SOME PERFORMANCE MEASURES FORVACATION MODELS WITHA BATCHMARKOVIAN ARRIVAL PROCESS Submitted to Journal ofAppliedMathematics andStochastic Analysis

We consider a single-server infinite capacity queueing system with a batch Markovian arrival process (BMAP).In particular, BMAP’s are the batch Poisson arrival process, Markovian arrival process (MAP’s), many batch arrival processes with correlated inter-arrival times and batch sizes, and superpositions ofthese processes. We note that the MAP includes phase-type (PH) renewal processes and non-renewal processes such as the Markov modulated Poisson process (MMPP). The server applies Kella’s vacation scheme, i.e., a vacation policy where the decision ofwhether to take a new vacation or not, when the system is empty, depends on the number of vacation already taken in the current inactive phase. This exhaustive service discipline includes the single vacation T-policy, T(SV), and the multiple vacation T-policy, T(M). The service times are i.i.d, random variables, independent of the inter-arrival times and the vacation durations. Some important performance measures such as the distribution functions and means ofthe virtual and the actual waiting time processes are given. Finally, a numerical example is presented.