Loss Queueing Systems with Limited Processor Sharing and Applications to Communication Networks

The paper considers a queueing system with limited processor sharing. No more than n jobs may be served simultaneously. This system may be used for modeling bandwidth sharing in wireless communication systems. If there are n jobs in the considered queueing system and a new job arrives, then the arriving job is lost or the service of a job is interrupted and this job is lost. We study two rules to choose the job to be lost. In accordance with one of these rules, the job with the shortest remaining length is lost. Relations are obtained between the state probabilities of considered system and the state probabilities of the corresponding unlimited processor sharing system. These relations allow to compute the state probabilities for considered system if the state probabilities for the unlimited processor sharing system are known. In the case of Poisson arrival process, the probability that the server capacity is exhausted is equal to the probability that a job is lost. We have obtained an explicit formulas for the stationary state probabilities and the loss probability for this case. The second considered rule is the First Come, First Displaced (FCFD) rule. If the service time is equal to a prescribed constant with probability 1, then the shortest remaining length job loss discipline is equivalent to the FCFD discipline. In the case of arbitrary length distribution, the loss probability for the shortest remaining length job loss discipline is greater than the loss probability for another discipline, and, in particular, the FCFD discipline. Hence we have an upper bound for the loss probability in the system with an arbitrary rule for choosing a job to be lost. We have found that, in the case of FCFD rule and constant service length, the stationary lost probability in a limited egalitarian processor sharing system can be not monotonically decreasing function of the maximum admissible number of jobs in the system for a fixed arrival rate. The considered queueing system may be used for modeling wireless communication network with channel capacity sharing.