\(\mathrm{M}/m(a, b)/(2,1)\) queueing system with servers repeated and delayed vacation (Q2920907)

The paper under review studies the following queueing system. Arrivals are assumed to be Poisson with parameter \(\lambda\) and there are two servers. The customers are served by batches as follows. The service starts when there are \(a\) customers in the system. If the queue size is not greater than \(b\) \((b\geq a)\), then the entire queue is served. If there are more than \(b\) customers in the queue, then only the first \(b\) customers are served, while the remaining part waits in the queue. The service time of a batch is exponentially distributed with parameter \(\mu\) (independent of the batch size). Upon the service completion, if there are fewer than \(a-1\) customers in the system and the other server is busy, the free server takes vacation for the exponentially distributed random time with parameter \(\theta\). However, if one of the servers is on vacation and the number of customers is less than \(a\), the second server cannot take vacation and must remain in the system to wait until the number of customers is enough to start the service (that is \(a\)). If after a service completion the number of customers exactly equals \(a-1\) and one of the servers is busy, then vacation is taken for an exponential random time with parameter \(\alpha\). A vacation is repeated if at the end of the earlier vacation there are fewer than \(a\) customers in the system. The paper studies the stationary queue-length process and calculates the expected queue-length.