Characterization of the stationary distribution of the batch-transfer networks as a product of biased geometric distributions

Consideration was given to the class of exponential queuing networks with batch arrivals and batch servicing. The messages arriving to the network nodes form random-size batches of customers associated with these nodes. Upon completion of servicing a current batch, it discharges the network and sends a message, if any, to the rest of the nodes according to some routing matrix. The necessary and sufficient conditions under which the stationary distribution of the network states is multiplicative were established in terms of the isolated nodes placed in a dummy random medium. These conditions underlie characterization of the stationary distribution in the form of biased geometric distributions.     

Invariance of the stationary distribution of states of networks with multimode service, different types of requests, and discriminatory processor sharing

Open and closed queuing networks with multimode strategies and several types of requests are studied. Single-line nodes can operate in several modes. The time spent in each mode has an exponential distribution. The queuing discipline of the server is the discriminatory processor sharing (DPS). The number of requests is a random variable with an arbitrary distribution function. The invariance is established of the stationary distribution of the probabilities of the network states with respect to the functional form of the distributions of the work required to service the requests.     

Stationary probability distribution for states of G-networks with constrained sojourn time

We consider an exponential queueing network that differs from a Gelenbe network (with the usual positive and so-called negative customers), first, in that the sojourn time of customers at the network nodes is bounded by a random value whose conditional distribution for a fixed number of customers in a node is exponential. Second, we significantly relax the conditions on possible values of parameters for incoming Poisson flows of positive and negative customers in Gelenbe’s theorem. Claims serviced at the nodes and customers leaving the nodes at the end of their sojourn time can stay positive, become negative, or leave the network according to different routing matrices. We prove a theorem that generalizes Gelenbe’s theorem.     

Measurement of liquid entrainment from a horizontal cylindrical vessel by a flow of air

An empirical formula has been derived for the time needed for removal of a liquid from a horizontal cylindrical chamber by an air flow.     

Queueing Networks with Dynamic Routing and Dynamic Stochastic Bypass of Nodes

We consider open exponential networks with routing matrices that depend on a network state. A customer entering a node is either independently of other customers queued with probability that depends on the network state or instantly bypasses the node with complementary probability. After bypassing a node, customers are routed according to a stochastic matrix that depends on the network state and may differ from the routing matrix. Under certain restrictions on parameters of the model, we establish a sufficient ergodicity condition and find the final stationary distribution.