How to start service only when other service had completed?

docker-compose version 1.29 comes with build in functionality: service_completed_successfully. According to spec:

service_completed_successfully - specifies that a dependency is expected to run to successful completion before starting a dependent service.

depends_on:
  <service-name>:
    condition: service_completed_successfully

Docker compose haven't any out-of-the-box solutions.
But exists a number of good implementations for solving this problem.

One of the better solutions is using service discovery as Consul and Autopilot model implemented in Joyent Containerpilot.

This model allows implementing dependencies between services even in docker swarm cluster where service dependency does not exist at all.

Containerpilot jobs model allow creating simple scripts to keep started services in the right order.

This is a supplement to @zooblin 's answer

Since docker-compose version 1.29, we can do it by condition: service_completed_successfully

In your scene, database service start will cost some time, so the migration scripts should be executed after database fully started.

And the application service should start after migration scripts executed successfully.

the docker-compose.yaml may be like following(here we use cassandra as exmaple, for other database, you can just modify healthcheck commands):

version: '3.8'
services:
  applicaion-service:
    image: your-applicaion-service:0.0.1
    depends_on:
      cassandra-init-keyspace:
        condition: service_completed_successfully


  cassandra:
    image: cassandra:4.0.1
    ports:
      - "9042:9042"
    healthcheck:
      test: ["CMD", "cqlsh", "-u cassandra", "-p cassandra" ,"-e describe keyspaces"]
      interval: 15s
      timeout: 10s
      retries: 10

  cassandra-init-keyspace:
    image: cassandra:4.0.1
    depends_on:
      cassandra:
        condition: service_healthy
    volumes:
      - ./src/main/resources/cassandra/init.cql:/init.cql
    command: /bin/bash -c "echo loading cassandra keyspace && cqlsh cassandra -f /init.cql"