Lindley equation
In
- 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]
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.
- 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.
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
- ISBN 0-387-00211-1.
- .
- MR 0047944.
- MR 0046597.
- ISBN 978-3-540-06763-4.