The transmission control protocol (TCP) is a transport protocol used in the Internet. In Ott (2005), a more general class of candidate transport protocols called ‘protocols in the TCP paradigm’ was introduced. The long-term objective of studying this class is to find protocols with promising performance characteristics. In this paper we study Markov chain models derived from protocols in the TCP paradigm. Protocols in the TCP paradigm, as TCP, protect the network from congestion by decreasing the ‘congestion window’ (i.e. the amount of data allowed to be sent but not yet acknowledged) when there is packet loss or packet marking, and increasing it when there is no loss. When loss of different packets are assumed to be independent events and the probability p of loss is assumed to be constant, the protocol gives rise to a Markov chain {Wn}, where Wn is the size of the congestion window after the transmission of the nth packet. For a wide class of such Markov chains, we prove weak convergence results, after appropriate rescaling of time and space, as p → 0. The limiting processes are defined by stochastic differential equations. Depending on certain parameter values, the stochastic differential equation can define an Ornstein-Uhlenbeck process or can be driven by a Poisson process.

We consider two weaker partial orderings among non-negative random variables. We derive some characterizations of the equivalence of two life distributions under the weaker orderings. These results lead to the characterizations of the equality of two non-negative random vectors in distributions by moments of extremes and sample mean. As the application of these results, we get various characterizations of the exponential distribution among HNBUE and HNWUE life distribution classes. The main results of Bhattacharjee and Sethuraman (1990), and Basu and Kirmani (1986) are special cases of these more general results.

We derive conditions under which the increments of a vector process are associated — i.e. under which all pairs of increasing functions of the increments are positively correlated. The process itself is associated if it is generated by a family of associated and monotone kernels. We show that the increments are associated if the kernels are associated and, in a suitable sense, convex. In the Markov case, we note a connection between associated increments and temporal stochastic convexity.

Our analysis is motivated by a question in variance reduction: assuming that a normalized process and its normalized compensator converge to the same value, which is the better estimator of that limit? Under some additional hypotheses we show that, for processes with conditionally associated increments, the compensator has smaller asymptotic variance.

We describe two chain-transformations which explain and extend identities for order statistics and quantiles proved by Wendel, Port and, more recently, by Dassios.

Let {Nt}t >0 be a renewal counting process (cf. Parzen (1962), p. 160) with underlying failure times let be a sequence of non-negative random variables and {Zt}t >0 an associated cumulative process, i.e. if Nt = 1, 2, …, and Zt = 0, if Nt = 0. By convention set Z0 = 0. Consider the maximum increment of the process {Zt}t >0 in [0, T] over a time K, 0 < K < T, divided by K. Under appropriate conditions it is shown that for a wide range of numbers a there exist constants C(a), uniquely determined by a and the distributions of the Xi's and Yj's, such that D(T, C log T) converges to a with probability 1. This result provides a renewal theoretic variant of Erdös and Rényi's (1970) ‘new law of large numbers’.

Recently, Elmes et al. (see [2]) proposed a definition of a quasistationary distribution to accommodate absorbing Markov chains for which absorption occurs with probability less than 1. We will show that the probabilistic interpretation pertaining to cases where absorption is certain (see [13]) does not hold in the present context. We prove that the state probabilities at time t conditional on absorption taking place after t, generally depend on t. Conditions are derived under which there is no initial distribution such that the conditional state probabilities are stationary.

In this paper a necessary and sufficient condition for the dispersive ordering in dilation sense is given by a convex function which is called the dispersive function and characterizes the distribution function. Some interesting properties of the ordering follow from this result.

The distribution theory for reward functions on semi-Markov processes has been of interest since the early 1960s. The relevant asymptotic distribution theory has been satisfactorily developed. On the other hand, it has been noticed that it is difficult to find exact distribution results which lead to the effective computation of such distributions. Note that there is no satisfactory exact distribution result for rewards accumulated over deterministic time intervals [0, t], even in the special case of continuous-time Markov chains. The present paper provides neat general results which lead to explicit closed-form expressions for the relevant Laplace transforms of general reward functions on semi-Markov and Markov additive processes.

In this paper a bisexual Moran model is introduced. The population consists of N pairs of individuals. At times t = 1, 2, ·· ·two individuals are born, who ‘choose their parents randomly' and independently of each other. Then one of the pairs is removed and replaced by the two individuals born at that instant.

The extinction probability of the descendants of a single pair and the number of ancestors of a whole generation are studied. A limit result for large population sizes has been derived by diffusion approximation methods.

We focus on supercritical decomposable (reducible) multitype branching processes. Types are partitioned into irreducible equivalence classes. In this context, extinction of some classes is possible without the whole process becoming extinct. We derive criteria for the almost-sure extinction of the whole process, as well as of a specific class, conditionally given the class of the initial particle. We give sufficient conditions under which the extinction of a class implies the extinction of another class or of the whole process. Finally, we show that the extinction probability of a specific class is the minimal nonnegative solution of the usual extinction equation but with added constraints.

It is shown that a Galton-Watson process in varying environment, unlike a classical Galton-Watson process, can have an infinite (though at most countable) number of distinct rates of growth.

We consider a birth, death and catastrophe process where the transition rates are allowed to depend on the population size. We obtain an explicit expression for the expected time to extinction, which is valid in all cases where extinction occurs with probability 1.

A general model for the growth and division of cells in which the growth rate and division probability at any instant depend only on their size at that time is introduced. Conditions under which (a) the distribution of cell-size at division converges ergodically, (b) the sizes tend to 0 or ∞, are exhibited, and bounds to the correlation between the sizes at division of sister cells are given in a wide class of cases.

A formula is derived for the supremum of a stationary Gaussian process which has a correlation function that is tent-like in shape, until it flattens out at a constant negative value. Examples and graphs are presented in the last section.

Consider a unknown realizable time-invariant linear filter driven by a point process. We are interested in the identification of this system, observing only the output random process. If the process is almost periodically correlated but not periodically correlated, we can identify the filter, using the second-order non-stationary spectrum of the process. We do not require the assumption that the filter is minimum phase.

Using an easy linear-algebraic method, we obtain spectral representations, without the need for eigenvector determination, of the transition probability matrices for completely general continuous time Markov chains with finite state space. Comparing the proof presented here with that of Brown (1991), who provided a similar result for a special class of finite Markov chains, we observe that ours is more concise.

A uniform recursive tree on n vertices is a random tree where each possible $(n-1)!$ labelled recursive rooted tree is selected with equal probability. We introduce and study weighted trees, a non-uniform recursive tree model departing from the recently introduced Hoppe trees. This class generalizes both uniform recursive trees and Hoppe trees, providing diversity among the nodes and making the model more flexible for applications. We analyse the number of leaves, the height, the depth, the number of branches, and the size of the largest branch in these weighted trees.

Asymptotic approximations and numerical computations are used to estimate the accuracy of the diffusion approximation for the expected time to extinction for some stochastic processes. The results differ for processes with a continuant transition matrix (e.g. a birth and death process), and those with a noncontinuant transition matrix (e.g. a non-linear branching process). In the latter case, the diffusion equation does not hold near the point of exit. Consequently, high-order corrections do not result in substantial improvement over the diffusion approximation.

By considering continuous-time Markov decision processes where decisions can be made at any time, we show in the case of M/M/1 queues with discounted costs that there exists a monotone optimal policy among all the regular policies.