In 1979, Melamed proved that, in an open migration process, the absence of ‘loops' is necessary and sufficient for the equilibrium flow along a link to be a Poisson process. In this paper, we prove approximation theorems with the same flavour: the difference between the equilibrium flow along a link and a Poisson process with the same rate is bounded in terms of expected numbers of loops. The proofs are based on Stein's method, as adapted for bounds on the distance of the distribution of a point process from a Poisson process in Barbour and Brown (1992b). Three different distances are considered, and illustrated with an example consisting of a system of tandem queues with feedback. The upper bound on the total variation distance of the process grows linearly with time, and a lower bound shows that this can be the correct order of approximation.

Nanosystems are devices that are in the size range of a billionth of a meter (1 x 10-9) and therefore are built necessarily from individual atoms. The one-dimensional nanosystems or linear nanosystems cover all the nanosized systems which possess one dimension that exceeds the other two dimensions, i.e. extension over one dimension is predominant over the other two dimensions. Here only two of the dimensions have to be on the nanoscale (less than 100 nanometers). In this paper we consider the structural relationship between a linear nanosystem and its atoms acting as components of the nanosystem. Using such information, we then assess the nanosystem's limiting reliability which is, of course, probabilistic in nature. We consider the linear nanosystem at a fixed moment of time, say the present moment, and we assume that the present state of the linear nanosystem depends only on the present states of its atoms.

Correction terms for the diffusion approximation to the maximum and ruin probabilities for a random walk with small negative drift, obtained by Siegmund (1979) in the exponential family case, are extended by different methods to some non-exponential family cases.

(1.1) In this paper, acquaintance will be assumed with the basic mechanics of phage/bacterium, antibody/virus interactions; for background reading see, for example, Adams (1959), Bouanchaud (1970), Durham and King (1969) and Fraser (1967).

As an extension of the discrete-time case, this note investigates the variance of the total cumulative reward for continuous-time Markov reward chains with finite state spaces. The results correspond to discrete-time results. In particular, the variance growth rate is shown to be asymptotically linear in time. Expressions are provided to compute this growth rate.

Here we introduce two simple models: simple cubic random packing and random packing by Hamming distance. Consider the packing density γ d of dimension d by cubic random packing. From computer simulations up to dimension 11, γ d+1/γ d seems to approach 1. Also, we give simulation results for random packing by Hamming distance and discuss the behavior of packing density when dimensionality is increased. For the case of Hamming distances of 2 or 3, d–α fits the simulation results of packing density where α is an empirical constant. The variance of packing density is larger when k is even and smaller when k is odd, where k represents Hamming distance.

The topic of the present paper is a generalized St Petersburg game in which the distribution of the payoff X is given by P(X = sr(k-1)/α) = pqk-1, k = 1, 2,…, where p + q = 1, s = 1 / p, r = 1 / q, and 0 < α ≤ 1. For the case in which α = 1, we extend Feller's classical weak law and Martin-Löf's theorem on convergence in distribution along the 2n-subsequence. The analog for 0 < α < 1 turns out to converge in distribution to an asymmetric stable law with index α. Finally, some limit theorems for polynomial and geometric size total gains, as well as for extremes, are given.

We study a family of Markov processes on P (k), the space of partitions of the natural numbers with at most k blocks. The process can be constructed from a Poisson point process on R + x ∏i=1 k P (k) with intensity dt ⊗ ϱν (k), where ϱν is the distribution of the paintbox based on the probability measure ν on P m, the set of ranked-mass partitions of 1, and ϱν (k) is the product measure on ∏i=1 k P (k). We show that these processes possess a unique stationary measure, and we discuss a particular set of reversible processes for which transition probabilities can be written down explicitly.

This paper considers the best-choice problem with a random number of objects having a known distribution. The optimality equation of the problem reduces to an integral equation by a scaling limit. The equation is explicitly solved under conditions on the distribution, which relate to the condition for an OLA policy to be optimal in Markov decision processes. This technique is then applied to three different versions of the problem and an exact value for the asymptotic optimal strategy is found.

We introduce and analyze a random tree model associated to Hoppe's urn. The tree is built successively by adding nodes to the existing tree when starting with the single root node. In each step a node is added to the tree as a child of an existing node, where these parent nodes are chosen randomly with probabilities proportional to their weights. The root node has weight ϑ>0, a given fixed parameter, all other nodes have weight 1. This resembles the stochastic dynamic of Hoppe's urn. For ϑ=1, the resulting tree is the well-studied random recursive tree. We analyze the height, internal path length, and number of leaves of the Hoppe tree with n nodes as well as the depth of the last inserted node asymptotically as n→∞. Mainly expectations, variances, and asymptotic distributions of these parameters are derived.

An urn contains black and red balls. Let Z n be the proportion of black balls at time n and 0≤L<U≤1 random barriers. At each time n, a ball b n is drawn. If b n is black and Z n-1<U, then b n is replaced together with a random number B n of black balls. If b n is red and Z n-1>L, then b n is replaced together with a random number R n of red balls. Otherwise, no additional balls are added, and b n alone is replaced. In this paper we assume that R n =B n . Then, under mild conditions, it is shown that Z n →a.s. Z for some random variable Z, and D n ≔√n(Z n -Z)→𝒩(0,σ2) conditionally almost surely (a.s.), where σ2 is a certain random variance. Almost sure conditional convergence means that ℙ(D n ∈⋅|𝒢n )→w 𝒩(0,σ2) a.s., where ℙ(D n ∈⋅|𝒢n ) is a regular version of the conditional distribution of D n given the past 𝒢n . Thus, in particular, one obtains D n →𝒩(0,σ2) stably. It is also shown that L<Z<U a.s. and Z has nonatomic distribution.

In this paper, we discuss the problem of the pricing of American-style options in the exponential Lévy security market model. This model is typically incomplete, and we derive the explicit bounds of the interval of no arbitrage prices and the related optimal stopping moments for American put options and American call options in both finite and infinite horizon time. We consider a large class of Lévy processes.

For a sequence ξ1, ξ2, · ·· of i.i.d. random variables let X0 = 0 and Xk = max(Xk–1 + ξ k, 0) for k = 1, 2, ···. Let . These stopping times are used in Page's (1954) one-sided cusum procedures and are also important in queueing theory. Various asymptotic properties of Nx are derived.

Investigations into time-slotted communication channels for transmission of data packets led us to analyze the stochastic behavior of the next-fit bin packing algorithm. In this paper we obtain results for general piece-size distributions and truncated distributions, we calculate explicit solutions for the case of the truncated exponential, and we apply the results to calculate the maximal throughput of the time-slotted communication channel.

Recently Gani (1965) has considered a stochastic model for the attachment of phages to bacteria. In this paper we study some of the basic properties of this model. These results will be used in a subsequent paper dealing with the estimation of the parameter which occurs in this model.

An adaptive-type exponential smoothing, motivated by an insurance tariff problem, is treated. We consider the process Zn = ß(Zn –1)Xn +(1 – ß (Zn–1))Zn–1, where Xn are i.i.d. taking values in the interval [0, M], M ≦ ∞ and ß is a monotonically increasing function [0, M] → [c, d], 0 < c < d < 1.

Together with (Zn), we consider the ordinary exponential smoothing Yn = αXn + (1 – α)Yn –1 where α is a constant, 0 < α < 1. We show that (Yn) and (Zn) are geometrically ergodic Markov chains (in the case of finite interval we even have uniform ergodicity) and that EYn, EZn converge to limits EY, EZ, respectively, with a geometric convergence rate. Moreover, we show that Ez is strictly less than EY = EXn.

This paper considers a semi-infinite storage model, of the type studied by Senturia and Puri [13] and Balagopal [2], defined on a Markov renewal process, {(Xn, Tn), n = 0, 1, ·· ·}, with 0 = T0 < T1 < · ··, almost surely, where Xn takes values in the set {1, 2, ·· ·}. If at Tn, Xn = j, then there is a random ‘input' Vn (j) (a negative input implying a demand) of ‘type' j, having distribution function Fj(·). We assume that {Vn (j)} is an i.i.d. sequence of random variables, taken to be independent of {(Xn, Tn)} and of {Vn (k)}, for k ≠ j, and that Vn (j) has first and second moments. Here the random variables Vn (j) represent instantaneous ‘inputs' (a negative value implying a demand) of type j for our storage model. Under these assumptions, we establish certain limit distributions for the joint process (Z(t), L(t)), where Z(t) (defined in (2)) is the level of storage at time t and L(t) (defined in (3)) is the demand lost due to shortage of supply during [0, t]. Different limit distributions are obtained for the cases when the ‘average stationary input' ρ, as defined in (5), is positive, zero or negative.

This study is concerned with finite Markov decision processes whose dynamics and reward structure are unknown but the state is observable exactly.

We establish a learning algorithm which yields an optimal policy and construct an adaptive policy which is optimal under the average expected reward criterion.

It is shown that, in a superposition of finitely many independent renewal processes, an observation from the limiting (when t →∞) joint distribution of backward and forward recurrence times at t can be simulated by simulating an observation of the pair (UW, (1 – U)W), where U and Ware independent random variables with U ~ uniform(0, 1) and W distributed according to the limiting total life distribution of the superposition process.

We consider a system that gradually deteriorates. At random times Tn the system is inspected and may or may not be replaced, but it must be if its state is too bad. The process of deterioration is described by a Markovian renewal process. Under some monotonicity conditions we derive a policy for the finite and the infinite horizon which minimizes the total discounted costs. This paper is a generalization of a result of Feldman (1977).