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 [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(B fwd k) obtain a larger share of the capacity, where dk is the cost associated to class k, and E(B fwd k) 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

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

Qualitative properties of α\alpha-fair policies in bandwidth-sharing networks

We consider a flow-level model of a network operating under an $\alpha$-fair bandwidth sharing policy (with $\alpha>0$) proposed by Roberts and Massouli\'{e} [Telecomunication Systems 15 (2000) 185-201]. This is a probabilistic model that captures the long-term aspects of bandwidth sharing between users or flows in a communication network. We study the transient properties as well as the steady-state distribution of the model. In particular, for $\alpha\geq1$, we obtain bounds on the maximum number of flows in the network over a given time horizon, by means of a maximal inequality derived from the standard Lyapunov drift condition. As a corollary, we establish the full state space collapse property for all $\alpha\geq1$. For the steady-state distribution, we obtain explicit exponential tail bounds on the number of flows, for any $\alpha>0$, by relying on a norm-like Lyapunov function. As a corollary, we establish the validity of the diffusion approximation developed by Kang et al. [Ann. Appl. Probab. 19 (2009) 1719-1780], in steady state, for the case where $\alpha=1$ and under a local traffic condition.Comment: Published in at http://dx.doi.org/10.1214/12-AAP915 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

A skill based parallel service system under FCFS-ALIS : steady state, overloads and abandonments

We consider a queueing system with servers S={m1,...,mJ}, and with customer types C={a,b,...}. A bipartite graph G describes which pairs of server-customer types are compatible. We consider FCFS-ALIS policy: A server always picks the first, longest waiting compatible customer, and a customer is always assigned to the longest idle compatible server. We assume Poisson arrivals and server dependent exponential service times. We derive an explicit product-form expression for the stationary distribution of this system when service capacity is sufficient. We also calculate fluid limits of the system under overload, to show that local steady state exists. We distinguish the case of complete resource pooling when all the customers are served at the same rate by the pooled servers, and the case when the system has a unique decomposition into subsets of customer types, each of which is served at its own rate by a pooled subset of the servers. Finally, we discuss possible behavior of the system with generally distributed abandonments, under many server scaling. This paper complements and extends previous results of Kaplan, Caldentey and Weiss [18], and of Whitt and Talreja [34], as well as previous results of the authors [4, 35] on this topic. Keywords: Service systems, multi type customers, multi type skill based servers, matching of infinite sequences, product form solution, first come first served policy, assign longest idle server policy, complete resource pooling, local steady state, overloaded queues, abandonment

On the effective bandwidth for resource management in ATM networks

Ankara : Department of Electrical and Electronics Engineering and the Institute of Engineering and Sciences of Bilkent University, 1997.Thesis (Master's) -- Bilkent University, 1997.Includes bibliographical references leaves 81-84.Chahed, TijaniM.S

Collaborating queues: large service network and a limit order book

E-thesis pagination differs from hardbound copy kept in the Manuscripts Department, Cambridge University Library.We analyse the steady-state behaviour of two different models with collaborating queues: that is, models in which "customers" can be served by many types of "servers", and "servers" can process many types of "customers". The first example is a large-scale service system, such as a call centre. Collaboration is the result of cross-trained staff attending to several different types of incoming calls. We first examine a load-balancing policy, which aims to keep servers in different pools equally busy. Although the policy behaves order-optimally over fixed time horizons, we show that the steady-state distribution may fail to be tight on the diffusion scale. That is, in a family of ever-larger networks whose arrival rates grow as O(r) (where r is a scaling parameter growing to infinity), the sequence of steady-state deviations from equilibrium scaled down by sqrt(r) is not tight. We then propose a different policy, for which we show that the sequence of invariant distributions is tight on the r^(1/2+epsilon) scale, for any epsilon > 0. For this policy we conjecture that tightness holds on the diffusion scale as well. The second example models a limit order book, a pricing mechanism for a single-commodity market in which buyers (respectively sellers) are prepared to wait for the price to drop (respectively rise). We analyse the behaviour of a simplified model, in which the arrival events are independent of each other and the state of the limit order book. The system can be represented by a queueing model, with "customers" and "servers" corresponding to bids and asks; the roles of customers and servers are symmetric. We show that, with probability 1, the price interval breaks up into three regions. At small (respectively large) prices, only finitely many bid (respectively ask) orders ever get fulfilled, while in the middle region all orders eventually clear. We derive equations which define the boundaries between these regions, and solve them explicitly in the case of iid uniform arrivals to obtain numeric values of the thresholds. We derive a heuristic for the distribution of the highest bid (respectively lowest ask), and present simulation data confirming it.This work was supported by the US National Science Foundation Graduate Research Fellowship

Control of multiclass queueing systems with abandonments and adversarial customers

This thesis considers the defensive surveillance of multiple public areas which are the open, exposed targets of adversarial attacks. We address the operational problem of identifying a real time decision-making rule for a security team in order to minimise the damage an adversary can inflict within the public areas. We model the surveillance scenario as a multiclass queueing system with customer abandonments, wherein the operational problem translates into developing service policies for a server in order to minimise the expected damage an adversarial customer can inflict on the system. We consider three different surveillance scenarios which may occur in realworld security operations. In each scenario it is only possible to calculate optimal policies in small systems or in special cases, hence we focus on developing heuristic policies which can be computed and demonstrate their effectiveness in numerical experiments. In the random adversary scenario, the adversary attacks the system according to a probability distribution known to the server. This problem is a special case of a more general stochastic scheduling problem. We develop new results which complement the existing literature based on priority policies and an effective approximate policy improvement algorithm. We also consider the scenario of a strategic adversary who chooses where to attack. We model the interaction of the server and adversary as a two-person zero-sum game. We develop an effective heuristic based on an iterative algorithm which populates a small set of service policies to be randomised over. Finally, we consider the scenario of a strategic adversary who chooses both where and when to attack and formulate it as a robust optimisation problem. In this case, we demonstrate the optimality of the last-come first-served policy in single queue systems. In systems with multiple queues, we develop effective heuristic policies based on the last-come first-served policy which incorporates randomisation both within service policies and across service policies