# Queueing Theory

### From 118wiki

## Introduction to Queueing Theory

Queueing theory is a mathematical basis for modelling the behavior of discrete "customers" and "servers" in real (continuous) time. That means time is a real-valued variable, whereas customers are finite and countable. Customers and servers in the real world behave somewhat unpredictably, though they usually follow statistical patterns. Therefore queueing theory relies on a probabilistic view of the behavior over time.

### Probability Distributions

A small background on Probability Distributions is helpful. The file media:Distribs.pdf also has some overview of the distributions. For modelling customer arrival rates, the Poisson distribution is most useful; for modelling service rates, the Exponential distribution is used (whereas, the most-used distribution in most sciences is the Normal distribution). The Poisson distribution is closely related to the exponential distribution; the exponential distribution is essentially the only Memoryless distribution.

### Models of Queues

The document media:Qintro.pdf is a brief introduction. Your goal in studying Queueing Theory is mainly to understand Little's law, and the relation between arrival rates, interarrival rates, and service rates. Important things to remember are (for so-called M/M/1 queueing systems):

- λ is the mean arrival rate (in customers per second)
- μ is the mean rate for a server (in how many customers per second it can serve)
- arrival rates are Poisson distributed
- "time gaps" between customers are exponentially distributed, and the mean gap time is 1/λ
- The formula for Little's law is N = λ T where N is the number of customers in the "system" (definition of system varies depending on the problem considered), and T is the mean total time a customer spends in the system.
- The "steady state" equation for mean queue length is N = λ / ( μ - λ )