1. 1 Part 2 & 3 Performance evaluation

2. 2 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

3. 3 Stochastic ? Stochastic: from Greek stokhastikos(conjectural), meaning results of hasard Stochastic phenomena : which is not deterministic

4. 4 Performance evaluation System Models Performances Modeling Analysis of the results ! Attention: the results are performances of the model and not those of the system!

5. 5 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

6. 6 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

7. 7 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.

8. 8 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 M1M1 M2M2 M3M3 M4M4 Raw material buffer Finished Good Inventory Machine

9. 9 Performance indicators M1M1 M2M2 M3M3 M4M4 Raw material buffer Finished Good Inventory Machine Mean response time Mean buffer level Utilization ratio of machine M 3 Production rate of M 3

10. 10 Stochastic processes A stochastic process {X t, 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

11. Assumptions We restrict ourselves to discrete event processes. Two types of processes will be considered: Discrete time stochastic process {X n } n  IN Example: inventory level at the beginning of each day. Continuous time stochastic process {X t } t > 0 Example: number of customers in a queue.