311 research outputs found
Analysis of a trunk reservation policy in the framework of fog computing
We analyze in this paper a system composed of two data centers with limited
capacity in terms of servers. When one request for a single server is blocked
at the first data center, this request is forwarded to the second one. To
protect the single server requests originally assigned to the second data
center, a trunk reservation policy is introduced (i.e., a redirected request is
accepted only if there is a sufficient number of free servers at the second
data center). After rescaling the system by assuming that there are many
servers in both data centers and high request arrival rates, we are led to
analyze a random walk in the quarter plane, which has the particularity of
having non constant reflecting conditions on one boundary of the quarter plane.
Contrary to usual reflected random walks, to compute the stationary
distribution of the presented random walk, we have to determine three unknown
functions, one polynomial and two infinite generating functions. We show that
the coefficients of the polynomial are solutions to a linear system. After
solving this linear system, we are able to compute the two other unknown
functions and the blocking probabilities at both data centers. Numerical
experiments are eventually performed to estimate the gain achieved by the trunk
reservation policy
Perturbation analysis of an M/M/1 queue in a diffusion random environment
We study in this paper an queue whose server rate depends upon the
state of an independent Ornstein-Uhlenbeck diffusion process so that
its value at time is , where is some bounded
function and . We first establish the differential system for the
conditional probability density functions of the couple in the
stationary regime, where is the number of customers in the system at
time . By assuming that is defined by for some positive real numbers
, and , we show that the above differential system has a
unique solution under some condition on and . We then show that this
solution is close, in some appropriate sense, to the solution to the
differential system obtained when is replaced with
for sufficiently small . We finally
perform a perturbation analysis of this latter solution for small
. This allows us to check at the first order the validity of the
so-called reduced service rate approximation, stating that everything happens
as if the server rate were constant and equal to \mu(1-\eps\E(X(t)))
Perturbation Analysis of a Variable M/M/1 Queue: A Probabilistic Approach
Motivated by the problem of the coexistence on transmission links of
telecommunication networks of elastic and unresponsive traffic, we study in
this paper the impact on the busy period of an M/M/1 queue of a small
perturbation in the server rate. The perturbation depends upon an independent
stationary process (X(t)) and is quantified by means of a parameter \eps \ll 1.
We specifically compute the two first terms of the power series expansion in
\eps of the mean value of the busy period duration. This allows us to study the
validity of the Reduced Service Rate (RSR) approximation, which consists in
comparing the perturbed M/M/1 queue with the M/M/1 queue where the service rate
is constant and equal to the mean value of the perturbation. For the first term
of the expansion, the two systems are equivalent. For the second term, the
situation is more complex and it is shown that the correlations of the
environment process (X(t)) play a key role
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