In this paper we consider the GI/M/1 queueing model with infinite waiting-room capacity. The customer arriving at t = 0 will find k — 1 customers waiting. The latter customers belong to a second priority class, whereas the ones arriving in [0,∞) belong to a first priority class and have the higher priority. Within each class we have a first-in-first-out queueing discipline. A customer, once at the service-point, remains there until his service is completed. Then the next customer for service is the one of highest priority among those queueing.

For this model we derive the transient waiting times for customers belonging to both priority classes. The results are of special interest in appointment systems where customers may not turn up.

Consider N sources, each of which transmits signals according to a renewal process with a common, known, underlying distribution. Sequential point estimators for N, which are asymptotically efficient in comparison to the best fixed stopping strategy, are derived. Confidence statements for N, and applications to the problem of estimating finite populations are discussed.

The exact limiting distribution, and moments, of the stopping time we use, which was suggested by Starr, are derived.

A target is assumed to move according to a Wiener process in ℝ1. The probability of detecting the target is computed in terms of the search effort which accumulates along the target's path. Under certain independence assumptions this probability is given by the expectation of an exponential functional of the process. It is shown in this note that the failure probability in a search for a Wiener target is asymptotically proportional to , where T is the accumulated time spent searching. The asymptotic failure probability is also shown to be independent of the position of the search in ℝ1. In a similar fashion, it is shown that the failure probability in a search for a Wiener target in ℝ2 is independent of the position of the search and asymptotically proportional to (c log T + l)–1, c >0

A well-known result in the theory of random walks states that E{X2} is finite if and only if E{Z+} and E{Z_} are both finite (Z+ and Z_ being the ladder heights and X a typical step-length) in which case E{X2} = 2E{Z+}E{Z_}. This paper contains results relating the existence of moments of X of order ß to the existence of the moments of Z+ and Z_ of order ß – 1. The main result is that if β > 2 E{|X|β} < ∞ if and only if and are both finite.

For correlated random walks a method of transition probability matrices as an alternative to the much-used methods of probability generating functions and difference equations has been investigated in this paper. To illustrate the use of transition probability matrices for computing the various probabilities for correlated random walks, the transition probability matrices for restricted/unrestricted one-dimensional correlated random walk have been defined and used to obtain some of the probabilities.

By making the time parameters of a birth and death process random variables, we create a continuous-time analogue of random walk in a random environment. Criteria for recurrence or transience are discussed and an a.s. convergence law is determined.

A linear process is generated by applying a linear filter to independent, identically distributed random variables. Only the modulus of the frequency response function can be estimated if only the linear process is observed and if the independent identically distributed random variables are Gaussian. In this case a number of distinct but related problems coalesce and the usual discussion of these problems assumes, for example, in the case of a moving average that the zeros of the polynomial given by the filter have modulus greater than one. However, if the independent identically distributed random variables are non-Gaussian, these problems become distinct and one can estimate the transfer function under appropriate conditions except for a possible linear phase shift by using higher-order spectral estimates.

We derive the representation of additive measures of information with the sum property under a regular generating function. This family of measures includes the Shanon entropy, directed/symmetric divergence and inaccuracy.

We adapt arguments from a paper of Seymour and Welsh concerning percolation probabilities on the infinite square lattice L to show that for certain regions R in L, if there is a positive probability of having an infinite open path in L starting at the origin then there is also a positive probability of having such a path within R.

On a circle of unit circumference arcs of length a are placed at random. Let Nα be equal to the necessary number of arcs to cover at least the length 1 − p, 0 ≦ p < 1, of the circumference at least m (≧1) times. In the present paper limit distributions of Nα are derived when α → 0. Some results for spacings are also obtained.

Many queueing models have embedded Markov chains, whose invariant probability vector is of a (modified) matrix-geometric form. The rate matrix R is shown to have the following probabilistic interpretation:

The element Rvj is the expected number of visits to the state (i + 1, j), before the first return to the set i = {(i, 1), …,(i, m)}, in a chain starting in the state (i, v).

This paper describes some sufficient conditions that ensure the convergence of successive random applications of a family of mappings {Γα : α ∈ A} on a compact metric space (X, d) to a stochastic fixed point. The results are similar in spirit to a recent result of Yahav (1975).