School of Mathematical Sciences

Tel-Aviv University

Office Hours: Sunday 11-12, Schreiber 326

+972-3-640-9637.

Spring semester 1999

Tuesday 16:00-19:00, Wolfson 238

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

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.

- Introduction
- The M/G/1 Queue
- Birth and Death Processes
- Capacity Allocation in Communication Networks
- Jackson Networks
- Priorities in Queueing Systems
- Reliability

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

Generating functions, Laplace transforms

Markov chains

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

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

CONWIP

Kanban

Applications to Computer and Communication Networks

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

- Kleinrock, L., Queueing Systems (Vol. 1: Theory, Vol.2: Computer applications), Wiley, 1975 & 1976.
- Cooper, R.B., Introd. to Queueing Theory, North Holland, 1981.
- Walrand, J., An introd. to Queueing Networks, Prentice-Hall, 1988.
- Hall, J.A., Queueing Methods for Services and Manufacturing, Prentice Hall, 1988
- Barlow,R.E. and Proschan, F., Statistical Theory of Reliability and Life Testing, Holt, Rinehart and Winston, 1975.
- Buzacott,J.A.and Shantikumar, J.G., Stochastic Models of Manufacturing Systems, Wiley, 1993.

Last update: March 11, 1999