A storage model with self-similar input process is studied. A relation coupling together the storage requirement, the achievable utilization and the output rate is derived. A lower bound for the complementary distribution function of the storage level is given. Keywords: Self-similar, fractional Brownian motion, Local Area Network traffic 1 Introduction In a series of papers (e.g. Leland [8], Leland and Wilson [7], Fowler and Leland [4], Leland et al. [9]), researchers from Bellcore have reported and analyzed remarkable Local Area Network (LAN) traffic measurements challenging traditional data traffic modelling. The Bellcore data are both very accurate and extensive in time, and their most striking feature is the tremendous burstiness of LAN traffic at, practically, any timescale. More than that, the statistical analysis has shown that the traffic is self-similar with a surprising accuracy (see Leland et al. [9]). Traditional traffic models based on the Poisson process or, more gener...

It is now known that the usual traffic condition (the nominal load being less than one at each station) is not sufficient for stability for a multiclass open queueing network. Although there has been some progress in establishing the stability conditions for a multiclass network, there is no unified approach to this problem. In this paper, we prove that a queueing network is positive Harris recurrent if the corresponding fluid limit model eventually reaches zero and stays there regardless of the initial system configuration. As an application of the result, we prove that single class networks, multiclass feedforward networks and first-buffer-first-served preemptive resume discipline in a re-entrant line are positive Harris recurrent under the usual traffic condition. AMS 1991 subject classification: Primary 60K25, 90B22; Secondary 60K20, 90B35. Key words and phrases: multiclass queueing networks, Harris positive recurrent, stability, fluid approximation Running title: Stability of mu...

Abstract: We study the implications of imperfect information for term structures of credit spreads on corporate bonds. We suppose that bond investors cannot observe the issuer’s assets directly, and receive instead only periodic and imperfect accounting reports. For a setting in which the assets of the firm are a geometric Brownian motion until informed equityholders optimally liquidate, we derive the conditional distribution of the assets, given accounting data and survivorship. Contrary to the perfect-information case, there exists a default-arrival intensity process. That intensity is calculated in terms of the conditional distribution of assets. Credit yield spreads are characterized in terms of accounting information. Generalizations are provided. 1 We are exceptionally grateful to Michael Harrison for his significant contributions to this paper, which are noted within. We are also grateful for insightful research assistance

Abstract. We study a model of controlled queueing network, which operates and makes control decisions in discrete time. An underlying random network mode determines the set of available controls in each time slot. Each control decision “produces ” a certain vector of “commodities”; it also has associated “traditional” queueing control effect, i.e., it determines traffic (customer) arrival rates, service rates at the nodes, and random routing of processed customers among the nodes. The problem is to find a dynamic control strategy which maximizes a concave utility function H(X), where X is the average value of commodity vector, subject to the constraint that network queues remain stable. We introduce a dynamic control algorithm, which we call Greedy Primal-Dual (GPD) algorithm, and prove its asymptotic optimality. We show that our network model and GPD algorithm accommodate a wide range of applications. As one example, we consider the problem of congestion control of networks where both traffic sources and network processing nodes may be randomly time-varying and interdependent. We also discuss a variety of resource allocation problems in wireless networks, which in particular involve average power consumption constraints and/or optimization, as well as traffic rate constraints.

This paper uses a variety of mathematical models to explore some of the consequences of rapidly growing communications capacity for the evolution of the Internet. It argues that queueing delays may become small in comparison with propagation delays, and that differentiation between traffic classes within the network may become redundant. Instead, a simple packet network may be able to support an arbitrarily differentiated and constantly evolving set of services, by conveying information on incipient congestion to intelligent end-nodes which themselves determine what should be their demands on the packet network.

Most capital structure models assume that the decision of how much debt to issue is a static choice. In practice, however, firms adjust outstanding debt levels in response to changes in firm value. In this article, we

This paper concerns a dynamic scheduling problem for a queueing system that has two streams of arrivals to infinite capacity buffers and two (non-identical) servers working in parallel. One server can only process jobs from one buffer, whereas the other server can process jobs from either buffer. The service time distribution may depend on the buffer being served and the server providing the service. The system manager dynamically schedules waiting jobs onto available servers. We consider a parameter regime in which the system satisfies both a heavy traffic condition and a resource pooling condition. Our cost function is a mean cumulative discounted cost of holding jobs in the system, where the (undiscounted) cost per unit time is a linear function of normalized (with heavy traffic scaling) queue length. We first review the analytic solution of the Brownian control problem (formal heavy traffic approximation) for this system. We "interpret" this solution by proposing a threshold contro...

: We show that several well-known optimization problems related to the optimal control of queues are provably intractable ---independently of any unproven conjecture such as P6=NP. In particular, we show that several versions of the problem of optimally controlling a simple network of queues with simple arrival and service distributions and multiple customer classes is complete for exponential time. This is perhaps the first such intractability result for a well-known optimization problem. We also show that the restless bandit problem (the generalization of the multi-armed bandit problem to the case in which the unselected processes are not quiescent) is complete for polynomial space. 1. INTRODUCTION The optimal control of a network of queues is a well-known, much studied, and notoriously difficult problem. We are given several servers, a set of customer classes, and class-dependent probability distributions for the service times. For each customer class, there is only one server tha...

. Let n particles move in standard Brownian motion in one dimension, with the process terminating if two particles collide. This is a specific case of Brownian motion constrained to stay inside a Weyl chamber; the Weyl group for this chamber is An\Gamma1 , the symmetric group. For any starting positions, we compute a determinant formula for the density function for the particles to be at specified positions at time t without having collided by time t. We show that the probability that there will be no collision up to time t is asymptotic to a constant multiple of t \Gamman(n\Gamma1)=4 as t goes to infinity, and compute the constant as a polynomial of the starting positions. We have analogous results for the other classical Weyl groups; for example, the hyperoctahedral group Bn gives a model of n independent particles with a wall at x = 0. We can define Brownian motion on a Lie algebra, viewing it as a vector space; the eigenvalues of a point in the Lie algebra correspond to a point ...

As is well known, the classic Black-Scholes option pricing model assumes that returns follow Brownian motion. It is widely recognized that return processes differ from this benchmark in at least three important ways. First, asset prices jump, leading to non-normal return innovations. Second, return volatilities vary stochastically over time. Third, returns and their volatilities are correlated, often negatively for equities. We propose that time-changed Lévy processes be used to simultaneously address these three facets of the underlying asset return process. We show that our framework encompasses almost all of the models proposed in the option pricing literature. Despite the generality of our approach, we show that it is straightforward to select and test a particular option pricing model through the use of characteristic function technology.