Lindley equation

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. ::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. ==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. ==Notes==