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 $M/M/1$ queue whose server rate depends upon the state of an independent Ornstein-Uhlenbeck diffusion process $(X(t))$ so that its value at time $t$ is $\mu \phi(X(t))$, where $\phi(x)$ is some bounded function and $\mu>0$. We first establish the differential system for the conditional probability density functions of the couple $(L(t),X(t))$ in the stationary regime, where $L(t)$ is the number of customers in the system at time $t$. By assuming that $\phi(x)$ is defined by $\phi(x) = 1-\varepsilon ((x\wedge a/\varepsilon)\vee(-b/\varepsilon))$ for some positive real numbers $a$, $b$ and $\varepsilon$, we show that the above differential system has a unique solution under some condition on $a$ and $b$. We then show that this solution is close, in some appropriate sense, to the solution to the differential system obtained when $\phi$ is replaced with $\Phi(x)=1-\varepsilon x$ for sufficiently small $\varepsilon$. We finally perform a perturbation analysis of this latter solution for small $\varepsilon$. 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