Efficient descriptor-vector multiplications in stochastic automata networks
Journal of the ACM (JACM)
Nonsymmetric Algebraic Riccati Equations and Wiener--Hopf Factorization for M-Matrices
SIAM Journal on Matrix Analysis and Applications
Single-Server Queue with Markov-Dependent Inter-Arrival and Service Times
Queueing Systems: Theory and Applications
Queueing Systems: Theory and Applications
Sojourn time distributions in the queue defined by a general QBD process
Queueing Systems: Theory and Applications
Journal of Computational and Applied Mathematics
Structured Markov chains solver: algorithms
SMCtools '06 Proceeding from the 2006 workshop on Tools for solving structured Markov chains
Queues in DOCSIS cable modem networks
Computers and Operations Research
A Joint Moments Based Analysis of Networks of MAP/MAP/1 Queues
QEST '08 Proceedings of the 2008 Fifth International Conference on Quantitative Evaluation of Systems
Q-MAM: a tool for solving infinite queues using matrix-analytic methods
Proceedings of the 3rd International Conference on Performance Evaluation Methodologies and Tools
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In this paper we study a broad class of semi-Markovian queues introduced by Sengupta. This class contains many classical queues such as the GI/M/1 queue, SM/MAP/1 queue and others, as well as queues with correlated inter-arrival and service times. Queues belonging to this class are characterized by a set of matrices of size m and Sengupta showed that its waiting time distribution can be represented as a phase-type distribution of order m. For the special case of the SM/MAP/1 queue without correlated service and inter-arrival times the queue length distribution was also shown to be phase-type of order m, but no derivation for the queue length was provided in the general case. This paper introduces an order m^2 phase-type representation (@k,K) for the queue length distribution in the general case and proves that the order m^2 of the distribution cannot be further reduced in general. A matrix geometric representation (@k,K,@n) is also established for the number of type @t@?{1,...,m} customers in the system, where a customer is of type @t if the phase in which it completes service belongs to @t. We derive these results in both discrete and continuous time and also discuss the numerical procedure to compute (@k,K,@n). When the arrivals have a Markovian structure, the numerical procedure is reduced to solving a Quasi-Birth-Death (for the discrete time case) or fluid queue (for the continuous time case). Finally, by combining a result of Sengupta and Ozawa, we provide a simple formula to compute the order m phase-type representation of the waiting time in a MAP/MAP/1 queue without correlated service and inter-arrival times, using the R matrix of a Quasi-Birth-Death Markov chain.