Lindley equation
Encyclopedia
In probability theory
, the Lindley equation, Lindley recursion or Lindley processes is a discrete time stochastic process An where n takes integer
values and
Processes of this form can be used to describe the waiting time of customers in a queue
or evolution of a queue length over time. The idea was first proposed in the discussion following Kendall
's 1951 paper.
's first paper on the subject the equation is used to describe waiting times experienced by customers in a queue.
where
The first customer does not need to wait so W1 = 0. Subsequent customers will have to wait if they arrive at a time before the previous customer has been served.
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
, the Lindley equation, Lindley recursion or Lindley processes is a discrete time stochastic process An where n takes integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
values and
-
- An + 1 = max(0,An + Bn).
Processes of this form can be used to describe the waiting time of customers in a queue
Queueing theory
Queueing theory is the mathematical study of waiting lines, or queues. The theory enables mathematical analysis of several related processes, including arriving at the queue, waiting in the queue , and being served at the front of the queue...
or evolution of a queue length over time. The idea was first proposed in the discussion following Kendall
David George Kendall
David George Kendall FRS was an English statistician, who spent much of his academic life in the University of Oxford and the University of Cambridge. He worked with M. S...
's 1951 paper.
Waiting times
In Dennis LindleyDennis Lindley
Dennis Victor Lindley is a British statistician, decision theorist and leading advocate of Bayesian statistics.Dennis Lindley grew up in the south-west London suburb of Surbiton. He was an only child and his father was a local building contractor...
's first paper on the subject the equation is used to describe waiting times experienced by customers in a queue.
-
- Wn + 1 = max(0,Wn + Un)
where
- Tn is the time between the nth and (n+1)th arrivals,
- Sn is the service time of the nth customer, and
- Un = Sn − Tn
- Wn is the waiting time of the nth customer.
The first customer does not need to wait so W1 = 0. Subsequent customers will have to wait if they arrive at a time before the previous customer has been served.