Non-Stationary Characteristics in MAP/G/1/ Infinity Queue with the Foreground-Background Processor Sharing Discipline

The MAP/G/1/infinity queue with FBPS discipline is under consideration. The mathematical relations for calculation of the non-stationary joint distribution of the number of served customers up to and the number of customers in the system at the moment t are obtained. For the customers which presented in the queue at the moment t, their served length are considered. The equations are obtained in the terms of triple transform—Laplace transformwith respect to the time and generating functions with respect to the number of customers

G-network with the route change

Queueing networks with negative customers (G-networks), Poisson flow of positive customers, non-exponential nodes, and dependent service at the different nodes are under consideration. Every customer arriving at the network is defined by a set of random parameters: customer route, the length of customer route, customer volume and its service time at each route stage as well. The arrival of a negative customer to a queuing system causes one of the ordinary (or positive) customers to be removed (or killed) if any is present. The killed customer continues its way along the new random route. For such G-networks, the multidimensional stationary distribution of the network state probabilities is shown to be representable in product form

Analysis of multi-server queueing system with semi-Markovian input flow and negative customers acted upon queue end

The multi-server queueing system with a finite of an infinite buffer, with semi-Markovian input flow (for positive and negative customers) and with Markovian Service Process (for positive customers) whose the number of the states of the process and the intensities of the transitions between phases depend on the number of the customers in the system is considered. An arriving negative customer kills the one positive customer at the end of the queue. The relations and algorithms for computation of the steady-state probabilities and for calculation of the steadystate distribution of waiting time of positive customer are received. It is shown how the multiserver queueing system with semi-Markovian input flow, the servicing of the phase type and the above mentioned order of act of the negative customers can be bring to the general queuing system

Multiplicative solution for exponential G-networks with dependent service and preemptive resume of service of killed customers

G-networks with 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. Product form solution for multidimensional stationary distribution of the network state is derived

Queueing Network with Negative Customers and the Route Change

A queueing network with negative customers (G-network) is considered with the Poisson flow of positive customers, four types of nodes, and dependent service at different nodes. Every customer arriving at the network is determined by a set of random parameters: customer route, the length of customer route, customer size and its service time at each route stage as well. The arrival of a negative customer to a queuing system causes one of ordinary (or “positive”) customers to be removed (or “killed”) if any is present. The “killed” customer continues its way along the new random route. For such G-network, the multidimensional stationary distribution of the network state probabilities is shown to be representable in the form of a product

Tandem queues with a Markov flow and blocking

A tandem queueing system with two phases and a Markov flow entering into the first phase is studied. Both phases are characterized by one server with a buffer of finite capacity. The service times have an arbitrary distribution function and the service process in the second phase is of Markov-type. An arriving customer who finds the first buffer full is lost. A customer served in the first phase blocks its operation if there is no free waiting place in the second phase at this moment. The stationary distribution of a Markov chain embedded at the instants of customer transitions from the first phase to the second one is obtained. A computing algorithm was derived for PH-distribution of service time in the first server. Numerical examples are given

Analysis of the Multi-server Markov Queueing 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

A Geo_m/G/1/n Queueing System with LIFO Discipline, Service Interruptions and Resumption, and Restrictions on the Total Volume of Demands

Consideration is given to a discretetime queueing system with inverse discipline, service interruption and resumption, second-order geometrical demand arrival, arbitrary (discrete) distribution of demand length and finite storage. Each demand entering the queue has random volume besides its length. The total volume of the demands in the queue is limited by a certain number. Formulae for the stationary probabilities of states and the stationary waiting time distribution in the queuing system are obtained