The Waiting Time Distribution of a TypekCustomer in a Discrete-Time MMAP[K]/PH[K]/c (c = 1, 2) Queue Using QBDs

Chris L. Blondia

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Advanced Queuing Theory Analysis Probability and Statistical Research Petri Nets in System Modeling

Abstract

This paper presents an improved method to calculate the delay distribution of a type k customer in a first-come-first-serve (FCFS) discrete-time queueing system with multiple types of customers, where each type has different service requirements, and c servers, with c = 1, 2 (the MMAP[K]/PH[K]/c queue).The first algorithms to compute this delay distribution, using the GI/M/1 paradigm, were presented in [9,10].The two most limiting properties of these algorithms are: (i) the computation of the rate matrix R related to the GI/M/1 type Markov chain, (ii) the amount of memory needed to store the transition matrices A l and B l .In this paper we demonstrate that each of the three GI/M/1 type Markov chains used to develop the algorithms in [9,10] can be reduced to a QBD with a block size which is only marginally larger than that of its corresponding GI/M/1 type Markov chain.As a results, the two major limiting factors of each of these algorithms are drastically reduced to computing the G matrix of the QBD and storing the 6 matrices that characterize the QBD.Moreover, these algorithms are easier to implement, especially for the system with c = 2 servers.We also include some numerical examples that further demonstrate the reduction in computational resources.

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References

1999Introduction to Matrix Analytic Methods in Stochastic Modeling 1998Markov chains with marked transitions11This research was supported in part by Grant No. DMI-930628 from the National Science Foundation and the K.C. Wang Education Foundation.1996Queues with marked customers 1994The workload in theMAP/G/1 queue with state-dependent services:its application to a queue with preemptive resume priority 2001Queue Length Distribution in a FIFO Single-Server Queue with Multiple Arrival Streams Having Different Service Time Distributions 2001The Versatility of MMAP[K] and the MMAP[K]/G[K]/1 Queue 2002The delay distribution of a type k customer in a first-come-first-served MMAP[K]/PH[K]/1 queue

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2005Age Process, Workload Process, Sojourn Times, and Waiting Times in a Discrete Time SM[K]/PH[K]/1/FCFS Queue 2012Analysis of a continuous time SM[K]/PH[K]/1/FCFS queue: Age process, sojourn times, and queue lengths 2018First-come-first-served queues with multiple servers and customer classes 2008Q-MAM: A Tool for Solving Infinite Queues using Matrix-Analytic Methods 2010Quasi-birth-and-death processes with restricted transitions and its applications

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