# Stochastic Methods for Industrial Engineers (0571.5225)

Professor Uri Yechiali
Department of Statistics and Operations Research
School of Mathematical Sciences
Tel-Aviv University
uriy@math.tau.ac.il

Office Hours:    Sunday 11-12,   Schreiber 326

+972-3-640-9637.

### Class Hours:

Spring semester 1999

Tuesday 16:00-19:00,    Wolfson 238

### Prerequisites

Probability Theory, Operations Research 1 (or an equivalent), Introd. to Stochastic Processes (desired)

### Course Content:

The course focuses on Stochastic methods, primarily Queueing and Reliability models, which are applicable to Industrial Engineers.

Queueing Theory deals with the analysis, design and optimization of complex systems where several 'servers' serve randomly-arriving 'customers' (e.g. jobs, calls, messages), each requiring service of random-duration length. While waiting for server(s) to become available, the customers form 'waiting lines', or queues.

Queueing theory is extensively applied in various areas, including Manufacturing, Communication and Computer Networks, Road Traffic Analysis, etc.

Reliability of a system is its probability of functioning properly for a given period of time.

Reliability theory deals with the probabilistic analysis of multi-component complex systems where the life-time of each component is random. Characterizations of such systems and their properties will be discussed in this part of the course.

The course is directed to graduate Industrial Engineers with background in Probability Theory and some knowledge in Stochastic Processes.

## Topics:

1. Introduction
2. Probability distribution functions (Bernoulli, Geometric, Binomial, Poisson, Exponential, Erlang (Gamma), Phase-type, etc.

Generating functions, Laplace transforms

Markov chains

3. The M/G/1 Queue
4. Discrete-time model (Geom/Geom/1); Continuous time formulation (Embedded Markov chain); Stationary distribution; Khintchine-Pollaczek formula; Burke's theorem; PASTA; Waiting times; Little's law; Busy period, Delay Busy Period

5. Birth and Death Processes
6. The M/M/1 and M/M/c queues

Truncated Models; Erlang's Loss Formula (The Parking Lot Problem); Repairman Problem (Machine interference)

Multi-Dimensional Birth and Death Processes

7. Capacity Allocation in Communication Networks
8. Jackson Networks
9. CONWIP

Kanban

10. Priorities in Queueing Systems
11. Applications to Computer and Communication Networks

12. Reliability
13. Structure and coherent functions

Systems in Seies, Parallel and k-out-of-n

Paths and Cuts

Life-time distributions

Failure-rate and Hazard functions

IFR and DFR functions

Design and Optimization

Maintenance and Replacement