103 research outputs found
Crossings states and sets of states in P\'olya random walks
We consider the P\'olya random walk in . The paper establishes
a number of results for the distributions and expectations of the number of
usual (undirected) and specifically defined in the paper up- and down-directed
state-crossings and different sets of states crossings. One of the most
important results of this paper is that the expected number of undirected
state-crossings is equal to 1 for any state
. As well, the results of the
paper are extended to -dimensional random walks, , in bounded areas.Comment: Dear readers. I made a tremendous work to revise this paper after
referee report. There are 30 pages of 11pt format, 4 figures and 1 tabl
A Large Closed Queueing Network Containing Two Types of Node and Multiple Customer Classes: One Bottleneck Station
The paper studies a closed queueing network containing two types of node. The
first type (server station) is an infinite server queueing system, and the
second type (client station) is a single server queueing system with autonomous
service, i.e. every client station serves customers (units) only at random
instants generated by strictly stationary and ergodic sequence of random
variables. It is assumed that there are server stations. At the initial
time moment all units are distributed in the server stations, and the th
server station contains units, , where all the values
are large numbers of the same order. The total number of client stations is
equal to . The expected times between departures in the client stations are
small values of the order ~ . After service
completion in the th server station a unit is transmitted to the th
client station with probability ~ (), and being served
in the th client station the unit returns to the th server station. Under
the assumption that only one of the client stations is a bottleneck node, i.e.
the expected number of arrivals per time unit to the node is greater than the
expected number of departures from that node, the paper derives the
representation for non-stationary queue-length distributions in non-bottleneck
client stations.Comment: 39 pages, 5 figure
Analysis of Multiserver Retrial Queueing System: A Martingale Approach and an Algorithm of Solution
The paper studies a multiserver retrial queueing system with servers.
Arrival process is a point process with strictly stationary and ergodic
increments. A customer arriving to the system occupies one of the free servers.
If upon arrival all servers are busy, then the customer goes to the secondary
queue, orbit, and after some random time retries more and more to occupy a
server. A service time of each customer is exponentially distributed random
variable with parameter . A time between retrials is exponentially
distributed with parameter for each customer. Using a martingale
approach the paper provides an analysis of this system. The paper establishes
the stability condition and studies a behavior of the limiting queue-length
distributions as increases to infinity. As , the paper
also proves the convergence of appropriate queue-length distributions to those
of the associated `usual' multiserver queueing system without retrials. An
algorithm for numerical solution of the equations, associated with the limiting
queue-length distribution of retrial systems, is provided.Comment: To appear in "Annals of Operations Research" 141 (2006) 19-52.
Replacement corrects a small number of misprint
Asymptotic Behavior of the Number of Lost Messages
The goal of the paper is to study asymptotic behavior of the number of lost
messages. Long messages are assumed to be divided into a random number of
packets which are transmitted independently of one another. An error in
transmission of a packet results in the loss of the entire message. Messages
arrive to the finite buffer model and can be lost in two cases as
either at least one of its packets is corrupted or the buffer is overflowed.
With the parameters of the system typical for models of information
transmission in real networks, we obtain theorems on asymptotic behavior of the
number of lost messages. We also study how the loss probability changes if
redundant packets are added. Our asymptotic analysis approach is based on
Tauberian theorems with remainder.Comment: 18 pages, The list of references and citations slightly differ from
these appearing in the journa
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