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A two-sided exit problem is solved for a difference of a compound Poissonprocess and a compound renewal process. More precisely, the Laplace transformsof the joint distribution of the first exit time, the value of the overshootand the value of a linear component at this instant are found. Further, westudy the process reflected in its supremum. We determine the main two-boundarycharacteristics of the process reflected in its supremum. These results arethen applied for studying the $ M^{\varkappa}|G^{\delta}|1|B$ system. We derivethe distribution of a busy period and the numbers of customers in the system intransient and stationary regimes. The advantage is that these results are in aclosed form, in terms of resolvent sequences of the process.

Busy period, time of the first loss of a customer and the number of customers in $ M^{\varkappa}|G^{\delta}|1|B$