Note on the GI/GI/1 queue with LCFS-PR observed at arbitrary times

Consider the GI/GI/1 queue with the Last-Come First-Served Preemptive-Resume service discipline. We give intuitive explanations for (i) the geometric nature of the stationary queue length distribution and (ii) the mutual independence of the residual service requirements of the customers in the queue, both considered at arbitrary time points. These distributions have previously been established in the literature by either first considering the system at arrival instants or using balance equations. Our direct arguments provide further understanding of (i) and (ii)

Heavy-traffic analysis of the M/PH/1 Discriminatory Processor Sharing queue with phase-dependent weights

We analyze a generalization of the Discriminatory Processor Sharing (DPS) queue in a heavy-traffic setting. Customers present in the system are served simultaneously at rates controlled by a vector of weights. We assume phase-type distributed service requirements and allow that customers have different weights in various phases of their service. We establish a state-space collapse for the queue length vector in heavy traffic. The result shows that in the limit, the queue length vector is the product of an exponentially distributed random variable and a deterministic vector. This generalizes a previous result by [12] who considered a DPS queue with exponentially distributed service requirements. We finally discuss some implications for residual service requirements and monotonicity properties in the ordinary DPS model

Scaling limits via excursion theory: Interplay between Crump-Mode-Jagers branching processes and processor-sharing queues

We study the convergence of the $M/G/1$ processor-sharing, queue length process in the heavy traffic regime, in the finite variance case. To do so, we combine results pertaining to L\'{e}vy processes, branching processes and queuing theory. These results yield the convergence of long excursions of the queue length processes, toward excursions obtained from those of some reflected Brownian motion with drift, after taking the image of their local time process by the Lamperti transformation. We also show, via excursion theoretic arguments, that this entails the convergence of the entire processes to some (other) reflected Brownian motion with drift. Along the way, we prove various invariance principles for homogeneous, binary Crump-Mode-Jagers processes. In the last section we discuss potential implications of the state space collapse property, well known in the queuing literature, to branching processes.Comment: Published in at http://dx.doi.org/10.1214/12-AAP904 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Heavy-traffic analysis of a multiple-phase network with discriminatory processor sharing

We analyze a generalization of the Discriminatory Processor Sharing (DPS) queue in a heavy-traffic setting. Customers present in the system are served simultaneously at rates controlled by a vector of weights. We assume that customers have phase-type distributed service requirements and allow that customers have different weights in various phases of their service. In our main result we establish a state-space collapse for the queue length vector in heavy traffic. The result shows that in the limit, the queue length vector is the product of an exponentially distributed random variable and a deterministic vector. This generalizes a previous result by Rege and Sengupta (1996) who considered a DPS queue with exponentially distributed service requirements. Their analysis was based on obtaining all moments of the queue length distributions by solving systems of linear equations. We undertake a more direct approach by showing that the probability generating function satisfies a partial differential equation that allows a closed-form solution after passing to the heavy-traffic limit. Making use of the state-space collapse result, we derive interesting properties in heavy traffic: (i) For the DPS queue we obtain that, conditioned on the number of customers in the system, the residual service requirements are asymptotically i.i.d. according to the forward recurrence times. (ii) We then investigate how the choice for the weights influences the asymptotic performance of the system. In particular, for the DPS queue we show that the scaled holding cost reduces as classes with a higher value for d_k/E(B_k^fwd) obtain a larger share of the capacity, where d_k is the cost associated to class k, and E(B_k^fwd) is the forward recurrence time of the class-k service requirement. The applicability of this result for a moderately loaded system is investigated by numerical experiments

Heavy-traffic analysis of a multiple-phase network with discriminatory processor sharing

We analyze a generalization of the Discriminatory Processor Sharing (DPS) queue in a heavy-traffic setting. Customers present in the system are served simultaneously at rates controlled by a vector of weights. We assume that customers have phase-type distributed service requirements and allow that customers have different weights in various phases of their service. In our main result we establish a state-space collapse for the queue length vector in heavy traffic. The result shows that in the limit, the queue length vector is the product of an exponentially distributed random variable and a deterministic vector. This generalizes a previous result by Rege and Sengupta (1996) who considered a DPS queue with exponentially distributed service requirements. Their analysis was based on obtaining all moments of the queue length distributions by solving systems of linear equations. We undertake a more direct approach by showing that the probability generating function satisfies a partial differential equation that allows a closed-form solution after passing to the heavy-traffic limit. Making use of the state-space collapse result, we derive interesting properties in heavy traffic: (i) For the DPS queue we obtain that, conditioned on the number of customers in the system, the residual service requirements are asymptotically i.i.d. according to the forward recurrence times. (ii) We then investigate how the choice for the weights influences the asymptotic performance of the system. In particular, for the DPS queue we show that the scaled holding cost reduces as classes with a higher value for d_k/E(B_k^fwd) obtain a larger share of the capacity, where d_k is the cost associated to class k, and E(B_k^fwd) is the forward recurrence time of the class-k service requirement. The applicability of this result for a moderately loaded system is investigated by numerical experiments