Lindley equation

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

A_n + 1 = max(0, A_n + B_n).

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]

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.

W_n + 1 = max(0,W_n + U_n)

where

• T_n is the time between the nth and (n+1)th arrivals,
• S_n is the service time of the nth customer, and
• U_n = S_n − T_n
• W_n 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.

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.

$F(x)=\int _{{0^{-}}}^{\infty }K(x-y)F({\text{d}}y)\quad x\geq 0$

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]

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 MR47944.
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.