You’ve probably seen this famous question on an Amazon site:

“**Want it delivered Saturday, October 15?** Order it in the next 7 hours and 10 minutes, and choose One-Day Shipping at checkout.”

But what if you want to order in 8 hours? Why can’t you get one-day shipping then? Answer: Amazon is not sure they will be able to get the order out of the warehouse in time to make the last UPS pickup of the night. If you order 7 hours from now, they think they can do it.

But how did they decide on 7 hours and 10 minutes?

Although they probably don’t think of it in these terms, Amazon has determined that, if you order in the next 7 hours 10 minutes, the probability of missing the last truck of the night is low enough to justify the risk that they won’t fulfill the service promise. This question is faced thousands—or hundreds of thousands—of times every night by large internet retailers.

We are developing mathematical models to help order fulfillment systems better coordinate their internal operations so that more customers get their orders sooner. We are particularly interested in systems that operate against a *deadline*, such as the Amazon system described above. Nearly all of existing research in order fulfillment has taken for its objective maximizing throughput or other objectives related to Little’s Law. We believe our work is the first to address this important service objective.

**Next Scheduled Deadline**

The models we are developing maximize a service objective called *Next Scheduled Deadline *(NSD), which measures the fraction of orders each day arriving during a specified 24-hour period that are processed before a specific truck departure. For example, if the cut-off time is 1300, and 1,000 orders arrive between 1300 on Day 1 and 1300 on Day 2, NSD is 90 percent if only 900 of those orders make it on the departing truck at 1700 on Day 2. The metric is currently in use at the Defense Logistics Agency.

Our first paper on this concept looked at how to set the cutoff time and the goal for NSD in the presence of motivation—workers are motivated to work harder on orders due at the next deadline, and not as motivated to work on orders not due at that time. The paper will appear in *Production and Operations Management.*

**Understanding the “sojourn time” of an order**

Sojourn time in a distribution center is the total time an order takes from arrival to completion (ready for shipment), and of course, this time varies according to workload and natural variation in the system. We have built two queueing-theoretic models to help managers plan their operations.

The first approximation model produces a steady-state sojourn time distribution sojourn time distribution of customers or jobs arriving to an acyclic multi-server queueing network. The model accepts general interarrival times and general service times, and is based on the characteristics of phase-type distributions. The model could be used by a retail distribution center to establish a profit-maximizing cutoff time for premium shipping (think Amazon, above).

The second model produces a *state-dependent *sojourn time distribution for orders arriving to a multi-stage, multi-server queueing network, where processing times can take on general distributions. Imagine arriving to a service system (customers to a bank teller, patrons to a ride at Disney World, or orders to an Amazon warehouse) with customers in front of you, what is the probability you will finish service before a deadline (the bank closes, the rides stop, or the FedEx truck leaves)? The answer requires a state-dependent sojourn time distribution, which describes the probability that waiting time in queue plus service time with the server will be less than a particular value, given the number of people in front of you. We show how to calculate that probability.

Below is an example distribution produced for an order arriving to a 4-stage, multi-server serial line. Upon arrival, the order finds 11 customers ahead of it at the first workstation, and there are 12, 8, and 13 orders ahead in the following 3 stages. The graphic shows results from a simulation model (dashed lines) and our approximation.

The model can also handle acyclic queueing networks. Below is the sojourn time distribution for an order arriving to a system having two paths—one with a fast path, the other with a relatively slow path. The resulting distribution tells the story (probabilistically) that there is a high probability of a low sojourn time and a low probability of a higher sojourn time.