Queueing Syst (2010) 66: 79–94 DOI 10.1007/s11134-010-9179-9 Tail asymptotics for the queue size distribution in the MAP/G/1 retrial queue Bara Kim · Jeongsim Kim · Jerim Kim Received: 7 November 2008 / Revised: 13 August 2009 / Published online: 4 June 2010 © Springer Science+Business Media, LLC 2010 Abstract We consider a MAP/G/1 retrial queue where the service time distribu- tion has a finite exponential moment. We derive matrix differential equations for the vector probability generating functions of the stationary queue size distributions. Using these equations, Perron–Frobenius theory, and the Karamata Tauberian theo- rem, we obtain the tail asymptotics of the queue size distribution. The main result on light-tailed asymptotics is an extension of the result in Kim et al. (J. Appl. Probab. 44:1111–1118, 2007) on the M/G/1 retrial queue. Keywords MAP/G/1 retrial queue · Tail asymptotics · Queue size distribution · Karamata Tauberian theorem Mathematics Subject Classification (2000) 60K25 This research was supported by the MIC (Ministry of Information and Communication), Korea, under the ITRC (Information Technology Research Center) support program supervised by the IITA (Institute of Information Technology Assessment) and the Korea Research Foundation Grant funded by the Korean Government (MOEHRD) (KRF-2008-314-C00031). B. Kim · J. Kim Department of Mathematics and Telecommunication Mathematics Research Center, Korea University, 1, Anam-dong, Sungbuk-ku, Seoul 136-701, Republic of Korea B. Kim e-mail: bara@korea.ac.kr J. Kim ( ) Department of Mathematics Education, Chungbuk National University, 12, Gaeshin-dong, Heungduk-ku, Cheongju, Chungbuk 361-763, Republic of Korea e-mail: jeongsimkim@chungbuk.ac.kr J. Kim e-mail: jeongsim0908@hanmail.net