- 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.
- Wn + 1 = max(0,Wn + Un)
- 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.
The evolution of the queue length process can also be written in the form of a Lindley 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.
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