Which cloud service model provides the consumer with the infrastructure to create applications and host them?

In the platform-as-a-service (PaaS) model, the consumer has access to the infrastructure to create applications and host them. Software-as-a-service (SaaS) supplies a particular application; infrastructure-as-a-service (IaaS) does not directly provide the ability to create applications, although this distinction is quickly blurring; and IDaaS is identity-as-a-service.