ISSN 0032-9460, Problems of Information Transmission, 2014, Vol. 50, No. 1, pp. 106–116.
Pleiades Publishing, Inc., 2014.
Original Russian Text
A.N. Starovoitov, 2014, published in Problemy Peredachi Informatsii, 2014, Vol. 50, No. 1, pp. 116–127.
COMMUNICATION NETWORK THEORY
Conditions for a Product-Form Stationary Distribution
of One Queueing System with Batch Transfers
and a Disaster Flow
A. N. Starovoitov
Belarusian State University of Transport, Gomel, Belarus
Received June 13, 2012; in ﬁnal form, January 16, 2014
Abstract—We consider an open exponential network with two types of arrival ﬂows at the
network nodes: a message ﬂow and a disaster ﬂow. Messages arriving at the nodes form batches
of customers of a random size. A disaster arrival at a node completely empties the queue at
the node if it is nonempty and has no eﬀect otherwise. Customers are served in batches of a
random size. After a batch is served at a node, the batch quits the network and, according
to a routing matrix, either sends a message or a disaster to another node or does not send
anything. We ﬁnd conditions for the stationary distribution of the network state probabilities
to be represented as a product of shifted geometric distributions.
In  a queueing network with batch arrivals and assemble-transfer batch service was considered.
This service can be described as follows. The service process starts when either a new customer
arrives at a node or the preceding service is completed and there are other customers in the node.
The service time is exponentially distributed. At the end of a service time, a random-size batch
of customers is chosen from the queue of the node, which is regarded to be served. Note that
in  a served batch changes its size and transfers to another node according to a routing matrix.
It was found that for a stationary distribution in the form of a product of geometric distributions
to exist it is required to introduce an auxiliary Poisson ﬂow with arrivals that enter a network node
if its queue is empty. In  a network with the same service was considered; however, unlike ,
arrivals at network nodes form batches of a random size. At the end of a service time, a random-
size batch of arrivals is removed from the node queue and, according to a routing matrix, either
sends a message to another node or does not send anything. Necessary and suﬃcient conditions
were established for the network stationary distribution to be represented as a product of shifted
geometric distributions. In  a closed network with batch transfers was considered.
In the present paper we consider a queueing network analogous to that considered in  but with
an additional simplest ﬂow of disasters (catastrophes). A disaster completely empties the queue at
a node if it is nonempty and has no eﬀect otherwise. Models of this type can be used to describe
computer networks or, for example, distributed databases. In this case a disaster can be a virus
or a command that aborts all currently executed transactions. Queueing systems and networks
with disasters were studied in [4–9]. In [4, 5], clearing systems were considered, which are similar
to systems with disasters. In  an M/G/1systemwiththesimplestdisaster ﬂow was studied and
a generalization of the Pollaczek–Khintchine formula was obtained. In [7, 8], more general models
of queueing systems were studied with various assumptions on both the arrival ﬂow, disaster ﬂow,
and service. In , for a Jackson network with the simplest disaster ﬂow, a product-form stationary