An invariance principle for semi-Markov processes

1985 ◽  
Vol 17 (1) ◽  
pp. 100-126
Author(s):  
D. McDonald
Keyword(s):  
Sojourn Time ◽  
Time Distribution ◽  
Invariant Probability Measure ◽  
Transition Kernel ◽  
Mixing Conditions ◽  
Sojourn Time Distribution ◽  
Semi Markov Process ◽  
Image Position ◽  
Expected Sojourn Time ◽  
Semi Markov Processes

Let (I(t))∞t = () be a semi-Markov process with state space II and recurrent probability transition kernel P. Subject to certain mixing conditions, where Δis an invariant probability measure for P and μb is the expected sojourn time in state b ϵΠ. We show that this limit is robust; that is, for each state b ϵ Πthe sojourn-time distribution may change for each transition, but, as long as the expected sojourn time in b is µb on the average, the above limit still holds. The kernel P may also vary for each transition as long as Δis invariant.

An invariance principle for semi-Markov processes

1985 ◽  
Vol 17 (01) ◽  
pp. 100-126
Author(s):  
D. McDonald
Keyword(s):  
Sojourn Time ◽  
Time Distribution ◽  
Invariant Probability Measure ◽  
Transition Kernel ◽  
Mixing Conditions ◽  
Sojourn Time Distribution ◽  
Semi Markov Process ◽  
Image Position ◽  
Expected Sojourn Time ◽  
Semi Markov Processes

Let (I(t))∞ t = () be a semi-Markov process with state space II and recurrent probability transition kernel P. Subject to certain mixing conditions, where Δis an invariant probability measure for P and μ b is the expected sojourn time in state b ϵΠ. We show that this limit is robust; that is, for each state b ϵ Πthe sojourn-time distribution may change for each transition, but, as long as the expected sojourn time in b is µ b on the average, the above limit still holds. The kernel P may also vary for each transition as long as Δis invariant.

Reward processes with nonlinear reward functions

1996 ◽  
Vol 33 (4) ◽  
pp. 1011-1017 ◽  
Author(s):  
A. Reza Soltani
Keyword(s):  
Markov Process ◽  
Explicit Formula ◽  
Sojourn Time ◽  
Time Distribution ◽  
Reward Function ◽  
Sojourn Time Distribution ◽  
Semi Markov Process ◽  
Reward Process ◽  
Reward Functions

Based on a semi-Markov process J(t), t ≧ 0, a reward process Z(t), t ≧ 0, is introduced where it is assumed that the reward function, p(k, x) is nonlinear; if the reward function is linear, i.e. ρ (k, x) = kx, the reward process Z(t), t ≧ 0, becomes the classical one, which has been considered by many authors. An explicit formula for E(Z(t)) is given in terms of the moments of the sojourn time distribution at t, when the reward function is a polynomial.

Reward processes with nonlinear reward functions

1996 ◽  
Vol 33 (04) ◽  
pp. 1011-1017 ◽  
Author(s):  
A. Reza Soltani
Keyword(s):  
Markov Process ◽  
Explicit Formula ◽  
Sojourn Time ◽  
Time Distribution ◽  
Reward Function ◽  
Sojourn Time Distribution ◽  
Semi Markov Process ◽  
Reward Process ◽  
Reward Functions

Based on a semi-Markov process J(t), t ≧ 0, a reward process Z(t), t ≧ 0, is introduced where it is assumed that the reward function, p(k, x) is nonlinear; if the reward function is linear, i.e. ρ (k, x) = kx, the reward process Z(t), t ≧ 0, becomes the classical one, which has been considered by many authors. An explicit formula for E(Z(t)) is given in terms of the moments of the sojourn time distribution at t, when the reward function is a polynomial.

scholarly journals The sojourn time distribution in an infinite server resequencing queue with dependent interarrival and service times

2002 ◽  
Vol 39 (03) ◽  
pp. 590-603 ◽  
Author(s):  
Tijs Huisman ◽  
Richard J. Boucherie
Keyword(s):  
Markov Process ◽  
General Model ◽  
Sojourn Time ◽  
Time Distribution ◽  
Arrival Process ◽  
Accurate Approximation ◽  
Infinite Server ◽  
Sojourn Time Distribution ◽  
Semi Markov Process ◽  
Service Times

We consider an infinite server resequencing queue, where arrivals are generated by jumps of a semi-Markov process and service times depend on the jumps of this process. The stationary distribution of the sojourn time, conditioned on the state of the semi-Markov process, is obtained both for the case of hyperexponential service times and for the case of a Markovian arrival process. For the general model, an accurate approximation is derived based on a discretisation of interarrival and service times.

scholarly journals The sojourn time distribution in an infinite server resequencing queue with dependent interarrival and service times

2002 ◽  
Vol 39 (3) ◽  
pp. 590-603 ◽  
Author(s):  
Tijs Huisman ◽  
Richard J. Boucherie
Keyword(s):  
Markov Process ◽  
General Model ◽  
Sojourn Time ◽  
Time Distribution ◽  
Arrival Process ◽  
Accurate Approximation ◽  
Infinite Server ◽  
Sojourn Time Distribution ◽  
Semi Markov Process ◽  
Service Times

We consider an infinite server resequencing queue, where arrivals are generated by jumps of a semi-Markov process and service times depend on the jumps of this process. The stationary distribution of the sojourn time, conditioned on the state of the semi-Markov process, is obtained both for the case of hyperexponential service times and for the case of a Markovian arrival process. For the general model, an accurate approximation is derived based on a discretisation of interarrival and service times.

On the time a Markov chain spends in a lumped state

1997 ◽  
Vol 34 (02) ◽  
pp. 340-345
Author(s):  
Tommy Norberg
Keyword(s):  
Markov Chain ◽  
State Space ◽  
Continuous Time ◽  
Sojourn Time ◽  
Time Distribution ◽  
Sojourn Time Distribution

The sojourn time that a Markov chain spends in a subset E of its state space has a distribution that depends on the hitting distribution on E and the probabilities (resp. rates in the continuous-time case) that govern the transitions within E. In this note we characterise the set of all hitting distributions for which the sojourn time distribution is geometric (resp. exponential).

A BMAP/SM/1 queueing system with Markovian arrival input of disasters

1999 ◽  
Vol 36 (03) ◽  
pp. 868-881
Author(s):  
Alexander Dudin ◽  
Shoichi Nishimura
Keyword(s):  
Sojourn Time ◽  
Queue Length ◽  
Time Distribution ◽  
Queueing System ◽  
Queuing System ◽  
Complicated System ◽  
Sojourn Time Distribution ◽  
Negative Arrivals

Disaster arrival in a queuing system with negative arrivals causes all customers to leave the system instantaneously. Here we obtain a queue-length and virtual waiting (sojourn) time distribution for the more complicated system BMAP/SM/1 with MAP input of disasters.

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