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In Mobile World Interoperability for Microwave Access (also known as IEEE802.16e), a standard power-saving mechanism (PSM) that alternates between sleep and wake modes is adopted to extend the lifetime of mobile stations (MSs), but it simultaneously induces a medium access control (MAC) service data unit (SDU) response delay. The standard PSM defines three power-saving classes (PSCs), among which Type I is based on a binary-exponential increase in the sleep window, and Type II is based on a constant sleep window. Over the years, most studies have mainly focused on PSC Type I, while only a few have considered Type II. This study derives several analytical models of PSC Type II and evaluates the power consumption and MAC SDU response delay performance. Based on the derived models, an optimised power under delay-bound mechanism (OPDBM) is proposed to find the optimal initial sleep window parameter that minimises the power consumption of an MS while satisfying a given MAC SDU response delay constraint. Both numerical analysis and simulation experiments show that the OPDBM effectively minimises the power consumption of an MS while guaranteeing any imposed/required MAC SDU response delay constraint in a wide variety of environments.