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Successive events and Energy Networks for Markovian deteriorating - - PowerPoint PPT Presentation

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First-passage time analysis Successive events and Energy Networks for Markovian deteriorating model Markovian deteriorating model Gejza Dohnal Introduction Model Gejza Dohnal One-way Model Repairable model Example Center of Quality and

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First-passage time analysis for Markovian deteriorating model Gejza Dohnal Introduction Model

One-way Model Repairable model Example

Lifetime of the system First-affect time Renewal period Conclusions

Gejza Dohnal

Center of Quality and Reliability of Production, CTU in Prague Czech Republic

Modeling Smart Grids - A new Challenge for Stochastics and Optimization September 10-11, Prague

Successive events and Energy Networks Markovian deteriorating model

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First-passage time analysis for Markovian deteriorating model Gejza Dohnal Introduction Model

One-way Model Repairable model Example

Lifetime of the system First-affect time Renewal period Conclusions

Introduction

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First-passage time analysis for Markovian deteriorating model Gejza Dohnal Introduction Model

One-way Model Repairable model Example

Lifetime of the system First-affect time Renewal period Conclusions

Introduction

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First-passage time analysis for Markovian deteriorating model Gejza Dohnal Introduction Model

One-way Model Repairable model Example

Lifetime of the system First-affect time Renewal period Conclusions

Introduction

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First-passage time analysis for Markovian deteriorating model Gejza Dohnal Introduction Model

One-way Model Repairable model Example

Lifetime of the system First-affect time Renewal period Conclusions

Introduction

Failure × Disaster

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First-passage time analysis for Markovian deteriorating model Gejza Dohnal Introduction Model

One-way Model Repairable model Example

Lifetime of the system First-affect time Renewal period Conclusions

Introduction

Failure × Disaster random occurrence in time random occurrence in time independent recurrence growing spread renewal process domino effect quick repair or renewal repairs slower than spreading

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First-passage time analysis for Markovian deteriorating model Gejza Dohnal Introduction Model

One-way Model Repairable model Example

Lifetime of the system First-affect time Renewal period Conclusions

Introduction

Failure × Disaster random occurrence in time random occurrence in time independent recurrence growing spread renewal process domino effect quick repair or renewal repairs slower than spreading equipment failure explosion, fire accidents, health damage disease, epidemic information system failure system breakdown, piracy attack motor vehicle accident traffic collapse natural accident natural disaster ........ .......

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First-passage time analysis for Markovian deteriorating model Gejza Dohnal Introduction Model

One-way Model Repairable model Example

Lifetime of the system First-affect time Renewal period Conclusions

Introduction

Failure × Disaster random occurrence in time random occurrence in time independent recurrence growing spread renewal process domino effect quick repair or renewal repairs slower than spreading equipment failure explosion, fire accidents, health damage disease, epidemic information system failure system breakdown, piracy attack motor vehicle accident traffic collapse natural accident natural disaster ........ ....... Model → Prediction → Prevention

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First-passage time analysis for Markovian deteriorating model Gejza Dohnal Introduction Model

One-way Model Repairable model Example

Lifetime of the system First-affect time Renewal period Conclusions

Introduction

Failure × Disaster random occurrence in time random occurrence in time independent recurrence growing spread renewal process domino effect quick repair or renewal repairs slower than spreading equipment failure explosion, fire accidents, health damage disease, epidemic information system failure system breakdown, piracy attack motor vehicle accident traffic collapse natural accident natural disaster ........ ....... Model → Prediction → Prevention Preventive Maintenance Policy Disaster Recovery Plan

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First-passage time analysis for Markovian deteriorating model Gejza Dohnal Introduction Model

One-way Model Repairable model Example

Lifetime of the system First-affect time Renewal period Conclusions

Model

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First-passage time analysis for Markovian deteriorating model Gejza Dohnal Introduction Model

One-way Model Repairable model Example

Lifetime of the system First-affect time Renewal period Conclusions

Model

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First-passage time analysis for Markovian deteriorating model Gejza Dohnal Introduction Model

One-way Model Repairable model Example

Lifetime of the system First-affect time Renewal period Conclusions

Model

State of the system: ω(t) = (ω1(t), ω2(t), . . . , ωn(t)), t ≥ 0,

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First-passage time analysis for Markovian deteriorating model Gejza Dohnal Introduction Model

One-way Model Repairable model Example

Lifetime of the system First-affect time Renewal period Conclusions

Model

State of the system: ω(t) = (ω1(t), ω2(t), . . . , ωn(t)), t ≥ 0, (i) at the beginning (at the time 0), the system is in the state ω0 = (0, 0, . . . , 0), (ii) the process starts by deterioration of an object i with probability πi, i = 1, 2, . . . , n, (iii) when at the time t an object i was affected, there was a random time period τ after which the event moved onto some

  • f the unaffected objects,

(iv) states of the system in time t create a stochastic process {X(t), t ≥ 0} in continuous time. Values of this process lie within the set Ω = {0, 1}n

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First-passage time analysis for Markovian deteriorating model Gejza Dohnal Introduction Model

One-way Model Repairable model Example

Lifetime of the system First-affect time Renewal period Conclusions

Model

State of the system: ω(t) = (ω1(t), ω2(t), . . . , ωn(t)), t ≥ 0, (i) at the beginning (at the time 0), the system is in the state ω0 = (0, 0, . . . , 0), (ii) the process starts by deterioration of an object i with probability πi, i = 1, 2, . . . , n, (iii) when at the time t an object i was affected, there was a random time period τ after which the event moved onto some

  • f the unaffected objects,

(iv) states of the system in time t create a stochastic process {X(t), t ≥ 0} in continuous time. Values of this process lie within the set Ω = {0, 1}n

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First-passage time analysis for Markovian deteriorating model Gejza Dohnal Introduction Model

One-way Model Repairable model Example

Lifetime of the system First-affect time Renewal period Conclusions

Model

State of the system: ω(t) = (ω1(t), ω2(t), . . . , ωn(t)), t ≥ 0, (i) at the beginning (at the time 0), the system is in the state ω0 = (0, 0, . . . , 0), (ii) the process starts by deterioration of an object i with probability πi, i = 1, 2, . . . , n, (iii) when at the time t an object i was affected, there was a random time period τ after which the event moved onto some

  • f the unaffected objects,

(iv) states of the system in time t create a stochastic process {X(t), t ≥ 0} in continuous time. Values of this process lie within the set Ω = {0, 1}n

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First-passage time analysis for Markovian deteriorating model Gejza Dohnal Introduction Model

One-way Model Repairable model Example

Lifetime of the system First-affect time Renewal period Conclusions

Model

State of the system: ω(t) = (ω1(t), ω2(t), . . . , ωn(t)), t ≥ 0, (i) at the beginning (at the time 0), the system is in the state ω0 = (0, 0, . . . , 0), (ii) the process starts by deterioration of an object i with probability πi, i = 1, 2, . . . , n, (iii) when at the time t an object i was affected, there was a random time period τ after which the event moved onto some

  • f the unaffected objects,

(iv) states of the system in time t create a stochastic process {X(t), t ≥ 0} in continuous time. Values of this process lie within the set Ω = {0, 1}n

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First-passage time analysis for Markovian deteriorating model Gejza Dohnal Introduction Model

One-way Model Repairable model Example

Lifetime of the system First-affect time Renewal period Conclusions

Model

Example (n = 7 π = (1, 0, 0, 0, 0, 0, 0)) From all 27 possible states only the following 14 are admissible: (1,0,0,0,0,0,0), (1,1,0,0,0,0,0), (1,0,1,0,0,0,0), (1,1,1,0,0,0,0), (1,0,1,1,0,0,0), (1,1,1,1,0,0,0), (1,0,1,1,1,0,0), (1,1,1,1,1,0,0), (1,0, 1,1,1,1,0), (1,0,1,1,1,0,1),(1,1,1,1,1,1,0), (1,1,1,1,1,0,1), (1,0,1,1,1,1,1), (1,1,1,1,1,1,1).

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First-passage time analysis for Markovian deteriorating model Gejza Dohnal Introduction Model

One-way Model Repairable model Example

Lifetime of the system First-affect time Renewal period Conclusions

Model 1

Model 1 – One-way Model Let us consider the following supplementary assumptions:

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First-passage time analysis for Markovian deteriorating model Gejza Dohnal Introduction Model

One-way Model Repairable model Example

Lifetime of the system First-affect time Renewal period Conclusions

Model 1

Model 1 – One-way Model Let us consider the following supplementary assumptions: (v) an event can only affect one object in one moment, (vi) an event can only occur once on a particular object, (vii) the process moves to the next object with a probability which depends only on the recent state, not on the path leading to the recent state (the time sequence of events).

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First-passage time analysis for Markovian deteriorating model Gejza Dohnal Introduction Model

One-way Model Repairable model Example

Lifetime of the system First-affect time Renewal period Conclusions

Model 1

Model 1 – One-way Model Let us consider the following supplementary assumptions: (v) an event can only affect one object in one moment, (vi) an event can only occur once on a particular object, (vii) the process moves to the next object with a probability which depends only on the recent state, not on the path leading to the recent state (the time sequence of events).

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First-passage time analysis for Markovian deteriorating model Gejza Dohnal Introduction Model

One-way Model Repairable model Example

Lifetime of the system First-affect time Renewal period Conclusions

Model 1

Model 1 – One-way Model Let us consider the following supplementary assumptions: (v) an event can only affect one object in one moment, (vi) an event can only occur once on a particular object, (vii) the process moves to the next object with a probability which depends only on the recent state, not on the path leading to the recent state (the time sequence of events).

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First-passage time analysis for Markovian deteriorating model Gejza Dohnal Introduction Model

One-way Model Repairable model Example

Lifetime of the system First-affect time Renewal period Conclusions

Model 1

Model 1 – One-way Model Let us consider the following supplementary assumptions: (v) an event can only affect one object in one moment, (vi) an event can only occur once on a particular object, (vii) the process moves to the next object with a probability which depends only on the recent state, not on the path leading to the recent state (the time sequence of events). ⇒ States of the system in time t create a Markovian process {X(t), t ≥ 0} in continuous time.

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First-passage time analysis for Markovian deteriorating model Gejza Dohnal Introduction Model

One-way Model Repairable model Example

Lifetime of the system First-affect time Renewal period Conclusions

Model 1 – One-way Model

Let us denote: |ω| = ω1 + ω2 + · · · + ωn the number of objects, on which a disastrous event holds when system is in the state ω (size of deterioration of the system)

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First-passage time analysis for Markovian deteriorating model Gejza Dohnal Introduction Model

One-way Model Repairable model Example

Lifetime of the system First-affect time Renewal period Conclusions

Model 1 – One-way Model

Let us denote: |ω| = ω1 + ω2 + · · · + ωn the number of objects, on which a disastrous event holds when system is in the state ω (size of deterioration of the system) Ωj = {ω ∈ Ω : |ω| = j} contains all states, in which the pursued event hold exactly on j objects. The number of such states equals to n

j

  • .
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First-passage time analysis for Markovian deteriorating model Gejza Dohnal Introduction Model

One-way Model Repairable model Example

Lifetime of the system First-affect time Renewal period Conclusions

Model 1 – One-way Model

Let us denote: |ω| = ω1 + ω2 + · · · + ωn the number of objects, on which a disastrous event holds when system is in the state ω (size of deterioration of the system) Ωj = {ω ∈ Ω : |ω| = j} contains all states, in which the pursued event hold exactly on j objects. The number of such states equals to n

j

  • .

Ω =

n

  • j=1

Ωj, Ωi ∩ Ωj = ∅ for all i = j

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First-passage time analysis for Markovian deteriorating model Gejza Dohnal Introduction Model

One-way Model Repairable model Example

Lifetime of the system First-affect time Renewal period Conclusions

Model 1 – One-way Model

Proposition Let us order states of the system in ascending order by the size of

  • deterioration. Then, under assumptions (i)-(vii), the transition

intensity matrix Q of the process {X(t), t ≥ 0} is upper triangular of the size 2n × 2n. The matrix Q can be written in the block-form as Q =        D0,0 P0,1 O0,2 O0,3 · · · O1,0 D1,1 P1,2 O1,3 · · · O1,n O2,0 O2,1 D2,2 P2,3 · · · O2,n . . . . . . . . . . . . ... . . . On,1 On,2 On,3 · · ·        where Pi,j is a rectangular matrix of size n

i

  • ×

n

j

  • , the symbol Oij

denotes a null matrix and Dii are square diagonal matrices for i = 1, . . . , n. Moreover, −Di,i = e.P′

i,i+1.Ii,i

where Ii,i is the unit matrix of size i, e is the row vector of n

j

  • nes.
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First-passage time analysis for Markovian deteriorating model Gejza Dohnal Introduction Model

One-way Model Repairable model Example

Lifetime of the system First-affect time Renewal period Conclusions

Model 2

Model 2 – Repairable Model Let us omit the assumption (vi): (v) an event can only affect one object in one moment, (vi) an event can only occur once on a particular object, (vii) the process moves to the next object with a probability which depends only on the recent state, not on the path leading to the recent state (the time sequence of events). ⇒ States of the system in time t create a Markovian process {X(t), t ≥ 0} in continuous time.

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First-passage time analysis for Markovian deteriorating model Gejza Dohnal Introduction Model

One-way Model Repairable model Example

Lifetime of the system First-affect time Renewal period Conclusions

Model 2

Model 2 – Repairable Model Let us omit the assumption (vi): (v) an event can only affect one object in one moment, (vi) an event can only occur once on a particular object, (vii) the process moves to the next object with a probability which depends only on the recent state, not on the path leading to the recent state (the time sequence of events). ⇒ States of the system in time t create a Markovian process {X(t), t ≥ 0} in continuous time.

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First-passage time analysis for Markovian deteriorating model Gejza Dohnal Introduction Model

One-way Model Repairable model Example

Lifetime of the system First-affect time Renewal period Conclusions

Model 2 – Repairable Model

The process X(t) can be considered as a random walk between sets Ω0, . . . , Ωn. When the process in a state ω ∈ Ωj, 0 < j < n, the

  • nly transitions to some states in Ωj−1 or Ωj+1 are allowed,

whereas Ω0 = {ω0} and ΩN = {ωN} are reflection states. In the model, all states are transient. The process can be understood as the process of event spreading in the system with repair.

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First-passage time analysis for Markovian deteriorating model Gejza Dohnal Introduction Model

One-way Model Repairable model Example

Lifetime of the system First-affect time Renewal period Conclusions

Model 2 – Repairable Model

Proposition In the model 2, the transition intensity matrix Q of the process {X(t), t ≥ 0} has the following block-form Q =        D0,0 P0,1 O0,2 O0,3 · · · R1,0 D1,1 P1,2 O1,3 · · · O1,n O2,0 R2,1 D2,2 P2,3 · · · O2,n . . . . . . . . . . . . ... . . . On,1 On,2 On,3 · · · Dn,n        where Pi,j is a rectangular matrix of size n

i

  • ×

n

j

  • , Rj,i is a

rectangular matrix of size n

j

  • ×

n

i

  • , the symbol Oij denotes a null

matrix and Dii are square diagonal matrices for i = 1, . . . , n. Moreover, −Di,i = (ei.R′

i−1,i + ei+1.P′ i,i+1).Ii,i

where Ii,i is the unit matrix of size i, ej is the row vector of n

j

  • nes.
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First-passage time analysis for Markovian deteriorating model Gejza Dohnal Introduction Model

One-way Model Repairable model Example

Lifetime of the system First-affect time Renewal period Conclusions

Example

Example From all 27 possible states only the following 14 are admissible: (1,0,0,0,0,0,0), (1,1,0,0,0,0,0), (1,0,1,0,0,0,0), (1,1,1,0,0,0,0), (1,0,1,1,0,0,0), (1,1,1,1,0,0,0), (1,0,1,1,1,0,0), (1,1,1,1,1,0,0), (1,0, 1,1,1,1,0), (1,0,1,1,1,0,1),(1,1,1,1,1,1,0), (1,1,1,1,1,0,1), (1,0,1,1,1,1,1), (1,1,1,1,1,1,1). The initial probability vector is equal to π = (1, 0, 0, 0, 0, 0, 0).

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First-passage time analysis for Markovian deteriorating model Gejza Dohnal Introduction Model

One-way Model Repairable model Example

Lifetime of the system First-affect time Renewal period Conclusions

Example

The transient intensity matrix S has the form                          u a b −c 0 c v d e −f 0 f w g h −i 0 i x j k −l l y m n z p q −r r −s s −t t 0 0                          .

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First-passage time analysis for Markovian deteriorating model Gejza Dohnal Introduction Model

One-way Model Repairable model Example

Lifetime of the system First-affect time Renewal period Conclusions

Example

Se = 0 ⇒ u = −a − b, v = −d − e, w = −g − h, x = −j − k, y = −m − n, z = −p − q. There remains 19 unknown parameters which correspond to conditional transient intensities: a = i(1 → 2|1); g = i(1 → 2|1, 3, 4); m = i(1 → 2|1, 3, 4, 5, 6); b = i(1 → 3|1); h = i(4 → 5|1, 3, 4); n = i(5 → 7|1, 3, 4, 5, 6); c = i(1 → 3|1, 2); i = i(4 → 5|1, 2, 3, 4); p = i(1 → 2|1, 3, 4, 5, 7); d = i(1 → 2|1, 3); j = i(1 → 2|1, 3, 4, 5); q = i(5 → 6|1, 3, 4, 5, 7); e = i(3 → 4|1, 2); k = i(5 → 6|1, 3, 4, 5); r = i(5 → 7|1, 2, 3, 4, 5, 6); f = i(3 → 4|1, 3); l = i(5 → 6|1, 2, 3, 4, 5); s = i(5 → 6|1, 2, 3, 4, 5, 7); t = (1 → 2|1, 3, 4, 5, 6, 7). In the case, where transitions between objects are independent of the previous path, we have a = d = g = j = m = p = t, b = c, e = f, h = i, k = l = q = s, n = r. The whole system can be reduced to 6 unknown parameters a, b, e, h, k, n.

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First-passage time analysis for Markovian deteriorating model Gejza Dohnal Introduction Model

One-way Model Repairable model Example

Lifetime of the system First-affect time Renewal period Conclusions

Lifetime of the system

One-way Model: The time of the total deterioration of the system is the time T, in which the process X(t) will attach the state ωN. During this time, the pursued events will pass through all objects of the system, the state ωN is absorbing.

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First-passage time analysis for Markovian deteriorating model Gejza Dohnal Introduction Model

One-way Model Repairable model Example

Lifetime of the system First-affect time Renewal period Conclusions

Lifetime of the system

Model 1

Proposition In the model 1, the time T to the system deterioration is a random variable, which has phase type probability distribution with initial probability vector π = (π1, π2, . . . , πn, 0, 0, . . . , 0) and upper triangular transition intensity matrix S =      D1,1 P1,2 O1,3 · · · O1,n−1 O2,1 D2,2 P2,3 · · · O2,n−1 . . . . . . . . . ... . . . On−1,1 On−1,2 On−1,3 · · · Dn−1,n−1      .

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First-passage time analysis for Markovian deteriorating model Gejza Dohnal Introduction Model

One-way Model Repairable model Example

Lifetime of the system First-affect time Renewal period Conclusions

Lifetime of the system

Model 1

Proposition In the model 1, the time T to the system deterioration is a random variable, which has phase type probability distribution with initial probability vector π = (π1, π2, . . . , πn, 0, 0, . . . , 0) and upper triangular transition intensity matrix S =      D1,1 P1,2 O1,3 · · · O1,n−1 O2,1 D2,2 P2,3 · · · O2,n−1 . . . . . . . . . ... . . . On−1,1 On−1,2 On−1,3 · · · Dn−1,n−1      . For PH-distribution, we know all moments E(T k) = (−1)kk!πS−ke, k ∈ N. where e is vector of ones.

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First-passage time analysis for Markovian deteriorating model Gejza Dohnal Introduction Model

One-way Model Repairable model Example

Lifetime of the system First-affect time Renewal period Conclusions

Lifetime of the system

Model 2

Proposition In model 2, the time T to the system deterioration is a random variable, which has phase type probability distribution with initial probability vector π = (1, 0, . . . , 0) and transition intensity matrix S =        D0,0 P0,1 O0,2 · · · O0,n−1 R1,0 D1,1 P1,2 · · · O1,n−1 02,0 R2,1 D2,2 · · · O2,n−1 . . . . . . . . . ... . . . On−1,0 On−1,1 On−1,2 · · · Dn−1,n−1        .

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First-passage time analysis for Markovian deteriorating model Gejza Dohnal Introduction Model

One-way Model Repairable model Example

Lifetime of the system First-affect time Renewal period Conclusions

Lifetime of the system

Model 2

Proposition In model 2, the time T to the system deterioration is a random variable, which has phase type probability distribution with initial probability vector π = (1, 0, . . . , 0) and transition intensity matrix S =        D0,0 P0,1 O0,2 · · · O0,n−1 R1,0 D1,1 P1,2 · · · O1,n−1 02,0 R2,1 D2,2 · · · O2,n−1 . . . . . . . . . ... . . . On−1,0 On−1,1 On−1,2 · · · Dn−1,n−1        . Particularly, E(T) = −πS−1e, Var(T) = 2πS−2e − (πS−1e)2.

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First-passage time analysis for Markovian deteriorating model Gejza Dohnal Introduction Model

One-way Model Repairable model Example

Lifetime of the system First-affect time Renewal period Conclusions

First-affect time

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First-passage time analysis for Markovian deteriorating model Gejza Dohnal Introduction Model

One-way Model Repairable model Example

Lifetime of the system First-affect time Renewal period Conclusions

First-affect time

For given i denote Si

0 = {ω ∈ Ω : ωi = 0}

Si

1 = {ω ∈ Ω : ωi = 1}

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First-passage time analysis for Markovian deteriorating model Gejza Dohnal Introduction Model

One-way Model Repairable model Example

Lifetime of the system First-affect time Renewal period Conclusions

First-affect time

For given i denote Si

0 = {ω ∈ Ω : ωi = 0} – the set of states of the system, in

which the i-th object is not affected Si

1 = {ω ∈ Ω : ωi = 1}

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First-passage time analysis for Markovian deteriorating model Gejza Dohnal Introduction Model

One-way Model Repairable model Example

Lifetime of the system First-affect time Renewal period Conclusions

First-affect time

For given i denote Si

0 = {ω ∈ Ω : ωi = 0}

Si

1 = {ω ∈ Ω : ωi = 1} – the set of states, in which the i-th

  • bject is affected
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First-passage time analysis for Markovian deteriorating model Gejza Dohnal Introduction Model

One-way Model Repairable model Example

Lifetime of the system First-affect time Renewal period Conclusions

First-affect time

For given i denote Si

0 = {ω ∈ Ω : ωi = 0}

Si

1 = {ω ∈ Ω : ωi = 1}

The sets Si

0 and Si 1 are disjoint and Si 0 = Ω − Si 1

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First-passage time analysis for Markovian deteriorating model Gejza Dohnal Introduction Model

One-way Model Repairable model Example

Lifetime of the system First-affect time Renewal period Conclusions

First-affect time

One-way Model Reordering states in both of these sets in ascending order according to |ω|, we can write the transition intensity matrix in the following block form: Si Si

1

Si Si

1

A C O B

  • where

A is upper triangular of the size (2n−1 − 1) × (2n−1 − 1), B is is upper triangular of the size (2n−1 × 2n−1),

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First-passage time analysis for Markovian deteriorating model Gejza Dohnal Introduction Model

One-way Model Repairable model Example

Lifetime of the system First-affect time Renewal period Conclusions

First-affect time

One-way Model Reordering states in both of these sets in ascending order according to |ω|, we can write the transition intensity matrix in the following block form: Si Si

1

Si Si

1

A C O B

  • where

A is upper triangular of the size (2n−1 − 1) × (2n−1 − 1), B is is upper triangular of the size (2n−1 × 2n−1), Repairable Model After reordering of states we obtain the transition intensity matrix Si Si

1

Si Si

1

A C G B

  • where G is not null matrix.
slide-46
SLIDE 46

First-passage time analysis for Markovian deteriorating model Gejza Dohnal Introduction Model

One-way Model Repairable model Example

Lifetime of the system First-affect time Renewal period Conclusions

First-affect time

Proposition The first affect time Ti for an object i in the system is the random variable with the phase type probability distribution with representation (πi, A).

slide-47
SLIDE 47

First-passage time analysis for Markovian deteriorating model Gejza Dohnal Introduction Model

One-way Model Repairable model Example

Lifetime of the system First-affect time Renewal period Conclusions

First-affect time

Proposition The first affect time Ti for an object i in the system is the random variable with the phase type probability distribution with representation (πi, A). πi = (π1, . . . , πi−1, πi+1, . . . , πn−1, 0, . . . , 0, πi),

slide-48
SLIDE 48

First-passage time analysis for Markovian deteriorating model Gejza Dohnal Introduction Model

One-way Model Repairable model Example

Lifetime of the system First-affect time Renewal period Conclusions

First-affect time

Proposition The first affect time Ti for an object i in the system is the random variable with the phase type probability distribution with representation (πi, A). πi = (π1, . . . , πi−1, πi+1, . . . , πn−1, 0, . . . , 0, πi), A is a matrix in the similar form as the matrix S in the previous Propositions: in the One-way Model A =      D1,1 P1,2 O1,3 · · · O1,n−1 O2,1 D2,2 P2,3 · · · O2,n−1 . . . . . . . . . ... . . . On−1,1 On−1,2 On−1,3 · · · Dn−1,n−1      where Pi,j is a rectangular matrix of size n

i

  • ×

n

j

  • , Oij is a null matrix,

Dii are square diagonal matrices for i = 1, . . . , n. Moreover Di,i = −(e.P′

i,i+1.Ii,i), where e is the row vector of

n

j

  • nes and

Ii,i is the unit matrix of size i.

slide-49
SLIDE 49

First-passage time analysis for Markovian deteriorating model Gejza Dohnal Introduction Model

One-way Model Repairable model Example

Lifetime of the system First-affect time Renewal period Conclusions

First-affect time

Proposition The first affect time Ti for an object i in the system is the random variable with the phase type probability distribution with representation (πi, A). πi = (π1, . . . , πi−1, πi+1, . . . , πn−1, 0, . . . , 0, πi), A is a matrix in the similar form as the matrix S in the previous Propositions: in the Repairable Model A =        D0,0 P0,1 O0,2 · · · O0,n−1 R1,0 D1,1 P1,2 · · · O1,n−1 02,0 R2,1 D2,2 · · · O2,n−1 . . . . . . . . . ... . . . On−1,0 On−1,1 On−1,2 · · · Dn−1,n−1        . where Rj,i are rectangular matrices of size n

j

  • ×

n

i

  • , and

Di,i = −(ei.R′

i−1,i + ei+1.P′ i,i+1).Ii,i

slide-50
SLIDE 50

First-passage time analysis for Markovian deteriorating model Gejza Dohnal Introduction Model

One-way Model Repairable model Example

Lifetime of the system First-affect time Renewal period Conclusions

Renewal period

Under the assumptions of model 2, the process can be understood as renewal process. The renewal occurs when the process reaches the state ω0 and the renewal period covers the time which the process needs to return to the state ω0 again, if it started in ω0.

slide-51
SLIDE 51

First-passage time analysis for Markovian deteriorating model Gejza Dohnal Introduction Model

One-way Model Repairable model Example

Lifetime of the system First-affect time Renewal period Conclusions

Renewal period

Under the assumptions of model 2, the process can be understood as renewal process. The renewal occurs when the process reaches the state ω0 and the renewal period covers the time which the process needs to return to the state ω0 again, if it started in ω0. For this purpose, we shall consider the state ω0 in two manners:

as a starting state (π0 = (1, 0, . . . , 0)) as an absorption state after the process will move to it form another state.

slide-52
SLIDE 52

First-passage time analysis for Markovian deteriorating model Gejza Dohnal Introduction Model

One-way Model Repairable model Example

Lifetime of the system First-affect time Renewal period Conclusions

Renewal period

Under the assumptions of model 2, the process can be understood as renewal process. The renewal occurs when the process reaches the state ω0 and the renewal period covers the time which the process needs to return to the state ω0 again, if it started in ω0. For this purpose, we shall consider the state ω0 in two manners:

as a starting state (π0 = (1, 0, . . . , 0)) as an absorption state after the process will move to it form another state.

slide-53
SLIDE 53

First-passage time analysis for Markovian deteriorating model Gejza Dohnal Introduction Model

One-way Model Repairable model Example

Lifetime of the system First-affect time Renewal period Conclusions

Renewal period

Under the assumptions of model 2, the process can be understood as renewal process. The renewal occurs when the process reaches the state ω0 and the renewal period covers the time which the process needs to return to the state ω0 again, if it started in ω0. For this purpose, we shall consider the state ω0 in two manners:

as a starting state (π0 = (1, 0, . . . , 0)) as an absorption state after the process will move to it form another state.

⇒ We can describe the distribution of renewal period using PH-distribution.

slide-54
SLIDE 54

First-passage time analysis for Markovian deteriorating model Gejza Dohnal Introduction Model

One-way Model Repairable model Example

Lifetime of the system First-affect time Renewal period Conclusions

Renewal period

Proposition In the model 2, the distribution of renewal period can be described by PH-distribution with the representation (π0, W), where W =        D0,0 P0,1 O0,2 O0,3 · · · O1,0 D1,1 P1,2 O1,3 · · · O1,n O2,0 R2,1 D2,2 P2,3 · · · O2,n . . . . . . . . . . . . ... . . . On,1 On,2 On,3 · · · Dn,n        where Pi,j is a rectangular matrix of size n

i

  • ×

n

j

  • , the symbol Oij

denotes a null matrix and Dii are square diagonal matrices for i = 1, . . . , n. Moreover, −Di,i = e.P′

i,i+1.Ii,i

where Ii,i is the unit matrix of size i, e is the row vector of n

j

  • nes.
slide-55
SLIDE 55

First-passage time analysis for Markovian deteriorating model Gejza Dohnal Introduction Model

One-way Model Repairable model Example

Lifetime of the system First-affect time Renewal period Conclusions

Conclusions

There has been proposed the model for deterioration spreading in two variants:

slide-56
SLIDE 56

First-passage time analysis for Markovian deteriorating model Gejza Dohnal Introduction Model

One-way Model Repairable model Example

Lifetime of the system First-affect time Renewal period Conclusions

Conclusions

There has been proposed the model for deterioration spreading in two variants:

One-way Model (for nonrepairable systems)

slide-57
SLIDE 57

First-passage time analysis for Markovian deteriorating model Gejza Dohnal Introduction Model

One-way Model Repairable model Example

Lifetime of the system First-affect time Renewal period Conclusions

Conclusions

There has been proposed the model for deterioration spreading in two variants:

One-way Model (for nonrepairable systems) Repairable Model (when we permit repairs of system components)

slide-58
SLIDE 58

First-passage time analysis for Markovian deteriorating model Gejza Dohnal Introduction Model

One-way Model Repairable model Example

Lifetime of the system First-affect time Renewal period Conclusions

Conclusions

There has been proposed the model for deterioration spreading in two variants:

One-way Model (for nonrepairable systems) Repairable Model (when we permit repairs of system components)

The model is supposed to be Markovian.

slide-59
SLIDE 59

First-passage time analysis for Markovian deteriorating model Gejza Dohnal Introduction Model

One-way Model Repairable model Example

Lifetime of the system First-affect time Renewal period Conclusions

Conclusions

There has been proposed the model for deterioration spreading in two variants:

One-way Model (for nonrepairable systems) Repairable Model (when we permit repairs of system components)

The model is supposed to be Markovian. The first-passage time is modelled using PH distribution.

slide-60
SLIDE 60

First-passage time analysis for Markovian deteriorating model Gejza Dohnal Introduction Model

One-way Model Repairable model Example

Lifetime of the system First-affect time Renewal period Conclusions

Conclusions

There has been proposed the model for deterioration spreading in two variants:

One-way Model (for nonrepairable systems) Repairable Model (when we permit repairs of system components)

The model is supposed to be Markovian. The first-passage time is modelled using PH distribution. Thanks for your attention.

Gejza Dohnal Centre for Quality and Reliability of Production, CTU Prague dohnal@nipax.cz