A. V. Pechinkin

The Stationary Characteristics of the G / MSP /1/ r Queueing System

A single-server queueing system with recurrent input flow and Markov service process is considered. Both the cases of finite and infinite buffers are investigated. The analysis of this system is based on the method of embedded Markov chain. The main stationary characteristics of system performance are derived.

Analysis of the multi-server Markov queuing system with unlimited buffer and negative customers

Consideration was given to the multi-server queuing system with unlimited buffer, Markov input flow, and Markov (general) process of servicing all customers on servers with the number of process states and intensities of the inter-phase passage depending on the number of customers in the system. Additionally, a Markov flow of negative customers arrives to the system, the arriving negative customer killing the last queued positive customer. A recurrent algorithm to calculate the stationary probabilities of system states was obtained, and a method of calculation of the stationary distribution of the waiting time before starting servicing of a positive customer was proposed.

Decomposition of Queueing Networks with Dependent Service and Negative Customers

Queueing networks with negative customers (G-networks) and dependent service at different nodes are studied. Every customer arriving at the network is defined by a set of random parameters: his route over the network (a sequence of nodes visited by the customers), route length, and volume and service length of the customer at every stage of the route. For G-networks, which are the analogs of BCMP-networks, the multidimensional stationary distribution of the network state probabilities is shown to be representable in product form.

A queueing system with negative claims and a bunker for superseded claims in discrete time

We consider a discrete time single-line queueing system with independent geometric streams of regular and negative claims, infinite buffer, and geometric service. A negative claim pushes a regular claim out of the buffer queue and moves it to a bunker of infinite capacity. If the buffer is empty, a negative claim leaves the system without any effect. After servicing a claim, the system receives the next claim from the buffer, if it is not empty, or from the bunker. We obtain relations that allow computing stationary distributions for queues in the buffer and the bunker.

Stationary time characteristics of the GI/M/n/∞ system with some variants of the generalized renovation discipline

Consideration was given to the queuing system with recurrent arrivals, exponential distribution of the service time, infinite buffers, and the following variants of the discipline of generalized renovation and service: inversive order of renovation with direct order of customer service and direct and inversive orders of renovation with inversive order of customer service. For the serviced and lost customers, the stationary time characteristics of sojourn in the buffer were determined

Product form solution for exponential G-networks with dependent service and completion of service of killed customers

Queueing networks with negative customers (G-networks), Poisson flow of positive customers, multi-server exponential nodes, and dependent service at the different nodes are studied. Every customer arriving at the network is defined by a set of random parameters: customer route, the length of customer route, customer volume and his service time at each route stage as well. A killed positive customer is removed at the last place in the queue and quits the network just after his remaining service time will be elaborated. For such G-networks, the multidimensional stationary distribution of the network state probabilities is shown to be representable in product form.

Product Form Solution For G-Networks with Dependent Service

We consider a G-network with Poisson flow of positive customers. Each positive customer entering the network is characterized by a set of stochastic parameters: customer route, the length of customer route, customer volume and his service length at each route stage as well. The following node types are considered: (0) an exponential node with cn servers, infinite buffer and FIFO discipline; (1) an infinite-server node; (2) a single-server node with infinite buffer and LIFO PR discipline; (3) a single-server node with infinite buffer and PS discipline. Negative customers arriving at each node also form a Poisson flow. A negative customer entering a node with k customers in service, with probability 1/k chooses one of served positive customer as a target. Then, if the node is of a type 0 the negative customer immediately kills (displaces from the network) the target customer, and if the node is of types 1-3 the negative customer with given probability depending on parameters of the target customer route kills this customer and with complementary probability he quits the network with no service. A product form for the stationary probabilities of underlying Markov process is obtained.

On temporal characteristics in an exponential queueing system with negative claims and a bunker for ousted claims

We consider a queueing system with Poisson input streams of positive and negative claims, an infinite collector, and exponential service. A negative claim ousts a positive claim out of the collector queue and moves it to a bunker of unbounded capacity. If the collector is empty then a negative claim leaves the system with no influence on it. After a claim is serviced, the device receives the next claim from the collector or, if the collector is empty, from the bunker. For different combinations of FIFO and LIFO orders of choosing a claim for service from the collector’s queue, choosing a claim for service from the bunker’s queue, and ousting claims from the collector to the bunker, we obtain formulas for computing the stationary waiting time distribution for a claim to begin service and other temporal characteristics.

Geom/G/1/n system with LIFO discipline without interrupts and constrained total amount of customers

Consideration was given to the discrete-time queuing system with inversive servicing without interrupts, second-order geometrical arrivals, arbitrary (discrete) distribution of the customer length, and finite buffer. Each arriving customer has length and random volume. The total volume of the customers sojourning in the system is bounded by some value. Formulas of the stationary state probabilities and stationary distribution of the time of customer sojourn in the system were established.

Markov queueing system with finite buffer and negative customers affecting the queue end

A multiserver queueing system with finite buffer, Markov input flow, and Markov (general) service process of all customers on servers with the number of process states and intensities of inter-phase transitions depending on the number of customers in the system is considered. A Markov flow of negative customers arrives to the system; one negative customer “kills” one positive customer at the end of the queue. A recurrent algorithm for computing stationary probabilities of system states is obtained; and a method for calculating stationary distribution of waiting time before starting service of a positive customer is proposed.