Heavy-Traffic Analysis of a Multiple-Phase Network with Discriminatory Processor Sharing

Abstract

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.

Markov-modulated M/G/1-type queue in heavy traffic and its application to time-sharing disciplines

“…The first term of the numerator is in accordance with the results of [30] and [33], the second term is a result of the random environment. The results in Sections 5 and 6.1 are based on the assumption that lim N →∞ 1 N M · 1 {Z=d} exists.…”

“…Proof. We follow closely the steps of the proof of Lemma 3 in [33]. The proof has 3 steps: (i) Show that F k,∞ (s) is parallel to the hyperplane.…”

“…S T is a generator corresponding to a finite-state Markov chain. From the proof of Lemma 4 in [33], it is easily seen that the Markov chain corresponding to S T is irreducible (since we assume that all λ k,d > 0 for all k and at least one d). Retracing the arguments stated there, for completeness, it follows that this Markov chain has a unique equilibrium distribution (column) vector, η, such that η T S T = 0.…”

Heavy-traffic analysis of the M/PH/1 discriminatory processor sharing queue with phase-dependent weights

“…In particular, Proposition 2.1 gives a proof of the state space collapse stated in [10,Theorem 5]. It also implies several other interesting new results for the standard DPS queue, which we briefly mention here, referring to [15,Chapter 2] and [16] for full details. For example, we can show that the (scaled) numbers of customers in the various classes and the remaining service requirements of any finite subset of customers are independent in a heavy-traffic setting.…”

“…Equation (2) turns out to be very useful to analyze the joint queue length distribution in heavy traffic, because it allows for an explicit solution in that asymptotic regime as we will indicate next. We refer to [15,Chapter 2] and [16] for full details. We writes = (s 1 , .…”