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Transaction management in Grids is responsible for ensuring the reliable execution of inherently distributed Grid applications. Grid transaction management is different from existing distributed transaction models because Grid resources are highly autonomous, dynamic and heterogeneous. This paper proposes a Grid transaction service (GridTS) and coordination algorithms that manage short-lived and long-lived Grid transactions respectively, providing reliability support for Grid applications. Unlike existing long-lived transaction models that require application programmers to develop compensating transactions, the GridTS can automatically generate compensating transactions during the execution of long-lived Grid transactions. The feasibility of GridTS and the effectiveness of proposed coordination algorithms are demonstrated through simulation studies.