Queueing Theory Definition, Equations, and Applications

Below is a Definition of Queueing Theory, a glossary, and vocabulary. Knowing the concepts below will help you better understand these articles on Queues.

• Queue: A line (or buffer or inventory) feeding a number of servers
• Server: An operation fed by a queue.
• Arrival rate (λ): Mean number of arrivals per unit time (usually per hour or day).
• Service rate (μ): Mean number of customers that can be served at 100% utilization by each individual server per unit time (usually per hour or day). At the individual workstation level, the service rate will equal capacity.
• Channels (M): The number of parallel operations connected to an individual queue. For example, if each queue has 2 operations then it will have two channels.
• Utilization (u): A measure of how “busy” the system is. It is generally defined as the ratio of throughput to capacity. Note that u = λ/(Μμ) if λ < Μμ, i.e. the utilization is less than 100%. (Also, note that while the Greek letter μ— or mu— looks a bit like u, they are in fact two different variables.)
• Phase: A queue and its connected servers, or routes to a server.
• Balking: When a person, who would otherwise have entered a line, decides not to enter it.
• Reneging: When a person, who has entered a line, later decides to leave it without being served.
• Interarrival Time: The time between when one customer arrives at a queue and when the next customer arrives.
• Service Time: The time it takes for one particular server to complete a customer’s service. The average service time will be the same as the cycle time.

And now for some equations you’ll need to know.

• CV: The coefficient of variation. This is a measure of a random variable’s variability. For a random variable x, CVx is defined as Standard Deviation (x) x mean (x) CV = .
• CVIAT: The coefficient of variation of the interarrival time. The greater the CVIAT, the “lumpier” the arrival rate.
• CVST: The coefficient of variation of the service time. The smaller the CVST, the more “consistent” a server is.
• Lq: The average number of people in a line awaiting service.
• Wq: The average length of time a customer waits before being served.

After you have become familiar with the terms above, go ahead and read the articles on Queueing Theory.