Send-time selection across time zones: a constrained, experimental design

1Context and motivation

A message has a content, a recipient and a moment. Systems that send marketing email, push, SMS and in-app messages at scale usually settle the moment with a rule: a fixed local hour for everyone, or the hour at which the customer last opened a message. Both rules are easy to run, and both ignore several things that matter when customers live in many time zones and countries: routines differ by day of the week and by country, the engagement history a rule learns from is itself shaped by past send times, outcomes such as purchases arrive hours or days after the send, and some hours are closed by law, by the merchant or by the customer.

We describe a design for send-time selection that treats the moment as a decision under constraints and under uncertainty. We give the reasons for each part and the evaluation a deployment would need. We do not report effects, because we have not measured any.

2Why timing is a decision problem

Three properties of the problem drive the design.

Observed engagement times are confounded. A customer who has only ever been sent messages at 10:00 can only have opened them after 10:00. A last-open rule learns the sending schedule back from its own output. Estimating the effect of sending at a given time requires sends that vary in time for reasons unrelated to the customer, which is randomisation. In uplift terms, the quantity of interest is the change in outcome caused by choosing one window over another, not the outcome rate observed in each window¹.

Outcomes are delayed. An open may follow within minutes, a purchase days later. A learner that counts a send as a failure because no purchase has arrived yet will be biased toward windows whose outcomes arrive quickly. In display advertising, conversions can follow an impression by up to a month, and modelling the delay separately lets a system tell a sample that has not yet converted from one that will not². In online learning, delay increases regret additively in stochastic problems, and algorithms built for immediate feedback can be adapted to delayed feedback³.

Attention is a shared resource. Notifications compete with each other and with the customer's other activities. A one-week field study of 15 mobile phone users recorded an average of 63.5 notifications per day, with more notifications associated with more negative emotions⁴, and the disruption a notification causes depends on its presentation, sender and the task the user is engaged in⁵. Two campaigns that each choose the same "good" window for a customer compete for that attention.

3Evidence on daily rhythms and receptivity

Published work supports modelling time in the customer's local clock, by day of the week and by country, rather than one global schedule.

• An analysis of millions of public Twitter messages identified individual-level diurnal mood rhythms in cultures across the globe: positive affect peaks in the morning and declines through the day, and on weekends the morning peak is delayed by two hours, which suggests that people wake later⁶. Weekday and weekend routines therefore need separate treatment.
• Smartphone data from users around the world showed that a country's average bedtime, but not its average wake time, predicts sleep duration, and that social pressures delay bedtime⁷. Evening hours that are free time in one country may be close to bedtime in another, so quiet-hour defaults and evening windows cannot be copied across countries.
• In a four-week field study with 337 mobile phone users, a model that used phone-use context predicted whether a user would click a notification and engage with the offered content with a 66.6% higher success rate than a naive baseline, and adding the user's past behaviour improved predictions fivefold over that baseline⁸. Receptivity is predictable from context, not only from the clock.

4Design

4.1Inputs

• Contact and outcome log with timestamps in UTC and in the customer's local time: sends, deliveries, opens, clicks, sessions, purchases, unsubscribes.
• Activity outside messaging: sessions, browsing and purchases by local hour and day.
• Customer context: country and time zone, language, device mix, lifecycle state, consent and channel preferences, customer-set quiet hours.
• Constraints: legal limits per jurisdiction, merchant rules, campaign deadlines, the contact envelope from the fatigue engine.

4.2Time-to-engagement model

For each candidate window, the engine estimates the distribution of time from send to engagement, not just whether engagement occurs. We model this as a survival problem with covariates (local hour and day of the send, channel, recent activity, recent contacts), using a proportional hazards form⁹. A survival model treats a send with no engagement yet as censored rather than as a failure, which handles messages still in flight at training time.

4.3Learning across windows

Window choice is a contextual bandit: the context is the customer and message, the actions are candidate windows, and the reward is engagement or purchase within a horizon¹⁰. The bandit keeps a share of sends randomised across eligible windows. That share is the source of causal estimates of timing effects; without it the model can only reproduce past schedules. Rewards that arrive late are handled with an explicit delay model, so that windows with slow but real outcomes are not penalised^{2, 3}.

4.4Constraint-aware scheduler

The scheduler turns scores into a send time. It removes windows that fall in legal limits, merchant rules or the customer's own quiet hours; it checks the contact envelope shared with other campaigns; it spreads sends for a single campaign across nearby windows when many customers share the same peak; and it falls back to a conservative default window when the model's uncertainty is high. Legal timing limits are hard constraints. For example, United States federal rules bar telephone solicitations to residential subscribers before 8 a.m. or after 9 p.m., local time at the called party's location¹¹. Rules like this are stated in local time and differ by jurisdiction, so the scheduler resolves every window in the recipient's time zone.

Table 1 maps each component to the problem it addresses and the evidence behind it.

Component	Problem it addresses	Evidence
Local time, weekday and weekend features	Routines shift on weekends	6
Country-specific evening and quiet-hour defaults	Bedtimes differ by country	7
Context features beyond the clock	Receptivity depends on context	8, 5
Survival model of time to engagement	Sends still in flight look like failures	9
Randomised share of sends across windows	Observed open times reflect past schedules	10, 1
Explicit delay model for late outcomes	Learners favour windows with fast outcomes	2, 3
Shared contact envelope	Campaigns compete for the same attention	4
Legal limits as hard constraints in local time	Timing rules are jurisdiction-specific	11
Randomised send-time experiments	Timing effects cannot be read from logs	12

Source: our design, with the works cited in the table.

5A worked example of window selection

Table 2 shows the scheduler's arithmetic for one customer and one message. The probabilities and the penalty are illustrative inputs, not estimates. The customer's quiet hours are 21:00 to 08:00 local time, and a promotion from another campaign is already scheduled for 12:00.

Window (local)	Probability of engagement within 24 h (input)	Inside quiet hours (input)	Conflict penalty (input)	Score (computed)
07:00	0.11	Yes	0.00	Excluded
08:30	0.10	No	0.00	0.10
12:30	0.08	No	0.03	0.05
19:00	0.13	No	0.00	0.13
22:30	0.16	Yes	0.00	Excluded

Source: Worked model; inputs in the table.

The window with the highest raw probability, 22:30, is excluded by the customer's quiet hours, and so is 07:00. Among the rest, 19:00 scores highest. The 12:30 window loses 0.03 to the conflict penalty because it sits next to another campaign's send. Two caveats apply. The probabilities here are engagement rates, not incremental effects; a deployed scorer would use effects estimated from the randomised share. And if the 19:00 estimate were based on few observations, the bandit might still send at 08:30 to learn.

6Evaluation protocol

Before any claim about outcomes could be made, the design would need randomised evidence.

• Randomised send-time experiments. Customers are randomised among a fixed local hour, the last-open rule and the engine's policy. Outcomes are measured over a horizon long enough to capture delayed purchases, and analysis is by intent to treat, including customers whose messages were deferred or suppressed. Pre-registration, guardrail metrics and sample size planning follow standard practice¹².
• Metrics. Clicks and purchases per message sent, time to engagement, total messages per customer, unsubscribes and complaints, and the same metrics by country and time zone.
• Exploration cost. The share of randomised sends is reported with its cost in forgone engagement, so that the value of learning can be weighed against it.

7Ethics, fairness and privacy

A window that maximises engagement can still intrude on rest or family time. The design treats quiet hours as constraints, not as features to be traded against engagement, and lets customers set their own. Inferring routines such as sleep times from activity data can feel invasive; the design uses coarse local-hour aggregates and does not infer or store sleep patterns as such. Optimising timing for engagement may favour customers with regular routines and leave others with poorly timed messages; reporting metrics by country, time zone and cohort makes such gaps visible.

8Limitations

• No measurements. We have not measured the effect of this design on any outcome. The benefits implied above are hypotheses for the evaluation protocol.
• Time zone and travel. Local time depends on knowing where the customer is; device time zones and addresses can be wrong or out of date.
• Small effects, noisy estimates. Timing effects for an individual customer may be small relative to noise, and per-customer windows may not be identifiable without many sends.
• Single-message focus. The design chooses a window per message and coordinates through a shared envelope; it does not jointly plan a sequence of messages.
• Legal coverage. The scheduler enforces only the timing rules that have been written into it. Rules differ by channel and jurisdiction and change over time.

9Conclusion

We have described a send-time design that scores windows in the customer's local time, learns timing effects from a randomised share of sends, models delayed outcomes, and schedules under quiet hours, legal limits and a shared contact envelope. Each part rests on published work summarised in Table 1. The next step is a randomised comparison of the engine against fixed-hour and last-open rules on one channel, with outcomes tracked long enough to capture delayed purchases.