Jeongsim Kim

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.

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.

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.

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.

Queue size distribution in a discrete-time D-BMAP/G/1 retrial queue

We consider a discrete-time batch Markovian arrival process (D-BMAP)/G/1 retrial queue. We find the light-tailed asymptotics for the stationary distributions of the number of customers at embedded epochs and at arbitrary time. Using these tail asymptotics we propose a method for calculating the stationary distributions of the number of customers at embedded epochs and at arbitrary time. Numerical examples are presented to illustrate our results.

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.

The processor-sharing queue with bulk arrivals and phase-type services

In this paper, we consider a queue with compound Poisson arrivals, phase type required service times in which a single processor serves according to the processor-sharing discipline. For this queue, we derive a system of equations for the transform of the queue-length and obtain the moments of the queue-length as a solution of linear equations. We also obtain a system of equations for the joint transforms of the sojourn time and the queue-length and find the moments of the sojourn time as a solution of linear equations. Numerical examples show that the smaller the variation of the required service times becomes, the larger the mean and variance of the sojourn times become.

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.

Comparison of DPS and PS systems according to DPS weights

In the discriminatory processor-sharing (DPS) system with a single processor and K job classes, all jobs present in the system are served simultaneously with rates controlled by a vector of weights {gj >0; j=1,middotmiddotmiddot, K}. When all gj is equal, the DPS system reduces to the egalitarian processor-sharing (PS) system. In this paper we show how the weights of DPS must be chosen in order to make DPS outperform PS.