Decomposing the queue length distribution of processor-sharing models into queue lengths of permanent customer queues

We obtain a decomposition result for the steady state queue length distribution in egalitarian processor-sharing (PS) models. In particular, for an egalitarian PS queue with $K$ customer classes, we show that the marginal queue length distribution for class $k$ factorizes over the number of other customer types. The factorizing coefficients equal the queue length probabilities of a PS queue for type $k$ in isolation, in which the customers of the other types reside \textit{ permanently} in the system. Similarly, the (conditional) mean sojourn time for class $k$ can be obtained by conditioning on the number of permanent customers of the other types. The decomposition result implies linear relations between the marginal queue length probabilities, which also hold for other PS models such as the egalitarian processor-sharing models with state-dependent system capacity that only depends on the total number of customers in the system. Based on the exact decomposition result for egalitarian PS queues, we propose a similar decomposition for discriminatory processor-sharing (DPS) models, and numerically show that the approximation is accurate for moderate differences in service weights. \u

Elastic Multi-resource Network Slicing: Can Protection Lead to Improved Performance?

In order to meet the performance/privacy requirements of future data-intensive mobile applications, e.g., self-driving cars, mobile data analytics, and AR/VR, service providers are expected to draw on shared storage/computation/connectivity resources at the network "edge". To be cost-effective, a key functional requirement for such infrastructure is enabling the sharing of heterogeneous resources amongst tenants/service providers supporting spatially varying and dynamic user demands. This paper proposes a resource allocation criterion, namely, Share Constrained Slicing (SCS), for slices allocated predefined shares of the network's resources, which extends the traditional alpha-fairness criterion, by striking a balance among inter- and intra-slice fairness vs. overall efficiency. We show that SCS has several desirable properties including slice-level protection, envyfreeness, and load driven elasticity. In practice, mobile users' dynamics could make the cost of implementing SCS high, so we discuss the feasibility of using a simpler (dynamically) weighted max-min as a surrogate resource allocation scheme. For a setting with stochastic loads and elastic user requirements, we establish a sufficient condition for the stability of the associated coupled network system. Finally, and perhaps surprisingly, we show via extensive simulations that while SCS (and/or the surrogate weighted max-min allocation) provides inter-slice protection, they can achieve improved job delay and/or perceived throughput, as compared to other weighted max-min based allocation schemes whose intra-slice weight allocation is not share-constrained, e.g., traditional max-min or discriminatory processor sharing

Heavy-traffic limits for Discriminatory Processor Sharing models with joint batch arrivals

We study the performance of Discriminatory Processor Sharing (DPS) systems, with exponential service times and in which batches of customers of different types may arrive simultaneously according to a Poisson process. We show that the stationary joint queue-length distribution exhibits state-space collapse in heavy traffic: as the load ρ tends to 1, the scaled joint queue-length vector (1−ρ)Q converges in distribution to the product of a determin

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

Higher response time moments for M/M/1 discriminatory processor sharing queues

© Copyright 2016 ICST.Obtaining response time moments in processor sharing (PS) queues is difficult due to serving of multiple jobs. Egaliatarian PS (EPS) queues are limited to one class of arriving jobs. Discriminatory PS (DPS) assigns weights to different job classes and offers more diverse modeling capabilities than EPS. It is known that response time is the representative metric for delay as specified in service level agreements (SLAs), which consider higher moments important. Hence, we build an automated numerical algorithm for calculating higher moments of response time in M/M/1-DPS queues for multiple job classes and test two different case studies