Lindley equation

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In probability theory, the Lindley equation, Lindley recursion or Lindley processes [1] is a discrete-time stochastic process An where n takes integer values and:

Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space. Any specified subset of these outcomes is called an event.

Integer Number in {..., –2, –1, 0, 1, 2, ...}

An integer is a number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, 5 1/2, and √2 are not.

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An + 1 = max(0, An + Bn).

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. [2] [3]

Queueing theory mathematical study of waiting lines, or queues

Queueing theory is the mathematical study of waiting lines, or queues. A queueing model is constructed so that queue lengths and waiting time can be predicted. Queueing theory is generally considered a branch of operations research because the results are often used when making business decisions about the resources needed to provide a service.

David George Kendall British statistician

David George Kendall FRS was an English statistician and mathematician, known for his work on probability, statistical shape analysis, ley lines and queueing theory. He spent most of his academic life in the University of Oxford (1946–1962) and the University of Cambridge (1962–1985). He worked with M. S. Bartlett during World War II, and visited Princeton University after the war.

Waiting times

In Dennis Lindley's first paper on the subject [4] the equation is used to describe waiting times experienced by customers in a queue with the First-In First-Out (FIFO) discipline.

Dennis Victor Lindley was an English statistician, decision theorist and leading advocate of Bayesian statistics.

Wn + 1 = max(0,Wn + Un)

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.

Queue lengths

The evolution of the queue length process can also be written in the form of a Lindley equation.

Integral equation

Lindley's integral equation is a relationship satisfied by the stationary waiting time distribution F(x) in a G/G/1 queue.

In queueing theory, a discipline within the mathematical theory of probability, the G/G/1 queue represents the queue length in a system with a single server where interarrival times have a general distribution and service times have a (different) general distribution. The evolution of the queue can be described by the Lindley equation.

Where K(x) is the distribution function of the random variable denoting the difference between the (k - 1)th customer's arrival and the inter-arrival time between (k - 1)th and kth customers. The Wiener–Hopf method can be used to solve this expression. [5]

The Wiener–Hopf method is a mathematical technique widely used in applied mathematics. It was initially developed by Norbert Wiener and Eberhard Hopf as a method to solve systems of integral equations, but has found wider use in solving two-dimensional partial differential equations with mixed boundary conditions on the same boundary. In general, the method works by exploiting the complex-analytical properties of transformed functions. Typically, the standard Fourier transform is used, but examples exist using other transforms, such as the Mellin transform.

Notes

  1. ↑ Asmussen, Søren (2003). Applied probability and queues. Springer. p. 23. doi:10.1007/0-387-21525-5_1. ISBN   0-387-00211-1.
  2. ↑ Kingman, J. F. C. (2009). "The first Erlang century—and the next". Queueing Systems . 63: 3–4. doi:10.1007/s11134-009-9147-4.
  3. ↑ Kendall, D. G. (1951). "Some problems in the theory of queues". Journal of the Royal Statistical Society, Series B. 13: 151–185. JSTOR   2984059. MR   0047944.
  4. ↑ Lindley, D. V. (1952). "The theory of queues with a single server". Mathematical Proceedings of the Cambridge Philosophical Society. 48 (2): 277–289. doi:10.1017/S0305004100027638. MR   0046597.
  5. ↑ Prabhu, N. U. (1974). "Wiener-Hopf Techniques in Queueing Theory". Mathematical Methods in Queueing Theory. Lecture Notes in Economics and Mathematical Systems. 98. pp. 81–90. doi:10.1007/978-3-642-80838-8_5. ISBN   978-3-540-06763-4.
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