Algebraic method for approximating joint stationary distribution in finite capacity queue with negative customers and two queues

Consideration is given to the single-server queueing system (QS) with a Poisson flow of (ordinary) customers and Poisson flow of negative customers. There is a queue of capacity $k$ ($0<k<\infty$), where ordinary customers wait for service. If an ordinary customer finds the queue full upon an arrival, it is considered to be lost. Each negative customer upon arrival moves one ordinary customer from the queue, if it not empty, to another queue (bunker) of capacity $r$ ($0<r<\infty$) and after that it leaves the system. If upon arrival of a negative customer the queue is not empty and the bunker is full, the negative customer and one ordinary customer from the queue leave the system. In all other cases, an arrival of a negative customer has no effect on the system. Customers from bunker are served with relative priority (i. e., a customer from bunker enters server if only there are no customers in the queue to be served). Service times of customers from both the queue and the bunker are exponentially distributed with the same parameter. Purely algebraic method based on generating functions, Chebyshev and Gegenbauer polynomials for approximate calculation of joint stationary probability distribution is presented for the case $k=r$. Numerical examples, showing both pros and cons of the method are provided.