under restrictions

–Quick computation time

–Allow better understanding of the system

•The two approaches are complementary in practice.

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Another example : a production line

•Examples of state variables :–Nb of parts in intermediate buffers (0, 1, 2,…, capacity of the buffer)–State of the machine (UP or DOWN)

•Examples of events :

–Completion of a part on a machine

–Failure of a machine

M1 M2 M3 M4

Raw material buffer

Finished Good Inventory

Machine

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Performance indicators

M1 M2 M3 M4

Raw material buffer

Finished Good Inventory

Machine

Mean response time

Mean buffer level

Utilization ratio of machine M3

Production rate of M3

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Stochastic processes

• A stochastic process {Xt, t T} is a sequence of random variables defined on the same state space E.

• It describes the evolution of a random variable over time.• The state space and time can be either discrete or

continuous.

E and T discrete E continuous and T discrete

E discrete and T continuous E and T continuous

Assumptions

We restrict ourselves to discrete event processes.

Two types of processes will be considered:

•Discrete time stochastic process {Xn}nIN

Example: inventory level at the beginning of each day.

•Continuous time stochastic process {Xt}t > 0

Example: number of customers in a queue.