Introduction to Matrix Analytic Methods in Stochastic Modeling
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Publication:4236284
DOI10.1137/1.9780898719734zbMath0922.60001OpenAlexW2062573385MaRDI QIDQ4236284
Guy Latouche, Vaidyanathan Ramaswami
Publication date: 23 March 1999
Full work available at URL: https://doi.org/10.1137/1.9780898719734
numerical algorithmsphase type distributionquasi-birth-death processmatrix analytical methodsmatrix geometric distribution
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and matrix-analytic solutions for stochastic chemical kinetics ⋮ Unnamed Item ⋮ A Compressed Cyclic Reduction for QBD processes with Low-Rank Upper and Lower Transitions ⋮ Acceptance-Rejection Methods for Generating Random Variates from Matrix Exponential Distributions and Rational Arrival Processes ⋮ Impact of Dampening Demand Variability in a Production/Inventory System with Multiple Retailers ⋮ A direct method for solving block-Toeplitz with near-circulant-block systems with applications to hybrid manufacturing systems ⋮ Markovian Trees Subject to Catastrophes: Transient Features and Extinction Probability ⋮ Shipment Consolidation by Private Carrier: The Discrete Time and Discrete Quantity Case ⋮ Some Structural Properties of Markov and Rational Arrival Processes ⋮ Level product form QSF processes and an analysis of queues with <scp>C</scp> oxian interarrival distribution ⋮ On matrix exponential distributions ⋮ ON CUSTOMERS ACTING AS SERVERS ⋮ On a Customer-Induced Interruption in a Service System ⋮ A Fast Newton's Iteration for M/G/1-Type and GI/M/1-Type Markov Chains ⋮ Quasi-birth-and-death processes, level-geometric distributions. An aggregation/disaggregation approach ⋮ Solving nonlinear matrix equations arising in tree-like stochastic processes. ⋮ Stochastic modeling on M/M/1/N inventory system with queue-dependent service rate and retrial facility ⋮ Markov chain representations of discrete distributions applied to queueing models ⋮ On a dual queueing system with preemptive priority service discipline ⋮ A stochastic epidemic model with two quarantine states and limited carrying capacity for quarantine ⋮ Moments and polynomial expansions in discrete matrix-analytic models ⋮ On level-dependent QBD processes with explosive state space ⋮ Bi-objective sequence optimization in reliability problems with a matrix-analytic approach ⋮ Analysis of semi-open queueing networks using lost customers approximation with an application to robotic mobile fulfilment systems ⋮ On a many-to-many matched queueing system with flexible matching mechanism and impatient customers ⋮ Comparison theorems for splittings of M-matrices in (block) Hessenberg form ⋮ Explicit results for the distribution of the number of customers served during a busy period for \(M^X/PH/1\) queue ⋮ A stochastic model for the patient-bed assignment problem with random arrivals and departures ⋮ On lattice path counting and the random product representation, with applications to the \(E_r/M/1\) queue and the \(M/E_r/1\) queue ⋮ Weak stability bounds for approximations of invariant measures with applications to queueing ⋮ The optimal allocation of server time slots over different classes of patients ⋮ Maintenance of a deteriorating single server system with Markovian arrivals and random shocks ⋮ Low-rank updates and divide-and-conquer methods for quadratic matrix equations ⋮ Analysis of queueing systems with customer interjections ⋮ Performance evaluation and analysis of a discrete queue system with multiple working vacations and non-preemptive priority ⋮ A nonhomogeneous quasi-birth-death process approach for an \((s,S)\) policy for a perishable inventory system with retrial demands ⋮ Analysis of discrete-time queues with batch renewal input and multiple vacations ⋮ Tail asymptotics for \(M/M/c\) retrial queues with non-persistent customers ⋮ SIR-type epidemic models as block-structured Markov processes ⋮ Towards a linear algebra of programming ⋮ Analysis of an M/PH/1 retrial queueing-inventory system with level dependent retrial rate ⋮ Kemeny's constant for countable Markov chains ⋮ Privacy and safety analysis of timed stochastic discrete event systems using Markovian trajectory-observers ⋮ Non-product form equilibrium probabilities in a class of two-station closed reentrant queueing networks ⋮ Markov decision processes in service facilities holding perishable inventory ⋮ The spectral matrices associated with the stochastic Darboux transformations of random walks on the integers ⋮ Time-dependent stress-strength reliability models based on phase type distribution ⋮ Strategic behavior of customers and optimal control for batch service polling systems with priorities ⋮ Join the shortest queue among \(k\) parallel queues: tail asymptotics of its stationary distribution ⋮ Asymptotics for the stationary distribution in a discrete-time two-dimensional quasi-birth-and-death process ⋮ On classes of Bitcoin-inspired infinite-server queueing systems ⋮ Many-server diffusion limits for \(G/Ph/n+GI\) queues ⋮ The residual time approach for \((Q, r)\) model under perishability, general lead times, and lost sales ⋮ The role of information in system stability with partially observable servers ⋮ Analysis of tollbooth systems with two tandem skill-based servers and two vehicle types ⋮ Analysis of G-queue with pseudo-fault and multiple working vacations ⋮ A structured Markov chain model to investigate the effects of pre-exposure vaccines in tuberculosis control ⋮ Numerical methods to analyses of queuing systems with instantaneous feedback, positive server setup time and impatient calls ⋮ A queueing system with probabilistic joining strategy for priority customers ⋮ Delayed capital injections for a risk process with Markovian arrivals ⋮ Algorithmic computation of \(MAP/PH/1\) queue with finite system capacity and two-stage vacations ⋮ Queue size distribution on a new ND policy \(Geo/G/1\) queue and its computation designs ⋮ Quality and pricing decisions in production/inventory systems ⋮ A fast Newton-Shamanskii iteration for a matrix equation arising from M/G/1-type Markov chains ⋮ Augmented truncation approximations to the solution of Poisson's equation for Markov chains ⋮ Fitting procedure for the two-state batch Markov modulated Poisson process ⋮ Coordinating lead times and safety stocks under autocorrelated demand ⋮ Light-tailed behavior of stationary distribution for state-dependent random walks on a strip ⋮ Deviation matrix and asymptotic variance for \(\mathrm{GI}/\mathrm{M}/1\)-type Markov chains ⋮ Appointments in care pathways: the \(\mathrm{Geo}^x/\mathrm{D}/1\) queue with slot reservations ⋮ A model for deceased-donor transplant queue waiting times ⋮ On a batch matching system with impatient servers and boundedly rational customers ⋮ Spectral properties of the tandem Jackson network, seen as a quasi-birth-and-death process ⋮ Queues with postponed work ⋮ On the subcritical birth-and-death processes in a random environment ⋮ Integrated planning of spare parts and service engineers with partial backlogging ⋮ The finite/infinite horizon ruin problem with multi-threshold premiums: a Markov fluid queue approach ⋮ Commuting birth-and-death processes ⋮ Analyzing retrial queues by censoring ⋮ A discrete time bulk service queue with accessible batch: \(Geo/ NB^{(L,K)}/1\) ⋮ Asymptotic behavior for MAP/PH/\(c\) queue with shortest queue discipline and jockeying ⋮ On time-discretized versions of the stochastic SIS epidemic model: a comparative analysis ⋮ A stochastic SIS epidemic model with heterogeneous contacts ⋮ Absorbing-reflecting factorizations for birth-death chains on the integers and their Darboux transformations ⋮ A sequential update algorithm for computing the stationary distribution vector in upper block-Hessenberg Markov chains ⋮ To clean or not to clean: malware removal strategies for servers under load ⋮ Strategic customer behavior in a queueing system with alternating information structure ⋮ A catastrophic queueing model with delayed action ⋮ On the MAP/PH/c retrial queueing system with two types of nonpersistent calls ⋮ Equilibrium threshold joining strategies in partially observable batch service queueing systems ⋮ Networks of interacting stochastic fluid models with infinite and finite buffers ⋮ Analysis of \(\mathrm{MAP}/\mathrm{PH}_1,\mathrm{PH}_2/1\) queue with vacations and optional secondary services ⋮ Equilibrium and optimal balking strategies of customers in Markovian queues with multiple vacations and \(N\)-policy ⋮ A queueing system with decomposed service and inventoried preliminary services ⋮ Strategic customer behavior in a two-stage batch processing system ⋮ On an unreliable-server retrial queue with customer feedback and impatience ⋮ Strategic behaviour in a tandem queue with alternating server ⋮ Asymptotic properties of the occupation measure in a multidimensional skip-free Markov-modulated random walk ⋮ Strategic customer behavior in an \(M/M/1\) feedback queue ⋮ An \(M/ PH /1\) queue with workload-dependent processing speed and vacations ⋮ On scale functions for Lévy processes with negative phase-type jumps ⋮ An exact algorithm for the redundancy allocation problem with heterogeneous components under the mixed redundancy strategy ⋮ Birth-death chains on a spider: spectral analysis and reflecting-absorbing factorization ⋮ Steady-state and first passage time distributions for waiting times in the \(MAP/M/s+G\) queueing model with generally distributed patience times ⋮ On first passage times in discrete skeletons and uniformized versions of a continuous-time Markov chain ⋮ An explicit polynomial to globalize algorithms for solving matrix polynomial equations ⋮ Analysis of \(MAP / PH_1\), \(PH_2/2\) queue with working breakdown, starting failure, and Bernoulli vacation ⋮ Extreme values in SIR epidemic models with two strains and cross-immunity ⋮ Tail asymptotics in any direction of the stationary distribution in a two-dimensional discrete-time QBD process ⋮ Service with a queue and a random capacity cart: random processing batches and E-limited policies ⋮ Cost optimization of an unreliable server queue with two stage service process under hybrid vacation policy ⋮ On the local convergence of a quasi-Newton method for solving matrix polynomial equations ⋮ A matrix-geometric approximation for tandem queues with blocking and repeated attempts ⋮ Discrete time queues and matrix-analytic methods. (With comments and rejoinder) ⋮ On the study of the running maximum and minimum level of level-dependent quasi-birth–death processes and related models ⋮ AN INFINITE-PHASE QUASI-BIRTH-AND-DEATH MODEL FOR THE NON-PREEMPTIVE PRIORITYM/PH/1 QUEUE ⋮ ON LATOUCHE–RAMASWAMI'S LOGARITHMIC REDUCTION ALGORITHM FOR QUASI-BIRTH-AND-DEATH PROCESSES ⋮ A Fixed Point Approach to the Classification of Markov Chains with a Tree Structure ⋮ β-Invariant Measures for Transition Matrices of GI/M/1 Type ⋮ A Markovian arrival stream approach to stochastic gene expression in cells ⋮ A new matrix-infinite-product-form solution for upper block-Hessenberg Markov chains and its quasi-algorithmic constructibility ⋮ Catalan generating functions for bounded operators ⋮ M/PH/1 queueing model with re-servicing ⋮ An EWMA sign chart for dispersion with exact run length properties ⋮ Proofs of conjectures on the competition between observable and unobservable servers ⋮ Measures to assess a warning vaccination level in a stochastic SIV model with imperfect vaccine ⋮ Stationary analyses for a double-ended queueing system with random service capacity and balking customers ⋮ QBD processes associated with Jacobi-Koornwinder bivariate polynomials and urn models ⋮ Design of attribute EWMA type control charts with reliable run length performance ⋮ Ruin problems for risk processes with dependent phase-type claims ⋮ The bilateral birth–death chain generated by the associated Jacobi polynomials ⋮ The GI/M/1 queue in a multi-phase service environment with working vacations and Bernoulli vacation interruption ⋮ Performance analysis of a multi server retrial queueing system with unreliable server, discouragement and vacation model ⋮ A Markovian epidemic model in a resource-limited environment ⋮ Modelling the variation and uncertainty problem of right-turn-on-red queue in a variety of conflicting environments ⋮ Comparing Some Algorithms for Solving QBD Processes Exhibiting Special Structures ⋮ A geometric convergence formula for the level-increment-truncation approximation of \(M/G/1\)-type Markov chains ⋮ Multiple allocation hub location with service level constraints for two shipment classes ⋮ Matrix-Analytic Methods for Solving Poisson’s Equation with Applications to Markov Chains of GI/G/1-Type ⋮ Asymmetric kernel method in the study of strong stability of the PH/M/1 queuing system ⋮ Markovian queues with Poisson control ⋮ A 3-queue polling system with join the shortest-serve the longest policy ⋮ Sufficient conditions for regularity, positive recurrence, and absorption in level‐dependent QBD processes and related block‐structured Markov chains ⋮ Iterative and doubling algorithms for Riccati‐type matrix equations: A comparative introduction ⋮ Three-level modeling of a speed-scaling supercomputer ⋮ Stability of queueing-inventory systems with customers of different priorities ⋮ Phase-type approximations of service-time distributions in \(M/G/1\) queues ⋮ On The Renewal Input Batch-arrival Queue Under Single And Multiple Working Vacation Policy With Application To EPON ⋮ The periodic steady-state solution for queues with Erlang arrivals and service and time-varying periodic transition rates ⋮ Stability of a cascade system with multiple stations ⋮ Continuous‐time multi‐type Ehrenfest model and related Ornstein–Uhlenbeck diffusion on a star graph ⋮ On queueing-inventory-location problems ⋮ Two-commodity queueing-inventory system with phase-type distribution of service times ⋮ Resource optimization in \(MMAP[2/PH[2]/S\) priority queueing model with threshold \(PH\) retrial times and the preemptive resume policy] ⋮ Impact of the degradation in service rate in \({MAP}/{PH}/1\) queueing system with phase type vacations, breakdowns, and repairs ⋮ Aggregate Markov models in life insurance: properties and valuation ⋮ Unnamed Item ⋮ Unnamed Item ⋮ Unnamed Item ⋮ Efficient algorithms for transient analysis of stochastic fluid flow models ⋮ Block-Structured Fluid Queues Driven by QBD Processes ⋮ Kronecker-Based Infinite Level-Dependent QBD Processes ⋮ Loss Rate Asymptotics in aGI/G/1 Queue with Finite Buffer ⋮ On the Moments of the Time of Ruin with Applications to Phase-Type Claims ⋮ A heuristic algorithm for the optimization of a retrial system with Bernoulli vacation ⋮ Computing Stationary Expectations in Level-Dependent QBD Processes ⋮ The analysis of the M/M/1 queue with two vacation policies (M/M/1/SWV+MV) ⋮ Approximate Performance Analysis of Production Lines with Continuous Material Flows and Finite Buffers ⋮ Stability for Fluid Queues: Characteristic Inequalities ⋮ The Algebraic Degree of Phase-Type Distributions ⋮ Tail asymptotics for M/G/1-type queueing processes with light-tailed increments ⋮ ALGORITHMS FOR RETURN PROBABILITIES FOR STOCHASTIC FLUID FLOWS ⋮ ANALYSIS OF A GI/GY/1 SYSTEM IN DISCRETE-TIME ⋮ A matrix-analytic approach to the N-player ruin problem ⋮ Solving matrix polynomial equations arising in queueing problems ⋮ On the impact of job size variability on heterogeneity-aware load balancing ⋮ Functionals of Markovian Branching D-BMAPS ⋮ Matching Three Moments with Minimal Acyclic Phase Type Distributions ⋮ Fitting with Matrix-Exponential Distributions ⋮ Analysis of Stochastic Online Bin Packing Processes ⋮ The Impact of Self-Generation of Priorities on Multi-Server Queues with Finite Capacity ⋮ Reliability of Supply Between Production Lines ⋮ Compliance of the Token-Bucket Model with Markovian Traffic ⋮ A Tandem Queue with Server Slow-Down and Blocking ⋮ Approximated Transient Queue Length and Waiting Time Distributions via Steady State Analysis ⋮ Approximating the ΣGI/G/squeue by Using Aggregation and Matrix Analytic Methods ⋮ A New Policy Evaluation Algorithm for Markov Decision Processes with Quasi Birth-Death Structure ⋮ Characterizing the BMAP/MAP/1 Departure Process via the ETAQA Truncation ⋮ Risk processes analyzed as fluid queues ⋮ The deficit at ruin in the stationary renewal risk model ⋮ Decompositions of binomial ideals ⋮ Light-tailed asymptotics of stationary probability vectors of Markov chains of GI/G/1 type ⋮ Slowing time: Markov-modulated Brownian motions with a sticky boundary ⋮ Spectral Polynomial Algorithms for Computing Bi-Diagonal Representations for Phase Type Distributions and Matrix-Exponential Distributions ⋮ Decay rates for quasi-birth-and-death processes with countably many phases and tridiagonal block generators ⋮ On Time Duality for Markov Chains ⋮ Several Types of Ergodicity for M/G/1-Type Markov Chains and Markov Processes ⋮ Transient analysis of piecewise homogeneous QBD process ⋮ Traffic lights, clumping and QBDs ⋮ Single-server queues under overdispersion in the heavy-traffic regime ⋮ User-perceived reliability of unrepairable shared protection systems with functionally identical units ⋮ A multi-server queueing model with retrial connection arrivals as a model for optimisation of the traffic control ⋮ FUNDAMENTAL MATRIX OF TRANSIENT QBD GENERATOR WITH FINITE STATES AND LEVEL DEPENDENT TRANSITIONS ⋮ Characterization of the conditional stationary distribution in Markov chains via systems of linear inequalities ⋮ MAP/PH/1 Retrial Queue with Abandonment, Flush Out and Search of Customers ⋮ A Retrial Queueing System with Multiple Hierarchial Orbits and Orbital Search ⋮ Steady-State and Transient Analysis of a Single Channel Cognitive Radio Model with Impatience and Balking ⋮ Analysis of a MAP Risk Model with Stochastic Incomes, Inter-Dependent Phase-Type Claims and a Constant Barrier ⋮ A Two-Stage Tandem Queue with Specialist Servers ⋮ Stationary Distribution of Discrete-Time Finite-Capacity Queue with Re-sequencing ⋮ Highly Accurate Latouche-Ramaswami Logarithmic Reduction Algorithm for Quasi-Birth-and-Death Process ⋮ Optimal designs of the exponentially weighted moving average (EWMA) median chart for known and estimated parameters based on median run length ⋮ Ruin problems for epidemic insurance ⋮ Recursive Calculation of the Dividend Moments in a Multi-threshold Risk Model ⋮ The efficiency of run rules schemes for the multivariate coefficient of variation: a Markov chain approach ⋮ A new model for the reliability-redundancy allocation problem with the mixed redundancy strategy ⋮ Adding edge dynamics to bipartite random-access networks ⋮ Simple analytical solutions for the , , and related queues ⋮ SCALING PROPERTIES OF QUEUES WITH TIME-VARYING LOAD PROCESSES: EXTENSIONS AND APPLICATIONS ⋮ EXPLICIT SOLUTIONS FOR CONTINUOUS-TIME QBD PROCESSES BY USING RELATIONS BETWEEN MATRIX GEOMETRIC ANALYSIS AND THE PROBABILITY GENERATING FUNCTIONS METHOD ⋮ A BMAP/PH/N SYSTEM WITH IMPATIENT REPEATED CALLS ⋮ Spectral analysis of bilateral birth–death processes: some new explicit examples ⋮ ON THE TIME-DEPENDENT BEHAVIOR OF A MARKOVIAN REENTRANT-LINE MODEL ⋮ The EWMA sign chart revisited: performance and alternatives without and with ties ⋮ On the performance of the adaptive EWMA chart for monitoring time between events ⋮ An EWMA-type chart based on signed ranks with exact run length properties ⋮ PERFORMANCE MEASURES FOR THE TWO-NODE QUEUE WITH FINITE BUFFERS ⋮ Water cycle algorithm for solving the reliability-redundancy allocation problem with a choice of redundancy strategies ⋮ Computational analysis of the queue with working breakdowns and delaying repair under a Bernoulli-schedule-controlled policy ⋮ Conditional design of the EWMA median chart with estimated parameters ⋮ Reward distributions associated with some block tridiagonal transition matrices with applications to identity by descent ⋮ Quasi-Birth-and-Death Processes, Lattice Path Counting, and Hypergeometric Functions ⋮ Perturbed MAP Risk Models with Dividend Barrier Strategies ⋮ Optimal design of one-sided exponential EWMA charts based on median run length and expected median run length ⋮ STATIONARY MARKOVIAN ARRIVAL PROCESSES: RESULTS AND OPEN PROBLEMS ⋮ System reliability-redundancy optimization with cold-standby strategy by fitness-distance balance stochastic fractal search algorithm ⋮ Guaranteed conditional performance of the median run length based EWMA X¯ chart with unknown process parameters ⋮ Algorithms for Queueing Systems with Reneging and Priorities Modeled as Quasi-Birth-Death Processes ⋮ Matrix-analytic methods for the analysis of stochastic fluid-fluid models ⋮ Strong convergence to two-dimensional alternating Brownian motion processes ⋮ On retrial queue with customer balking and feedback subject to server breakdowns ⋮ An urn model for the Jacobi-Piñeiro polynomials ⋮ Numerical Solution of a Matrix Integral Equation Arising in Markov-Modulated Lévy Processes ⋮ The adaptive EWMA median chart for known and estimated parameters ⋮ One-sided synthetic control charts for monitoring the multivariate coefficient of variation ⋮ Conditional design of the CUSUM median chart for the process position when process parameters are unknown ⋮ DES AND RES PROCESSES AND THEIR EXPLICIT SOLUTIONS ⋮ THE RUNNING MAXIMUM OF A LEVEL-DEPENDENT QUASI-BIRTH-DEATH PROCESS ⋮ FINITE TWO LAYERED QUEUEING SYSTEMS ⋮ Solving Quadratic Matrix Equations Arising in Random Walks in the Quarter Plane ⋮ A Polynomial Time Algorithm for Computing Extinction Probabilities of Multitype Branching Processes ⋮ A Retrial Queueing System with Orbital Search of Customers Lost from an Offer Zone ⋮ An Infinite-Server Queueing $$MMAP_k|G_k|\infty $$ Model in Semi-Markov Random Environment Subject to Catastrophes ⋮ Analysis of an Infinite-Server Queue $$MAP_k|G_k|\infty $$ in Random Environment with k Markov Arrival Streams and Random Volume of Customers ⋮ Retrial Queue with Search of Interrupted Customers from the Finite Orbit ⋮ Discussion on “On the Laplace Transform of the Aggregate Discounted Claims with Markovian Arrivals,” by Jiandong Ren, Volume 12(2) ⋮ The probabilities of extinction in a branching random walk on a strip ⋮ The UnreliableM/M/1 Retrial Queue in a Random Environment ⋮ Yaglom limit for stochastic fluid models ⋮ A Multi-Server Retrial Queueing Model with Poisson Signals ⋮ DISCRETE-TIME GIX/GEO/1/N QUEUE WITH WORKING VACATIONS AND VACATION INTERRUPTION ⋮ THE ISRAELI QUEUE WITH INFINITE NUMBER OF GROUPS ⋮ Signatures in Markov-Modulated Processes ⋮ TAIL PROBABILITIES IN QUEUEING PROCESSES ⋮ AN EFFICIENT METHOD FOR PERFORMANCE ANALYSIS OF BLENDED CALL CENTERS WITH REDIAL ⋮ A New Look at Urban Water Storage in a Series of Connected Dams ⋮ Costing Mixed Coxian Phase-type Systems with Poisson Arrivals ⋮ The Batch Markovian Arrival Process Subject to Renewal Generated Geometric Catastrophes ⋮ Newton-Shamanskii Method for a Quadratic Matrix Equation Arising in Quasi-Birth-Death Problems ⋮ The surplus prior to ruin and the deficit at ruin for a correlated risk process ⋮ On the Distribution of the Deficit at Ruin when Claims are Phase-type ⋮ A RETRIAL QUEUEING MODEL WITH THRESHOLDS AND PHASE TYPE RETRIAL TIMES ⋮ Quantifying the performance effects of idle time utilization in multiserver systems ⋮ Performance evaluation of a production/inventory system with periodic review and endogenous lead times ⋮ Tri-Layered QBD Processes with Boundary Assistance for Service Resources ⋮ PH-Invariant Polytopes and Coxian Representations of Phase Type Distributions ⋮ A Poisson Limit Theorem for Reliability Models Based on Markov Chains ⋮ On the ergodicity of a class of level-dependent quasi-birth-and-death processes ⋮ Invariant Measures for Fluid Queues ⋮ Bilateral Phase Type Distributions ⋮ Computing absorbing times via fluid approximations ⋮ An ODE for an Overloaded X Model Involving a Stochastic Averaging Principle ⋮ Many-server queues with customer abandonment: numerical analysis of their diffusion models ⋮ The multi-server retrial system with Bernoulli feedback and starting failures ⋮ Fitting Matrix Geometric Distributions by Model Reduction ⋮ On the Laplace Transform of the Aggregate Discounted Claims with Markovian Arrivals ⋮ Eulerian polynomials and Quasi-Birth-Death processes with time-varying-periodic rates ⋮ Analysis of a Generalized Penalty Function in a Semi-Markovian Risk Model
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