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Kohsaku MASUDA, "Approximations of State Transition Probabilities in Finite Birth-Death Processes" in IEICE TRANSACTIONS on Fundamentals,
vol. E74-A, no. 4, pp. 715-721, April 1991, doi: .
Abstract: This paper approximations to the transient probabilities Pij (t) (i, j=0, 1, 2, , n) for a transition from state i at t=0 to state j at time t in the n-channel birthdeath processes. First, P0n(t) is considered as an extension of Gnedenko's approximate expressions when state n is regarded as an absorbing state for the models M/M/1/n/, M/M/1/n/N, M/M/n/n/, and M/M/n/n/N. That is to say, if Pn is the steady-state probability of state n, the approximations P0n(t)/Pn when state n is not an absorbing state can be obtained from the function 1-exp{-Q(t)} (Q(t)0 is an analytic function). Based on these considerations, transition diagrams are derived to obtain P0n (t) for the models M/M/S/n/ and M/M/S/n/N. Finally, Pij(t) can be expressed with this P0n(t). Several examples show that the approximations of the transient probabilities are nearly equal to the exact values calculated numerically using the Runge-Kutta method on a personal computer. As the approximations in this paper are very precise and calculable instantaneously on a personal computer, they may be applicable for time-dependent traffic theory which will be useful, for instance, in real-time network management technology.
URL: https://global.ieice.org/en_transactions/fundamentals/10.1587/e74-a_4_715/_p
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@ARTICLE{e74-a_4_715,
author={Kohsaku MASUDA, },
journal={IEICE TRANSACTIONS on Fundamentals},
title={Approximations of State Transition Probabilities in Finite Birth-Death Processes},
year={1991},
volume={E74-A},
number={4},
pages={715-721},
abstract={This paper approximations to the transient probabilities Pij (t) (i, j=0, 1, 2, , n) for a transition from state i at t=0 to state j at time t in the n-channel birthdeath processes. First, P0n(t) is considered as an extension of Gnedenko's approximate expressions when state n is regarded as an absorbing state for the models M/M/1/n/, M/M/1/n/N, M/M/n/n/, and M/M/n/n/N. That is to say, if Pn is the steady-state probability of state n, the approximations P0n(t)/Pn when state n is not an absorbing state can be obtained from the function 1-exp{-Q(t)} (Q(t)0 is an analytic function). Based on these considerations, transition diagrams are derived to obtain P0n (t) for the models M/M/S/n/ and M/M/S/n/N. Finally, Pij(t) can be expressed with this P0n(t). Several examples show that the approximations of the transient probabilities are nearly equal to the exact values calculated numerically using the Runge-Kutta method on a personal computer. As the approximations in this paper are very precise and calculable instantaneously on a personal computer, they may be applicable for time-dependent traffic theory which will be useful, for instance, in real-time network management technology.},
keywords={},
doi={},
ISSN={},
month={April},}
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TY - JOUR
TI - Approximations of State Transition Probabilities in Finite Birth-Death Processes
T2 - IEICE TRANSACTIONS on Fundamentals
SP - 715
EP - 721
AU - Kohsaku MASUDA
PY - 1991
DO -
JO - IEICE TRANSACTIONS on Fundamentals
SN -
VL - E74-A
IS - 4
JA - IEICE TRANSACTIONS on Fundamentals
Y1 - April 1991
AB - This paper approximations to the transient probabilities Pij (t) (i, j=0, 1, 2, , n) for a transition from state i at t=0 to state j at time t in the n-channel birthdeath processes. First, P0n(t) is considered as an extension of Gnedenko's approximate expressions when state n is regarded as an absorbing state for the models M/M/1/n/, M/M/1/n/N, M/M/n/n/, and M/M/n/n/N. That is to say, if Pn is the steady-state probability of state n, the approximations P0n(t)/Pn when state n is not an absorbing state can be obtained from the function 1-exp{-Q(t)} (Q(t)0 is an analytic function). Based on these considerations, transition diagrams are derived to obtain P0n (t) for the models M/M/S/n/ and M/M/S/n/N. Finally, Pij(t) can be expressed with this P0n(t). Several examples show that the approximations of the transient probabilities are nearly equal to the exact values calculated numerically using the Runge-Kutta method on a personal computer. As the approximations in this paper are very precise and calculable instantaneously on a personal computer, they may be applicable for time-dependent traffic theory which will be useful, for instance, in real-time network management technology.
ER -
English
