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.

[1]  A. M. Cohen.  Numerical Methods for Laplace Transform Inversion , 2007 .

[2]  Michel Mandjes.  Large Deviations for Gaussian Queues , 2007 .

[3]  Gábor Horváth,et al.  A Joint Moments Based Analysis of Networks of MAP/MAP/1 Queues , 2008, 2008 Fifth International Conference on Quantitative Evaluation of Systems.

[4]  Quan-Lin Li.  Constructive Computation in Stochastic Models with Applications: The RG-Factorizations , 2010 .

[5]  David M. Lucantoni,et al.  New results for the single server queue with a batch Markovian arrival process , 1991 .

[6]  C. Blondia,et al.  The delay distribution of a type k customer in a first-come-first-served MMAP[K]/PH[K]/1 queue , 2002, Journal of Applied Probability.

[7]  Barbara Haas Margolius,et al.  Transient and periodic solution to the time-inhomogeneous quasi-birth death process , 2007, Queueing Syst. Theory Appl..

[8]  Beatrice Meini,et al.  Numerical methods for structured Markov chains , 2005 .

[9]  Lothar Breuer.  From Markov jump processes to spatial queues , 2003 .

[10]  B. Sengupta.  Markov processes whose steady state distribution is matrix-exponential with an application to the GI/PH/1 queue , 1989, Advances in Applied Probability.

[11]  A. Krishnamoorthy,et al.  Analysis of a multi-server retrial queue with search of customers from the orbit , 2006, Perform. Evaluation.

[12]  R. H. Norden.  On the distribution of the time to extinction in the stochastic logistic population model , 1982, Advances in Applied Probability.

[13]  Søren Asmussen,et al.  Matrix‐analytic Models and their Analysis , 2000 .

[14]  Gábor Horváth,et al.  A minimal representation of Markov arrival processes and a moments matching method , 2007, Perform. Evaluation.

[15]  M. Neuts.  The burstiness of point processes , 1993 .

[16]  Qi-Ming He,et al.  Quasi-birth-and-death Markov processes with a tree structure and the MMAP[K]/PH[K]/N/LCFS non-preemptive queue , 2000, Eur. J. Oper. Res..

[17]  Che Soong Kim,et al.  The BMAP/PH/N retrial queueing system operating in Markovian random environment , 2010, Comput. Oper. Res..

[18]  Marcel F. Neuts.  Models Based on the Markovian Arrival Process , 1992 .

[19]  John A. Buzacott,et al.  Optimal and near-optimal inventory control policies for a make-to-order inventory-production system , 2002, Eur. J. Oper. Res..

[20]  Jesús R. Artalejo,et al.  Retrial Queueing Systems: A Computational Approach , 2008 .

[21]  Tetsuya Takine,et al.  The workload in the MAP/G/1 queue with state-dependent services: its application to a queue with preemptive resume priority , 1994 .

[22]  David M. Lucantoni,et al.  The BMAP/G/1 QUEUE: A Tutorial , 1993, Performance/SIGMETRICS Tutorials.

[23]  Jesús R. Artalejo,et al.  A state‐dependent Markov‐modulated mechanism for generating events and stochastic models , 2009 .

[24]  Ho Woo Lee,et al.  A factorization property for BMAP/G/1 vacation queues under variable service speed , 2008, Ann. Oper. Res..

[25]  Srinivas R. Chakravarthy.  The Batch Markovian Arrival Process: A Review and Future Work , 2001 .

[26]  Jing-Sheng Song,et al.  Optimal Policies for Multiechelon Inventory Problems with Markov-Modulated Demand , 2001, Oper. Res..

[27]  Claire Milne.  Transient Behaviour of the Interrupted Poisson Process , 1982 .

[28]  Mogens Bladt,et al.  Point processes with finite-dimensional conditional probabilities , 1999 .

[29]  H. Okamura,et al.  Markovian Arrival Process Parameter Estimation With Group Data , 2009, IEEE/ACM Transactions on Networking.

[30]  M. Neuts,et al.  A single-server queue with server vacations and a class of non-renewal arrival processes , 1990, Advances in Applied Probability.

[31]  Rafael Pérez-Ocón,et al.  Reliability of a system under two types of failures using a Markovian arrival process , 2006, Oper. Res. Lett..

[32]  C. Blondia,et al.  Queues with Correlated Service and Inter-Arrival Times and Their Application to Optical Buffers , 2006 .

[33]  Marcel F. Neuts,et al.  Local poissonification of the markovian arrival process , 1992 .

[34]  Montoro-Cazorla Delia,et al.  A maintenance model with failures and inspection following Markovian arrival processes and two repair modes , 2008 .

[35]  Y. W. Yang Woo Shin,et al.  BMAP/G/1 queue with correlated arrivals of customers and disasters , 2004, Oper. Res. Lett..

[36]  Alexander N. Dudin,et al.  A BMAP/SM/1 queueing system with Markovian arrival input of disasters , 1999 .

[37]  Attahiru Sule Alfa,et al.  Modelling Vehicular Traffic Using the Discrete Time Markovian Arrival Process , 1995, Transp. Sci..

[38]  Wai-Ki Ching.  Markov-modulated Poisson processes for multi-location inventory problems , 1997 .

[39]  R. Tweedie.  Operator-geometric stationary distributions for markov chains, with application to queueing models , 1982, Advances in Applied Probability.

[40]  Vaidyanathan Ramaswami,et al.  Introduction to Matrix Analytic Methods in Stochastic Modeling , 1999, ASA-SIAM Series on Statistics and Applied Mathematics.

[41]  V. Ramaswami.  The N/G/1 queue and its detailed analysis , 1980, Advances in Applied Probability.

[42]  Qi-Ming He,et al.  Queues with marked customers , 1996, Advances in Applied Probability.

[43]  E. Seneta,et al.  On quasi-stationary distributions in absorbing continuous-time finite Markov chains , 1967, Journal of Applied Probability.

[44]  Chris Blondia,et al.  Statistical Multiplexing of VBR Sources: A Matrix-Analytic Approach , 1992, Perform. Evaluation.

[45]  Attahiru Sule Alfa,et al.  Computational analysis ofMMAP[K]/PH[K]/1 queues with a mixed FCFS and LCFS service discipline , 2000 .

[46]  Esther Frostig,et al.  Availability of inspected systems subject to shocks - A matrix algorithmic approach , 2009, Eur. J. Oper. Res..

[47]  Eric C. K. Cheung,et al.  Perturbed MAP Risk Models with Dividend Barrier Strategies , 2009, Journal of Applied Probability.

[48]  Bo Friis Nielsen,et al.  Higher Order Moments and Conditional Asymptotics of the Batch Markovian Arrival Process , 2007 .

[49]  Yutaka Takahashi,et al.  Algorithmic Computation of the Transient Queue Length Distribution in the BMAP/D/c Queue , 2007 .

[50]  Marcel F. Neuts,et al.  Algorithmic probability - a collection of problems , 1995, Stochastic modeling series.

[51]  Evgenia Smirni,et al.  Versatile models of systems using map queueing networks , 2008, 2008 IEEE International Symposium on Parallel and Distributed Processing.

[52]  Soohan Ahn,et al.  On the analysis of the Gerber–Shiu discounted penalty function for risk processes with Markovian arrivals , 2007 .

[53]  Tetsuya Takine,et al.  The Nonpreemptive Priority MAP/G/1 Queue , 1999, Oper. Res..

[54]  Naishuo Tian,et al.  Vacation Queueing Models: Theory and Applications (International Series in Operations Research & Management Science) , 2006 .

[55]  M. Rudemo.  Point processes generated by transitions of Markov chains , 1973 .

[56]  Marcel Neuts,et al.  An Algorithm for the P(n,t) Matrices of a Continuous BMAP , 1996 .

[57]  L. Allen.  An introduction to stochastic processes with applications to biology , 2003 .

[58]  H. Johnson,et al.  A comparison of 'traditional' and multimedia information systems development practices , 2003, Inf. Softw. Technol..

[59]  W. Whitt,et al.  The transient BMAP/G/l queue , 1994 .

[60]  António Pacheco,et al.  Markov-additive Processes of Arrivals , 1994 .

[61]  Søren Asmussen,et al.  Calculation of the Steady State Waiting Time Distribution in GI/PH/c and MAP/PH/c Queues , 2001, Queueing Syst. Theory Appl..

[62]  V. Ramaswami.  ALGORITHMS FOR A CONTINUOUS-REVIEW (s, S) INVENTORY SYSTEM , 1981 .

[63]  Bo Friis Nielsen,et al.  A Markovian approach for modeling packet traffic with long-range dependence , 1998, IEEE J. Sel. Areas Commun..

[64]  Marcel F. Neuts,et al.  Markov chains with marked transitions , 1998 .

[65]  Paul Manuel,et al.  A perishable inventory system with service facilities, MAP arrivals and PH — Service times , 2007 .

[66]  Vidyadhar G. Kulkarni,et al.  A New Class of Multivariate Phase Type Distributions , 1989, Oper. Res..

[67]  Attahiru Sule Alfa,et al.  Discrete‐time analysis of MAP/PH/1 multiclass general preemptive priority queue , 2003 .

[68]  Yiqiang Q. Zhao,et al.  The stationary tail asymptotics in the GI/G/1-type queue with countably many background states , 2004, Advances in Applied Probability.

[69]  Masakiyo Miyazawa.  A Markov Renewal Approach to M/G/1 Type Queues with Countably Many Background States , 2004, Queueing Syst. Theory Appl..

[70]  Lothar Breuer,et al.  An EM algorithm for platoon arrival processes in discrete time , 2005, Oper. Res. Lett..

[71]  Bara Kim,et al.  Queue size distribution in a discrete-time D-BMAP/G/1 retrial queue , 2010, Comput. Oper. Res..

[72]  Quan-Lin Li,et al.  A BMAP/G/1 Retrial Queue with a Server Subject to Breakdowns and Repairs , 2006, Ann. Oper. Res..

[73]  Wei Li,et al.  Infinite block-structured transition matrices and their properties , 1998, Advances in Applied Probability.

[74]  Naoki Makimoto,et al.  GEOMETRIC DECAY OF THE STEADY-STATE PROBABILITIES IN A QUASI-BIRTH-AND-DEATH PROCESS WITH A COUNTABLE NUMBER OF PHASES , 2001 .

[75]  Srinivas R. Chakravarthy,et al.  Markovian Arrival Processes , 2010 .

[76]  Qi-Ming He.  The Versatility of MMAP[K] and the MMAP[K]/G[K]/1 Queue , 2001, Queueing Syst. Theory Appl..

[77]  Wai-Ki Ching.  Iterative Methods for Queuing and Manufacturing Systems , 2001 .

[78]  Evgenia Smirni,et al.  KPC-Toolbox: Best recipes for automatic trace fitting using Markovian Arrival Processes , 2010, Perform. Evaluation.

[79]  Jesus R. Artalejo,et al.  Performance Analysis of a Block-Structured Discrete-Time Retrial Queue with State-Dependent Arrivals , 2010, Discret. Event Dyn. Syst..

[80]  Miklós Telek,et al.  Moment characterization of matrix exponential and Markovian arrival processes , 2008, Ann. Oper. Res..

[81]  Marcel F. Neuts,et al.  The first two moment matrices of the counts for the Markovian arrival process , 1992 .

[82]  Michel Mandjes.  Large Deviations for Gaussian Queues: Modelling Communication Networks , 2007 .

[83]  Steve Drekic,et al.  Analysis of a threshold dividend strategy for a MAP risk model , 2007 .

[84]  Lothar Breuer,et al.  An EM Algorithm for Batch Markovian Arrival Processes and its Comparison to a Simpler Estimation Procedure , 2002, Ann. Oper. Res..

[85]  P. Taylor,et al.  Some properties of the rate operators in level dependent quasi-birth-and-death processes with a countable number of phases , 1996 .

[86]  Tien Van Do,et al.  The MM sum(k=1 to K of CPPk/GE/c/L) G-queue with heterogeneous servers: Steady state solution and an application to performance evaluation , 2007, Perform. Evaluation.

[87]  M. Neuts.  A Versatile Markovian Point Process , 1979 .

[88]  Naishuo Tian,et al.  Vacation Queueing Models Theory and Applications , 2006 .

[89]  Bara Kim,et al.  MAP/M/c Queue with Constant Impatient Time , 2004, Math. Oper. Res..

[90]  S. Asmussen,et al.  Marked point processes as limits of Markovian arrival streams , 1993 .