Delay analysis of a queue with re-sequencing buffer and Markov environment

There are simple service disciplines where the system time of a tagged customer depends only on the customers arrived to the system earlier (e.g. FIFO) or later (e.g. LIFO) than the tagged one. In this paper we consider single server queueing system with two infinite queues in which the system time of a tagged customer may depend on both the customers arrived to the system earlier and later than the tagged one. New regular customers arrive at the system according to MAP flow, occupy one place in buffer and receive service in FIFO order. External re-sequencing signals also arrive at the system according to (different) MAP flow. Each re-sequencing signal transforms one regular customer into delayed one by moving it to another queue (resequencing buffer), wherefrom it is served with lower priority than the regular ones. Service times of customers from both queues also have MAP distribution different from those which govern arrivals. Similar queueing system has been analysed with memoryless ingredients (arrival, service, re-sequencing). In this paper we investigate how the essential analytical properties of scalar functions, which made the analysis of the memoryless system feasible, can be extended to the case of Markov environment.