Markovian arrivals in stochastic modelling: a survey and some new results

This paper aims to provide a comprehensive review on Markovian arrival processes (MAPs), which constitute a rich class of point processes used extensively in stochastic modelling. Our starting point is the versatile process introduced by Neuts (1979) which, under some simplified notation, was coined as the batch Markovian arrival process (BMAP). On the one hand, a general point process can be approximated by appropriate MAPs and, on the other hand, the MAPs provide a versatile, yet tractable option for modelling a bursty flow by preserving the Markovian formalism. While a number of well-known arrival processes are subsumed under a BMAP as special cases, the literature also shows generalizations to model arrival streams with marks, nonhomogeneous settings or even spatial arrivals. We survey on the main aspects of the BMAP, discuss on some of its variants and generalizations, and give a few new results in the context of a recent state-dependent extension.Peer Reviewe

A fast EM algorithm for fitting marked Markovian arrival processes with a new special structure

This paper presents an EM algorithm for fitting traces with Markovian arrival processes (MAPs). The proposed algorithm operates on a special subclass of MAPs. This special structure enables the efficient implementation of the EM algorithm; it is more orders of magnitudes faster than methods operating on the general MAP class while providing similar or better likelihood values. An other important feature of the algorithm is that it is able to fit multi-class traces with marked Markovian arrival processes as well. Several numerical examples demonstrate the efficiency of the procedure

Matching marginal moments and lag autocorrelations with MAPs

This paper presents a procedure that constructs a Markovian Arrival Process (MAP) based on the mean, the squared coefficient of variation and the lag-1 autocorrelation of the inter-arrival times. This method always provides a valid MAP without posing any restrictions on the three input parameters. Besides matching these three parameters, it is possible to match the third moment of the inter-arrival times and the decay of the autocorrelation function as well, if they fall into the given (very wide) bounds

Bayesian analysis of the stationary MAP2

In this article we describe a method for carrying out Bayesian estimation for the two-state stationary Markov arrival process (MAP(2)), which has been proposed as a versatile model in a number of contexts. The approach is illustrated on both simulated and real data sets, where the performance of the MAP(2) is compared against that of the well-known MMPP2. As an extension of the method, we estimate the queue length and virtual waiting time distributions of a stationary MAP(2)/G/1 queueing system, a matrix generalization of the M/G/1 queue that allows for dependent inter-arrival times. Our procedure is illustrated with applications in Internet traffic analysis.Research partially supported by research grants and projects MTM2015-65915-R, ECO2015- 66593-P (Ministerio de Economía y Competitividad, Spain) and P11-FQM-7603, FQM-329 (Junta de Andalucía, Spain). The authors thank both the Associate Editor and referee for their constructive comments from which the paper greatly benefited

Markovian arrivals in stochastic modelling : a survey and some new results

This paper aims to provide a comprehensive review on Markovian arrival processes (MAPs), which constitute a rich class of point processes used extensively in stochastic modelling. Our starting point is the versatile process introduced by Neuts (1979) which, under some simplified notation, was coined as the batch Markovian arrival process (BMAP). On the one hand, a general point process can be approximated by appropriate MAPs and, on the other hand, the MAPs provide a versatile, yet tractable option for modelling a bursty flow by preserving the Markovian formalism. While a number of well-known arrival processes are subsumed under a BMAP as special cases, the literature also shows generalizations to model arrival streams with marks, nonhomogeneous settings or even spatial arrivals. We survey on the main aspects of the BMAP, discuss on some of its variants and generalizations, and give a few new results in the context of a recent state-dependent extension

On finite-time ruin probabilities with reinsurance cycles influenced by large claims

Market cycles play a great role in reinsurance. Cycle transitions are not independent from the claim arrival process : a large claim or a high number of claims may accelerate cycle transitions. To take this into account, a semi-Markovian risk model is proposed and analyzed. A refined Erlangization method is developed to compute the finite-time ruin probability of a reinsurance company. As this model needs the claim amounts to be Phase-type distributed, we explain how to fit mixtures of Erlang distributions to long-tailed distributions. Numerical applications and comparisons to results obtained from simulation methods are given. The impact of dependency between claim amounts and phase changes is studied.