# Queueing theory basics

This post captures and reflects some of the thoughts we shared. The versions here are based primarily on my personal application of it in the software development context. This assumption is a very good approximation for arrival process in real systems that meet the following rules: The number of customers in the system is very large. Impact of a single customer on the performance of the system is very small, i. All customers are independent, i. Cars on a Highway As you can see these assumptions are fairly general, so they apply to a large variety of systems.

Lets consider the example of cars entering a highway. Lets see if the above rules are met. Total number of cars driving on the highway is very large. A single car uses a very small percentage of the highway resources. Decision to enter the highway Queueing theory basics independently made by each car driver.

The above observations mean that assuming a Poisson arrival process will be a good approximation of the car arrivals on the highway. If any one of the three conditions is not met, we cannot assume Poisson arrivals. For example, if a car rally is being conducted on a highway, we cannot assume that each car driver is independent of each other.

In this case all cars had a common reason to enter the highway start of the race. Telephony Arrivals Lets take another example.

Consider arrival of telephone calls to a telephone exchange.

## Queueing theory - Wikipedia

Putting our rules to test we find: Total number of customers that are served by a telephone exchange is very large. A single telephone call takes a very small fraction of the systems resources. Decision to make a telephone call is independently made by each customer.

Again, if all the rules are not met, we cannot assume telephone arrivals are Poisson. If the telephone exchange is a PABX catering to a few subscribers, the total number of customers is small, thus we cannot assume that rule 1 and 2 apply.

If rule 1 and 2 do apply but telephone calls are being initiated due to some disaster, calls cannot be considered independent of each other. This violates rule 3.

Poisson Arrival Process Now that we have established scenarios where we can assume an arrival process to be Poisson. Lets look at the probability density distribution for a Poisson process.

This equation describes the probability of seeing n arrivals in a period from 0 to t. Negative Exponential Arrivals We have seen the Poisson probability distribution.

This equation gives information about how the probability is distributed over a time interval. Unfortunately it does not give an intuitive feel of this distribution. To get a good grasp of the equation we will analyze a special case of the distribution, the probability of no arrivals taking place over a given interval.

Its easy to see that by substituting n with 0, we get the following equation:more detail on specific models that are commonly used, a textbook on queueing theory such as Hall () is recommended.

Before discussing past and potential uses of queueing models in healthcare, it’s important to first understand some queueing theory fundamentals. Queueing Fundamentals A basic queueing system is a service system .

OR-Notes are a series of introductory notes on topics that fall under the broad heading of the field of operations research (OR). Queuing theory. Queuing theory deals with problems which involve queuing (or waiting). Typical examples might be: In order to get answers to the above questions there are two basic approaches: analytic.

Since this book was published in , and since queueing theory has expanded enormously since then, one might think that this book (Queueing Systems, Volume 1) would be hopelessly out of date.

The use of multi-programming for spooling can be ascribed to the Atlas computer in the early s. The Atlas project was a joint effort between Manchester University and Ferranti Ltd. Development of computer science. Computer science as an independent discipline dates to only about , although the electronic digital computer that is the object of its study was invented some two decades earlier.

The roots of computer science lie primarily in the related fields of electrical engineering and mathematics. ﬁrst because the ﬁrst problems of queueing theory was raised by calls and Erlang was the ﬁrst who treated congestion problems in the beginning of 20th century, see Erlang [21,22].

Queueing Theory Basics