On the pathwise optimal Bernoulli routing policy for homogeneous parallel servers

A long-standing conjecture on the optimal Bernoulli routing policy is proven to be true. For the case of equal exponential service times it is shown that splitting equally among the queues minimizes the departure times in a stochastic pathwise sense. A new technique is used, showing that certain distributional properties related to Schur convexity propagate forward in time

Routing in multi-class queueing networks

PhD ThesisWe consider the problem of routing (incorporating local scheduling) in a distributed network. Dedicated jobs arrive directly at their specified station for processing. The choice of station for generic jobs is open. Each job class has an associated holding cost rate. We aim to develop routing policies to minimise the long-run average holding cost rate. We first consider the class of static policies. Dacre, Glazebrook and Nifio-Mora (1999) developed an approach to the formulation of static routing policies, in which the work at each station is scheduled optimally, using the achievable region approach. The achievable region approach attempts to solve stochastic optimisation problems by characterising the space of all possible performances and optimising the performance objective over this space. Optimal local scheduling takes the form of a priority policy. Such static routing policies distribute the generic traffic to the stations via a simple Bernoulli routing mechanism. We provide an overview of the achievements made in following this approach to static routing. In the course of this discussion we expand upon the study of Becker et al. (2000) in which they considered routing to a collection of stations specialised in processing certain job classes and we consider how the composition of the available stations affects the system performance for this particular problem. We conclude our examination of static routing policies with an investigation into a network design problem in which the number of stations available for processing remains to be determined. The second class of policies of interest is the class of dynamic policies. General DP theory asserts the existence of a deterministic, stationary and Markov optimal dynamic policy. However, a full DP solution may be unobtainable and theoretical difficulties posed by simple routing problems suggest that a closed form optimal policy may not be available. This motivates a requirement for good heuristic policies. We consider two approaches to the development of dynamic routing heuristics. We develop an idea proposed, in the context of simple single class systems, by Krishnan (1987) by applying a single policy improvement step to some given static policy. The resulting dynamic policy is shown to be of simple structure and easily computable. We include an investigation into the comparative performance of the dynamic policy with a number of competitor policies and of the performance of the heuristic as the number of stations in the network changes. In our second approach the generic traffic may only access processing when the station has been cleared of all (higher priority) jobs and can be considered as background work. We deploy a prescription of Whittle (1988) developed for RBPs to develop a suitable approach to station indexation. Taking an approximative approach to Whittle's proposal results in a very simple form of index policy for routing the generic traffic. We investigate the closeness to optimality of the index policy and compare the performance of both of the dynamic routing policies developed here

Topics in queueing theory

There are three topics in the thesis. In the first topic, we addressed a control problem for a queueing system, known as the ``$N$-system\u27\u27, under the Halfin-Whitt heavy traffic regime and a static priority policy was proposed and is shown to be asymptotically optimal, using weak convergence techniques. In the second topic, we focused on the hospitals, where faster servers(nurses), though work more efficiently, have the heavier workload, and the Randomized Most-Idle (RMI) routing policy was proposed to tackle this unfairness issue, trying to reward faster servers who serve more with less workload. we extended the existing result to show that this desirable property of the RMI policy holds under a system with multiple customer classes using theoretical exact analysis as well as numerical simulations. In the third topic, the problem was to decide an appropriate number of representatives over time according to the prescribed service quality level in the call center. We examined the stability of two methods which were designed to generate appropriate staffing functions on a simulated data and real call center data from an actual bank

Dynamic power allocation and routing for satellite and wireless networks with time varying channels

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, February 2004.Includes bibliographical references (p. 283-295).This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Satellite and wireless networks operate over time varying channels that depend on attenuation conditions, power allocation decisions, and inter-channel interference. In order to reliably integrate these systems into a high speed data network and meet the increasing demand for high throughput and low delay, it is necessary to develop efficient network layer strategies that fully utilize the physical layer capabilities of each network element. In this thesis, we develop the notion of network layer capacity and describe capacity achieving power allocation and routing algorithms for general networks with wireless links and adaptive transmission rates. Fundamental issues of delay, throughput optimality, fairness, implementation complexity, and robustness to time varying channel conditions and changing user demands are discussed. Analysis is performed at the packet level and fully considers the queueing dynamics in systems with arbitrary, potentially bursty, arrival processes. Applications of this research are examined for the specific cases of satellite networks and ad-hoc wireless networks. Indeed, in Chapter 3 we consider a multi-beam satellite downlink and develop a dynamic power allocation algorithm that allocates power to each link in reaction to queue backlog and current channel conditions. The algorithm operates without knowledge of the arriving traffic or channel statistics, and is shown to achieve maximum throughput while maintaining average delay guarantees. At the end of Chapter 4, a crosslinked collection of such satellites is considered and a satellite separation principle is developed, demonstrating that joint optimal control can be implemented with separate algorithms for the downlinks and crosslinks.(cont.) Ad-hoc wireless networks are given special attention in Chapter 6. A simple cell- partitioned model for a mobile ad-hoc network with N users is constructed, and exact expressions for capacity and delay are derived. End-to-end delay is shown to be O(N), and hence grows large as the size of the network is increased. To reduce delay, a transmission protocol which sends redundant packet information over multiple paths is developed and shown to provide O(vN) delay at the cost of reducing throughput. A fundamental rate- delay tradeoff curve is established, and the given protocols for achieving O(N) and O(vN) delay are shown to operate on distinct boundary points of this curve. In Chapters 4 and 5 we consider optimal control for a general time-varying network. A cross-layer strategy is developed that stabilizes the network whenever possible, and makes fair decisions about which data to serve when inputs exceed capacity. The strategy is decoupled into separate algorithms for dynamic flow control, power allocation, and routing, and allows for each user to make greedy decisions independent of the actions of others. The combined strategy is shown to yield data rates that are arbitrarily close to the optimally fair operating point that is achieved when all network controllers are coordinated and have perfect knowledge of future events. The cost of approaching this fair operating point is an end-to-end delay increase for data that is served by the network.by Michael J. Neely.Ph.D

Stability and asymptotic optimality of generalized maxweight policies

Abstract It is shown that stability of the celebrated MaxWeight or back pressure policies is a consequence of the following interpretation: either policy is myopic with respect to a surrogate value function of a very special form, in which the "marginal disutility" at a buffer vanishes for vanishingly small buffer population. This observation motivates the h-MaxWeight policy, defined for a wide class of functions h. These policies share many of the attractive properties of the MaxWeight policy: (i) Arrival rate data is not required in the policy. (ii) Under a variety of general conditions, the policy is stabilizing when h is a perturbation of a monotone linear function, a monotone quadratic, or a monotone Lyapunov function for the fluid model. (iii) A perturbation of the relative value function for a workload relaxation gives rise to a myopic policy that is approximately average-cost optimal in heavy traffic, with logarithmic regret. The first results are obtained for a general Markovian network model. Asymptotic optimality is established for a general Markovian scheduling model with a single bottleneck, and homogeneous servers

Routing of airplanes to two runways: monotonicity of optimal controls

We consider the problem of routing incoming airplanes to two runways of an airport. Due to air turbulence, the necessary separation time between two successive landing operations depends on the types of the airplanes. When viewed as a queueing problem, this means that we have dependent service times. The aim is to minimise waiting times of aircrafts. We consider here a model where arrivals form a stochastic process and where the decision maker does not know anything about future arrivals. We formulate this as a problem of stochastic dynamic programming and investigate monotonicity of optimal routing strategies with respect e.g. to the workload of the runways. We show that an optimal strategy is monotone (i.e. of switching type) only in a restricted case where decisions depend on the state of the runways only and not on the type of the arriving aircraft. Surprisingly, in the more realistic case where this type is also known to the decision maker, monotonicity need not hold