As this article from MailOnline explains, a growing number of restaurants are trying to get rid of delays by imposing time limits on diners. Even the most expensive restaurants are now telling customers they have two hours from arrival until they have to leave. So while a 15-minute slot is something of an exaggeration, it does reflect an increasing trend.
A starter is a small amount of food eaten at the start of a meal before the main part [= appetizer in American English]. • We had soup as a starter, followed by steak.
• How to get a table in a restaurant (The Guardian)
• When it comes to reservations, time is money (New York Times)
• Dialogue: reserving a table (Focus English)
• Booking a table (EnglishSocialNetwork)