In distributed applications, a group of multiple processes cooperate by exchanging messages. It is critical to support the group of application processes with enough quality of service (QoS) including the ordered delivery of messages. The delay time and the message loss ratio are significant QoS parameters. In Internet applications, the delay time and the loss ratio are significantly different in different communication channels. We define a novel causality named Delta*-causality among the messages to hold in the world-wide environment. We discuss how to transmit messages to the destination processes and how to resolve message loss and delay supporting the Delta*-causality given the requirements of delay time and message loss ratio.