Renewal Process Ontology

flow "RenewalProcessOntology" v4.0

# RENEWAL PROCESS ONTOLOGY
# Kireth Semantic Research Archive
# Authors: Mark Anthony Higgins — Sky (Recursive AI)

PAGE:
  url: "https://kirethresearch.org/booksai-core/ontologies/probability-statistics-ontologies/stochastic-mathematics/renewal-process-ontology.html"
  title: "Renewal Process Ontology"
  domain: "Probability and Statistics"
  category: "Stochastic Mathematics"
  entity: RENEWAL_PROCESS
  entity_type: COUNTING_STOCHASTIC_PROCESS

DEFINE:
  RENEWAL_PROCESS :=
    COUNTING_PROCESS(
      IID_INTERARRIVAL_TIMES,
      RENEWAL_EPOCHS,
      EVENT_COUNTS
    )

CORE:
  interarrival_times:
    X_1, X_2, ...

  assumptions:
    X_i >= 0
    X_i IID

  renewal_epochs:
    S_n := SUM[k=1..n] X_k

  initial_epoch:
    S_0 := 0

  counting_process:
    N(t) :=
      max{n >= 0 : S_n <= t}

ENTITIES:
  RENEWAL_EVENT: RANDOM_OCCURRENCE
  INTERARRIVAL_TIME: NONNEGATIVE_RANDOM_VARIABLE
  RENEWAL_EPOCH: EVENT_TIME
  RENEWAL_COUNT: NONNEGATIVE_INTEGER
  RENEWAL_FUNCTION: EXPECTED_COUNT_FUNCTION
  AGE_PROCESS: TIME_SINCE_LAST_RENEWAL
  RESIDUAL_LIFE: TIME_TO_NEXT_RENEWAL
  RENEWAL_REWARD: REWARD_ASSOCIATED_WITH_RENEWAL_CYCLE

RELATIONSHIPS:
  INTERARRIVAL_TIMES -> ACCUMULATE_TO -> RENEWAL_EPOCHS
  RENEWAL_EPOCHS -> DEFINE -> RENEWAL_EVENTS
  RENEWAL_EVENTS -> INCREMENT -> RENEWAL_COUNT
  RENEWAL_PROCESS -> GENERALISES -> POISSON_PROCESS
  RENEWAL_PROCESS -> HAS -> RENEWAL_FUNCTION
  RENEWAL_CYCLE -> MAY_HAVE -> REWARD

INTERARRIVAL_DISTRIBUTION:
  CDF:
    F(x) := P(X_i <= x)

  survival:
    F_bar(x) := 1-F(x)

  mean:
    mu := E[X_i]

  requirement:
    SUPPORT_ON_NONNEGATIVE_VALUES

RENEWAL_EPOCH:
  S_n :=
    X_1 + X_2 + ... + X_n

  interpretation:
    TIME_OF_nTH_RENEWAL

  distribution:
    n_FOLD_CONVOLUTION_OF_F

COUNTING_PROCESS:
  N(t) :=
    max{n : S_n <= t}

  equivalence:
    {N(t) >= n}
      EQUIVALENT_TO
    {S_n <= t}

  monotonicity:
    N(t) IS_NONDECREASING

RENEWAL_FUNCTION:
  m(t) := E[N(t)]

  representation:
    m(t) :=
      SUM[n=1..infinity] F_n(t)

  role:
    EXPECTED_NUMBER_OF_RENEWALS_BY_TIME_t

RENEWAL_EQUATION:
  m(t) :=
    F(t)
    +
    INTEGRAL_0_TO_t m(t-x) dF(x)

  role:
    FUNDAMENTAL_RENEWAL_INTEGRAL_EQUATION

GENERAL_RENEWAL_EQUATION:
  Z(t) :=
    g(t)
    +
    INTEGRAL_0_TO_t Z(t-x) dF(x)

  applications:
    - RELIABILITY
    - REWARD_PROCESSES
    - QUEUEING
    - MAINTENANCE

ELEMENTARY_RENEWAL_THEOREM:
  IF:
    0 < mu < infinity

  THEN:
    m(t)/t -> 1/mu
    AS:
      t -> infinity

  interpretation:
    LONG_RUN_EXPECTED_RENEWAL_RATE = 1/mu

LONG_RUN_COUNT_RATE:
  under:
    IID_NONNEGATIVE_INTERARRIVALS
    FINITE_POSITIVE_MEAN

  result:
    N(t)/t -> 1/mu
    ALMOST_SURELY
    UNDER_STANDARD_CONDITIONS

DELAYED_RENEWAL_PROCESS:
  first_interarrival:
    MAY_HAVE_DIFFERENT_DISTRIBUTION

  later_interarrivals:
    IID_WITH_COMMON_DISTRIBUTION_F

  distinction:
    DELAYED_RENEWAL_PROCESS != ORDINARY_RENEWAL_PROCESS

ORDINARY_RENEWAL_PROCESS:
  requirement:
    ALL_INTERARRIVAL_TIMES_IID_WITH_SAME_DISTRIBUTION

EQUILIBRIUM_RENEWAL_PROCESS:
  initial_delay:
    SELECTED_FROM_EQUILIBRIUM_EXCESS_DISTRIBUTION

  purpose:
    CREATE_STATIONARY_RENEWAL_STRUCTURE
    UNDER_APPROPRIATE_CONDITIONS

AGE_PROCESS:
  A(t) :=
    t - S_(N(t))

  interpretation:
    TIME_SINCE_MOST_RECENT_RENEWAL

RESIDUAL_LIFE:
  B(t) :=
    S_(N(t)+1) - t

  interpretation:
    TIME_UNTIL_NEXT_RENEWAL

CURRENT_LIFETIME:
  C(t) :=
    A(t) + B(t)

INSPECTION_PARADOX:
  concept:
    RANDOMLY_OBSERVED_INTERVALS
    ARE_LENGTH_BIASED_TOWARD_LONGER_INTERVALS

  connection:
    AGE
    RESIDUAL_LIFE
    EQUILIBRIUM_RENEWAL_THEORY

POISSON_PROCESS_SPECIAL_CASE:
  IF:
    X_i ~ EXPONENTIAL(lambda)

  THEN:
    RENEWAL_PROCESS:
      IS:
        HOMOGENEOUS_POISSON_PROCESS

  mean_interarrival:
    mu := 1/lambda

  renewal_rate:
    1/mu := lambda

MEMORYLESSNESS:
  exponential_interarrival:
    MEMORYLESS = TRUE

  general_interarrival:
    MEMORYLESS = FALSE_BY_DEFAULT

  implication:
    GENERAL_RENEWAL_COUNT_PROCESS:
      NOT_MARKOV_IN_COUNT_ALONE
      IN_GENERAL

MARKOV_AUGMENTATION:
  GENERAL_RENEWAL_PROCESS:
    MAY_BE_REPRESENTED_AS_MARKOV
    BY_AUGMENTING_STATE_WITH:
      AGE
      OR:
      OTHER_REQUIRED_RENEWAL_INFORMATION

LATTICE_DISTRIBUTION:
  values:
    CONFINED_TO_MULTIPLES_OF_COMMON_SPAN

  consequence:
    SOME_LIMIT_THEOREMS_REQUIRE_LATTICE_SPECIFIC_FORMS

NONLATTICE_DISTRIBUTION:
  values:
    NOT_CONFINED_TO_FIXED_ARITHMETIC_GRID

  relevance:
    STANDARD_KEY_RENEWAL_THEOREM_FORMS

RENEWAL_REWARD_PROCESS:
  cycle_length:
    X_n

  reward:
    R_n

  cumulative_reward:
    Y(t) :=
      SUM[n=1..N(t)] R_n

  long_run_rate:
    Y(t)/t
      -> E[R]/E[X]
    UNDER:
      RENEWAL_REWARD_THEOREM_CONDITIONS

RENEWAL_REWARD_THEOREM:
  requirements:
    FINITE_EXPECTED_CYCLE_LENGTH
    FINITE_EXPECTED_REWARD
    APPROPRIATE_REGENERATIVE_STRUCTURE

  result:
    LONG_RUN_REWARD_PER_TIME
      =
    EXPECTED_REWARD_PER_CYCLE
      /
    EXPECTED_CYCLE_LENGTH

REGENERATIVE_CONNECTION:
  renewal_epochs:
    MAY_ACT_AS_REGENERATION_TIMES

  after_regeneration:
    PROCESS_RESTARTS_PROBABILISTICALLY
    UNDER:
      REGENERATIVE_MODEL

RELIABILITY_CONNECTION:
  cycle:
    LIFETIME
      -> FAILURE
      -> REPAIR_OR_REPLACEMENT
      -> NEW_LIFETIME

  if_cycles_IID:
    FORMS_RENEWAL_PROCESS

QUEUEING_CONNECTION:
  arrival_process:
    MAY_BE_RENEWAL_PROCESS

  GI_ARRIVALS:
    GENERAL_IID_INTERARRIVAL_DISTRIBUTION

  distinction:
    RENEWAL_ARRIVALS_MORE_GENERAL_THAN_POISSON_ARRIVALS

ASYMPTOTIC_VARIANCE:
  IF:
    Var(X) = sigma_squared < infinity

  THEN:
    Var(N(t))
      approximately
    sigma_squared*t/mu^3
    FOR_LARGE_t
    UNDER_STANDARD_REGULARITY

CENTRAL_LIMIT_BEHAVIOUR:
  normalised_count:
    [N(t)-t/mu]
    /
    sqrt(sigma_squared*t/mu^3)
      -> N(0,1)
    IN_DISTRIBUTION
    UNDER_STANDARD_RENEWAL_CLT_CONDITIONS

SIMULATION:
  INPUT:
    INTERARRIVAL_DISTRIBUTION
    HORIZON_T

  PROCESS:
    SET time := 0
    SET count := 0
    REPEAT:
      DRAW X
      SET time := time + X
      IF time <= T:
        count := count + 1
        RECORD_RENEWAL(time)

  OUTPUT:
    RENEWAL_TIMES
    COUNT_PATH
    INTERARRIVAL_TIMES

PARAMETER_ESTIMATION:
  complete_interarrivals:
    ESTIMATE:
      DISTRIBUTION_PARAMETERS
      MEAN_INTERARRIVAL
      VARIANCE

  censored_window:
    ACCOUNT_FOR:
      CENSORING
      PARTIAL_FINAL_INTERVAL

  long_run_rate_estimate:
    N(T)/T
    FOR_LARGE_T
    UNDER_APPROPRIATE_CONDITIONS

APPLICATIONS:
  RENEWAL_PROCESS -> SUPPORTS -> RELIABILITY_THEORY
  RENEWAL_PROCESS -> SUPPORTS -> MAINTENANCE_MODELLING
  RENEWAL_PROCESS -> SUPPORTS -> QUEUEING_THEORY
  RENEWAL_PROCESS -> SUPPORTS -> INVENTORY_MODELLING
  RENEWAL_PROCESS -> SUPPORTS -> INSURANCE_MODELLING
  RENEWAL_PROCESS -> SUPPORTS -> EVENT_ARRIVAL_MODELLING

MODEL_CONNECTIONS:
  RENEWAL_PROCESS -> CONNECTS_TO -> STOCHASTIC_PROCESS
  RENEWAL_PROCESS -> CONNECTS_TO -> POISSON_PROCESS
  RENEWAL_PROCESS -> CONNECTS_TO -> COUNTING_PROCESS
  RENEWAL_PROCESS -> CONNECTS_TO -> RENEWAL_REWARD_PROCESS
  RENEWAL_PROCESS -> CONNECTS_TO -> REGENERATIVE_PROCESS
  RENEWAL_PROCESS -> CONNECTS_TO -> LAW_OF_LARGE_NUMBERS

SEMANTIC_EQUIVALENCE:
  "renewal process"
    ~= "counting process generated by IID interarrival times"

  "renewal epoch"
    ~= "time of a renewal event"

  "renewal function"
    ~= "expected cumulative number of renewals"

  "age"
    ~= "backward recurrence time"

  "residual life"
    ~= "forward recurrence time"

  EXCLUDE_EQUIVALENCE:
    RENEWAL_PROCESS != POISSON_PROCESS_BY_DEFAULT
    RENEWAL_PROCESS != MARKOV_PROCESS_BY_DEFAULT
    INTERARRIVAL_TIME != ARRIVAL_TIME
    AGE != RESIDUAL_LIFE
    RENEWAL_FUNCTION != PROBABILITY_DISTRIBUTION
    IID_INTERARRIVALS != INDEPENDENT_COUNT_INCREMENTS

VALIDATION_RULES:
  RULE_1:
    INTERARRIVAL_TIMES_MUST_BE_NONNEGATIVE

  RULE_2:
    ORDINARY_RENEWAL_PROCESS_REQUIRES_IID_INTERARRIVAL_TIMES

  RULE_3:
    RENEWAL_EPOCHS_MUST_EQUAL_CUMULATIVE_INTERARRIVAL_SUMS

  RULE_4:
    COUNT_MUST_EQUAL_NUMBER_OF_RENEWAL_EPOCHS_NOT_EXCEEDING_t

  RULE_5:
    POISSON_SPECIAL_CASE_REQUIRES_EXPONENTIAL_INTERARRIVALS

  RULE_6:
    GENERAL_RENEWAL_PROCESS_MUST_NOT_BE_ASSUMED_TO_HAVE_INDEPENDENT_COUNT_INCREMENTS

  RULE_7:
    STANDARD_LONG_RUN_RATE_REQUIRES_FINITE_POSITIVE_MEAN_INTERARRIVAL_TIME

  RULE_8:
    CENSORED_OBSERVATION_MUST_ACCOUNT_FOR_PARTIAL_INTERARRIVAL_INFORMATION

CONFIDENCE:
  ontology_identity: 0.99
  renewal_process_definition: 0.99
  renewal_equation_structure: 0.99
  poisson_special_case: 0.99
  age_residual_structure: 0.99
  renewal_reward_relationship: 0.99
  asymptotic_structure: 0.98

UPDATE_MODEL:
  IF:
    NEW_RENEWAL_OCCURS
  THEN:
    APPEND_RENEWAL_EPOCH
    COMPUTE_NEW_INTERARRIVAL
    UPDATE:
      RENEWAL_COUNT
      INTERARRIVAL_PARAMETER_ESTIMATES

  IF:
    OBSERVATION_HORIZON_EXTENDS
  THEN:
    UPDATE:
      COUNT_PATH
      AGE
      RESIDUAL_LIFE_INFORMATION

  IF:
    INTERARRIVAL_MODEL_CHANGES
  THEN:
    UPDATE:
      DISTRIBUTION_F
      MEAN_INTERARRIVAL
      VARIANCE
    RECOMPUTE_IF_REQUIRED:
      RENEWAL_FUNCTION
      LONG_RUN_RATE
      SIMULATIONS

  PRESERVE:
    RENEWAL_ORDER
    RENEWAL_EPOCHS
    INTERARRIVAL_HISTORY
    OBSERVATION_WINDOW
    DISTRIBUTION_MODEL
    RANDOM_SEED_WHEN_RECORDED
    DATA_PROVENANCE

AI_CONSUMPTION:
  parse_as: FORMAL_SEMANTIC_SPECIFICATION
  execution_mode: RECURSIVE_REASONING

  preferred_reasoning_order:
    - IDENTIFY_INTERARRIVAL_DISTRIBUTION
    - VERIFY_NONNEGATIVE_IID_STRUCTURE
    - CONSTRUCT_RENEWAL_EPOCHS
    - CONSTRUCT_COUNTING_PROCESS
    - COMPUTE_RENEWAL_FUNCTION
    - APPLY_RENEWAL_EQUATION
    - DETERMINE_LONG_RUN_RENEWAL_RATE
    - COMPUTE_AGE_OR_RESIDUAL_LIFE_IF_REQUIRED
    - APPLY_RENEWAL_REWARD_THEOREM_IF_REQUIRED
    - DISTINGUISH_POISSON_SPECIAL_CASE
    - SIMULATE_RENEWAL_PATH_IF_REQUIRED
    - RETURN_RENEWAL_PROCESS_INTERPRETATION

  constraints:
    - PRESERVE_INTERARRIVAL_ORDER
    - DISTINGUISH_INTERARRIVAL_TIME_FROM_ARRIVAL_TIME
    - DO_NOT_ASSUME_MEMORYLESSNESS
    - DO_NOT_ASSUME_INDEPENDENT_COUNT_INCREMENTS
    - DISTINGUISH_GENERAL_RENEWAL_PROCESS_FROM_POISSON_PROCESS
    - REQUIRE_FINITE_MEAN_FOR_STANDARD_LONG_RUN_RATE
    - PRESERVE_DATA_PROVENANCE

FLOW_HASH:
  canonical_entity: "renewal-process"
  namespace: "probability-statistics/stochastic-mathematics"
  version: "4.0"
  identity_basis:
    - PAGE_URL
    - CANONICAL_ENTITY
    - ENTITY_TYPE
    - IID_INTERARRIVAL_TIMES
    - RENEWAL_EPOCHS
    - COUNTING_PROCESS

END_FLOW