CONSTRUCTION OF CONTINUOUS TIME MARKOVIAN ARRIVAL PROCESSES

Markovian arrival processes were introduced by Neuts in 1979 （Neuts 1979） and have been used extensively in the stochastic modeling of queueing, inventory, reliability, risk, and telecommunications systems. In this paper, we introduce a constructive approach to define continuous time Markovian arrival processes. The construction is based on Poisson processes, and is simple and intuitive. Such a construction makes it easy to interpret the parameters of Markovian arrival processes. The construction also makes it possible to establish rigorously basic equations, such as Kolmogorov differential equations, for Markovian arrival processes, using only elementary properties of exponential distributions and Poisson processes. In addition, the approach can be used to construct continuous time Markov chains with a finite number of states

SINGLE SERVER QUEUES WITH A BATCH MARKOVIAN ARRIVAL PROCESS AND BULK RENEWAL OR NON-RENEWAL SERVICE

We first consider an infinite-buffer single server queue where arrivals occur according to a batch Markovian arrival process （BMAP）. The server serves customers in batches of maximum size ＇b＇ with a minimum threshold size ＇a＇. The service time of each batch follows general distribution independent of each other as well as the arrival process. The proposed analysis is based on the use of matrix-analytic procedure to obtain queue-length distribution at a post-departure epoch. Next we obtain queue-length distributions at various other epochs such as, pre-arrival, arbitrary and pre-service using relations with post-departure epoch. Later we also obtain the system-length distributions at post-departure and arbitrary epochs using queue-length distribution at post-departure epoch. Some important performance measures, like mean queue-lengths and mean waiting times have been obtained Total expected cost function per trait time is also derived to determine the locally optimal values of a and b. Secondly, we perform similar analysis for the corresponding infinite-buffer single server queue where arrivals occur according to a BMAP and service process in this case follows a non-renewal one, namely, Markovian service process （MSP）.

On a risk model with Markovian arrivals and tax

A risk model with Markovian arrivals and tax payments is considered.When the insurer is in a profitable situation,the insurer may pay a certain proportion of the premium income as tax payments.First,the Laplace transform of the time to cross a certain level before ruin is discussed.Second,explicit formulas for a generalized Gerber-Shiu function are established in terms of the＇original＇Gerber-Shiu function without tax and the Laplace transform of the first passage time before ruin.Finally,the differential equations satisfied by the expected accumulated discounted tax payments until ruin are derived.An explicit expression for the discounted tax payments is also given.

The MAP/PH(PH/PH)/1 Discrete-time Queuing System with Repairable Server

In this paper, we discuss a discrete time repairable queuing system with Markovian arrival process, where lifetime of server, service time and repair time of server are all discrete phase type random variables. Using the theory of matrix geometric solution, we give the steady state distribution of queue length and waiting time. In addition, the stable availability of the system is also provided.

Performance Evaluation of Mobility Anchor Point with Guard Load Reservation in Hierarchical Mobile IPv6

Hierarchical mobile IPv6 （HMIPv6） introduces a mobility anchor point to reduce the signaling overhead and handoff latency. In this paper, we apply the matrix-analytical approach to explore the performance measures of the ongoing mobile nodes （MNs） drop and new MNs block probabilities of mobility anchor point with a guard bandwidth reservation scheme. We apply the Markovian arrival process （MAP） to model ongoing MNs and new MNs. Five related performance measures are derived, including the long-term new MN block and ongoing MN drop probabilities, and the three short-term measures of average length of a block period and a non-block period, as well as the conditional ongoing MN drop probability during a block period. These performance measures greatly assist the guard bandwidth reservation mechanism in determining a proper threshold guard bandwidth. The results presented in this paper can provide guidelines for designing adaptive algorithms to adjust the threshold in the guard bandwidth reservation scheme.

Total duration of negative surplus for a MAP risk model

In this paper, we study the risk model with Markovian arrivals where we allow the surplus process to continue if the surplus falls below zero. We first derive expressions for the severity of ruin. Then by using the strong Markovian property of a two-dimensional Markov process and the expression for the severity of ruin, we obtain the Laplace transform of the total duration of negative surplus.

The MAP/PH/N Retrial Queue in a Random Environment

We consider the MAP/PH/N retrial queue with a finite number of sources operating in a finite state Markovian random environment. Two different types of multi-dimensional Markov chains are investigated describing the behavior of the system based on state space arrangements. The special features of the two formulations are discussed. The algorithms for calculating the stationary state probabilities are elaborated, based on which the main performance measures are obtained, and numerical examples are presented as well.

On a BMAP/G/1 G-queue with Setup Times and Multiple Vacations

In this paper, we consider a BMAP/G/1 G-queue with setup times and multiple vacations. Arrivals of positive customers and negative customers follow a batch Markovian arrival process （BMAP） and Markovian arrival process （MAP） respectively. The arrival of a negative customer removes all the customers in the system when the server is working. The server leaves for a vacation as soon as the system empties and is allowed to take repeated （multiple） vacations. By using the supplementary variables method and the censoring technique, we obtain the queue length distributions. We also obtain the mean of the busy period based on the renewal theory.

A Stochastic Inventory System with Postponed Demands and Infinite Pool in Discrete-Time Setup

In this article,we consider a discrete-time inventory model in which demands arrive according to a discrete Markovian arrival process.The inventory is replenished according to an es;ST policy,and the lead time is assumed to follow a discrete phase-type distribution.The demands that occur during stock-out periods either enter a pool which has an infinite capacity or leave the system with a predefined probability.The demands in the pool are selected one by one,if the on-hand inventory level is above s t 1;and the interval time between any two successive selections is assumed to have a discrete phase-type distribution.The joint probability distribution of the number of customers in the pool and the inventory level is obtained in the steady-state case.We derive the system performance measures under steady state and using these measures,the total expected cost rate of the system is calculated.The impacts of arrival rate on the performance measures are graphically illustrated.Finally,we study the impact of cost on the optimal values of the total expected cost rate,inventory level and the reorder point.