Jerim Kim

Tail asymptotics of the queue size distribution in the M/M/m retrial queue

We consider an M/M/m retrial queue and investigate the tail asymptotics for the joint distribution of the queue size and the number of busy servers in the steady state. The stationary queue size distribution with the number of busy servers being fixed is asymptotically given by a geometric function multiplied by a power function. The decay rate of the geometric function is the offered load and independent of the number of busy servers, whereas the exponent of the power function depends on the number of busy servers. Numerical examples are presented to illustrate the result.

Analysis of the M/G/1 queue with discriminatory random order service policy

We consider an M/G/1 queue with different classes of customers and discriminatory random order service (DROS) discipline. The DROS discipline generalizes the random order service (ROS) discipline: when the server selects a customer to serve, all customers waiting in the system have the same selection probability under ROS discipline, whereas customers belonging to different classes may have different selection probabilities under DROS discipline. For the M/G/1 queue with DROS discipline, we derive equations for the joint queue length distributions and for the waiting time distributions of each class. We also obtain the moments of the queue lengths and the waiting time of each class. Numerical results are given to illustrate our results.

Stability of flow-level scheduling with Markovian time-varying channels

We consider the flow-level scheduling in wireless networks. The time is slotted and in each time slot the base station selects flows/users to serve. There are multi-class users and channel conditions vary over time. The channel state for each class user is assumed to be modeled as a finite state Markov chain. Using the fluid limit approach, we find the necessary and sufficient conditions for the stability of best rate (BR) scheduling policies. As a result, we show that any BR policy is maximally stable. Our result generalizes the result of Ayesta et al. (in press) [13] and solves the conjecture of Jacko (2011) [16]. We introduce a correlated channel state model and investigate the stability condition for BR policy in this model.

Tail asymptotics for the queue size distribution in the MAP/G/1 retrial queue

We consider a MAP/G/1 retrial queue where the service time distribution has a finite exponential moment. We derive matrix differential equations for the vector probability generating functions of the stationary queue size distributions. Using these equations, Perron–Frobenius theory, and the Karamata Tauberian theorem, we obtain the tail asymptotics of the queue size distribution. The main result on light-tailed asymptotics is an extension of the result in Kim et al. (J. Appl. Probab. 44:1111–1118, 2007) on the M/G/1 retrial queue.

Regularly varying tail of the waiting time distribution in M/G/1 retrial queue

We consider an M/G/1 retrial queue where the service time distribution has a regularly varying tail with index −β, β>1. The waiting time distribution is shown to have a regularly varying tail with index 1−β, and the pre-factor is determined explicitly. The result is obtained by comparing the waiting time in the M/G/1 retrial queue with the waiting time in the ordinary M/G/1 queue with random order service policy.

M/PH/1 QUEUE WITH DETERMINISTIC IMPATIENCE TIME

We consider an M/PH/1 queue with deterministic impatience time in which customers have phase-type service requirements. We find a related Markov process by using Markovian structure of the phase type distribution for services, and then obtain the stationary distribution of the Markov process. By using the results of the stationary distribution of the Markov process, we obtain performance measures such as the loss probability, the waiting time distribution and the queue size distribution.

Moments of the queue size distribution in the MAP/G/1 retrial queue

We consider an MAP/G/1 retrial queue. A necessary and sufficient condition is obtained for the existence of the moments of the queue size distribution. The condition is expressed in terms of the moment condition for a service time distribution. In addition, we provide recursive formulas for the moments of the queue size distribution. Numerical examples are given to illustrate our results.

WAITING TIME DISTRIBUTION IN THE M/M/M RETRIAL QUEUE

In this paper, we are concerned with the analysis of the waiting time distribution in the M/M/m retrial queue. We give expressions for the Laplace-Stieltjes transform (LST) of the waiting time distribution and then provide a numerical algorithm for calculating the LST of the waiting time distribution. Numerical inversion of the LSTs is used to calculate the waiting time distribution. Numerical results are presented to illustrate our results.

Iterative algorithm for the first passage time distribution in a jump-diffusion model with regime-switching, and its applications

For a regime-switching model with a finite number of regimes and double phase-type jumps, Jiang and Pistorius (2008) derived matrix equations with real parameters for the Wiener-Hopf factorization. The Laplace transform of the first passage time distribution is expressed in terms of the solution of the matrix equations. In this paper we provide an iterative algorithm for solving the matrix equations of Jiang and Pistorius (2008) with complex parameters. This makes it possible to obtain numeric values of the Laplace transform with complex parameters for the first passage time distribution. The Laplace transform with complex parameters can be inverted by numerical inversion algorithms such as the Euler method. As an application, we compute the prices of defaultable bonds under a structural model with regime switching and double phase-type jumps.

Analysis of a Markovian feedback queue with multi-class customers

We consider an M/G/1 Markovian feedback queue with multiclass customers. We derive functional equations for the stationary distribution of the queue size and the total response time. A system of linear equations is also derived for the moments of the queue size and the total response time distributions. The mean and the variance of the queue size and the total response time can be computed by solving the system of linear equations.