### Optimal and Threshold Approximation of the Hysteretic GI/M/1 Queue

Session : SS4-2 / SS4 : Modélisation Markovienne : méthodes et outils
Jeudi 11 février 15:00 - 16:40 Salle : RP10
Aicha Bareche, Mouloud Cherfaoui et Djamil Aïssani

In this work, the modeling of the $GI/M/1$ system with two-stage service policy by a Markov chain with two states $E_1$ and $E_2$ ($E_1$: the system is governed by a service rate $mu_1$ and $E_2$: the system is governed by a service rate $mu_2$) to determine the probability $theta$ ($theta$: the probability that the system is in the state $E_1$, $1 - theta$ is the probability that the system is in the state $E_2$), allowed us to calculate the global transition operator of the embedded Markov chain associated to the $GI/M/1$ system with two-stage service policy. By exploiting the global transition operator and using the strong stability method, we have determined the approximation conditions of the stationary characteristics of the $GI/M/1$ system with two-stage service policy by those of the standard $GI/M/1$ system when it is governed by one of the policies of the previous system and when it is governed by an optimal policy and then we have given the deviation between the stationary distributions for each case. Finally, the numerical applications carried out allowed us to conclude that: When $lambda / mu_1$ or $lambda / mu_2$ is large enough (respectively small enough), the $GI/M/1$ system with two-stage service policy tends to behave in the same manner as the standard $GI/M/1$ system having a service rate $mu_2$ (respectively $mu_1$). When $N$ is small enough (respectively large enough), the $GI/M/1$ system with two-stage service policy tends to behave in the same manner as the standard $GI/M/1$ system having a service rate $mu_2$ (respectively $mu_1$). The approximation of the stationary characteristics of the $GI/M/1$ system with two-stage service policy by those of the $GI/M/1$ system with a threshold policy (having a service rate $mu_1$ or $mu_2$) has not always a sense. Within the meaning of strong stability method, we always can define a $GI/M/1$ system with one-stage service policy, having a service rate $mu^*$ such that $mu_1leq mu^* leq mu_2$ and its stationary characteristics are best approximations of those of the $GI/M/1$ system with two-stage service policy, having service rate $(mu_1,mu_2)$.

Mots clés : hysteretic queue, Markov chain, strong stability, constrained nonlinear optimization