A Markov renewal theorem necessary for the derivation of the moment formulas for a filtered Markov renewal process stated by Marcus (1974) is proved and its applications are outlined.

Let (A 1A 2, · ··, An ) be a set of n events on a probability space. Let be the sum of the probabilities of all intersections of r events, and Mn the number of events in the set which occur. The classical Bonferroni inequalities provide upper and lower bounds for the probabilities P(Mn = m), and equal to partial sums of series of the form which give the exact probabilities. These inequalities have recently been extended by J. Galambos to give sharper bounds.

Here we present straightforward proofs of the Bonferroni inequalities, using indicator functions, and show how they lead naturally to new simple proofs of the Galambos inequalities.

Strong mixing is a condition which is often assumed to prove limit theorems for strictly stationary processes. Leadbetter's condition D(un ) is used to prove limit theorems for maxima of stationary processes.

A sufficient condition for strong mixing to hold is given for the case where the process satisfies a pth-order Markov property. This condition can be easy to check for when p is small. This point is illustrated by two examples of first-order autoregressive processes.

The condition D(un ) is shown to hold for any stationary Markov process.

An algorithm for calculating the probability distribution of ladder epochs is derived. Two theorems are given for bounds on the distribution function of ladder epoch probabilities.

A multichannel loss system with heterogeneous servers operating in parallel is analyzed. The sum of the service rates of all servers is assumed constant. The optimal service rates that minimize the probability of losing a customer are obtained, and are shown to be different from each other. The percentage reduction in the probability of losing a customer in the homogeneous and the best heterogeneous system for a few representative values are included.

Consider three independent Poisson processes of point events of rates λ1, λ 2 and λ12. Suppose there are two electronic counters, the first recording events from the first and third Poisson processes, and the second recording events from the second and third Poisson processes. Both counters are subject to dead times of Type II, i.e. where all events, whether recorded or not, cause dead times. The covariance of the numbers of events recorded by the two counters in a suitable time period is derived. This extends earlier work of Cox and Isham (1977) and Kingman (1977) who considered the same problem for Type I counters, i.e. where only recorded events cause dead times.