Time-shared Systems: a theoretical treatment
Journal of the ACM (JACM)
Sharing a Processor Among Many Job Classes
Journal of the ACM (JACM)
Operating Systems Theory
Asymptotic regimes and approximations for discriminatory processor sharing
ACM SIGMETRICS Performance Evaluation Review
Sojourn time asymptotics in processor-sharing queues
Queueing Systems: Theory and Applications
A survey on discriminatory processor sharing
Queueing Systems: Theory and Applications
ACM SIGMETRICS Performance Evaluation Review
Bandwidth-sharing networks in overload
Performance Evaluation
Monotonicity Properties for Multi-Class Queueing Systems
Discrete Event Dynamic Systems
A note on comparing response times in the M/GI/1/FB and M/GI/1/PS queues
Operations Research Letters
Heavy traffic analysis of the discriminatory randomorderofservice discipline
ACM SIGMETRICS Performance Evaluation Review - Special Issue on IFIP PERFORMANCE 2011- 29th International Symposium on Computer Performance, Modeling, Measurement and Evaluation
Heavy traffic approximation of equilibria in resource sharing games
WINE'11 Proceedings of the 7th international conference on Internet and Network Economics
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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 [Rege, K. M., B. Sengupta. 1996. Queue length distribution for the discriminatory processor-sharing queue. Oper. Res.44(4) 653--657], 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 independent and distributed 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 dk/E(Bkfwd) obtain a larger share of the capacity, where dk is the cost associated to class k, and E(Bkfwd) 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.