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On excess‐, current‐, and total‐life distributions of phase‐type renewal processes

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  • Edward P. C. Kao
  • Marion Spokony Smith

Abstract

Consider a renewal process whose interrenewal‐time distribution is phase type with representation (α, T). We show that the (time‐dependent) excess‐life distribution is phase type with representation (α′, T), where α′ is an appropriately modified initial probability vector. Using this result, we derive the (time‐dependent) distributions for the current life and the total life of the phase‐type renewal process. They in turn enable us to obtain the equilibrium distributions for the three random variables. These results simplify the computation of the respective distribution functions and consequently enhance the potential use of renewal theory in stochastic modeling—particularly in inventory, queueing, and reliability applications. © 1992 John Wiley & Sons, Inc.

Suggested Citation

  • Edward P. C. Kao & Marion Spokony Smith, 1992. "On excess‐, current‐, and total‐life distributions of phase‐type renewal processes," Naval Research Logistics (NRL), John Wiley & Sons, vol. 39(6), pages 789-799, October.
    • Handle: RePEc:wly:navres:v:39:y:1992:i:6:p:789-799
    DOI: 10.1002/1520-6750(199210)39:63.0.CO;2-A
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    References listed on IDEAS

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    1. Marcel F. Neuts, 1988. "Computation and applications of geometric mixtures of convolutions," Naval Research Logistics (NRL), John Wiley & Sons, vol. 35(4), pages 529-537, August.
    2. D. G. Nguyen & D. N. P. Murthy, 1984. "Cost analysis of warranty policies," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 31(4), pages 525-541, December.
    3. Yves Balcer & Izzet Sahin, 1986. "Replacement Costs Under Warranty: Cost Moments and Time Variability," Operations Research, INFORMS, vol. 34(4), pages 554-559, August.
    4. Marcel F. Neuts, 1978. "Renewal processes of phase type," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 25(3), pages 445-454, September.
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