An analysis of queueing systems with multi-task servers

We study a Markovian queueing system with multi-task servers. Each server can perform two types of job – serving the queue as primary jobs and taking vacations as secondary jobs. In such a system with c>1 servers, if at a service completion instant the server finds no customer waiting in line and the number of servers already on vacations is less than d(⩽c), he or she will take a vacation. At any time, the number of servers attending the queue or staying idle is at least c − d. By changing the parameter d, the queueing manager can better allocate the servers' time to performing the primary jobs and the secondary jobs. Using the matrix analytic method, we provide a new computational algorithm for the stationary distributions of the queue length and waiting time. The conditional stochastic decomposition properties have been established for such a system.