Chesoong Kim

A Tandem Queue with Group Occupation of Servers at the Second Station

We consider a two-stage tandem queue with single-server first station and multiserver second station. Customers arrive to Station 1 according to a batch Markovian arrival process (BMAP). A batch may consist of heterogeneous customers. The type of a customer is determined upon completion of a service at Station 1. The customers type is classified based on the number of servers required to process the request of the customer at Station 2. If the required number of servers is not available, the customer may leave the system forever or block Station 1 by waiting for the required number of servers. We determine the stationary distribution of the system states at embedded epochs and derive the Laplace-Stieltjes transform of the sojourn time distribution. Some key performance measures are calculated, and illustrative numerical results are presented.

Queueing System with Heterogeneous Customers as a Model of a Call Center with a Call-Back for Lost Customers

A multiserver queueing system with infinite and finite buffers, two types of customers, and two types of servers as a model of a call center with a call-back for lost customers is investigated. Type 1 customers arrive to the system according to a Markovian arrival process. All rejected type 1 customers become type 2 customers. Type , , servers serve type customers if there are any in the system and serve type , , customers if there are no type r customers in the system. The service times of different types of customers have an exponential distribution with different parameters. The steady-state distribution of the system is analyzed. Some key performance measures are calculated. The Laplace-Stieltjes transform of the sojourn time distribution of type 2 customers is derived. The problem of optimal choice of the number of each type servers is solved numerically.

A dual tandem queueing system with a finite intermediate buffer and cross traffic

We analyze a dual tandem queue having a single server queueing system with infinite buffer at the first station and a multi-server queueing system with a finite buffer at the second station. Arrival flow is described by the Batch Markovian Arrival Process (BMAP). Service time at the first station is generally distributed while at the second station it is exponentially distributed. In situation when the intermediate buffer between the stations is full at the service completion of a customer at the first station, this customer is lost or blocks the server until service completion in one of the servers at the second station. Besides the customers, which got service at the first station, an additional MAP flow (cross traffic) arrives to the second station directly, not entering the first station. Ergodicity condition for this system is derived. Stationary state distribution of the system at embedded and arbitrary time epochs is computed as well as the main performance measures of the system. Numerical results show possibility of optimization of the system operation by means of appropriate choosing the capacity of an intermediate buffer and the intensity of cross traffic. Necessity of the account of correlation in the arrival processes is illustrated.

Analysis of BMAP(r)/M(r)/N(r) Type Queueing System Operating in Random Environment

A multi-server queueing system with an infinite buffer and impatient customers is analyzed. The system operates in the finite state Markovian random environment. The number of available servers, the parameters of the batch Markovian arrival process, the rate of customers’ service, and the impatience intensity depend on the current state of the random environment and immediately change their values at the moments of jumps of the random environment. Dynamics of the system is described by the multi-dimensional asymptotically quasi-Toeplitz Markov chain. The ergodicity condition is derived. The main performance measures of the system are calculated. Numerical results are presented.

Multi-threshold control by a single-server queuing model with a service rate depending on the amount of harvested energy

Abstract We consider a single-server queuing model with energy harvesting. Customers and energy units arrive in the marked Markovian arrival flow. Buffers for customers and energy are finite. There are several service modes distinguished by the number of required units of energy, the service rate and the quality of service. The choice of the service mode for the next customer depends on the current number of energy units in the system. The quality of system operation is assumed to be defined by a customer’s loss probability. Loss can occur due to buffer overflow, low-quality service or impatience during the waiting time. The control strategy is of a multi-threshold type with thresholds defined by the amount of available energy. Under the fixed values of the thresholds, the behaviour of the system is described by a multi-dimensional Markov chain. The generator of this chain is derived. The main performance measures are calculated. The distribution of the waiting time is derived in explicit form. A numerical study of the model via extensive experiments that provide insight into the quantitative behaviour of the system is presented.

Analysis of Two-Server Queueing Model with Phase-Type Service Time Distribution and Common Phases of Service

We consider two server queueing system with an infinite buffer. Customers arrive to the system according to the Markovian Arrival Process. Service time of a customer has a phase-type distribution. The servers use the same equipment (phases of PH) for customers processing. So, if service of a customer transits to the phase, at which another server is currently providing the service, the service of the customer is suspended until the phase will become available. Behavior of the system is described by the multi-dimensional Markov chain. The generator of this Markov chain is derived. Expressions for computation of the main performance measures are derived.