On the loss process in a batch arrival queue

In this paper, we investigate the loss process in a finite-buffer queue with batch arrivals and total rejection discipline. In such a model, if the buffer has insufficient capacity to accept all the customers included in an arriving batch, the whole batch is blocked and lost. This scheme is especially useful in performance evaluation of buffering processes in IP (internet protocol) networks. The main result of this paper is a closed-form formula for the joint distribution of the length of the first lost series of batches and the time of the first loss. Moreover, the limiting distribution (as the buffer size grows to infinity) is shown.     

A batch arrival retrial queue with two phases of service and Bernoulli vacation schedule

We consider an M X /G/1 queueing system with two phases of heterogeneous service and Bernoulli vacation schedule which operate under a linear retrial policy. In addition, each individual customer is subject to a control admission policy upon the arrival. This model generalizes both the classical M/G/1 retrial queue with arrivals in batches and a two phase batch arrival queue with a single vacation under Bernoulli vacation schedule. We will carry out an extensive stationary analysis of the system , including existence of the stationary regime, embedded Markov chain, steady state distribution of the server state and number of customer in the retrial group, stochastic decomposition and calculation of the first moment.     

The randomized vacation policy for a batch arrival queue

This paper examines an M^[x]/G/1${\text{M}}^{[\text{x}]}/\text{G}/1$ queueing system with a randomized vacation policy and at most J vacations. Whenever the system is empty, the server immediately takes a vacation. If there is at least one customer found waiting in the queue upon returning from a vacation, the server will be immediately activated for service. Otherwise, if no customers are waiting for service at the end of a vacation, the server either remains idle with probability p   or leaves for another vacation with probability 1-p$1-p$. This pattern continues until the number of vacations taken reaches J. If the system is empty by the end of the J  th vacation, the server is dormant idly in the system. If there is one or more customers arrive at server idle state, the server immediately starts his services for the arrivals. For such a system, we derive the distributions of important characteristics, such as system size distribution at a random epoch and at a departure epoch, system size distribution at busy period initiation epoch, idle period and busy period, etc. Finally, a cost model is developed to determine the joint suitable parameters (p^∗,J^∗)$(p^{\ast }\text{,}{J}^{\ast })$ at a minimum cost, and some numerical examples are presented for illustrative purpose.     

The infinite-buffer single server queue with a variant of multiple vacation policy and batch Markovian arrival process

We consider an infinite-buffer single server queue where arrivals occur according to a batch Markovian arrival process (BMAP). The server serves until system emptied and after that server takes a vacation. The server will take a maximum number H of vacations until either he finds at least one customer in the queue or the server has exhaustively taken all the vacations. We obtain queue length distributions at various epochs such as, service completion/vacation termination, pre-arrival, arbitrary, departure, etc. Some important performance measures, like mean queue lengths and mean waiting times, etc. have been obtained. Several other vacation queueing models like, single and multiple vacation model, queues with exceptional first vacation time, etc. can be considered as special cases of our model.     

An optimal service rate in a Poisson arrival queue with two-stage service policy*

We consider a two-stage service policy for a Poisson arrival queueing system. The idle server starts to work with ordinary service rate when a customer arrives. If the number of customers in the system reaches N, the service rate gets faster and continues until the system becomes empty. Otherwise, the server finishes the busy period with ordinary service rate. After assigning various operating costs to the system, we show that there exists a unique fast service rate minimizing the long-run average cost per unit time.This work was supported by Korea Research Foundation Grant(KRF-2002-070-C00021).     

Threshold control by a single-server retrial queue with batch arrivals and group services

A controlled single-server retrial queueing system is investigated. Customers arrive according to batch Markovian arrival process. The system has several operation modes which are controlled by means of a threshold strategy. The stationary distribution is calculated. Optimization problem is considered and a numerical example is presented.     

Analyzing the finite buffer batch arrival queue under Markovian service process: GIX/MSP/1/N

We consider a batch arrival finite buffer single server queue with inter-batch arrival times are generally distributed and arrivals occur in batches of random size. The service process is correlated and its structure is presented through Markovian service process (MSP). The model is analyzed for two possible customer rejection strategies: partial batch rejection and total batch rejection policy. We obtain steady-state distribution at pre-arrival and arbitrary epochs along with some important performance measures, like probabilities of blocking the first, an arbitrary, and the last customer of a batch, average number of customers in the system, and the mean waiting times in the system. Some numerical results have been presented graphically to show the effect of model parameters on the performance measures. The model has potential application in the area of computer networks, telecommunication systems, manufacturing system design, etc.      

Optimal multithreshold control for a BMAP/G/1 queue with N service modes

This paper deals with the problem of the optimal service rate control in the system with BMAP (Batch Markovian Arrival Process) arrival stream. An algorithm for the computation of the embedded stationary queue length distribution is developed. The procedure for the cost criteria calculation is elaborated for any fixed parameters of the multithreshold control policy. This revised version was published online in June 2006 with corrections to the Cover Date.     

A BMAP/SM/1 queue with service times depending on the arrival process

We study a BMAP/>SM/1 queue with batch Markov arrival process input and semi‐Markov service. Service times may depend on arrival phase states, that is, there are many types of arrivals which have different service time distributions. The service process is a heterogeneous Markov renewal process, and so our model necessarily includes known models. At first, we consider the first passage time from level {κ+1} (the set of the states that the number of customers in the system is κ+1) to level {κ} when a batch arrival occurs at time 0 and then a customer service included in that batch simultaneously starts. The service descipline is considered as a LIFO (Last‐In First‐Out) with preemption. This discipline has the fundamental role for the analysis of the first passage time. Using this first passage time distribution, the busy period length distribution can be obtained. The busy period remains unaltered in any service disciplines if they are work‐conserving. Next, we analyze the stationary workload distribution (the stationary virtual waiting time distribution). The workload as well as the busy period remain unaltered in any service disciplines if they are work‐conserving. Based on this fact, we derive the Laplace–Stieltjes transform for the stationary distribution of the actual waiting time under a FIFO discipline. In addition, we refer to the Laplace–Stieltjes transforms for the distributions of the actual waiting times of the individual types of customers. Using the relationship between the stationary waiting time distribution and the stationary distribution of the number of customers in the system at departure epochs, we derive the generating function for the stationary joint distribution of the numbers of different types of customers at departures. This revised version was published online in June 2006 with corrections to the Cover Date.     

Finite buffer vacation models under E-limited with limit variation service and Markovian arrival process

We consider a finite-buffer single server queue with single (multiple) vacation(s) and Markovian arrival process. The service discipline is E-limited with limit variation (ELV). Several other service disciplines like, Bernoulli scheduling, nonexhaustive and E-limited service can be treated as special cases of the ELV service.     

A single-server batch arrival queue with returning customers

We consider a new class of batch arrival retrial queues. By contrast to standard batch arrival retrial queues we assume if a batch of primary customers arrives into the system and the server is free then one of the customers starts to be served and the others join the queue and then are served according to some discipline. With the help of Lyapunov functions we have obtained a necessary and sufficient condition for ergodicity of embedded Markov chain and the joint distribution of the number of customers in the queue and the number of customers in the orbit in steady state. We also have suggested an approximate method of analysis based on the corresponding model with losses.     

Equilibrium arrival times to a queue with order penalties

Suppose customers need to choose when to arrive to a congested queue with some desired service at the end, provided by a single server that operates only during a certain time interval. We study a model where the customers incur not only congestion (waiting) costs but also penalties for their index of arrival. Arriving before other customers is desirable when the value of service decreases with every admitted customer. This may be the case for example when arriving at a concert or a bus with unmarked seats or going to lunch in a busy cafeteria. We provide game theoretic analysis of such queueing systems with a given number of customers, specifically we characterize the arrival process which constitutes a symmetric Nash equilibrium.     

The BMAP/PH/N retrial queue with Markovian flow of breakdowns

We consider a multi-server retrial queue with the Batch Markovian Arrival Process (BMAP). The servers are identical and independent of each other. The service time distribution of a customer by a server is of the phase (PH) type. If a group of primary calls meets idle servers the primary calls occupy the corresponding number of servers. If the number of idle servers is insufficient the rest of calls go to the orbit of unlimited size and repeat their attempts to get service after exponential amount of time independently of each other. Busy servers are subject to breakdowns and repairs. The common flow of breakdowns is the MAP. An event of this flow causes a failure of any busy server with equal probability. When a server fails the repair period starts immediately. This period has PH type distribution and does not depend on the repair time of other broken-down servers and the service time of customers occupying the working servers. A customer whose service was interrupted goes to the orbit with some probability and leaves the system with the supplementary probability. We derive the ergodicity condition and calculate the stationary distribution and the main performance characteristics of the system. Illustrative numerical examples are presented.     

Continuous time control of the arrival process in an m/g/1 queue

The problem of continuously controlling the arrival process in an M/G/1 queue is studied. The control is exercised by keeping the facility open or closed for potential arrivals, and is based on the residual workload process. The reward structure includes a reward rate R when the server is busy, and a holding cost rate cx when the residual workload is x. The economic criterion used is long run average return. A control limit policy is shown to be optimal. An iterative method for calculating this control limit policy is suggested.     

On the characterization of departure rules for discrete-time queueing networks with batch movements and its applications

This paper supplements and generalizes the results of sawa [11] in this special issue from the viewpoint of discrete-time networks of queues with batch arrivals and batch departures, due to Henderson and Taylor [7]. We first note that the D-rule of sawa [11] is equivalent to the specific form for the release rate function, introduced in [7]. Such forms have widely appeared in the literature, too. sawa [11] found that the D-rule can be characterized in terms of the reversed-time process of a certain vector-valued process. He obtained this characterization for a single node model. We generalize this result for networks of queues with batch arrivals and batch departures. This reveals why the specific form of the release rate function is common in the literature. Furthermore, the characterization is useful to consider traffic flows in a discrete-time queueing network.This research is partially supported by NEC C&C Laboratories.     

The minimal solution of a Markov renewal equation

Given a killed Markov process, one can use a procedure of Ikedaet al. to revive the process at the killing times. The revived process is again a Markov process and its transition function is the minimal solution of a Markov renewal equation. In this paper we will calculate such solutions for a class of revived processes.     

Structure-reversibility and departure functions of queueing networks with batch movements and state dependent routing

We consider characterizations of departure functions in Markovian queueing networks with batch movements and state-dependent routing in discrete-time and in continuous-time. For this purpose, the notion of structure-reversibility is introduced, which means that the time-reversed dynamics of a queueing network corresponds with the same type of queueing network. The notion is useful to derive a traffic equation. We also introduce a multi-source model, which means that there are different types of outside sources, to capture a wider range of applications. Characterizations of the departure functions are obtained for any routing mechanism of customers satisfying a recurrent condition. These results give a unified view to queueing network models with linear traffic equations. Furthermore, they enable us to consider new examples as well as show limited usages of this kind of queueing networks. This revised version was published online in June 2006 with corrections to the Cover Date.     

Discrete-time queue with Bernoulli bursty source arrival and generally distributed service times

In this contribution, a discrete-time single-server infinite-capacity queue with correlated arrivals and general service times is investigated. Arrivals of cells are modelled as an on/off source process with geometrically distributed on-periods and off-periods, which is called Bernoulli bursty source. Based on the probability generating function technique, closed-form expression of some performance measures of system, such as average buffer content, unfinished work, cell delay and so on, are obtained. Finally, the effects of system parameters on performance measures are illustrated by some numerical examples.     

Discrete NT-policy single server queue with Markovian arrival process and phase type service

We consider a discrete time single server queueing system in which arrivals are governed by the Markovian arrival process. During a service period, all customers are served exhaustively. The server goes on vacation as soon as he/she completes service and the system is empty. Termination of the vacation period is controlled by two threshold parameters N and T, i.e. the server terminates his/her vacation as soon as the number waiting reaches N or the waiting time of the leading customer reaches T units. The steady state probability vector is shown to be of matrix-geometric type. The average queue length and the probability that the server is on vacation (or idle) are obtained. We also derive the steady state distribution of the waiting time at arrivals and show that the vacation period distribution is of phase type.