## All Graduate Plan B and other Reports

1-1-1969

Report

#### Degree Name

Master of Science (MS)

#### Department

Mathematics and Statistics

Ronald V. Canfield

#### Abstract

Just as the probability theory is regarded as the study of mathematical models of random phenomena, the theory of stochastic processes plays an important role in the investigation of random phenomena depending on time. A random phenomenon that arises through a process which is developing in time and controlled by some probability law is called a stochastic process. Thus, stochastic processes can be referred to as the dynamic part of the probability theory. We will now give a formal definition of a stochastic process.

Let T be a set which is called the index set (thought of as time), then, a collection or family of random variables {X(t), t ε T} is called a stochastic process.

If T is a denumerable infinite sequence then {X(t)} is called a stochastic process with discrete parameter. If T is a finite or infinite interval, then {X(t)} is called a stochastic process with continuous parameter. In the definition above, T is the time interval involved and {X(t)} is the observation at time t.

The theory of stochastic processes has developed very rapidly and has found application in a large number of fields; for example, the study of fluctuations and noise in the physical system, in the information theory of communication and control, in operations research, in biology, in astronomy, and so on. No attempt has been made to investigate all applications in this report, as we are especially interested in the study of the theory of stochastic processes in application to operations research. Since stochastic processes provides a method of quantitative study through the mathematical model, it plays an important role in the modern discipline or operations research. The waiting-line analysis or queueing problem of operations research is the most important part in which the theory of stochastic processes applies most often. A brief description follows.

The waiting-line problem is described by a flow of customers requiring service when there is some restriction on the service that can be provided. The group waiting to receive service is called a queue; for example: Patients arriving at a clinic to see a doctor; students waiting at a window for registration packages; persons waiting in a Greyhound bus station to buy tickets; numerous problems connected with telephone exchanges; machines which stop from time to time and require attention by an operator before restarting, the operator being able to attend to only one machine at a time, etc. All these form a queue. In order to study the nature of the waiting-line problem, the following three aspects should be specified.

1. Input process or the arrival pattern is the probability law with both the average rate or customers and the statistical pattern of the arrivals. One of the common arrival patterns is that of Poisson input.

2. The service mechanism describes when service is available, how many customers can be served at a time, and how long service takes; for examples, the exponential service time distribution, the constant service time distribution, etc.

3. The Queue discipline is the manner in which a customer is selected for service out of all those awaiting service. One of the possible ways is "first come, first served."

The queuing theory is concerned with the effect that each of the three aspects has on various quantities of interest; such as the length of the queue, the service time distributions, and the average waiting time.

We shall first deal with the theoretical developments. The applications will then follow.

COinS