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Degradation of services arises in practice due to a variety of reasons including wear-and-tear of machinery and fatigue. In this paper, we look at MAP/PH/1-type queueing models in which degradation is introduced. There are several ways to incorporate degradation into a service system. Here, we model the degradation in the form of the service rate declining (i.e., the service rate decreases with the number of services offered) until the degradation is addressed. The service rate is reset to the original rate either after a fixed number of services is offered or when the server becomes idle. We look at two models. In the first, we assume that the degradation is instantaneously fixed, and in the second model, there is a random time that is needed to address the degradation issue. These models are analyzed in steady state using the classical matrix-analytic methods. Illustrative numerical examples are provided. Comparisons of both the models are drawn.