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In this paper we analyze a single server queue with batch arrivals andsemi-Markovian service times. We also include the feature that the firstservice of each busy period might have a different distribution than subsequentservice times. Our generating function based approach allows us to determinethe heavy traffic limit of the scaled queue-length distribution. It turns outthat this distribution converges to an exponential distribution.Nonsurprisingly, the exceptional first service does not influence this limitingdistribution. We identify a sufficient and necessary condition under which thedependence between successive service times disappears in the limit, which weillustrate in a numerical example.

Heavy-traffic analysis of the $M^X/\text{semi-Markov}/1$ queue