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We develop a general framework for a novel switch architecture, the cross-path switch, to provide per-session statistical quality of service (QoS) guarantees. With characterizing the service each session receives by service curves, we derive a set of statistical bounds on the delay, backlog, and departure processes at the switch on a per-session manner using exponential bounded burstiness processes as source session traffic models. These bounds show that the service guarantees offered by the cross-path switch depend on the way of token assignment in the central stage of the switch. To provide better performance guarantees, we determine the criteria for designing a token assignment algorithm for the cross-path switch. Also, we quantify the service guaranteed by the cross-path switch with the central stage implemented in optical domain, which is important for the provision of QoS guarantees to each session in semioptical networks.