Bara Kim

M/M/1 Queue with Impatient Customers of Higher Priority

We introduce a simple approach for the analysis of the M/M/c queues with a single class of customers and constant impatience time by finding simple Markov processes (see (2.1) and (2.15) below), and then by applying this approach we analyze the M/M/1 queues with two classes of customers in which class 1 customers have impatience of constant duration, and class 2 customers have no impatience and lower priority than class 1 customers.

MAP/M/c Queue with Constant Impatient Time

We consider aMAP/M/c queue where a customer who cannot begin to receive his service for a fixed time is lost. We find a simple Markov process by using a concept of virtual waiting time and then obtain the stationary distribution of the Markov process. We find several performance measures such as loss probability, waiting time distribution, mean waiting time, and mean queue size by using the results of the stationary distribution of the Markov process.

Asymptotic Behavior of Loss Probability in GI/M/1/K Queue as K Tends to Infinity

We obtain an asymptotic behavior of the loss probability for the GI/M/1/K queue as K→∞ for cases of ρ<1, ρ>1 and ρ=1.

A survey of retrial queueing systems

Retrial queueing systems have been extensively studied because of their applications in telephone systems, call centers, telecommunication networks, computer systems, and in daily life. This survey deals with various retrial queueing models. The main focus of this survey is to show analytic results for queue length distributions, waiting time distributions, and tail asymptotics for the queue length and waiting time distributions. This survey also considers the stability analysis of retrial queueing models.

Pricing of geometric Asian options under Heston's stochastic volatility model

In this work, it is assumed that the underlying asset price follows Hestons stochastic volatility model and explicit solutions for the prices of geometric Asian options with fixed and floating strikes are derived. This approach has to deal with the derivation of the generalized joint Fourier transform of a square-root process and of three different weighted integrals of the square-root process with constant, linear and quadratic weights. Numerical implementation results for the complicated expressions are presented, together with the computational stability and efficiency of the method.

Sojourn time distribution in the M/M/ 1 queue with discriminatory processor-sharing

In this paper, we consider a queue with multiple K job classes, Poisson arrivals, exponentially distributed required service times in which a single processor serves according to the DPS discipline. More precisely, if there are ni class i jobs in the system, i = 1,...,K, each class j job receives a fraction αj/Σi=1k αini of the processor capacity. For this queue, we obtain a system of equations for joint transforms of the sojourn time and the number of jobs. Using this system of equations we find the moments of the sojourn time as a solution of linear simultaneous equations, which solves an open problem.

An M/G/ 1 queue with multiple types of feedback, gated vacations and FCFS policy

We consider an M/G/1 queueing system with multiple types of feedback, gated vacations and FCFS policy where the first service of a new customer is either successful (and then the customer leaves the system) or unsuccessful (and then the customer joins to the end of the queue for another service as old customer with different Bernoulli feedback parameter and different service distribution), and customers are served in the order of joining the tail of the queue. By applying a new method developed by authors (Queueing system with fixed feedback policy, J. Aust. Math. Soc. B, to appear, Comput. Oper. Res. 27 (2000) 269) we obtain joint probability generating function of system sizes of new and old customers at steady state and Laplace Stieltjes transform of total response time. We also give algorithms for calculation of moments of system size and total response time.

On the M/G/1 Bernoulli feedback queue with multi-class customers

Abstract We consider an M/G/1 queue with Bernoulli feedback policy and multi-class customers. We assume that different classes of customers have different arrival rates, service time distributions and feedback parameters. We obtain the joint probability generating function of system size of each class and also obtain the Laplace–Stieltjes transform of the total response time of a customer in each class. Using these results we obtain the moments of system size of each class and total response time of a customer in each class. Scope and purpose We obtain the joint distribution of system size of each class and distribution of total response time of a customer in each class for M/G/1 Bernoulli feedback queue with multi-class customers which has been an open problem since Takacs, Bell System Technical Journal 1963;42:509–19, analyzed the classical M/G/1 Bernoulli feedback queue with single-class customers. We also obtain the mean and higher moments of the system size of each class and total response time for a customer in each class.

Stability of join-the-shortest-queue networks

Abstract This paper investigates stability behavior in a variant of a generalized Jackson queueing network. In our network, some customers use a join-the-shortest-queue policy when entering the network or moving to the next station. Furthermore, we allow interarrival and service times to have general distributions. For networks with two stations we derive necessary and sufficient conditions for positive Harris recurrence of the network process. These conditions involve only the mean values of the network primitives. We also provide counterexamples showing that more information on distributions and tie-breaking probabilities is needed for networks with more than two stations, in order to characterize the stability of such systems. However, if the routing probabilities in the network satisfy a certain homogeneity condition, then we show that the stability behavior can be explicitly determined, again using the mean value parameters of the network. A byproduct of our analysis is a new method for using the fluid model of a queueing network to show non-positive recurrence of a process. In previous work, the fluid model was only used to show either positive Harris recurrence or transience of a network process.

A single server queue with Markov modulated service rates and impatient customers

Abstract We consider a single server queue in which the customers wait for service for a fixed time and leave the system if the service has not begun within that time. The customers arrive according to a Poisson process and each arriving customer brings in a certain amount of phase-type distributed work. The service rate of a server varies according to the underlying continuous time Markov process with finite states. We construct a Markov process by using the age process and then obtain the stationary distribution of the Markov process. By using the results of the stationary distribution of the Markov process, we obtain the loss probability, the waiting time distribution and the system size distribution.