Bibliography on G-networks, negative customers and applications

We give a brief survey of literature devoted to studying queueing systems with Markovian and batch Markovian arrival processes and their application to modeling telecommunication networks.

Queueing systems with correlated arrival flows and their applications to modeling telecommunication networks

Erlang loss queueing system with batch arrivals operating in a random environment

A disaster queue with Markovian arrivals and impatient customers

Complete analysis of finite and infinite buffer GI/MSP/ 1 queue-A computational approach

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.

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

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.

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

The open exponential queuing network with negative customers (G-network) was consideredFor each arriving customer, given was a set of its random parameters such as the route defining the sequence of network nodes passed by the customer, route length, size, and servicing duration at each stage of the route It was assumed that the negative customer arriving to the sth node with the probabilities ωs and ωs+ either kills the positive customer in a randomly chosen server or does not affect it at all and with the probability ωs– =1 – ωs – ωs+ transforms it into a negative customer which after an exponentially distributed time arrives to the s′th node with the given probability The multidimensional stationary probability distribution of the network states was proved to be representable in the multiplicative form

Exponential Queuing Network with Dependent Servicing, Negative Customers, and Modification of the Customer Type

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.

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Product form solution for g-networks with dependent service

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.

P. P. Bocharov

Papers

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

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

Exponential Queuing Network with Dependent Servicing, Negative Customers, and Modification of the Customer Type

Product form solution for g-networks with dependent service

Decomposition of Queueing Networks with Dependent Service and Negative Customers