# Operations Research 2 (0365.3531)

Professor Uri Yechiali
Department
of Statistics and Operations Research

School of Mathematical Sciences

Tel-Aviv University
uriy@post.tau.ac.il

Office Hours: Wednesday 16:00 - 17:00 (and by appointment),
Schreiber 326

+972-3-640-9637.

### Class Hours:

Spring semester 2005

Sunday 13:15 - 14:30 and 14:45 - 16:00, Dan David 204

[Note: The lecture on 17.4.2005 will be replaced by a lecture on
Friday, 4.3.2005, from 9:00 to 12:00 at Schreiber 008]

### Prerequisites

Probability Theory, Operations Research 1.

### Course Content:

The course concentrates on Stochastic models in Operations Research (OR)
and it complements the Operations Research 1 course in which the main
discussion involves Deterministic OR models (i.e. Linear Programming,
Transportation models, Flows in Networks, etc). Since many real-life
systems contain stochastic elements, the aim of the course is to
present and study OR methods that can be applied to analyse and solve
such problems. Examples of stochastic systems include all sorts of Queueing
and Communication
networks in which random arriving jobs request random processing times
from a limited number of 'servers'; Inventory systems where demand is
random; Reliability models where life times of the components of systems
are stochastic, and alike.

The course is directed to 3rd year Undergraduate students
with solid background in Probability Theory and with basic knowledge
of Deterministic OR models.

## Topics:

- Markovian queueing models.
1.1. Schematic description of queueing systems; Examples; Classification of
queues; The Poisson process; Exponential
and Erlang (Gamma) probability distribution functions; The M/M/1 queue:
time-dependent differential equations, steady state, balance equations,
the traffic intensity, mean queue size, waiting and sojourn times;
General Birth-and-Death processes; M/M/1/N-1 model and the
Parking-Lot problem; The M/M/c queue; Trancated models; Erlang's
loss formula.

1.2. Jackson Networks.

- Inventory models.
The EOQ formula; Backlogs; The Newsboy problem; Stochastic
multi-period models.

- Reliability models.

- Introduction to Markov chains (MCs), Examples.
One-step transition probabilities, Stationary distributions, Embedded MCs.

- Markov Decision Processes.
Sequential decision processes; State space; Action space; One-step
expected costs; Howards algorithm; Formulation by Linear Programming;
Existance of pure stationary policies; Examples.

### Helpful Reference Books

- Hillier, F.S. and Lieberman, G.J., Introduction to Operations
Research (5th or higher Edition), McGraw-Hill.
- Winston, W.L., Operations Research (3rd or higher Edition),
Duxbury Press.

Last update: January 2005