396 research outputs found
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
On transient queue-size distribution in the batch arrival system with the N-policy and setup times
In the paper the queueing system with the -policy and setup times is considered. An explicit formula for the Laplace
transform of the transient queue-size distribution is derived using
the approach consisting of few steps. Firstly, a "special\u27\u27
modification of the original system is investigated and, using the
formula of total probability, the analysis is reduced to the case
of the corresponding system without limitation in the service. Next,
a renewal process generated by successive busy cycles is used to
obtain the general result. Sample numerical computations
illustrating theoretical results are attached as well
USING SINGULARITY ANALYSIS TO APPROXIMATE TRANSIENT CHARACTERISTICS IN QUEUEING SYSTEMS
In this article, we develop a simple method to approximate the transient behavior of queueing systems. In particular, it is shown how singularity analysis of a known generating function of a transient sequence of some performance measure leads to an approximation of this sequence. To illustrate our approach, several specific transient sequences are investigated in detail. By means of some numerical examples, we validate our approximations and demonstrate the usefulness of the technique
On transient queue-size distribution in the batch arrival system with the N-policy and setup times
In the paper the queueing system with the -policy and setup times is considered. An explicit formula for the Laplace
transform of the transient queue-size distribution is derived using
the approach consisting of few steps. Firstly, a "special\u27\u27
modification of the original system is investigated and, using the
formula of total probability, the analysis is reduced to the case
of the corresponding system without limitation in the service. Next,
a renewal process generated by successive busy cycles is used to
obtain the general result. Sample numerical computations
illustrating theoretical results are attached as well
Charting the landscape of stochastic gene expression models using queueing theory
Stochastic models of gene expression are typically formulated using the
chemical master equation, which can be solved exactly or approximately using a
repertoire of analytical methods. Here, we provide a tutorial review of an
alternative approach based on queuing theory that has rarely been used in the
literature of gene expression. We discuss the interpretation of six types of
infinite server queues from the angle of stochastic single-cell biology and
provide analytical expressions for the stationary and non-stationary
distributions and/or moments of mRNA/protein numbers, and bounds on the Fano
factor. This approach may enable the solution of complex models which have
hitherto evaded analytical solution.Comment: 24 pages, 6 figure
Rare-event analysis of mixed Poisson random variables, and applications in staffing
A common assumption when modeling queuing systems is that arrivals behave
like a Poisson process with constant parameter. In practice, however, call
arrivals are often observed to be significantly overdispersed. This motivates
that in this paper we consider a mixed Poisson arrival process with arrival
rates that are resampled every time units, where and a
scaling parameter. In the first part of the paper we analyse the asymptotic
tail distribution of this doubly stochastic arrival process. That is, for large
and i.i.d. arrival rates , we focus on the evaluation of
, the probability that the scaled number of arrivals exceeds .
Relying on elementary techniques, we derive the exact asymptotics of :
For we identify (in closed-form) a function
such that tends to as .
For and we find a partial
solution in terms of an asymptotic lower bound. For the special case that the
s are gamma distributed, we establish the exact asymptotics across all . In addition, we set up an asymptotically efficient importance sampling
procedure that produces reliable estimates at low computational cost. The
second part of the paper considers an infinite-server queue assumed to be fed
by such a mixed Poisson arrival process. Applying a scaling similar to the one
in the definition of , we focus on the asymptotics of the probability
that the number of clients in the system exceeds . The resulting
approximations can be useful in the context of staffing. Our numerical
experiments show that, astoundingly, the required staffing level can actually
decrease when service times are more variable
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