The MMAP[K]/PH[K]/1 queues with a last-come-first-served preemptive service discipline
Queueing Systems: Theory and Applications
The Versatility of MMAP[K] and the MMAP[K]/G[K]/1 Queue
Queueing Systems: Theory and Applications
Queueing Systems: Theory and Applications
Q-MAM: a tool for solving infinite queues using matrix-analytic methods
Proceedings of the 3rd International Conference on Performance Evaluation Methodologies and Tools
Structured Markov chains solver: tool extension
Proceedings of the Fourth International ICST Conference on Performance Evaluation Methodologies and Tools
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In this paper, we study a discrete time queueing system with multiple types of customers and a first-come-first-served (FCFS) service discipline. Customers arrive according to a semi-Markov arrival process and the service times of individual customers have PH-distributions. A GI/M/1 type Markov chain for a generalized age process of batches of customers is introduced. The steady state distribution of the GI/M/1 type Markov chain is found explicitly and, consequently, the steady state distributions of the age of the batch in service, the total workload in the system, waiting times, and sojourn times of different batches and different types of customers are obtained. We show that the generalized age process and a generalized total workload process have the same steady state distribution. We prove that the waiting times and sojourn times have PH-distributions and find matrix representations of those PH-distributions. When the arrival process is a Markov arrival process with marked transitions, we construct a QBD process for the age process and the total workload process. The steady state distributions of the waiting times and the sojourn times, both at the batch level and the customer level, are obtained from the steady state distribution of the QBD process. A number of numerical examples are presented to gain insight into the waiting processes of different types of customers.