Lindley equation

In probability theory, the Lindley equation, Lindley recursion or Lindley processes 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.

Waiting times

In Dennis Lindley's first paper on the subject 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.