### A bulk queueing system under $N$-policy with bilevel service delay discipline and start-up time.

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This paper concerns a discrete time Geo[X]/G/1 retrial queue with general retrial time in which all the arriving customers require first essential service with probability ${\alpha}_{0}$ while only some of them demand one of other optional services: type − r (r = 1, 2, 3,...M) service with probability ${\alpha}_{r}$. The system state distribution, the orbit size and the system size distributions are obtained in terms of generating functions. The stochastic decomposition law holds for the proposed model. Performance measures...

We consider the steady-state behavior of random walks in the quarter-plane, in particular, the expected value of performance measures that are component-wise linear over the state space. Since the stationary distribution of a random walk is in general not readily available we establish upper and lower bounds on performance in terms of another random walk with perturbed transition probabilities, for which the stationary distribution is a geometric product-form. The Markov reward approach as developed...