 next new packet (here is (K+1)-th) arrives, the K-th packet process leaves the transient state and immediately goes into absorbing state. At the same time, the (K+1)-th packet process starts transitions and repeats the same behavior. K-th Packet arrival (K+1)-thPacketarrival (K+2)-th Packet arrival k-th Arrival Epoch (k+1)-th Arrival Epoch absorbing absorbing absorbing state transition if m=1, Markovian Arrival Process (MAP) with an arrival packet type m, transient state transition, if m >1, BMAP Represented by D(m) Represented by D(0) Figure 6.56: Packet arrival process (BMAP model) The BMAP is characterized by a finite and absorbing Markov chain [Luc91]. Considering a two-dimensional Markov process {N(t),J(t)} on the state space {(i,j); i>0,1 <j<n}, N(t) 6 Dimensioning for Single ATM-Based lub counts the number of arrivals in the interval (0,t], and J(t) represents the underlying Markov chain. Transitions from a state (i,j) to a state (i+m,k), m > 1,1 j,k n, correspond to batch arrivals of size m, and thus the batch size can depend on j and k. An infinitesimal generator Q has the structure: (6.64) Here, D(0) is the matrix (the dimension equals to the number states of J(t)), which has negative diagonal elements and nonnegative off-diagonal elements, representing a state transition without arrivals (batch size = 0). D(m) (m >1) is the matrix (the dimension equals to the number states of J(t)), which has nonnegative elements representing a state transition with an arrival batch size m The basic characteristics of a BMAP are: (1) the density function: (6.65) (2) the complementary cumulative distribution function: (6.66) If we define r as the initial probability vector of the underlying Markov chain {J(t)}, T satisfies (6.67) Here, and the 1 is a column vector of ones. The inter-arrival time distribution function for the batch size m is [KKSC02]: (6.68) Here, 1 is the unit matrix. The arrival rate of the process is: (6.69) The importance of BMAP lies in its ability to be more effective and powerful traffic model th