We consider a single server queueing system with repeated attempts in which customers arrive according a Markov Arrival Process (MAP) and with a LCFS PR discipline. The service times are independent and have a common general distribution. After service completion time the server initiates his search time with an arbitrary distribution function. We consider two cases where the maximum number of repeated customers waiting in the orbit to seek service again is limited by r(r < ∞) or can be unlimited (r = ∞). We derive the steady state probabilities of the embedded Markov chain at service completion times of the process and also the steady state probabilities of the underlying Markov linear process. RESUMEN Consideramos un servicio de cola con un solo servidor con intentos repetidos en el cual los usuarios arriban de acuerdo a un Proceso de Arribos de Markov (PAM) y con disciplina LCFS PR. Los tiempos de servicio son independientes y poseen la misma distribución general. Después del terminar el servicio el servidor inicia su tiempo de búsqueda con una distribución arbitraria. Consideramos dos casas donde el número máximo de usuarios repetidos que esperan en la órbita buscando servicio nuevamente esta limitado por r (r < ∞). Nosotros derivamos las probabilidades de estado estable de la cadena de Markov inmersa en los tiempos de completamiento del proceso y también las probabilidades de estabilidad del estado del proceso linear de Markov subyacente. MSC: 60K25.

We consider a single-server retrial queueing system with K(K ≥ 1) Poisson input flows. The service times have a common arbitrary distribution function Bi(x) for customer of type i. An arriving customers of type i, , K , 1 i = who finds the server free begins to get service inmediately and leaves the system after completion. Otherwise, if the server is busy, the customer with probability 1 - Hi leaves the system without service and with probability Hi > 0 joins an orbit of repeated customer but conserves its own type. The intervals separating two succesive repeated attempts of each customers from the orbit are exponentially distributed with rate γ. The orbit is finite or infinite. In case of a finite orbit an arriving customer who finds the server busy and the orbit completely full is lost. We derive the steady state probabilities of the multidimensional Markov process underlying the considered queueing system.

A retrial single-server queueing system with finite buffer is considered. The primary incoming flow is Poissonian. If the buffer is overflown, a call entering the system becomes a repeat call and joins the group of repeat calls referred to as an orbit. The maximum number of calls that can simultaneously be contained in the orbit is limited. A call from the orbit makes new attempts to enter the system until a vacancy occurs. Time between repeat attempts for each call is an exponentially distributed random variable. At the initial moment of service, a type of a call is defined: with probability a i it becomes a call of type i and its service time in this case has distribution function B i (x), i = \(\overline {1,K}\). For this system, the stationary joint distribution of queues in the buffer and orbit is found. Numerical examples are given.

type i, , K , 1 i = who finds the server free begins to get service inmediately and leaves the system after completion. Otherwise, if the server is busy, the customer with probability 1 Hi leaves the system without service and with probability Hi > 0 joins an orbit of repeated customer but conserves its own type. The intervals separating two succesive repeated attempts of each customers from the orbit are exponentially distributed with rate γ. The orbit is finite or infinite. In case of a finite orbit an arriving customer who finds the server busy and the orbit completely full is lost. We derive the steady state probabilities of the multidimensional Markov process underlying the considered queueing system.

The queue with Markov input flow, retrying customers, and limited queues both of the primary and retrying customers was discussed. A new arriving customer is queued with the primary customers, or is sent to the queue of retrying customers if all places on the queue of primary customers are occupied, or leaves the system if the Queue of retrying customers also has no vacant places. A customer from the queue of retrying customers can be sent only to a free server. For this system, the stationary probabilities of states, as well as the stationary time distributions of the numbers of retrying and primary customers in the system were determined for the embedded Markov chain which is generated by the instants of server release.

Consideration is given to the queuing system with one server, finite-length buffer, and Markov flow of customers. The servicing limes have arbitrary distributions that depend on the number of customers in the system al the instant when servicing of the current customer begins. For the stationary distribution of slates of the Markov linear-wise process that describes the system behavior, recurrent matrix formulas have been obtained. For the stationary state probabilities of the system that is considered at the instants of customer arrival and servicing completion, expressions have been obtained as well.

A single-server queue system with limited waiting room is studied. Two Markov inputs arrive at the system. Their service times have arbitrary distribution functions, different for different input types. The customers of the first input have absolute priority, and a customer whose service is interrupted is served anew. A matrix algorithm is designed for computing the stationary probabilities of the states at arbitrary instants, as well as at customer arrival or service termination instants.