We consider a model of random access (slotted-aloha-type) communication networks of general topology. Assuming that network links receive exogenous arrivals of packets for transmission, we seek dynamic distributed random access strategies whose goal is to keep all network queues stable. We prove that two dynamic strategies, which we collectively call queue length based random access (QRA), ensure stability as long as the rates of exogenous arrival flows are within the network saturation rate region. The first strategy, QRA-I, can be viewed as a random-access-model counterpart of the max-weight scheduling rule, while the second strategy, QRA-II, is a counterpart of the exponential (EXP) rule. The two strategies induce different dynamics of the queues in the fluid scaling limit, which can be exploited for the quality-of-service control in applications.

Pharmaceutical companies have to face huge risks and enormous costs of production before they can produce a drug. Efficient allocation of resources is essential to help in maximizing profits. Yu and Gittins (2007) described a model and associated software for determining efficient allocations for a preclinical research project. This is the starting point for this paper. We provide explicit optimal policies for the selection of successive candidate drugs for two restricted versions of the Yu and Gittins (2007) model. To some extent these policies are likely to be applicable to the unrestricted model.

We give a new method for simulating the time average steady-state distribution of a continuous-time queueing system, by extending a ‘read-once’ or ‘forward’ version of the coupling from the past (CFTP) algorithm developed for discrete-time Markov chains. We then use this to give a new proof of the ‘Poisson arrivals see time averages’ (PASTA) property, and a new proof for why renewal arrivals see either stochastically smaller or larger congestion than the time average if interarrival times are respectively new better than used in expectation (NBUE) or new worse than used in expectation (NWUE).

Hu et al. [“A boundary problem for group testing”, SIAM J. Algebraic Discrete Meth.2 (1981), 81–87] conjectured that the minimax test number to find d defectives in 3d items is 3d−1, a surprisingly difficult combinatorial problem about which very little is known. In this article we state three more conjectures and prove that they are all equivalent to the conjecture of Hu et al. Notably, as a byproduct, we also obtain an interesting upper bound for M(d,n).

The TCP window size process can be modeled as a piecewise-deterministic Markov process that increases linearly and experiences downward jumps at Poisson times. We present a transient analysis of this window size process. Our main result is the Laplace transform of the transient moments. Formulae for the integer and fractional moments are derived, as well as an explicit characterization of the speed of convergence to steady state. Central to our approach are the infinitesimal generator and Dynkin's martingale.

In designing a network to link n points in a square of area n, we might be guided by the following two desiderata. First, the total network length should not be much greater than the length of the shortest network connecting all points. Second, the average route length (taken over source-destination pairs) should not be much greater than the average straight-line distance. How small can we make these two excesses? Speaking loosely, for a nondegenerate configuration, the total network length must be at least of order n and the average straight-line distance must be at least of order n 1/2, so it seems implausible that a single network might exist in which the excess over the first minimum is o(n) and the excess over the second minimum is o(n 1/2). But in fact we can do better: for an arbitrary configuration, we can construct a network where the first excess is o(n) and the second excess is almost as small as O(log n). The construction is conceptually simple and uses stochastic methods: over the minimum-length connected network (Steiner tree) superimpose a sparse stationary and isotropic Poisson line process. Together with a few additions (required for technical reasons), the mean values of the excess for the resulting random network satisfy the above asymptotics; hence, a standard application of the probabilistic method guarantees the existence of deterministic networks as required (speaking constructively, such networks can be constructed using simple rejection sampling). The key ingredient is a new result about the Poisson line process. Consider two points a distance r apart, and delete from the line process all lines which separate these two points. The resulting pattern of lines partitions the plane into cells; the cell containing the two points has mean boundary length approximately equal to 2r + constant(log r). Turning to lower bounds, consider a sequence of networks in satisfying a weak equidistribution assumption. We show that if the first excess is O(n) then the second excess cannot be

Consider an infinite-source marked Poisson process to model end user inputs to a data network. At Poisson times, connections are initated. The connection is characterized by a triple (F, L, R) denoting the total quantity of transmitted data in a connection, the length or duration of the connection, and the transmission rate; the three quantities are related by F = LR. How critical is the dependence structure of the mark for network characteristics such as burstiness, distribution tails of cumulative input, and long-range dependence properties of traffic measured in consecutive time slots? In a previous publication (D'Auria and Resnick (2006)) we assumed that F and R were independent. Here we assume that L and R are independent. The change in dependence assumptions means that the model properties change dramatically: tails of cumulative input per time slot are dramatically heavier, traffic cannot be approximated by a Gaussian distribution, and the decay of dependence cannot be measured in the traditional way using correlation functions. Different network applications are likely to have different mark dependence structure. We argue that the present independence assumption on L and R is likely to be appropriate for network applications such as streaming media or peer-to-peer networks. Our conclusion is that it is desirable to separate network traffic by application and to model each application with its own appropriate dependence structure.

The failure rate of a mixture of even the most standard distributions used in reliability can have a complicated shape. However, failure rates of mixtures of two carefully selected distributions will have the well-known bathtub shape. Here we show that mixtures of whole families of distribtions can have a bathtub-shaped failure rate.

Consider a sales contract, called a swing contract, between a seller and a buyer concerning some underlying commodity, with the contract specifying that during some future time interval the buyer will purchase an amount of the commodity between some specified minimum and maximum values. The purchase price and capacity at each time point is also prespecified in the contract. Assuming a random market price process and ignoring the possibility of storage, we look for the maximal expected net gain for the buyer of such a contract, and the strategy that achieves this maximal expected net gain. We study the effects that various contract constraints and market price processes have on the optimal strategy and on the contract value. We show how we can reduce the general swing contract to a multiple exercising of American (Bermudan) style options. Also, in important special cases, we give explicit expressions for the optimal contract value function and the optimal strategy.

Aven and Jensen (1999) proposed an approximation to the downtime distribution for arbitrary coherent systems with binary and independent repairable components, where each component follows an alternating renewal process. The approximation is based on a mixture of the duration distributions of the cut sets in the system. We extend this approach to coherent systems composed by multistate independent components, where every component follows a semi-Markov process and where we introduce the concept of a minimal cut set in the multistate setting. We test our proposal on a simple power demand-generation system, by comparing the analytic approximation with simulation results, and we find it is accurate when the system is highly available.

Owen's multilinear extension (MLE) of a game is a very important tool in game theory and particularly in the field of simple games. Among other applications it serves to efficiently compute several solution concepts. In this paper we provide bounds for the MLE. Apart from its self-contained theoretical interest, the bounds offer the means in voting system studies of approximating the probability that a proposal is approved in a particular simple game having a complex component arrangement. The practical interest of the bounds is that they can be useful for simple games having a tedious MLE to evaluate exactly, but whose minimal winning coalitions and minimal blocking coalitions can be determined by inspection. Such simple games are quite numerous.

We deal with the problem of seating an airplane's passengers optimally, namely in the fastest way. Under several simplifying assumptions, whereby the passengers are infinitely thin and react within a constant time to boarding announcements, we are able to rewrite the asymptotic problem as a calculus of variations problem with constraints. This problem is solved in turn using elementary methods. While the optimal policy is not unique, we identify a rigid discrete structure which is common to all solutions. We also compare the (nontrivial) optimal solutions we find with some simple boarding policies, one of which is shown to be near-optimal.

We develop a conservation law for a multi-class GI/GI/1 queue operating under a general work-conserving scheduling discipline. For single-class single-server queues, conservation laws have been obtained for both nonanticipating and anticipating disciplines with general service time distributions. For multi-class single-server queues, conservation laws have been obtained for (i) nonanticipating disciplines with exponential service time distributions and (ii) nonpreemptive nonanticipating disciplines with general service time distributions. The unifying conservation law we develop generalizes already existing conservation laws. In addition, it covers popular nonanticipating multi-class time-sharing disciplines such as discriminatory processor sharing (DPS) and generalized processor sharing (GPS) with general service time distributions. As an application, we show that the unifying conservation law can be used to compare the expected unconditional response time under two scheduling disciplines.

Consider a sequence X = (X n : n ≥ 1) of independent and identically distributed random variables, and an independent geometrically distributed random variable M with parameter p. The random variable S M = X 1 + ∙ ∙ ∙ + X M is called a geometric sum. In this paper we obtain asymptotic expansions for the distribution of S M as p ↘ 0. If EX 1 > 0, the asymptotic expansion is developed in powers of p and it provides higher-order correction terms to Renyi's theorem, which states that P(pS M > x) ≈ exp(-x/EX 1). Conversely, if EX 1 = 0 then the expansion is given in powers of √p. We apply the results to obtain corrected diffusion approximations for the M/G/1 queue. These expansions follow in a unified way as a consequence of new uniform renewal theory results that are also developed in this paper.

We study a service facility modeled as a queueing system with finite or infinite capacity. Arriving customers enter if there is room in the facility and if they are willing to pay the price posted by the service provider. Customers belong to one of a finite number of classes that have different willingnesses-to-pay. Moreover, there is a penalty for congestion in the facility in the form of state-dependent holding costs. The service provider may advertise class-specific prices that may fluctuate over time. We show the existence of a unique optimal stationary pricing policy in a continuous and unbounded action space that maximizes the long-run average profit per unit time. We determine an expression for this policy under certain conditions. We also analyze the structure and the properties of this policy.

We explicitly calculate the dynamic allocation indices (i.e. the Gittins indices) for multi-armed Bandit processes driven by superdiffusive noise sources. This class of model generalizes former results derived by Karatzas for diffusive processes. In particular, the Gittins indices do, in this soluble class of superdiffusive models, explicitly depend on the noise state.

We show in this paper that if a stationary traffic source is regulated by a leaky bucket with leak rate ρ and bucket size σ, then the amount of information generated in successive time intervals is dominated, in the increasing convex ordering sense, by that of a Poisson arrival process with rate ρ/σ, with each arrival bringing an amount of information equal to σ. By exploiting this property, we then show that the mean value in the stationary regime of the content of a buffer drained at constant rate and fed with the superposition of regulated flows is less than the mean value of the same buffer fed with an adequate Poisson process, whose characteristics depend upon the regulated input flows.

We consider a constant rate traffic which shares a buffer with a random cross traffic. A first come first served or priority service discipline is applied at the buffer. After service at the first buffer the constant rate traffic moves to a play-out buffer. Both buffers provide service at constant rate and infinite waiting room. We investigate logarithmic large and moderate deviation asymptotics for the tail probabilities of the steady-state queue length distribution at the play-out buffer for long-range dependent cross traffic in critical loading. We characterize the asymptotic behavior of the cross traffic which leads to a large queue length at the play-out buffer and compare it to the one for renewal cross traffic.

The symmetric rendezvous problem on a network Q asks how two players, forced to use the same mixed strategy, can minimize their expected meeting time, starting from a known initial distribution on the nodes of Q. This minimum is called the (symmetric) ‘rendezvous value’ of Q. Traditionally, the players are assumed to receive no information while playing the game. We consider the effect on rendezvous times of giving the players some information about past actions and chance moves, enabling each of them to apply Bayesian updates to improve his knowledge of the other's whereabouts. This technique can be used to give lower bounds on the rendezvous times of the original game (without any revealed information). We consider the case in which they are placed a known distance apart on the line graph Q (known as ‘symmetric rendezvous on the line’). Our approach is to concentrate on a general analysis of the effect of revelations, rather than compute the best bounds possible with our technique.

A large dam model is the object of study of this paper. The parameters L lower and L upper define its lower and upper levels,L = L upper - L lower is large, and if the current level of water is between these bounds, the dam is assumed to be in a normal state. Passage across one or other of the levels leads to damage. Let J 1 and J 2 denote the damage costs of crossing the lower and, respectively, the upper levels. It is assumed that the input stream of water is described by a Poisson process, while the output stream is state dependent. Let L t denote the dam level at time t, and let p 1 = limt→∞P{L t = L lower} and p 2 = limt→∞P{L t > L upper} exist. The long-run average cost,J = p 1 J 1 + p 2 J 2, is a performance measure. The aim of the paper is to choose the parameter controlling the output stream so as to minimize J.