Dependence patterns related to the BMAP Rosa E. Lillo, Joanna Rodr - PowerPoint PPT Presentation

Dependence patterns related to the BMAP Rosa E. Lillo, Joanna Rodr´ ıguez, Pepa Ram´ ırez-Cobo Department of Statistics Universidad Carlos III de Madrid The Ninth International Conference on Matrix-Analytic Methods in Stochastic Models Budapest June 30, 2016 Rosa E. Lillo Dependence patterns related to the BMAP

Dependence? Why? In real life scenarios, there exist data sets that display significant and complex correlations structures in both the times of consecutive events and in the size of the consecutive events. Events ≡ Failures of a system, arrivals of a packet of bytes, claims in an insurance company, calls in a call center... The event ocurrence can be understood as a single event or batch event. The models used in the literature to fit these types of data sets ignore the dependence. Rosa E. Lillo Dependence patterns related to the BMAP

Example I: teletraffic data set Bellcore LAN trace files (named BC-pAug89) found in http://ita.ee.lbl.gov/html/contrib/BC.html . The data file consists of the time in seconds of the packet arrival, and the Ethernet data length in bytes. Bellcore inter−event ACF 1 Sample Autocorrelation 0.5 0 −0.5 0 2 4 6 8 10 12 14 16 18 20 Lag Bellcore batches ACF 1 Sample Autocorrelation 0.5 0 −0.5 0 2 4 6 8 10 12 14 16 18 20 Lag Rosa E. Lillo Dependence patterns related to the BMAP

Example II: call center The data archive of Mandelbaum (2012), collected daily from March 26, 2001 to October 26, 2003 from an American banking call center. Call−center inter−event ACF 1 Sample Autocorrelation 0.5 0 −0.5 0 2 4 6 8 10 12 14 16 18 20 Lag Call−center batches ACF 1 Sample Autocorrelation 0.5 0 −0.5 0 2 4 6 8 10 12 14 16 18 20 Lag Rosa E. Lillo Dependence patterns related to the BMAP

Our proposal = ⇒ The BMAP Versatile Markovian point process (Neuts, 1979). Batch Markovian Arrival process or BMAP (Lucantoni, 1991). Stationary BMAP s are dense in the family of stationary point processes. 1 Tractability of the Poisson process. 2 Dependent interarrival times. 3 Non-exponential interarrival times. 4 Correlated batch sizes. 5 Special cases: A MAP with i.i.d. batch arrivals. 1 Batch PH-renewal processes. 2 Batch Markov-modulated Poisson process. 3 Rosa E. Lillo Dependence patterns related to the BMAP

The BMAP as a generalization of the Batch Poisson proccess Batch Poisson process:   − λ λ p 1 λ p 2 λ p 3 · · · · · · 0 − λ λ p 1 λ p 2 · · · · · ·     Q B − POISSON = 0 0 − λ λ p 1 · · · · · · .     0 0 0 − λ · · · · · ·   · · · · · · · · · · · · · · · · Consider now m × m matrices for the rates instead of numbers....   D 0 D 1 D 2 D 3 · · · 0 D 0 D 1 D 2 · · ·     Q BMAP = 0 0 D 0 D 1 · · · ,     0 0 0 D 0 · · ·   · · · · · · · · · · · · · Rosa E. Lillo Dependence patterns related to the BMAP

How does a BMAP m ( k ) work? � m ≡ order of the matrix D b , with 1 ≤ b ≤ k , Notation: k ≡ the maximum batch arrival size. The BMAP m ( k ) behaves as follows: The Initial state i 0 ∈ S = { 1 , 2 ..., m } is given by an initial probability vector θ = ( θ 1 , ..., θ m ). At the end of an exponentially distributed sojourn time in state i , with rate λ i , two possible state transitions can occur: With probability p ij 0 , no arrival occurs and the BMAP m enters in a different 1 state j � = i . With probability p ijb , with 1 ≤ l ≤ k , a transition to state j with a batch 2 arrival of size b occurs. Rosa E. Lillo Dependence patterns related to the BMAP

How does a BMAP m ( k ) work? � m ≡ order of the matrix D b , with 1 ≤ b ≤ k , Notation: k ≡ the maximum batch arrival size. The BMAP m ( k ) behaves as follows: The Initial state i 0 ∈ S = { 1 , 2 ..., m } is given by an initial probability vector θ = ( θ 1 , ..., θ m ). At the end of an exponentially distributed sojourn time in state i , with rate λ i , two possible state transitions can occur: With probability p ij 0 , no arrival occurs and the BMAP m enters in a different 1 state j � = i . With probability p ijb , with 1 ≤ l ≤ k , a transition to state j with a batch 2 arrival of size b occurs. Rosa E. Lillo Dependence patterns related to the BMAP

How does a BMAP m ( k ) work? � m ≡ order of the matrix D b , with 1 ≤ b ≤ k , Notation: k ≡ the maximum batch arrival size. The BMAP m ( k ) behaves as follows: The Initial state i 0 ∈ S = { 1 , 2 ..., m } is given by an initial probability vector θ = ( θ 1 , ..., θ m ). At the end of an exponentially distributed sojourn time in state i , with rate λ i , two possible state transitions can occur: With probability p ij 0 , no arrival occurs and the BMAP m enters in a different 1 state j � = i . With probability p ijb , with 1 ≤ l ≤ k , a transition to state j with a batch 2 arrival of size b occurs. Rosa E. Lillo Dependence patterns related to the BMAP

How does a BMAP m ( k ) work? � m ≡ order of the matrix D b , with 1 ≤ b ≤ k , Notation: k ≡ the maximum batch arrival size. The BMAP m ( k ) behaves as follows: The Initial state i 0 ∈ S = { 1 , 2 ..., m } is given by an initial probability vector θ = ( θ 1 , ..., θ m ). At the end of an exponentially distributed sojourn time in state i , with rate λ i , two possible state transitions can occur: With probability p ij 0 , no arrival occurs and the BMAP m enters in a different 1 state j � = i . With probability p ijb , with 1 ≤ l ≤ k , a transition to state j with a batch 2 arrival of size b occurs. Rosa E. Lillo Dependence patterns related to the BMAP

The BMAP 2 ( k ) example b 0 b 1 b 2 b 3 b 4 λ 1 λ 2 λ 1 λ 2 λ 1 λ 2 λ 1 λ 1 λ 2 λ 1 λ 2 λ 2 λ 1 2 1 2 2 1 2 1 2 D b 2 D b 4 D b 1 D b 3 1 1 1 2 1 D 0 D 0 D 0 The rate matrices D 0 , D 1 , ..., D k are defined in terms of the transitions probabilities as: ( D 0 ) ii = − λ i , i ∈ S , ( D 0 ) ij = i , j ∈ S , i � = j , λ i p ij 0 , ( D l ) ij = i , j ∈ S , 1 ≤ b ≤ k . λ i p ijb , Rosa E. Lillo Dependence patterns related to the BMAP

The BMAP 2 ( k ) example b 0 b 1 b 2 b 3 b 4 λ 1 λ 2 λ 1 λ 2 λ 1 λ 2 λ 1 λ 1 λ 2 λ 1 λ 2 λ 2 λ 1 2 1 2 2 1 2 1 2 D b 2 D b 4 D b 1 D b 3 1 1 1 2 1 D 0 D 0 D 0 The rate matrices D 0 , D 1 , ..., D k are defined in terms of the transitions probabilities as: ( D 0 ) ii = − λ i , i ∈ S , ( D 0 ) ij = i , j ∈ S , i � = j , λ i p ij 0 , ( D l ) ij = i , j ∈ S , 1 ≤ b ≤ k . λ i p ijb , Rosa E. Lillo Dependence patterns related to the BMAP

The BMAP 2 ( k ) example b 0 b 1 b 2 b 3 b 4 λ 1 λ 2 λ 1 λ 2 λ 1 λ 2 λ 1 λ 1 λ 2 λ 1 λ 2 λ 2 λ 1 2 1 2 2 1 2 1 2 D b 2 D b 4 D b 1 D b 3 1 1 1 2 1 D 0 D 0 D 0 The rate matrices D 0 , D 1 , ..., D k are defined in terms of the transitions probabilities as: ( D 0 ) ii = − λ i , i ∈ S , ( D 0 ) ij = i , j ∈ S , i � = j , λ i p ij 0 , ( D l ) ij = i , j ∈ S , 1 ≤ b ≤ k . λ i p ijb , Rosa E. Lillo Dependence patterns related to the BMAP

The BMAP 2 ( k ) example b 0 b 1 b 2 b 3 b 4 λ 1 λ 2 λ 1 λ 2 λ 1 λ 2 λ 1 λ 1 λ 2 λ 1 λ 2 λ 2 λ 1 2 1 2 2 1 2 1 2 D b 2 D b 4 D b 1 D b 3 1 1 1 2 1 D 0 D 0 D 0 The rate matrices D 0 , D 1 , ..., D k are defined in terms of the transitions probabilities as: ( D 0 ) ii = − λ i , i ∈ S , ( D 0 ) ij = i , j ∈ S , i � = j , λ i p ij 0 , ( D l ) ij = i , j ∈ S , 1 ≤ b ≤ k . λ i p ijb , Rosa E. Lillo Dependence patterns related to the BMAP

The BMAP 2 ( k ) example b 0 b 1 b 2 b 3 b 4 λ 1 λ 2 λ 1 λ 2 λ 1 λ 2 λ 1 λ 1 λ 2 λ 1 λ 2 λ 2 λ 1 2 1 2 2 1 2 1 2 D b 2 D b 4 D b 1 D b 3 1 1 1 2 1 D 0 D 0 D 0 The rate matrices D 0 , D 1 , ..., D k are defined in terms of the transitions probabilities as: ( D 0 ) ii = − λ i , i ∈ S , ( D 0 ) ij = i , j ∈ S , i � = j , λ i p ij 0 , ( D l ) ij = i , j ∈ S , 1 ≤ b ≤ k . λ i p ijb , Rosa E. Lillo Dependence patterns related to the BMAP

The BMAP 2 ( k ) example b 0 b 1 b 2 b 3 b 4 λ 1 λ 2 λ 1 λ 2 λ 1 λ 2 λ 1 λ 1 λ 2 λ 1 λ 2 λ 2 λ 1 2 1 2 2 1 2 1 2 D b 2 D b 4 D b 1 D b 3 1 1 1 2 1 D 0 D 0 D 0 The rate matrices D 0 , D 1 , ..., D k are defined in terms of the transitions probabilities as: ( D 0 ) ii = − λ i , i ∈ S , ( D 0 ) ij = i , j ∈ S , i � = j , λ i p ij 0 , ( D l ) ij = i , j ∈ S , 1 ≤ b ≤ k . λ i p ijb , Rosa E. Lillo Dependence patterns related to the BMAP

The BMAP 2 ( k ) example b 0 b 1 b 2 b 3 b 4 λ 1 λ 2 λ 1 λ 2 λ 1 λ 2 λ 1 λ 1 λ 2 λ 1 λ 2 λ 2 λ 1 2 1 2 2 1 2 1 2 D b 2 D b 4 D b 1 D b 3 1 1 1 2 1 D 0 D 0 D 0 The rate matrices D 0 , D 1 , ..., D k are defined in terms of the transitions probabilities as: ( D 0 ) ii = − λ i , i ∈ S , ( D 0 ) ij = i , j ∈ S , i � = j , λ i p ij 0 , ( D l ) ij = i , j ∈ S , 1 ≤ b ≤ k . λ i p ijb , Rosa E. Lillo Dependence patterns related to the BMAP

The BMAP 2 ( k ) example b 0 b 1 b 2 b 3 b 4 λ 1 λ 2 λ 1 λ 2 λ 1 λ 2 λ 1 λ 1 λ 2 λ 1 λ 2 λ 2 λ 1 2 1 2 2 1 2 1 2 D b 2 D b 4 D b 1 D b 3 1 1 1 2 1 D 0 D 0 D 0 The rate matrices D 0 , D 1 , ..., D k are defined in terms of the transitions probabilities as: ( D 0 ) ii = − λ i , i ∈ S , ( D 0 ) ij = i , j ∈ S , i � = j , λ i p ij 0 , ( D l ) ij = i , j ∈ S , 1 ≤ b ≤ k . λ i p ijb , Rosa E. Lillo Dependence patterns related to the BMAP