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On Numerical Solution of the Markov Renewal Equation: Tight Upper and Lower Kernel Bounds
 D. A. Elkins,
 M. A. Wortman
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We develop tight bounds and a fast parallel algorithm to compute the Markov renewal kernel. Knowledge of the kernel allows us to solve Markov renewal equations numerically to study nonsteady state behavior in a finite state Markov renewal process. Computational error and numerical stability for computing the bounds in parallel are discussed using wellknown results from numerical analysis. We use our algorithm and computed bounds to study the expected number of departures as a function of time for a two node overflow queueing network.
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 Title
 On Numerical Solution of the Markov Renewal Equation: Tight Upper and Lower Kernel Bounds
 Journal

Methodology And Computing In Applied Probability
Volume 3, Issue 3 , pp 239253
 Cover Date
 20010901
 DOI
 10.1023/A:1013767704349
 Print ISSN
 13875841
 Online ISSN
 15737713
 Publisher
 Kluwer Academic Publishers
 Additional Links
 Topics
 Keywords

 Markov renewal process
 Markov renewal equation
 Markov renewal kernel
 Toeplitz matrix
 convolution integral equation
 Authors

 D. A. Elkins ^{(1)}
 M. A. Wortman ^{(2)}
 Author Affiliations

 1. General Motors R&D Center, 30500 Mound Road, Mail Code 480106359, Warren, MI, 480909055
 2. Department of Industrial Engineering, Texas A&M University, College Station, Texas, 778433131