We calculate the mean throughput, number of collisions, successes, and idle slots for random tree algorithms with successive interference cancellation. Except for the case of the throughput for the binary tree, all the results are new. We furthermore disprove the claim that only the binary tree maximizes throughput. Our method works with many observables and can be used as a blueprint for further analysis.

Let q\ge2$q\ge2$ be an integer, \{X_n\}_{n\geq 1}$\{X_n\}_{n\geq 1}$ a stochastic process with state space \{0,\ldots,q-1\}$\{0,\ldots,q-1\}$ , and F the cumulative distribution function (CDF) of \sum_{n=1}^\infty X_n q^{-n}$\sum_{n=1}^\infty X_n q^{-n}$ . We show that stationarity of \{X_n\}_{n\geq 1}$\{X_n\}_{n\geq 1}$ is equivalent to a functional equation obeyed by F, and use this to characterize the characteristic function of X and the structure of F in terms of its Lebesgue decomposition. More precisely, while the absolutely continuous component of F can only be the uniform distribution on the unit interval, its discrete component can only be a countable convex combination of certain explicitly computable CDFs for probability distributions with finite support. We also show that \mathrm{d} F$\mathrm{d} F$ is a Rajchman measure if and only if F is the uniform CDF on [0, 1].

This paper gives an equivalent form of Picard’s theorem via entire solutions of the functional equation {{f}^{2}}\,+\,{{g}^{2}}\,=\,1${{f}^{2}}\,+\,{{g}^{2}}\,=\,1$ and then its improvements and applications to certain nonlinear (ordinary and partial) differential equations.

A class of games for finding a leader among a group of candidates is studied in detail. This class covers games based on coin tossing and rock-paper-scissors as special cases and its complexity exhibits similar stochastic behaviors: either of logarithmic mean and bounded variance or of exponential mean and exponential variance. Many applications are also discussed.

We consider a broad class of fair leader election algorithms, and study the duration of contestants (the number of rounds a randomly selected contestant stays in the competition) and the overall cost of the algorithm. We give sufficient conditions for the duration to have a geometric limit distribution (a perpetuity built from Bernoulli random variables), and for the limiting distribution of the total cost (after suitable normalization) to be a perpetuity. For the duration, the proof is established via convergence (to 0) of the first-order Wasserstein distance from the geometric limit. For the normalized overall cost, the method of proof is also convergence of the first-order Wasserstein distance, augmented with an argument based on a contraction mapping in the first-order Wasserstein metric space to show that the limit approaches a unique fixed-point solution of a perpetuity distributional equation. The use of these two steps is commonly referred to as the contraction method.

For two real characters ψ,ψ′ of conductor dividing 8 define where and the subscript 2 denotes the fact that the Euler factor at 2 has been removed. These double Dirichlet series can be extended to possessing a group of functional equations isomorphic to D12. The convexity bound for Z(s,w;ψ,ψ′) is |sw(s+w)|1/4+ε for ℜs=ℜw=1/2. It is proved that Moreover, the following mean square Lindelöf-type bound holds: for any Y1,Y2≥1.

We give the general solution of the n-dimensional mixed-type linear and quadratic functional equation, where , and 1<m<n are integers.

We formulate an explicit conjecture for the leading term at s=1 of the equivariant Dedekind zeta-function that is associated to a Galois extension of number fields. We show that this conjecture refines well-known conjectures of Stark and Chinburg, and we use the functional equation of the zeta-function to compare it to a natural conjecture for the leading term at s=0.

We describe a search problem that has arisen in the context of network monitoring. Abstractly, a known (very large) region may contain one or more ‘agents’. Starting with just one agent, we search until another agent is found; this new agent can assist in the remaining search, and so on recursively.

Let T = (T1, T2,…) be a sequence of real random variables with ∑j=1∞1|Tj|>0 < ∞ almost surely. We consider the following equation for distributions μ: W ≅ ∑j=1∞TjWj, where W, W1, W2,… have distribution μ and T, W1, W2,… are independent. We show that the representation of general solutions is a mixture of certain infinitely divisible distributions. This result can be applied to investigate the existence of symmetric solutions for Tj ≥ 0: essentially under the condition that E ∑j=1∞Tj2 log+Tj2 < ∞, the existence of nontrivial symmetric solutions is exactly determined, revealing a connection with the existence of positive solutions of a related fixed-point equation. Furthermore, we derive results about a special class of canonical symmetric solutions including statements about Lebesgue density and moments.

Let {Ai : i ≥ 1} be a sequence of non-negative random variables and let M be the class of all probability measures on [0,∞]. Define a transformation T on M by letting Tμ be the distribution of ∑i=1∞AiZi, where the Zi are independent random variables with distribution μ, which are also independent of {Ai}. Under first moment assumptions imposed on {Ai}, we determine exactly when T has a non-trivial fixed point (of finite or infinite mean) and we prove that all fixed points have regular variation properties; under moment assumptions of order 1 + ε, ε > 0, we find all the fixed points and we prove that all non-trivial fixed points have stable-like tails. Convergence theorems are given to ensure that each non-trivial fixed point can be obtained as a limit of iterations (by T) with an appropriate initial distribution; convergence to the trivial fixed points δ0 and δ∞ is also examined, and a result like the Kesten-Stigum theorem is established in the case where the initial distribution has the same tails as a stable law. The problem of convergence with an arbitrary initial distribution is also considered when there is no non-trivial fixed point. Our investigation has applications in the study of: (a) branching processes; (b) invariant measures of some infinite particle systems; (c) the model for turbulence of Yaglom and Mandelbrot; (d) flows in networks and Hausdorff measures in random constructions; and (e) the sorting algorithm Quicksort. In particular, it turns out that the basic functional equation in the branching random walk always has a non-trivial solution.

Given a random integer N ≧ 0 and a sequence of random variables Ai ≧ 0, we define a transformation T on the class of probability measures on [0, ∞) by letting Tμ be the distribution of , where {Zi} are independent random variables with distribution μ, which are independent of N and of {Ai} as well. We obtain the optimal conditions for the functional equation μ = Tμ to have a non-trivial solution of finite mean, and we study the existence of moments of the solutions. The work unifies the corresponding theorems of Kesten-Stigum concerning the Galton-Watson process, of Biggins for branching random walks, of Kahane-Peyrière for a model of turbulence of Yaglom made precise by Mandelbrot, and of Durrett-Liggett for the study of invariant measures for certain infinite particle systems.

We consider an increasing supercritical branching process in a random environment and obtain bounds on the Laplace transform and distribution function of the limiting random variable. There are two possibilities that can be distinguished depending on the nature of the component distributions of the environment. If the minimum family size of each is 1, the growth will be as a power depending on a parameter α. If the minimum family sizes of some are greater than 1, it will be exponential, depending on a parameter γ. We obtain bounds on the distribution function analogous to those found for the simple Galton-Watson case. It is not possible to obtain exact estimates and we are only able to obtain bounds to within ε of the parameters.

We analyse a stack protocol of the Capetanakis–Tsybakov–Mikhailov type for resolving collisions in a random multiple-access channel. We obtain a functional equation for the generating function of the expected collision resolution interval (CRI) durations, which is non-local with a non-commutative iteration semigroup. Using Mellin transform techniques and geometric properties of the iteration semigroup we show that for arrival rates smaller than a fixed threshold, the mean CRI duration for n initial colliders is asymptotically proportional to n. Ergodicity conditions are also demonstrated.

A proof is given of the Euler-Maclaurin sum formula, on a Banach space of differentiable vector-valued functions of bounded exponential growth, using the Laplace transformation. Some related summation formulae are proved by the same methods. Properties of the standard summation operator are proved, namely spectral properties and boundedness, continuity and differentiability results.

Let F(·) be a c.d.f. on [0,∞), f(s) = ∑∞0pjsi a p.g.f. with p0 = 0, < 1 < m = Σjpj < ∞ and 1 < ρ <∞. For the functional equation for a c.d.f. H(·) on [0,∞] we establish that if 1 – F(x) = O(x–θ) for some θ > α =(log m)/(log p) there exists a unique solution H(·) to (∗) in the class C of c.d.f.’s satisfying 1 – H(x) = o(x–α).

We give a probabilistic construction of this solution via branching random walks with discounting. We also show non-uniqueness if the condition 1 – H(x) = o(x–α) is relaxed.

We consider a single-server queueing system in which a customer gives up whenever his waiting time is larger than a random threshold, his patience time. In the case of a GI/GI/1 queue with i.i.d. patience times, we establish the extensions of the classical GI/GI/1 formulae concerning the stability condition and the relation between actual and virtual waiting-time distribution functions. We also prove that these last two distribution functions coincide in the case of a Poisson input process and determine their common law.