Method for Approximating Joint Stationary Distribution in Finite Capacity Queue with Negative Customers and Bunker for Ousted Customers

Queueing systems and networks with negative customers have been a subject of extensive research for two decades providing means to describe work-removal phenomena in different communication and information processing systems. This paper is the continuation of research devoted to elaboration of mathematical techniques for performance evaluation of queueing systems with different types of negative customers. Specifically we consider the Markovian single server queueing system with two finite 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 Poisson flow, occupy one place in buffer (if it is not full) and receive service in FIFO order. External negative signals also arrive at the system according to Poisson flow with different parameter. Each negative signal transforms one regular customer into delayed one by moving it to another finite-capacity queue (bunker), wherefrom, if it was accepted, it is served with lower priority than the regular ones. We propose new method based on Chebyshev and Gegenbauer polynomials for approximate calculation of joint stationary probability distribution of queues in buffer and bunker. Numerical example is provided.